We provide continuity and differentiability practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on continuity and differentiability skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.
List of continuity and differentiability Questions
| Question No | Questions | Class |
|---|---|---|
| 1 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left(frac{mathbf{1}-boldsymbol{x}}{mathbf{1}+boldsymbol{x}}right)= ) A ( cdot frac{-1}{1+x^{2}} ) B ( cdot frac{1}{1+x^{2}} ) c. ( frac{1+x}{1-x} ) D. ( frac{2}{1+x^{2}} ) |
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| 2 | ( frac{d sin x^{2}}{d x} ) A. ( 2 x cos x^{2} ) B . ( 4 x cos x^{2} ) c. ( 2 x sin x^{2} ) D. ( -2 x sin x^{2} ) |
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| 3 | 30. Iff: R R is a function defined by f (x) = [x] (2x-1) – Tt, where [x] denotes the greatest integer COS cos2 function, then fis. [2012] (a) continuous for every real x. (b) discontinuous only at x = 0 (c) discontinuous only at non-zero integral values of x. (d) continuous only at x =0. |
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| 4 | ( left{begin{array}{ccc}text { Find } lim _{x rightarrow 0} f(x) & text { where } f(x)= \ x-1 & text { if } & x0end{array}right. ) | 12 |
| 5 | Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem: ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{x}+boldsymbol{2} ) on [-1,2] |
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| 6 | (1+x,0 SX S2 (1983 – 2 Marks) 13 – x,2 5×53 Determine the form of g(x)=ff(x) and hence find the points of discontinuity of g, if any |
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| 7 | If ( u=tan ^{-1}left(frac{x^{2}+y^{2}}{x+y}right), ) then ( x frac{d u}{d x}+ ) ( boldsymbol{y} frac{boldsymbol{d} boldsymbol{u}}{boldsymbol{d} boldsymbol{y}}= ) ( mathbf{A} cdot sin 2 u ) B. ( frac{1}{2} sin 2 u ) c. ( frac{1}{3} sin 2 u ) D. ( 2 sin 2 u ) |
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| 8 | If ( int f(x) d x=frac{3}{55} sqrt[3]{tan ^{5} x}left(5 tan ^{2} x+right. ) 11) ( +C ) then ( f(x) ) is equal to This question has multiple correct options A ( cdot sqrt[3]{sin ^{2} x cos ^{-14} x} ) B. ( sqrt[3]{tan ^{2} xleft(1+tan ^{2} xright)^{6}} ) c. ( sqrt[3]{cos ^{2} x sin ^{-14} x} ) D. ( frac{7}{3} sqrt[3]{sin ^{2} x cos ^{-14} x} ) |
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| 9 | ( operatorname{Let} mathbf{f}(mathbf{x})=mathbf{x}+tan ^{-1} mathbf{x}, mathbf{g}(mathbf{x})= ) ( frac{x}{1+x^{2}}(x>0) ) Then A ( cdot mathrm{f}(mathrm{x})0 ) B. ( f(x)>g(x), x>0 ) c. ( f(x)<g(x) ) in ( [1, infty) ) D. None of these |
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| 10 | Evaluate ( int_{-6}^{0}|x+3| d x . ) What does this integral represent on the graph? | 12 |
| 11 | ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) in [-1,1] verify Rolle’s theorem. |
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| 12 | State if the given statement is True or False
Derivative of ( y=cos x ) with respect to ( x ) |
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| 13 | Differentiate: ( y=frac{x+2}{3 x+5} ) w.r.t ( x ) | 12 |
| 14 | Differentiate the following w.r.t. ( x: ) ( sin left(tan ^{-1} e^{-x}right) ) |
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| 15 | Let ( boldsymbol{f}(boldsymbol{x}) ) be a differentiable function in ( [2,7] . ) If ( f(2)=3 ) and ( f^{prime}(x) leq 5 ) for all ( x ) in ( (2,7), ) then the maximum possible value of ( f(x) ) at ( x=7 ) is ( A cdot 7 ) B. 15 c. 28 D. 14 |
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| 16 | ( boldsymbol{f}(boldsymbol{x})left{begin{aligned}=& frac{left|boldsymbol{x}^{2}-boldsymbol{x}right|}{boldsymbol{x}^{2}-boldsymbol{x}}, quad boldsymbol{x} neq boldsymbol{0}, boldsymbol{x} neq mathbf{1} \=& boldsymbol{x}=mathbf{0} \=-mathbf{1}, & boldsymbol{x}=mathbf{1} end{aligned} ) Discus right. its continuity in ( 0<x leq 1 ) This question has multiple correct options A. continuous at ( x=0 ) B. dis-continuous at ( x=1 ) c. dis-continuous at ( x=0 ) D. continuous at ( x=1 ) |
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| 17 | Assertion ( (A): f(x)=sin (pi[x]) ) is differentiable every where [] is greatest integer function Reason ( (mathrm{R}): ) If ( mathbf{x}=mathbf{n} boldsymbol{pi} Rightarrow sin boldsymbol{x}=mathbf{0} forall mathbf{n} in ) ( mathbf{Z} ) then A. Both (A) and (R) are true and R is correct explanation for A B. Both (A) and (R) are true and R is not correct explanation for c. (A) is true (R) is false D. (A) is false (R) is true |
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| 18 | If ( boldsymbol{y}=tan ^{-1}left(frac{1+boldsymbol{x}^{2}}{1-boldsymbol{x}^{2}}right) ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? ) A. ( frac{2 x}{left(1+x^{4}right)} ) B. ( frac{-2}{left(1+x^{4}right)} ) c. ( frac{x}{left(1+x^{4}right)} ) D. none of these |
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| 19 | Examine the continuity of: [ begin{aligned} boldsymbol{f}(boldsymbol{x}) &=boldsymbol{x}^{2}-boldsymbol{x}+boldsymbol{9} text { for } boldsymbol{x} leq mathbf{3} \ &=boldsymbol{4} boldsymbol{x}+mathbf{3} quad text { for } boldsymbol{x}>mathbf{3}, boldsymbol{a} boldsymbol{t} boldsymbol{x}=boldsymbol{3} end{aligned} ] |
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| 20 | Derivative of ( tan ^{3} theta ) with respect to ( sec ^{3} theta ) at ( theta=frac{pi}{3} ) is A ( cdot frac{3}{2} ) B. ( frac{sqrt{3}}{2} ) ( c cdot frac{1}{2} ) D. ( -frac{sqrt{3}}{2} ) |
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| 21 | If ( boldsymbol{y}=log _{10} boldsymbol{x}+log _{x} mathbf{1 0}+log _{x} boldsymbol{x}+ ) ( log _{10} 10, ) then ( frac{d y}{d x}= ) A ( cdot frac{1}{x log _{e} 10}-frac{log _{e} 10}{xleft(log _{e} xright)^{2}} ) B. ( frac{1}{log _{e} 10}-frac{log _{e} 10}{xleft(log _{e} xright)^{2}} ) c. ( frac{1}{x log _{e} 10}-frac{log _{e} 10}{x^{2}left(log _{e} xright)^{2}} ) D. None of these |
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| 22 | A function ( f(x) ) is defined by ( f(x)= ) ( left{begin{array}{cl}frac{left[x^{2}-1right]}{x^{2}-1} & text { for } x^{2} neq 1 \ 0 & text { for } x^{2}=1end{array} ) Discuss the right. contiuuity of ( f(x) ) at ( x=1 ) |
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| 23 | If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{a} cos ^{-1} boldsymbol{x}} ) then ( left(boldsymbol{1}-boldsymbol{x}^{2}right) boldsymbol{y}^{prime prime}-boldsymbol{x} boldsymbol{y}^{prime}= ) A . ( a y ) B. ( -a^{2} y ) c. ( -a y ) D ( cdot a^{2} y ) |
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| 24 | Differentiate: ( sin left(tan ^{-1} e^{x}right) ) | 12 |
| 25 | Identify whether ( f(x)=frac{x^{2}-4}{x-2} ) is continuous at ( x=2 ) or not |
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| 26 | Show that ( f(x)=frac{cos 3 x-cos 4 x}{x sin 2 x} ) for ( x neq 0, f(0)=frac{7}{4} ) is continuous at ( x= ) 0 |
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| 27 | Differentiate w.r.t ( x ) ( boldsymbol{y}=boldsymbol{x}^{2} sin 2 boldsymbol{x} ) |
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| 28 | Differentiate: ( mathbf{5}^{mathbf{5}^{5 x}}=boldsymbol{t} ) | 12 |
| 29 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}frac{x^{2}-2 x-3}{x+1^{2}}, & text { when } x neq-1 \ k, & text { when } x=-1end{array}right. ) If ( f(x) ) is continuous at ( x=-1 ) then ( boldsymbol{k}=? ) A . 4 B. -4 c. -3 D. |
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| 30 | If ( sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}}=4 ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{1} ) | 12 |
| 31 | Find the values of ( a ) and ( b ) so that the function, ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{1-sin ^{2} x}{3 cos ^{2} x}, quad xpi / 2end{array}right. ) continuous |
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| 32 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=tan ^{-1}left(frac{mathbf{5} boldsymbol{x}+mathbf{1}}{mathbf{3}-boldsymbol{x}-mathbf{6} boldsymbol{x}^{2}}right) ) | 12 |
| 33 | ( f(x)=frac{x^{2}-16}{x-4}+a ) for ( x4 ] continuous at ( x=4, ) find ( a ) and ( b ) |
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| 34 | The velocity of a particle is given by ( v= ) ( 12+3left(t+7 t^{2}right) . ) What is the acceleration of the particle? A ( .3+21 t ) B. ( 3+42 t ) ( c .42 t ) D. ( 4 t ) |
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| 35 | ( operatorname{Let} sqrt{x}+sqrt{x+sqrt{x+ldots ldots infty}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A. ( frac{1}{2 y-1} ) в. ( frac{x}{x+2 y} ) c. ( frac{1}{sqrt{1+4 x}} ) D. ( frac{y}{2 x+y} ) |
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| 36 | 3. The function f(x)=1+ sin xis (1986-2 Marks) (a) continuous nowhere (b) continuous everywhere © differentiable nowhere (d) not differentiable at x=0 (e) not differentiable at infinite number of points. |
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| 37 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of ( boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}=mathbf{1 0 0} ) | 12 |
| 38 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( log sqrt{frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}} ) |
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| 39 | Find the derivatives of ( x cos x ) | 12 |
| 40 | Find the derivative of ( x^{4}+4 ) | 12 |
| 41 | 3. (1994) If y = (sin x)tanx, then is equal to (a) (sin x)tan (1 + sec?x log sin x) (b) tan x (sin x)tan x-1.cos x (c) (sin x)tan x sec2x log sin x (d) tan x (sin x)tan x-1 |
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| 42 | 4. If r2 + y2=1 then (a) “-26”)2 + 1 = 0 (c) “+’)? – 1 = 0 (2000) (b) yy”+(2+1=0 (d) yy”+26′)2+1 = 0 |
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| 43 | 9. 3 If f(x + y) = f(x).f(y)Vx.y and f(5) = 2, f ‘(0) = 3, then f'(5) is [2002] (a) o (6) 1 (c) 6 (d) 2 0 0*=2110-12 them |
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| 44 | If ( x^{y}=e^{x-y}, ) then ( frac{d y}{d x} ) is equal to A ( cdot frac{log x}{1+log x} ) B. ( frac{log x}{1-log x} ) c. ( frac{log x}{(1+log x)^{2}} ) D. ( frac{y log x}{x(1+log x)^{2}} ) |
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| 45 | [ f(x)=left{begin{array}{ll} frac{1-sin ^{2} x}{3 cos ^{2} x}, & xfrac{pi}{2} end{array}right. ] then ( f(x) ) is continuous at ( x=frac{pi}{2} ) |
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| 46 | f ( boldsymbol{y}=tan (2 boldsymbol{x}+mathbf{3}) ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 47 | Assertion Let ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}1+x & x<0 \ 1+[x]+sin x & 0 leq x leq pi / 2 \ 3 & x geq pi / 2end{array}right. ) is continuous on ( mathrm{R}-{1} ) Reason The greatest integer function is discontinuous at every integer. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
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| 48 | 13. If x=2 cost-cos 2t, y=2 sin t-sin 2t, then at t = I dy (a) V2 +1 (b) V2+1 (d) None of these 2 (dy)² |
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| 49 | If ( y=log left(frac{1-x^{2}}{1+x^{2}}right), ) then ( frac{d y}{d x} ) is equal to A ( cdot frac{-4 x}{1-x^{4}} ) в. ( frac{4 x^{3}}{1-x^{4}} ) c. ( frac{1}{4-x^{4}} ) D. ( frac{-4 x^{3}}{1-x^{4}} ) |
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| 50 | Let ( R ) be the set of all real numbers and ( boldsymbol{f}:[-mathbf{1}, mathbf{1}] rightarrow boldsymbol{R} ) be defined by ( quad boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{c}x sin frac{1}{x}, x neq 0 \ 0, x=0end{array} . ) Then right. A. ( f ) satisfies the conditions of Rolle’s theorem on [-1,1] B. ( f ) satisfies conditions of Lagrange’s Mean Value Theorem on [-1,1] c. ( f ) satisfies the conditions of Rolle’s theorem on [0,1] D. ( f ) satisfies the conditions of Lagrange’s Mean Value Theorem on [0,1 |
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| 51 | Assertion Derivative of ( frac{x^{n}-a^{n}}{x-a} ) for some constant ( n ) is ( frac{(n-1) x^{n}-n a x^{n-1}+a^{n}}{(x-a)^{2}} ) Reason ( frac{boldsymbol{d}}{boldsymbol{x}}left(frac{boldsymbol{u}}{boldsymbol{v}}right)=frac{boldsymbol{u}^{prime} boldsymbol{v}-boldsymbol{u} boldsymbol{v}^{prime}}{boldsymbol{v}^{2}} ) where ( u ) and ( v ) are two distinct functions. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct |
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| 52 | If ( boldsymbol{x}=boldsymbol{y}(log boldsymbol{x} boldsymbol{y}) ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 53 | If ( f(x)=sin x ) and ( g(x)=cos x ) then ( D *(f circ g) ) is equal to ( mathbf{A} cdot-sin 2(cos x) sin x ) B. – sin (cos ( x ) ) ( sin x ) c. ( -sin ^{2}(cos x) sin x ) D. ( -sin (cos x) sin ^{2} x ) |
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| 54 | 28. The values of p and q for which the function 120111 sin(p+1)x+sin x reo x = 0 is continuous for all x in R, are 3/2 (b) P ( p=34= © p=29= mp4= ) p=5.q= (a |
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| 55 | Differentiate the following functions with respect to ( x ) : ( cos ^{-1}left{frac{x}{sqrt{x^{2}+a^{2}}}right} ) |
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| 56 | The solution of the differential equation ( left(frac{d y}{d x}right)^{2}-3 xleft(frac{d y}{d x}right)-2 y=8 ) A ( cdot y=2 x^{2}+4 ) B . ( y=2 x^{2}-4 ) ( mathbf{c} cdot y=2 x+4 ) D. ( y=2 x-4 ) |
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| 57 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, boldsymbol{i f} boldsymbol{x}^{frac{2}{3}}+boldsymbol{y}^{frac{boldsymbol{2}}{3}}=boldsymbol{a}^{frac{boldsymbol{2}}{3}} ) | 12 |
| 58 | The value of ( f(2) ) is ( A cdot 2 ) B. 4 ( c .6 ) D. 8 |
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| 59 | Show that ( f(x)=|x-3| ) is continuous but not differentiable at ( x=3 ) |
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| 60 | If ( 3 x^{2}+4 x y-7 y^{2}=0 ) Find ( (a) frac{d y}{d x} ) and (b) ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} ) |
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| 61 | Suppose that ( f(0)=-3 ) and ( f^{prime}(x) leq 5 ) for all values of ( x ). Then the largest value which ( f(2) ) can assume is ( ldots . ) A . 5 B. 6 ( c cdot 7 ) D. |
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| 62 | If ( y=frac{sin ^{2} x}{1+cot x}+frac{cos ^{2} x}{1+tan x} ) then ( y^{prime} ) is equal to? |
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| 63 | Differentiate with respect to ( x ) ( boldsymbol{y}=sin 2 boldsymbol{x}-boldsymbol{4} boldsymbol{e}^{boldsymbol{3}} boldsymbol{x} ) |
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| 64 | ( operatorname{Let} y=sec left(frac{theta}{2}-1right) ) then find ( frac{d y}{d theta} ) | 12 |
| 65 | Find ( frac{d y}{d x} ) when ( x^{2}+y^{2}=c^{2} ) | 12 |
| 66 | Find the derivative of ( frac{2^{x} cot x}{sqrt{x}} ) A ( cdot frac{2^{x}}{sqrt{x}}left{log 2 cot x-csc ^{2} x-frac{cot x}{2 x}right} ) B. ( frac{2(x+1)}{sqrt{x}}left{log 2 cot x-csc ^{2} x-frac{cot x}{2 x}right} ) C ( frac{2^{x}}{sqrt{x}}left{log 2 cot x-csc ^{2} x-frac{cot x}{x^{2}}right} ) D. None of these |
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| 67 | The difference of slopes of lines represent by ( y^{2}-2 x y sec ^{2} alpha+ ) ( left(3+tan ^{2} alpharight)left(tan ^{2} alpha-1right) x^{2}=0 ) is ( A cdot 3 ) B. 4 ( c cdot 0 ) D. |
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| 68 | f ( boldsymbol{y}=(mathbf{1}+boldsymbol{x})left(mathbf{1}+boldsymbol{x}^{2}right)left(mathbf{1}+boldsymbol{x}^{4}right) dots(mathbf{1}+ ) ( left.x^{2^{n}}right), ) then ( frac{d y}{d x} ) at ( x=0 ) is ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot 0 ) D. none of these |
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| 69 | If ( x^{x}+x^{y}+y^{x}=a^{b}, ) then find ( frac{d y}{d x} ) | 12 |
| 70 | If the function ( u=f(x) ) is continuous at the point ( x=a ) and the function ( y=g(u) ) is continuous at the point ( u=f(a), ) then the composite function ( boldsymbol{y}=(boldsymbol{g} boldsymbol{o} boldsymbol{f})(boldsymbol{x})= ) ( g(f(x)) ) is A. continuous at the point ( x=f(a) ) B. continuous at the point ( x=a ) c. discontinuous at the point ( x=a ) D. continuous at the point ( x=g(a) ) |
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| 71 | If we apply the mean value theorem to ( f(x)=2 sin x+sin 2 x ) then ( c= ) This question has multiple correct options A . ( pi ) в. ( pi / 4 ) c. ( pi / 2 ) D. ( pi / 3 ) |
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| 72 | 19. Determine the values of x for which the following function fails to be continuous or differentiable: (1997 – 5 Marks) (1-x, xxl f(x)= (1-x)(2-x), 1sx52 Justify your answer. [3-x, x>2 |
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| 73 | If ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{t}+frac{mathbf{1}}{boldsymbol{t}} ) and ( boldsymbol{x}^{4}+boldsymbol{y}^{4}=boldsymbol{t}^{2}+ ) ( frac{1}{t^{2}} ) then ( frac{d y}{d x}= ) A ( cdot-frac{x}{y} ) в. ( frac{-y}{x} ) c. ( frac{x^{2}}{y^{2}} ) D. ( frac{y^{2}}{x^{2}} ) |
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| 74 | Solve: ( frac{x cos ^{-1} x}{sqrt{1-x^{2}}} d x ) |
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| 75 | Function ( f(x)=|x-2|-2|x-4| ) is discontinous at: A . ( x=2,4 ) B. ( x=2 ) C. No where D. Except ( x=2 ) |
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| 76 | Solve ( : frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(sin boldsymbol{3} boldsymbol{x})=? ) | 12 |
| 77 | Find the derivative of the following functions(it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers): ( frac{a x+b}{c x+d} ) |
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| 78 | Find the intervals in which the following functions are increasing or decreasing ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}-mathbf{5} ) | 12 |
| 79 | Differentiate the function with respect to ( boldsymbol{x} ) ( f(x)=x^{2 / 3}+7 e^{x}-frac{5}{x}+7 tan x ) |
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| 80 | Assertion Statement-1: ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{{boldsymbol{x}}} ) is discontinuous for integral values of ( boldsymbol{x} ) where ( vartheta ) denotes the fractional part function. Reason Statement-2: For integral values of ( boldsymbol{x} ) ( f(x) ) is not defined. A. Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1 B. Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1 c. statement-1 is true, Statement-2 is false. D. Statement-1 is false, Statement-2 is true. |
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| 81 | If ( boldsymbol{y}=frac{1}{mathbf{3 – 4 x}} ) then ( boldsymbol{y}_{n}(1) ) equals A . B ( cdot(-1)^{n+1} n ! ) c. ( n ! 4^{n}(-1)^{n+1} ) D. None of these |
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| 82 | Solve: ( frac{d}{d x} x sin ^{2} x ) | 12 |
| 83 | Find the derivative by first principle ( cos 5 x ) |
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| 84 | Differentiate ( sin boldsymbol{h}^{-1}left(frac{boldsymbol{x}}{mathbf{3}}right) ) with respect to ( x ). Find out the solution of the integration ( int frac{1}{left(x^{2}+9right)} d x ) Further find out the value of the integral ( int frac{1}{left(x^{2}+49right)} d x ? ) |
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| 85 | The function ( f(x)=frac{tan left{pileft[x-frac{pi}{2}right]right}}{2+[x]^{2}} ) where ( [x] ) denotes the greatest integer ( leq x, ) is A. continuous for all values of ( x ) B. Discontinuous at ( x=frac{pi}{2} ) c. Not differentiable for some values of ( x ) D. Discontinuous at ( x=-2 ) |
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| 86 | If the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{1-cos 4 x}{8 x^{2}}, x neq 0 \ k, x=0end{array} ) is continuous at right. ( boldsymbol{x}=mathbf{0} ) then ( boldsymbol{k}=? ) A . 1 B . 2 ( c cdot frac{1}{2} ) D. ( frac{-1}{2} ) |
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| 87 | If ( y= ) ( (x+sqrt{x^{2}-a^{2}})^{n} quad ) then ( quadleft(x^{2}-a^{2}right)left(frac{d y}{d x}right. ) A ( cdot n^{2} y ) B. ( -n^{2} y ) ( c cdot n y^{2} ) D cdot ( n^{2} y^{2} ) |
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| 88 | If ( y=cos ^{-1}(sqrt{x}), ) then find ( frac{d y}{d x} ) using first principle. A ( cdot-frac{1}{sqrt{1-x}} ) в. ( frac{1}{sqrt{1-x}} ) c. ( -frac{1}{2 sqrt{x} sqrt{1-x}} ) D. ( frac{1}{2 sqrt{x} sqrt{1-x}} ) |
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| 89 | Given that, ( y=sin xleft(x^{2}right) e^{x} ), find ( y^{prime} ) at ( boldsymbol{x}=mathbf{0} ) |
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| 90 | Derivative of ( tan ^{-1}left(frac{x}{sqrt{1-x^{2}}}right) ) with respect to ( sin ^{-1}left(3 x-4 x^{3}right) ) is A ( cdot frac{1}{sqrt{1-x^{2}}} ) в. ( frac{3}{sqrt{1-x^{2}}} ) ( c .3 ) D. |
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| 91 | If ( x=a t^{2} ) and ( y=2 a t, ) then ( frac{d y}{d x} ) is equal to ( mathbf{A} cdot t ) B. ( c . ) D. ( t^{2} ) |
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| 92 | If ( x sin y=3 sin y+4 cos y, ) then ( frac{d y}{d x}= ) ( mathbf{A} cdot frac{-sin ^{2} y}{4} ) B. ( frac{sin ^{2} y}{4} ) ( mathbf{c} cdot frac{-cos ^{2} y}{4} ) D. ( frac{cos ^{2} y}{4} ) |
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| 93 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{y}=log left(boldsymbol{4} boldsymbol{x}-boldsymbol{x}^{5}right) ) | 12 |
| 94 | ff ( y=x^{2} cos x ) then ( y_{8}(0) ) is A . 72 B . 56 ( c cdot 0 ) D. – 56 |
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| 95 | Find the derivative of ( boldsymbol{y}=frac{1}{boldsymbol{x}}+frac{mathbf{1}}{boldsymbol{x}^{2}}+ ) ( frac{mathbf{3}}{boldsymbol{x}^{mathbf{3}}} ) |
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| 96 | A function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) satisfies ( boldsymbol{f}(boldsymbol{x})= ) ( boldsymbol{f}(boldsymbol{2} boldsymbol{a}-boldsymbol{x}) . ) Suppose ( boldsymbol{f}(boldsymbol{x}) ) is differentiable at ( x=a ) then This question has multiple correct options A ( cdot f^{prime}(a)=0 ) B . ( f^{prime}left(a^{+}right)=-f^{prime}left(a^{-}right) ) c. ( f^{prime}left(a^{+}right)=f^{prime}left(a^{-}right)=0 ) D. None of these |
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| 97 | Find the derivative of ( sin x ) with respect to ( x ) from first principles. |
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| 98 | 3. If f(x) = x” then the value of 20031 0 1 ) 21 (b) 21 (1) 3! (-1)”/”(1) : n! 21-1 (a) I (c) (d) 0 |
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| 99 | If ( boldsymbol{x}^{m} cdot boldsymbol{y}^{n}=(boldsymbol{x}+boldsymbol{y})^{boldsymbol{m}+boldsymbol{n}}, ) then ( frac{d boldsymbol{y}}{d boldsymbol{x}} ) is : A ( cdot frac{y}{2 x} ) в. ( frac{2 y}{x} ) ( c cdot-frac{y}{x} ) D. ( frac{y}{x} ) |
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| 100 | Find the derivative of ( f(x) ) from the first principles, where ( boldsymbol{f}(boldsymbol{x}) ) is ( sin x+cos x ) |
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| 101 | The values of ( f^{prime}(1) ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 3 |
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| 102 | f ( sin theta+2 cos theta=1 ) then prove that ( 2 sin theta-cos theta=0 ) |
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| 103 | Discuss the continuity and differentiability of the function, ( boldsymbol{f}(boldsymbol{x})=left[begin{array}{cc}frac{boldsymbol{x}}{mathbf{1}+|boldsymbol{x}|} & |boldsymbol{x}|>mathbf{1} \ frac{boldsymbol{x}}{mathbf{1}-|boldsymbol{x}|} & |boldsymbol{x}| leq mathbf{1}end{array}right] ) | 12 |
| 104 | Derivative of ( left(tan ^{-1} xright)^{2} ) wrt to ( x ) | 12 |
| 105 | Assertion Consider the polynomial function ( f(x)=frac{x^{7}}{7}-frac{x^{6}}{6}+frac{x^{5}}{5}-frac{x^{4}}{4}+frac{x^{3}}{3}- ) ( frac{x^{2}}{2}+x . ) The equation ( f(x)=0 ) cannot have two or more roots. Reason Rolle’s theorem is not applicable for ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) on any interval ( [boldsymbol{a}, boldsymbol{b}], ) where ( boldsymbol{a}, boldsymbol{b} boldsymbol{epsilon} boldsymbol{R} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
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| 106 | ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{sin (a+1) x+sin x}{x} text { if } x0end{array}right. ) ( f(0)=c ) is continuous at ( x=0 ) then A ( cdot a=frac{-3}{2}, c=frac{1}{2}, b neq 0 ) B cdot ( a=b=frac{1}{2}, c=0 ) C ( cdot a=b=frac{1}{2}, c=0 ) D ( cdot a=frac{1}{2} b neq 0 c=1 ) |
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| 107 | If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}boldsymbol{x}-boldsymbol{3}, quad boldsymbol{x}<mathbf{0} \ boldsymbol{x}^{2}-mathbf{3} boldsymbol{x}+mathbf{2}, quad boldsymbol{x} geq mathbf{0}end{array} ) and right. ( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(|boldsymbol{x}|)+|boldsymbol{f}(boldsymbol{x})|, ) then ( boldsymbol{g}(boldsymbol{x}) ) is This question has multiple correct options A . continuous is ( R-{0} ) B. Continuous in ( R ) C . Differentiable in ( R-{0,1,2} ) D. Differentiable in ( R-{1,2} ) |
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| 108 | Differentiate the following functions w.r.t. ( boldsymbol{x} ) ( e^{operatorname{cosec}^{2} x} ) |
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| 109 | If ( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}+boldsymbol{2} sqrt{boldsymbol{2} boldsymbol{x}-boldsymbol{4}}}+ ) ( sqrt{x-2 sqrt{2 x-4}} ), then the value of ( mathbf{1 0} boldsymbol{f}^{prime}left(mathbf{1 0 2}^{+}right) ) is A . -1 B. c. 1 D. Does not exist |
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| 110 | ( boldsymbol{g}(boldsymbol{x}+boldsymbol{y})=boldsymbol{g}(boldsymbol{x})+boldsymbol{g}(boldsymbol{y})+mathbf{3} boldsymbol{x} boldsymbol{y}(boldsymbol{x}+ ) ( boldsymbol{y}) forall boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon} boldsymbol{R} ) and ( boldsymbol{g}^{prime}(mathbf{0})=-4 . ) For which of the following values of ( x ) is ( sqrt{g(x)} ) not defined? A ( cdot[-2,0] ) в. ( [-2, infty] ) c. [-1,1] D. none of these |
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| 111 | Find the differential equation of the following. ( tan ^{-1}left(frac{1-cos x}{sin x}right) ) | 12 |
| 112 | ( f y^{prime}=-3 xleft(2-x^{2}right)^{frac{1}{2}} ) then find ( y^{prime prime} ) | 12 |
| 113 | Assertion Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+mathbf{7} boldsymbol{x}+boldsymbol{4} ) be a polynomial function, then ( boldsymbol{f}^{prime}(mathbf{2})=mathbf{1 1} ) Reason A polynomial function is differentiable everywhere A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct |
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| 114 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( log left(frac{x^{2}+x+1}{x^{2}-x+1}right) ) |
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| 115 | If ( f(x)=x sin x, ) then ( f^{prime}left(frac{pi}{2}right) ) is equal to: A. B. c. -1 D. |
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| 116 | If ( y=A sin 5 x, ) then ( frac{d^{2} y}{d x^{2}}= ) begin{tabular}{l} A. ( -25 y ) \ hline end{tabular} в. ( 25 y ) c. ( 5 y ) D. ( -5 y ) |
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| 117 | Differentiate with respect to ( x ) : ( boldsymbol{y}=boldsymbol{e}^{-mathbf{3} boldsymbol{x}}+sin mathbf{2} boldsymbol{x} ) |
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| 118 | If ( f(x) ) and ( g(x) ) are differentiable functions for ( 0 leq x leq 1 ) such that ( boldsymbol{f}(mathbf{0})=mathbf{2}, boldsymbol{g}(mathbf{0})=mathbf{0}, boldsymbol{f}(mathbf{1})=mathbf{6}, boldsymbol{g}(mathbf{1})=mathbf{2} ) then in the interval ( (mathbf{0}, mathbf{1}) ) ( mathbf{A} cdot f^{prime}(x)=0 ) for all ( x ) B . ( f^{prime}(x)=2 g^{prime}(x) ) for at least one ( x ) ( mathbf{C} cdot f^{prime}(x)=2 g^{prime}(x) ) for at most one ( x ) D. None of these |
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| 119 | 1 – 2 2at 9. If x= 2 and y=- 1+1² 1+ dx las 24 (c) a(t+1) 10 24 (d) a(t? – 1) |
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| 120 | If ( boldsymbol{y}=log left[tan left(frac{boldsymbol{pi}}{boldsymbol{4}}+frac{boldsymbol{x}}{2}right)right] ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( A cdot sec x ) B. ( sin x ) c. ( operatorname{cosec} x ) D. ( sec frac{x}{2} ) |
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| 121 | ( lim _{x rightarrow 0} frac{tan ^{4} x-sin ^{4} x}{x^{4}}= ) A . 0 B. ( c cdot frac{2}{3} ) D. |
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| 122 | 11. ху Let y be an implicit function of x defined by X- it function of x defined by x2x2x4 coty -1=0. Then y'(1) equals (a) 1 (b) log 2 (C) -log 2 (d) -1 [20091 |
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| 123 | f the following function is continuous at ( x=frac{pi}{2}, ) then find ( a ) and ( b ) ( f(x)=left{begin{array}{l}frac{1-sin ^{2} x}{3 cos ^{2} x}, text { if } xfrac{pi}{2}end{array}right. ) | 12 |
| 124 | 6. If y=x* sin x + int , then it will be (a) 2x sin x + r cos + 3 tan x – 3x sec- x tan” x (b) 2x sin x + 3x sec? x – 3 tan x (b) 2x sin x + tan” x x? cos x + (3 tan x – 3x sec? x) tan” x x? cos x – 2x sin x – (3 tan x – 3x sec? x) tan? x |
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| 125 | State true or false: If ( u(x) ) and ( v(x) ) are differentiable functions such that ( frac{u}{v}(x)=7 ) ( frac{boldsymbol{u}^{prime}(boldsymbol{x})}{boldsymbol{v}^{prime}(boldsymbol{x})}=boldsymbol{p} ) and ( left(frac{boldsymbol{u}(boldsymbol{x})}{boldsymbol{v}(boldsymbol{x})}right)^{prime}=boldsymbol{q}, ) then ( frac{boldsymbol{p}+boldsymbol{q}}{boldsymbol{p}-boldsymbol{q}}=mathbf{1} ) A. True B. False |
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| 126 | Show that ( f(x)=e^{2 x} ) is increasing on ( R ) | 12 |
| 127 | Differentiate ( mathbf{3} boldsymbol{x}^{mathbf{1} / mathbf{3}}+frac{mathbf{6}}{mathbf{7}} boldsymbol{x}^{mathbf{7} / mathbf{6}}+mathbf{3} boldsymbol{x}^{mathbf{2} / mathbf{3}}+boldsymbol{C} ) |
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| 128 | If a function is everywhere continuous and differentiable such that ( f^{prime}(x) geq 6 ) for all ( boldsymbol{x} epsilon[mathbf{2}, mathbf{4}] ) and ( boldsymbol{f}(mathbf{2})=-mathbf{4}, ) then ( mathbf{A} cdot f(4)<8 ) в. ( f(4) geq 8 ) c. ( f(4) geq 2 ) D. none of these |
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| 129 | If ( f(x) ) is a polynomial function and ( boldsymbol{f}^{prime}(boldsymbol{x})>boldsymbol{f}(boldsymbol{x}), forall boldsymbol{x} geq 1 ) and ( boldsymbol{f}(1)=mathbf{0} ) then A. ( f(x) geq 0, forall x geq 1 ) B. ( f(x)<0, forall x geq 1 ) c. ( f(x)=0, forall x geq 1 ) D. None of the above |
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| 130 | Let ( f(x) ) be a real valued function not identically zero, such that ( boldsymbol{f}left(boldsymbol{x}+boldsymbol{y}^{n}right)=boldsymbol{f}(boldsymbol{x})+(boldsymbol{f}(boldsymbol{y}))^{n} quad forall boldsymbol{x}, boldsymbol{y} in ) ( boldsymbol{R} ) where ( n in N(n neq 1) ) and ( f^{prime}(0) geq 0 . ) We may get an explicit form of the function ( boldsymbol{f}(boldsymbol{x}) ) ( int_{0}^{1} f(x) d x ) is equal to A ( cdot frac{1}{2 n} ) B. ( 2 n ) ( c cdot frac{1}{2} ) D. 2 |
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| 131 | ( f(x)=left{begin{array}{c}a sin frac{pi}{2}(x+1), x leq 0 \ frac{tan x-sin x}{x^{3}}, x>0end{array}right. ) continuous at ( x=0 . ) Find the value of ( boldsymbol{a} ) |
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| 132 | If ( f(x)=frac{e^{x^{2}}-cos x}{x^{2}}, ) for ( x neq 0, ) is continuous at ( boldsymbol{x}=mathbf{0}, ) find ( boldsymbol{f}(mathbf{0}) ) |
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| 133 | Let ( y=sin ^{-1}(cos x) ) then find ( frac{d y}{d x} ) | 12 |
| 134 | If ( cos y=x cos (a+y), ) find ( frac{d y}{d x} ) | 12 |
| 135 | The value of ( K ) which the Function [ boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc} frac{tan 4 boldsymbol{x}}{tan 5 x}, & 0<x<frac{pi}{2} \ boldsymbol{k}+frac{2}{5}, & boldsymbol{x}=frac{pi}{2} end{array}right. ] continuous at ( x=frac{pi}{2}, ) is |
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| 136 | Consider the piecewise defined [ text { function }left{begin{array}{ll} sqrt{-x}, & text { if } x4 end{array}right. ] the answer which best describes the continuity of this function- A. the function is unbounded and therefore cannot be continuous B. the function is right continuous at ( x=0 ) c. the function has a removable discontinuity at 0 and 4 but is continuous on the rest of the real line D. the function is continuous on the entire real line |
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| 137 | ff ( y=fleft(x^{2}+2right) ) and ( f^{prime}(3)=5, ) then ( frac{d y}{d x} ) at ( x=1 ) is ( mathbf{A} cdot mathbf{5} ) B . 25 c. 15 D. 10 |
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| 138 | If ( f(x)=frac{1+tan x}{1-tan x} ) then ( fleft(frac{pi}{6}right) ) | 12 |
| 139 | Differentiate: ( sin ^{2} boldsymbol{y}+cos boldsymbol{x} boldsymbol{y}=boldsymbol{K} ) |
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| 140 | ff ( boldsymbol{x}=boldsymbol{a} sin boldsymbol{theta}+boldsymbol{b} cos boldsymbol{theta}, boldsymbol{y}=boldsymbol{a} cos boldsymbol{theta} ) ( -b sin theta ) then show that ( (a x+a y)^{2}+(b x-a y)^{2} ) ( =left(a^{2}+b^{2}right)^{2} ) |
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| 141 | Differentiation of ( (2 x+3)^{6} ) with respect to ( x ) is A ( cdot 12(2 x+3)^{5} ) B. ( 6(2 x+3)^{5} ) c. ( 3(2 x+3)^{5} ) D. ( 6(2 x+3)^{6} ) |
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| 142 | Discuss the continuity of the following function at the indicated point(s): ( boldsymbol{f}(boldsymbol{x})left{begin{array}{l}|boldsymbol{x}| cos left(frac{1}{boldsymbol{x}}right), boldsymbol{x} neq mathbf{0} \ mathbf{0}, quad boldsymbol{x}=mathbf{0}end{array} quad, text { at } boldsymbol{x}=mathbf{0}right. ) |
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| 143 | State True or False. If ( frac{e^{y}}{e^{x}}=x y, ) then ( y^{prime}=frac{2-log x}{(1-log x)^{2}} ) A. True B. False |
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| 144 | Examine the continuity of the following function at given points: (i) ( boldsymbol{f}(boldsymbol{x})= ) [ begin{array}{ll} frac{e^{5 x}-e^{2 x}}{sin 3 x}, & text { for } x neq 0 \ = & text { for } x neq 0 end{array} ] |
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| 145 | greatest integer function and (1993 – 1 Mark) 10. Let [:] denote the greatest integer f(x) = [tan x], then: (a) limo f(x) does not exist (b) f(x) is continuous at x=0 © f(x) is not differentiable at x=0 (d) f'(0=1 |
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| 146 | If ( x sqrt{1+y}+y sqrt{1+x}=0 ) and ( x neq y ) show that ( frac{d y}{d x}=frac{-1}{(1+x)^{2}} ) |
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| 147 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: ) ( sin boldsymbol{x}-boldsymbol{3} boldsymbol{x}=mathbf{5} boldsymbol{y} ) |
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| 148 | ( sqrt{1+left(frac{d^{2} y}{d x^{2}}right)^{3}}=left(2+frac{d y}{d x}right)^{1 / 3} ) Find it’s order and degree. A .2,3 в. 2,9 ( c cdot 2,6 ) D. 2,2 |
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| 149 | Differentiable ( log _{7}(log x) ) with respect to ( boldsymbol{x} ) |
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| 150 | If ( S_{1} ) and ( S_{2} ) are respectively the sets of local minimum and local maximum points of the functions, ( f(x)=9 x^{4}+ ) ( 12 x^{3}-36 x^{2}+25, x in R, ) then A ( . S_{1}={-2,1} ; S_{2}={0} ) B. ( S_{1}={-2,0} ; S_{2}={1} ) c. ( S_{1}={-2,} ; S_{2}={0,1} ) D. ( S_{1}={-1} ; S_{2}={0,2} ) |
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| 151 | If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cl}frac{1-sqrt{2} sin x}{pi-4 x}, & text { if } x neq frac{pi}{4} \ a, & text { if } x=frac{pi}{4}end{array}right. ) is continuous at ( frac{n}{4} ) then ( a= ) ( A ) B. 2 c. 1 ( 0 . frac{1}{1} ) |
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| 152 | Find the equation of the tangent to the curve ( y=frac{x-y}{(x-2)(x-3)} ) at the point where it cuts the ( x ) -axis. |
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| 153 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: ) ( boldsymbol{y}=boldsymbol{e}^{left(1+log _{e} boldsymbol{x}right)} ) |
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| 154 | If ( boldsymbol{x y}=boldsymbol{e}^{boldsymbol{x}-boldsymbol{y}} ) then This question has multiple correct options A ( cdot frac{d y}{d x} ) doesn’t exist at ( x=0 ) B. ( frac{d y}{d x}=0 ) when ( x=1 ) C. ( frac{d y}{d x}=frac{1}{2} ) when ( x=0 ) D. none of these |
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| 155 | Differentiate ( x^{sin x}+(sin x)^{cos x} ) w.r.t ( x ) | 12 |
| 156 | ( f(x)=frac{left(e^{k x}-1right)(sin k x)}{4 x^{2}}, x neq 0 ) ( boldsymbol{f}(mathbf{0})=mathbf{9}, ) is continuous at ( boldsymbol{x}=mathbf{0}, ) then ( mathbf{k} ) ( =? ) ( mathbf{A} cdot pm 2 ) B. ±6 ( c .pm 4 ) D. None of the above |
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| 157 | Assertion Derivative of ( 3 cot x+5 operatorname{cosec} x ) is ( -operatorname{cosec} x(3 operatorname{cosec} x+5 cot x) ) Reason ( boldsymbol{f}^{prime}(boldsymbol{a})=lim _{boldsymbol{h} rightarrow mathbf{0}} frac{boldsymbol{f}(boldsymbol{a}+boldsymbol{h})-boldsymbol{f}(boldsymbol{a})}{boldsymbol{h}} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct |
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| 158 | ff ( y=log left(sqrt{x}+frac{1}{sqrt{x}}right), ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}-mathbf{1}}{boldsymbol{2} boldsymbol{x}(boldsymbol{x}+mathbf{1})} ) |
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| 159 | If ( y=sqrt{frac{1-sin ^{-1} x}{1+sin ^{-1} x}} ) then ( y^{prime}(0) ) is equal to ( mathbf{A} cdot mathbf{1} ) B. ( 1 / 2 ) ( c cdot-1 ) D. ( sqrt{2} / 3 ) |
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| 160 | 7. Iff(x)= x-1, then on the interval [0, ] (1989-2 Marks) (a) tan (x)] and 1/f(x) are both continuous (b) tan f(x)] and 1/f(x) are both discontinuous (c) tan [fx)) and s-‘(x) are both continuous (d) tan fx)] is continuous but 1/f(x) is not. |
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| 161 | Find the value ( : frac{d}{d x}left{cos x^{0}right}=? ) | 12 |
| 162 | If ( f(x)=cos ^{-1}left[frac{1-(log x)^{2}}{1+(log x)^{2}}right], ) then ( boldsymbol{f}^{prime}(boldsymbol{e})= ) ( A cdot frac{1}{e} ) B. ( frac{2}{e^{2}} ) ( c cdot frac{2}{e} ) D. None of these |
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| 163 | 16. Differentiation of x2 w.r.t. x is… | 12 |
| 164 | Discuss the countinuity of the following function at ( boldsymbol{x}=mathbf{0} ) ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}frac{1-cos x}{x^{2}}, & x neq 0 \ frac{1}{2}, & x=0end{array}right. ) |
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| 165 | Find the value ( c ) in mean value theorem for the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x} boldsymbol{epsilon}[mathbf{1}, boldsymbol{3}] ) |
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| 166 | ( f(x)=left{begin{array}{lc}frac{x^{2}}{a} quad ; & 0 leq x<1 \ -1 & 1 leq x<sqrt{2} \ frac{2 b^{2}-4 b}{x^{2}} & ; sqrt{2} leq x<inftyend{array}right. ) then find the value of ( a ) and ( b ) if ( f(x) ) is continuous in ( [mathbf{0}, infty) . ) Find ( boldsymbol{a}+boldsymbol{b} ) |
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| 167 | 0 1 21. Let f be differentiable for all x. Iff(1) =-2 and f ‘(x) 2 2 for x 6 [1, 6], then [2005] @ f6 28 (b) f(6<8 (c) f(6)<5 (d) f(6=5 |
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| 168 | Differentiate with respect to ( x ) : ( mathbf{3}^{e^{x}} ) |
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| 169 | If ( y sin x=x+y ) then ( left(frac{d y}{d x}right)_{x=0} ) equals ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot 0 ) D. 2 |
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| 170 | Let ( f(x)=tan 2 x cdot tan 3 x cdot tan 5 x, ) then ( boldsymbol{f}^{prime}(boldsymbol{pi}) ) equals A . 10 B. -10 ( c .0 ) D. |
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| 171 | Solution of differential equation ( x^{2}= ) ( 1+left(frac{x}{y}right)^{-1} frac{d y}{d x}+frac{left(frac{x}{y}right)^{-2}left(frac{d y}{d x}right)^{2}}{2 !}+ ) ( frac{left(frac{x}{y}right)^{-3}left(frac{d y}{d x}right)^{3}}{3 !}+ldots ) |
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| 172 | For what value of ( a ), the function [ boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll} frac{1-cos 4 x}{boldsymbol{x}^{2}}, & boldsymbol{i f} quad boldsymbol{x}0 end{array}right. ] continuous at ( boldsymbol{x}=mathbf{0} ) |
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| 173 | Let ( h(x) ) be differentiable for all ( x ) and let ( boldsymbol{f}(boldsymbol{x})=left(boldsymbol{k} boldsymbol{x}+boldsymbol{e}^{boldsymbol{x}}right) boldsymbol{h}(boldsymbol{x}) ) where ( boldsymbol{k} ) is some constant. If ( h(0)=5, h^{prime}(0)=-2 ) and ( f^{prime}(0)=18, ) then the value of ( k ) is equal to ( A cdot 3 ) B. 4 c. D. |
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| 174 | Find the derivative of ( boldsymbol{y}= ) ( tan ^{-1}left(frac{boldsymbol{a} boldsymbol{x}-boldsymbol{b}}{boldsymbol{b} boldsymbol{x}+boldsymbol{a}}right) ) | 12 |
| 175 | ( boldsymbol{g}(boldsymbol{x})=lim _{m rightarrow infty} frac{boldsymbol{x}^{m} boldsymbol{f}(boldsymbol{x})+boldsymbol{h}(boldsymbol{x})+boldsymbol{3}}{boldsymbol{2} boldsymbol{x}^{m}+boldsymbol{4} boldsymbol{x}+mathbf{1}} ) when ( x neq 1 ) and ( g(1)=e^{3} ) such that ( f(x), g(x) ) and ( h(x) ) are continuous function at ( boldsymbol{x}=mathbf{1} ) and ( boldsymbol{f}(mathbf{1})-boldsymbol{h}(mathbf{1})= ) ( a(b-g(1)) ) then ( a+b ) is |
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| 176 | If ( f(x)=frac{x^{3}+x^{2}-16 x+20}{(x-2)^{2}}, ) if ( x neq 2 ) [ =k, quad text { if } x=2 ] is continuous at ( x=0 ) then ( mathbf{A} cdot k=2 ) B. ( k=0 ) c. ( k=20 ) ( mathbf{D} cdot k=7 ) |
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| 177 | ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{a}+boldsymbol{b}^{frac{1}{x}}}{boldsymbol{c}+boldsymbol{d}^{frac{1}{x}}}, boldsymbol{b}>1, boldsymbol{d}>1, boldsymbol{c} neq mathbf{0} ) ( boldsymbol{f}(mathbf{0})=mathbf{1} ) is left continuous at ( boldsymbol{x}=mathbf{0} ) then ( mathbf{A} cdot a=0 ) B . ( a=2 c ) c. ( a=c ) D. ( a neq c ) |
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| 178 | ( lim _{x rightarrow 1} 2 x+1 ) | 12 |
| 179 | If ( f^{prime prime}(x)<0, forall x epsilon(a, b), ) and ( (c, f(c)) ) is point of maxima, where ( c epsilon(a, b), ) then ( f^{prime}(c) ) is A ( cdot frac{f(b)-f(a)}{b-a} ) в. ( left[frac{f(b)-f(a)}{b-a}right] ) c. ( 2left[frac{f(b)-f(a)}{b-a}right] ) D. |
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| 180 | 6. If f(x) = mx + c,fo) =f(0) = 1 then f(2)= (a) 1 (6) 2 (c) 3 (d) – 3 |
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| 181 | The function ( y=f(x) ) is ? A . odd B. even c. increasing D. decreasing |
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| 182 | Identify a possible graph for function given by ( f(x)=-(x-2)^{3}+1 ) A. graph a B. graph b c. grpah c D. grpah d |
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| 183 | Differentiate: ( log (log x), x>1 ) | 12 |
| 184 | Diff: ( cos ^{-1}left(frac{2 x}{1+x^{2}}right) ) w.r.t. ( x ) | 12 |
| 185 | Let ( boldsymbol{f}:[mathbf{0}, mathbf{1}] rightarrow boldsymbol{R} ) be a continuous function then the maximum value of ( int_{0}^{1} f(x) cdot x^{2} d x-int_{0}^{1} x cdot(f(x))^{2} d x ) for all such function(s) is ( A cdot frac{1}{8} ) B. ( frac{1}{20} ) c. ( frac{1}{12} ) D. ( frac{1}{16} ) |
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| 186 | Given, ( boldsymbol{f}(boldsymbol{x})=-frac{boldsymbol{x}^{3}}{mathbf{3}}+boldsymbol{x}^{2} sin mathbf{1 . 5} boldsymbol{a}- ) ( x sin a cdot sin 2 a-5 a r c sin left(a^{2}-8 a+17right) ) then A. ( f(x) ) is not defined at ( x=sin 8 ) B. ( f^{prime}(sin 8)>0 ) c. ( f^{prime}(x) ) is not defined at ( x=sin 8 ) D. ( f^{prime}(sin 8)<0 ) |
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| 187 | If ( f(x)=(a x+b) cos x+(c x+d) sin x ) and ( f^{prime}(x)=x cos x, ) for all values of ( x in ) ( R, ) then ( a, b, c, d ) are given by A ( a=b=c=d ) ( d ) в. 0,1,-1,0 c. 1,0,-1,0 D. 0,1,1,0 |
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| 188 | [ text { If } f(x)=frac{x^{3}+x^{2}-16 x+20}{(x-2)^{2}}, x neq 2 ] ( =k, x=2 ) is continuous at ( x=2 ) find the value of k. |
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| 189 | 3. There exist a function f(x), satisfying f(0) = 1,7 nction f (x), satisfying f(0) = 1, f'(0)=-1, f(x) > 0 for allx, and (1982 – 2 Marks) (a) “(x) > 0 for all x (b) -1<f"(x) <0 for all x (c) -2 "(x) S-1 for all (d) F"(x)<-2 for all x |
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| 190 | ( operatorname{Let} h(x)=min left{x, x^{2}right} ) for ( x in R ) Then which of the following is correct A. ( h ) is continuous for all ( x ) B. ( h ) is differentiable for all ( x ) C ( cdot h(x)=1 ) for all ( x>1 ) D. ( h ) is not a differentiable at 2 values of ( x ) |
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| 191 | If the function ( f ) is continuous at ( x=0 ) find ( boldsymbol{f}(mathbf{0}) ) where ( boldsymbol{f}(boldsymbol{x})=frac{cos 3 boldsymbol{x}-cos boldsymbol{x}}{boldsymbol{x}^{2}}, boldsymbol{x} neq mathbf{0} ) |
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| 192 | ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(boldsymbol{x}-1) ) in the interval ( [mathbf{1}, mathbf{2}] ) if ( boldsymbol{f}^{prime}(boldsymbol{c})=boldsymbol{f}(1) . ) Find ( c ) |
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| 193 | If ( y=tan ^{-1}left(frac{1}{1+x+x^{2}}right)+ ) ( tan ^{-1}left(frac{1}{x^{2}+3 x+2}right)+ ) ( tan ^{-1}left(frac{1}{x^{2}+5 x+6}right)+ldots+ ) upto ( n ) terms then ( frac{d y}{d x} ) at ( x=0 ) and ( n=1 ) is equal to A ( frac{1}{2} d ) B. ( -frac{1}{2} ) ( c ) ( D ) |
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| 194 | Find the derivative of ( left(x^{2}+cos xright) ) ( mathbf{A} cdot 2 x+sin x ) B. ( 2 x^{2}+sin x ) ( mathbf{c} cdot 2 x-sin x ) D. ( 2 x-cos x ) |
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| 195 | If ( boldsymbol{y}=frac{boldsymbol{x}}{boldsymbol{x}+mathbf{1}}+frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}}, ) then ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} ) at ( boldsymbol{x}= ) 1 is equal to ( A cdot frac{7}{4} ) B. ( frac{7}{8} ) ( c cdot frac{1}{4} ) D. ( frac{-7}{8} ) E ( frac{-7}{4} ) |
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| 196 | If ( boldsymbol{y}=(mathbf{1}+boldsymbol{x})left(mathbf{1}+boldsymbol{x}^{mathbf{2}}right)left(mathbf{1}+boldsymbol{x}^{mathbf{4}}right) dots mathbf{.} ) ( left.x^{2^{n}}right), ) then ( left(frac{d y}{d x}right)_{x=0}= ) ( mathbf{A} cdot mathbf{0} ) в. ( frac{1}{2} ) ( c cdot 1 ) ( D ) |
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| 197 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) ( boldsymbol{y}=sin boldsymbol{x} . cos boldsymbol{x} ) |
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| 198 | Let ( x^{k}+y^{k}=a^{k},(a, k>0) ) and ( frac{d y}{d x}+left(frac{y}{x}right)^{1 / 3}=0, ) then ( k ) is : A ( cdot frac{1}{3} ) B. ( frac{2}{3} ) ( c cdot frac{4}{3} ) ( D cdot 3 ) ( overline{2} ) |
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| 199 | Assertion Consider the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x} ) and ( boldsymbol{g}(boldsymbol{x})=-|boldsymbol{x}| ) The composite function ( boldsymbol{F}(boldsymbol{x})= ) ( boldsymbol{f}(boldsymbol{g}(boldsymbol{x})) ) is not derivable at ( boldsymbol{x}=mathbf{0} ) Reason ( boldsymbol{f}^{prime}left(mathbf{0}^{+}right)=mathbf{2} ) and ( boldsymbol{f}^{prime}left(mathbf{0}^{-}right)=-mathbf{2} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
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| 200 | If Rolle’s theorem is applicable to the function, ( boldsymbol{f}(boldsymbol{x})=frac{ln boldsymbol{x}}{boldsymbol{x}} ) over the interval ( [a, b], ) where ( a, b in I^{+} ) then the value of ( a+b ) is |
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| 201 | If ( y=tan ^{-1}left(frac{a cos x-b sin x}{b cos x+a sin x}right), ) then ( frac{d y}{operatorname{isequal}} ) to ( d x ) ( A cdot 2 ) B. – ( c cdot a ) ( D ) |
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| 202 | 21. If y = sin x, then will be … |
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| 203 | Evaluate ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{e}^{boldsymbol{x}} boldsymbol{operatorname { s i n }} boldsymbol{x}}{left(boldsymbol{x}^{2}+boldsymbol{2}right)^{3}} ) |
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| 204 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(tan ^{2} boldsymbol{a} boldsymbol{x}right) ) A ( cdot 2 a tan a x sec ^{2} a x ) B. ( -2 a tan a x sec ^{2} a x ) ( mathbf{c} cdot a tan a x sec ^{2} a x ) D. ( 2 a cot a x sec ^{2} a x ) |
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| 205 | If for ( boldsymbol{x} inleft(mathbf{0}, frac{mathbf{1}}{mathbf{4}}right), ) the derivative ( tan ^{-1}left(frac{6 x sqrt{x}}{1-9 x^{3}}right) ) is ( sqrt{x} . g(x), ) then ( g(x) ) equals: A ( cdot frac{3}{1+9 x^{3}} ) в. ( frac{9}{1+9 x^{3}} ) c. ( frac{3 x sqrt{x}}{1-9 x^{3}} ) D. ( frac{3 x}{1-9 x^{3}} ) |
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| 206 | Differentiate the following with respect to ( boldsymbol{x} ) ( cos ^{-1}[sqrt{frac{1+x}{2}}],-1<x<1 ) |
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| 207 | If ( boldsymbol{y}=sqrt{boldsymbol{x}+boldsymbol{y}}, ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( frac{1}{(2 y-1)} ) |
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| 208 | Find ( frac{d y}{d x}, ) if ( y=sqrt{1+sin 2 x} ) | 12 |
| 209 | Find the derivative of ( y=sqrt{x^{2}+1} ) | 12 |
| 210 | Which one of the following function is continuous everywhere in its domain but has at least one point where it is not differentiable? A ( . f(x)=x^{1 / 3} ) в. ( f(x)=frac{|x|}{x} ) ( mathbf{c} cdot f(x)=e^{-x} ) D. ( f(x)=tan x ) |
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| 211 | If ( boldsymbol{y}=(boldsymbol{A}+boldsymbol{B} boldsymbol{x}) e^{m boldsymbol{x}}+(boldsymbol{m}-mathbf{1})^{-2} boldsymbol{e}^{boldsymbol{x}} ) ( operatorname{then} frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}-2 boldsymbol{m} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{m}^{2} boldsymbol{y} ) is equal to: A ( cdot e^{x} ) B. ( e^{m} ) ( mathbf{c} cdot e^{-m x} ) D. ( e^{(1-m) x} ) |
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| 212 | Find the second derivative of the function ( log x ) |
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| 213 | The solution set of ( f^{prime}(x)>g^{prime}(x) ) where ( f(x)=left(frac{1}{2}right) 5^{2 x+1} ) and ( g(x)=5^{x}+ ) ( 4 x log 5 ) is ( A cdot(1, infty) ) B ( cdot(0,1) ) ( c cdot[2, infty) ) D. ( (0, infty) ) |
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| 214 | If ( frac{cos ^{4} theta}{x}+frac{sin ^{4} theta}{y}=frac{1}{x+y} ) then ( frac{d y}{d x}= ) A . ( x y ) B. ( tan ^{2} theta ) ( c cdot 0 ) D. ( left(x^{2}+y^{2}right) sec ^{2} theta ) |
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| 215 | ( boldsymbol{f}(boldsymbol{x})=mathbf{1}+frac{mathbf{1}}{boldsymbol{x}} ; boldsymbol{g}(boldsymbol{x})=frac{mathbf{1}}{mathbf{1}+boldsymbol{f}(boldsymbol{x})} Rightarrow ) ( boldsymbol{g}^{prime}(mathbf{2})= ) ( A cdot frac{1}{5} ) в. ( frac{1}{25} ) c. 5 D. ( frac{1}{16} ) |
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| 216 | If ( y=a^{frac{1}{2} log _{a} cos x}, ) find ( frac{d y}{d x} ) | 12 |
| 217 | ( f(x)=cot ^{-1}left(frac{x^{x}-x^{-x}}{2}right) ) then ( f^{1}(1)= ) ( mathbf{A} cdot-log 2 ) B. ( log 2 ) ( c cdot 1 ) D. – |
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| 218 | ( frac{e^{1 / x}-e^{-1 / x}}{e^{1 / x}+e^{-1 / x}} ) check continuity at ( x= ) ( mathbf{0}^{-} ) |
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| 219 | 1-cos 4x xco 14. Let f(x) = {a, x=0 (1990 – 4 Marks) >0 | 16+ √x – 4 Determine the value of a, if possible, so that the function is continuous at x = 0 |
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| 220 | Solve ( : boldsymbol{y}=sin ^{-1}left(frac{1-boldsymbol{x}^{2}}{1+boldsymbol{x}^{2}}right), boldsymbol{0}<boldsymbol{x}<1 ) | 12 |
| 221 | Find the derivative of ( cos ^{2} x, ) by using first principle of derivatives. |
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| 222 | Find the derivative of ( x^{-4}left(3-4 x^{-5}right) ) |
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| 223 | Find the value of ( k, ) so that the function ( f(x) ) is continuous at the indicated point ( left.boldsymbol{f}(boldsymbol{x}) begin{array}{l}=frac{8^{x}-2^{x}}{k^{x}-1} text { for } boldsymbol{x} neq boldsymbol{o} \ =boldsymbol{2} quad boldsymbol{x}=mathbf{0}end{array}right} boldsymbol{a} boldsymbol{t} boldsymbol{x}=0 ) |
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| 224 | If ( y=5^{x} x^{5}, ) then ( frac{d y}{d x} ) is A ( cdot 5^{x}left(x^{5} log 5-5 x^{4}right) ) B . ( x^{5} log 5-5 x^{4} ) c. ( x^{5} log 5+5 x^{4} ) D. ( 5^{x}left(x^{5} log 5+5 x^{4}right) ) |
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| 225 | differentiate : ( boldsymbol{y} log boldsymbol{x} ) |
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| 226 | If ( y=frac{f(x)}{phi(x)} ) and ( z=frac{f^{prime}(x)}{phi^{prime}(x)}, ) then ( frac{f^{prime prime}}{f}- ) ( frac{phi^{prime prime}}{phi}+frac{2(y-z)}{f phi}left(phi^{prime}right)^{2}= ) A ( cdot frac{d^{2} y}{d x^{2}} ) B. ( frac{1}{y} cdot frac{d^{2} y}{d x^{2}} ) c. ( y cdot frac{d^{2} y}{d x^{2}} ) D. None of these |
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| 227 | Find the derivative of ( f(log x) ) with respect to ( x ) where ( f(x)=log x ) |
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| 228 | Identify the graph of the polynomial function ( boldsymbol{f} ) ( f(x)=x^{4}-2 x^{3}-x^{2}+2 x ) A. graph a B. graph b c. graph c D. graoh d |
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| 229 | Xis 01+ 1 x D (1987-2 Marks) The set of all points where the function f(x) differentiable, is (a) (-00,00) (b) [0,00) (c) (-0,0 (0,0) (d) (0,00) (e) None |
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| 230 | Say true or false. The derivative of a constant function is always non-zero. A. True B. False |
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| 231 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(cos ^{-1}left(4 boldsymbol{x}^{3}-mathbf{3} boldsymbol{x}right)right),=therefore frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( frac{boldsymbol{m}}{sqrt{mathbf{1}-boldsymbol{x}^{k}}} . ) Find ( boldsymbol{k}-boldsymbol{m} ) ? |
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| 232 | A curve passing through the point (1,1) is such that the intercept made by a tangent to it on ( x ) -axis is three times the x co-ordinate of the point of tangency, then the equation of the curve is: A ( cdot y=frac{1}{x^{2}} ) в. ( y=sqrt{x} ) c. ( y=frac{1}{sqrt{x}} ) D. none |
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| 233 | If ( boldsymbol{y}=cos sqrt{boldsymbol{x}} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 234 | 2. If y = 1 -. then (a) (-a) (e) (3 + a)2 (6) -(2-a)2 (d) -(z + a)2 |
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| 235 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}{log (boldsymbol{x}+sqrt{boldsymbol{a}^{2}+boldsymbol{x}^{2}})}= ) A. ( frac{1}{(x+sqrt{a^{2}+x^{2}})} ) в. ( frac{x}{sqrt{a^{2}+x^{2}}} ) c. ( frac{1}{x(x+sqrt{a^{2}+x^{2}})} ) D. ( frac{1}{sqrt{a^{2}+x^{2}}} ) |
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| 236 | Find the derivative of the following functions(it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers): os as ( 1+sin ) |
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| 237 | Differentiate wirh respect to ( sqrt{x+frac{1}{x}} ) |
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| 238 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=(sin boldsymbol{x})^{boldsymbol{x}}+sin ^{-1} sqrt{boldsymbol{x}} ) |
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| 239 | If ( boldsymbol{y}=frac{mathbf{1}}{mathbf{1}+boldsymbol{x}^{boldsymbol{beta}-boldsymbol{alpha}}+boldsymbol{x}^{boldsymbol{gamma}-boldsymbol{alpha}}}+ ) ( frac{1}{1+x^{alpha-beta}+x^{gamma-beta}}+frac{1}{1+x^{alpha-gamma}+x^{beta-gamma}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{r}} ) is equal to A . 0 B. 1 C ( cdot(a+beta+gamma) X^{alpha+beta+gamma-1} ) D. None of these |
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| 240 | If ( f(x)=x^{frac{1}{x}} ) then ( f^{prime prime}(e) ) is equal to B ( cdot e^{1 / e} ) ( mathbf{c} cdot e^{1 /(e-2)} ) D. ( left.-e^{((1 / e)-3}right) ) |
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| 241 | Differentiate ( frac{x^{2}+1}{x} ) w.r.t ( x ) | 12 |
| 242 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) where ( boldsymbol{x}^{mathbf{3}}+boldsymbol{y}^{mathbf{3}}+mathbf{3} boldsymbol{x} boldsymbol{y}=mathbf{7} ) | 12 |
| 243 | 25. Iff(x) = xa log x and f(0) = 0, then the value of a for which Rolle’s theorem can be applied in [0, 1] is (20045 (a) -2 (6) -1 (c) o (d) 1/2 |
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| 244 | Diffrentiate w.r.t ( x: ) ( boldsymbol{y}=e^{2 x}(boldsymbol{a}+boldsymbol{b} boldsymbol{x}) ) |
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| 245 | Let y be an implicit function of ( mathbf{x} ) defined by ( mathbf{x}^{2 mathbf{x}}-mathbf{2} mathbf{x}^{mathbf{x}} cot boldsymbol{y}-mathbf{1}=mathbf{0} ) Then ( y^{prime}(1) ) equals A . -1 B. ( c cdot log 2 ) D. ( -log 2 ) |
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| 246 | If ( y=sqrt{frac{1+tan x}{1-tan x}} ) then ( frac{d y}{d x} ) is equal to A ( cdot frac{1}{2} sqrt{frac{1-tan x}{1+tan x}} sec ^{2}left(frac{pi}{4}+xright) ) в. ( sqrt{frac{1-tan x}{1+tan x}} sec ^{2}left(frac{pi}{4}+xright) ) c. ( frac{1}{2} sqrt{frac{1-tan x}{1+tan x}} sec left(frac{pi}{4}+xright) ) D. None of these |
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| 247 | Find the derivative of ( operatorname{cosec}^{2} x, ) by using first principle of derivatives? |
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| 248 | The function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}3 x-1, & text { if } x2end{array} ) continuous on right. A ( cdot(-infty, 1) ) B. ( (2, infty) ) C ( .(-infty, 1) cup(2, infty) ) D. (1,2) |
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| 249 | If the function ( mathbf{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{sin 3 x}{x} & (x neq 0) \ frac{k}{2} & (x=0)end{array} ) is continuous at right. ( x=0, ) then ( k ) is: ( A cdot 3 ) B. 6 ( c cdot 9 ) ( D ) |
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| 250 | Let ( boldsymbol{f}:[mathbf{1}, infty] rightarrow[mathbf{2}, infty] ) if ( boldsymbol{f}(mathbf{1})=mathbf{2} . ) be differentiable function such that ( 6 int_{1}^{x} f(t) d t=3 x f(x)-x^{3} ) then the value of ( boldsymbol{f}(mathbf{2}) ) is…. |
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| 251 | Differentiate w.r.t. ( boldsymbol{x} ) ( boldsymbol{y}=boldsymbol{e}^{cos (boldsymbol{6} boldsymbol{x}-mathbf{1})} ) |
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| 252 | If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) satisfies ( |boldsymbol{f}(boldsymbol{x})-boldsymbol{f}(boldsymbol{y})| leq ) ( |x-y|^{3} ) and ( f(4)=192 ) then ( f(7) ) is equal to |
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| 253 | ( boldsymbol{g}(boldsymbol{x}+boldsymbol{y})=boldsymbol{g}(boldsymbol{x})+boldsymbol{g}(boldsymbol{y})+mathbf{3} boldsymbol{x} boldsymbol{y}(boldsymbol{x}+ ) ( boldsymbol{y}) forall boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon} boldsymbol{R} ) and ( boldsymbol{g}^{prime}(mathbf{0})=-4 . ) The value of ( boldsymbol{g}^{prime}(1) ) is A . 0 B. c. -1 D. none of these |
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| 254 | Show that the function ( f(x) ) defined as ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} cos frac{1}{x}, boldsymbol{x} neq mathbf{0},=mathbf{0}, boldsymbol{x}=mathbf{0} ) is continuous at ( x=0 ) but not differentiable at ( x=0 ) |
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| 255 | Let ( boldsymbol{f} ) be a function satisfying ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) ) for all ( boldsymbol{x} ) and ( boldsymbol{y} ) and ( boldsymbol{f}(mathbf{0})=boldsymbol{f}^{prime}(mathbf{0})=mathbf{1} ) then This question has multiple correct options A. ( f ) is differentiable for all ( x ) B . ( f^{prime}(x)=f(x) ) ( mathbf{c} cdot f(x)=e^{x} ) D. ( f ) is continuous for all ( x ) |
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| 256 | 28. Let a + b = 4, where a 0 for all x, prove that so g(x) dx + So g(x)dx dx increases as (b-a) increases. (1997- 5 Marks) |
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| 257 | Differentiate w.r.t ( boldsymbol{x}: boldsymbol{x}^{boldsymbol{y}}+boldsymbol{y}^{boldsymbol{x}}=mathbf{1} ) | 12 |
| 258 | ff ( (x)=3 e^{x^{2}}, ) then ( f^{prime}(x)-2 x f(x)+ ) ( frac{1}{3} f(0)-f^{prime}(0) ) is equal to A. B. ( mathrm{c} cdot frac{7}{3} mathrm{e}^{x} ) D. ex ( ^{x^{2}} ) |
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| 259 | f ( p^{2}=a^{2} cos ^{2} theta+b^{2} sin ^{2} theta ) then ( frac{d^{2} p}{d theta^{2}}+ ) ( p ) is equal to ( (a neq b) ) A ( cdot frac{a^{2} b^{2}}{p^{4}} ) в. ( frac{a^{2} b^{2}}{p^{2}} ) c. ( frac{a b}{p} ) D. ( frac{a^{2} b^{2}}{p^{3}} ) |
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| 260 | 5. If x=ete . (20041 . x > 0, then 1+X 1-* (b) (d) |
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| 261 | If ( y=tan ^{-1}left(frac{2^{x^{prime}}}{1+2^{2 x+1}}right) ) then ( frac{d y}{d x} ) at ( boldsymbol{x}=mathbf{0} ) is? A ( cdot frac{1}{10} log 2 ) в. ( frac{1}{5} log 2 ) c. ( -frac{1}{10} log 2 ) D. ( log 2 ) |
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| 262 | Find the differentiation of ( sec left(tan ^{-1} xright) ) w.r.t. ( boldsymbol{x} ) |
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| 263 | Ify is a function of x and log (x + y) – 2xy=0, then the value of y’ (O) is equal to (2004S) (a) 1 (b) -1 (c) 2 (d) O |
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| 264 | ( frac{d^{2} x}{d y^{2}}=-frac{d^{2} y}{d x^{2}} cdotleft(frac{d x}{d y}right)^{3} ) | 12 |
| 265 | Find the value of ( f(2), ) so that the function ( boldsymbol{f}(boldsymbol{x})=frac{12 boldsymbol{x}-mathbf{2 4}}{(mathbf{4}+mathbf{2} boldsymbol{x})^{1 / 3}-mathbf{2}}, boldsymbol{x} neq mathbf{2} ) is continuous everywhere |
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| 266 | If ( f(x) ) is continuous and ( fleft(frac{9}{2}right)=frac{2}{9} ) then ( lim _{x rightarrow 0} fleft(frac{1-cos 3 x}{x^{2}}right) ) is equal to: A ( cdot frac{9}{2} ) в. ( frac{2}{9} ) ( c cdot 0 ) D. ( frac{8}{9} ) |
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| 267 | ( y=frac{x^{2}}{1+x} ) Find ( frac{d y}{d x} ) | 12 |
| 268 | ff ( y=sin ^{-1} x, ) show that ( left(1-x^{2}right) cdot frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}=0 ) |
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| 269 | 7. [20021 fis defined in (-5, 5] as f(x)=x ifx is rational = -x ifx is irrational. Then (a) f(x) is continuous at every x, except x = 0 (b) f(x) is discontinuous at every x, except x = 0 (c) f(x) is continuous everywhere (d) f(x) is discontinuous everywhere |
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| 270 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left(frac{1-boldsymbol{x}}{mathbf{1}+boldsymbol{x}}right)= ) A ( cdot frac{2}{1+x^{2}} ) B ( cdot frac{-1}{1+x^{2}} ) c. ( frac{1}{1+x^{2}} ) D. ( frac{-2}{1+x^{2}} ) |
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| 271 | If ( y=frac{(1-x)^{2}}{x^{2}} ) where ( x neq ) ( 0, ) then ( frac{d y}{d x} i s ) A ( cdot frac{2}{x^{2}}+frac{2}{x^{3}} ) B. ( -frac{2}{x^{3}}+frac{2}{x^{2}} ) ( c cdot-frac{2}{x^{2}}+frac{2}{x^{3}} ) D. ( -frac{2}{x^{2}}-frac{2}{x^{3}} ) |
12 |
| 272 | Differentiate ( y=sin b x^{2} ) w.r.t ( x ) | 12 |
| 273 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( frac{e^{x} sin x}{left(x^{2}+2right)^{3}} ) |
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| 274 | Let ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) ) for all ( boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon} boldsymbol{R} ) and suppose that ( f ) is differentiable at 0 and ( f^{prime}(0)=4 . ) If ( fleft(x_{0}right)=8 ) then ( f^{prime}left(x_{0}right) ) is equal to |
12 |
| 275 | Let ( f(x) ) be defined by ( f(x)= ) ( left{begin{array}{cl}sin 2 x & text { if } 0<x leq frac{pi}{6} \ a x+b & text { if } frac{pi}{6}<x leq 1end{array} . text { The values of } aright. ) and ( b ) such that ( f ) and ( f^{prime} ) are continuous, are A ( cdot a=1, b=frac{1}{sqrt{2}}+frac{pi}{6} ) в. ( a=frac{1}{sqrt{2}}, b=frac{1}{sqrt{2}} ) c. ( _{a=1, b}=frac{sqrt{3}}{2}-frac{pi}{6} ) D. None of these |
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| 276 | The derivative of ( sin ^{-1} frac{2 x}{1+x^{2}} ) with respect to ( cos ^{-1} frac{1-x^{2}}{1+x^{2}} ) is A . -1 B. ( c cdot 2 ) D. 4 |
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| 277 | Find the derivative of ( f(x) ) from the first principle. ( sin x div cos x ) |
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| 278 | If ( (cos x)^{y}=(sin y)^{x} ) then ( frac{d y}{d x}= ) A ( cdot frac{log (sin y)+y tan x}{log (cos x)-x cot y} ) B. ( frac{log (sin y)-y tan x}{log (cos x)+cot y} ) c. ( log (sin y) ) D. ( frac{log (cos x)}{log (sin y)} ) |
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| 279 | Evaluate ( frac{d}{d x} 3^{log _{3} sqrt{x}}=dots dots dots ) ( A cdot frac{1}{sqrt{x}} ) B. ( sqrt{x} ) c. ( frac{1}{2 sqrt{x}} ) D. ( -frac{1}{sqrt{x}} ) |
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| 280 | Prove that [ boldsymbol{f}(boldsymbol{x})=left{begin{array}{rl} frac{boldsymbol{x}^{2}-boldsymbol{2 5}}{boldsymbol{x}-mathbf{5}}, & boldsymbol{w} boldsymbol{h} boldsymbol{e} boldsymbol{n} quad boldsymbol{x} neq mathbf{5} \ boldsymbol{1 0}, boldsymbol{w h e n} & boldsymbol{x}=mathbf{5} end{array}right. ] continuous at ( boldsymbol{x}=mathbf{5} ) |
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| 281 | Evaluate ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{n} ) in ( [-mathbf{1}, mathbf{1}], boldsymbol{n} boldsymbol{epsilon} boldsymbol{Z}^{+} ) | 12 |
| 282 | ( operatorname{Let} f(x)=frac{1}{a x+b} ) then ( f^{prime prime}(0)= ) A ( cdot frac{2 a^{3}}{b^{2}} ) в. ( frac{2 a^{2}}{b^{3}} ) c. ( frac{2 a^{3}}{b^{3}} ) D. none of these |
12 |
| 283 | f ( y=tan ^{-1}left[frac{5 cos x-12 sin x}{12 cos x+5 sin x}right] ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot-2 ) D. |
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| 284 | Find the derivative of ( e^{x}+e^{y}=e^{x+y} ) ( mathbf{A} cdot-e^{x-y} y^{y} ) B . ( e^{x-y} ) ( mathbf{c} cdot-e^{y-x} ) D. ( e^{y-x} ) |
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| 285 | Write the value of the derivative of ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-1|+|boldsymbol{x}-mathbf{3}| ) at ( boldsymbol{x}=mathbf{2} ) |
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| 286 | [ begin{aligned} boldsymbol{f}(boldsymbol{x})=& frac{|boldsymbol{x}|}{boldsymbol{x}} boldsymbol{x} neq mathbf{0} \ mathbf{0} boldsymbol{x}=mathbf{0} end{aligned} ] Check whether ( f(x) ) is continous or not at ( x=0 ) |
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| 287 | If ( y=cos ^{-1} cos x, ) then ( frac{d y}{d x} ) at ( x=frac{5 pi}{4} ) A . 1 в. – 1 c. ( frac{1}{sqrt{2}} ) D. ( frac{5 pi}{4} ) |
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| 288 | Discuss the continuity of [ boldsymbol{f}(boldsymbol{x})= ] ( left{begin{array}{l}frac{sin 2 x}{sqrt{1-cos 2 x}} text { for } 0<x leq pi / 2 \ frac{cos }{pi-2 x} quad text { for } frac{pi}{2}<x<piend{array} ) at right. ( boldsymbol{x}=boldsymbol{pi} / 2 ) |
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| 289 | If ( f ) is a differentiable function at a point ‘a’ and ( f^{prime}(a) neq 0 ) then which of the following is true. A ( cdot f^{prime}(a)=lim _{h rightarrow 0} frac{f(a)-f(a-h)}{h} ) B. ( frac{1}{2} f^{prime}(a)=lim _{h rightarrow 0} frac{f(a+2 h)-f(a-h)}{2 h} ) c. ( f^{prime}(a)=lim _{h rightarrow 0} frac{f(a+2 h)-f(a)}{h} ) D. none of these |
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| 290 | (0) – If y = y(x) and it follows then y” (0)= and it follows the relation x cos y + y cos x = Tt (2005) (b) 1 (C) T-1 (d) (a) 1 |
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| 291 | Verify LMVT : ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}} ) for ( boldsymbol{x}=[mathbf{1}, boldsymbol{3}] ) |
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| 292 | The value of ( f(0) ) so that the function ( f(x)=frac{sqrt{1+x}-sqrt[3]{1+x}}{x} ) becomes continuous, is equal to A ( cdot frac{1}{6} ) B. ( frac{1}{4} ) ( c cdot 2 ) D. |
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| 293 | If ( f(x) ) is differentiable in ( [a, b] ) such that ( f(a)=2, f(b)=6, ) then there exists at least one ( c, a<c<b, ) such ( operatorname{that}left(b^{3}-a^{3}right) f^{prime}(c)= ) A ( cdot c^{2} ) B . ( 2 c^{2} ) ( c .-3 c^{2} ) D. ( 12 c^{2} ) |
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| 294 | f ( boldsymbol{y}=tan ^{-1}left(frac{sqrt{1+boldsymbol{a}^{2} boldsymbol{x}^{2}}-mathbf{1}}{boldsymbol{a} boldsymbol{x}}right), ) then ( left(1+a^{2} x^{2}right) y^{prime prime}+2 a^{2} x y^{1}= ) A . ( -2 a^{2} ) B ( cdot a^{2} ) ( c cdot 2 a^{2} ) ( D ) |
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| 295 | If ( e^{x}+e^{y}=e^{x+y}, ) find ( frac{d y}{d x} ) | 12 |
| 296 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( frac{boldsymbol{x}(mathbf{1}+boldsymbol{a} cos boldsymbol{x})-boldsymbol{b} sin boldsymbol{x}}{boldsymbol{x}^{3}}, boldsymbol{x} neq ) 0 ( operatorname{and} f(0)=1, ) then values if ‘a’ and ‘b’ so that ‘f’ is continuous are A ( cdot frac{5}{2}, frac{3}{2} ) B. ( frac{5}{2}, frac{-3}{2} ) c. ( -frac{5}{2}-frac{3}{2} ) D. ( frac{1}{2}-frac{3}{2} ) |
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| 297 | Given a function ‘g’ whcih has a derivative ( g^{prime}(x) ) for every real ‘ ( x ) ‘ and which satisfy ( g^{prime}(0)=2 ) and ( g(x+y)= ) ( e^{y} cdot g(x)+e^{x} cdot g(y) ) for all ( x, y . ) Find ( g(x) ) ( A cdot 2 x e^{x} ) В. ( x e^{x} ) c. ( x+e^{x} ) D. ( x-e^{x} ) |
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| 298 | If ( f(x)=int_{0}^{x} t(sin x-sin t) d t ) then? A ( cdot f^{prime prime prime}(x)+f^{prime}(x)=cos x-2 x sin x ) B . ( f^{prime prime prime}(x)+f^{prime prime}(x)-f^{prime}(x)=cos x ) C. ( f^{prime prime prime}(x)-f^{prime prime}(x)=cos x-2 x sin x ) D. ( f^{prime prime prime}(x)+f^{prime prime}(x)=sin x ) |
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| 299 | Find differentiation of ( sec ^{-1} tan x ) | 12 |
| 300 | If ( frac{x+a}{2}=b cot ^{-1}(b ln y), b>0, ) then value of ( boldsymbol{y} boldsymbol{y}^{prime prime}+boldsymbol{y} boldsymbol{y}^{prime} ln boldsymbol{y} ) equals A ( cdot y^{prime} ) B . ( y^{prime} ) ( c cdot 0 ) D. |
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| 301 | If ( cos (x+y)=y sin x, ) then find ( frac{d y}{d x} ) | 12 |
| 302 | If ( f(x)=x^{n} ln x ) and ( f(0)=0 ) then value of ( alpha ) for which Rolle’s Theorem can be applied in ( [mathbf{0}, mathbf{1}] ) A . -2 B. – c. 0 D. |
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| 303 | Range of ( boldsymbol{y}=log _{frac{3}{4}}(boldsymbol{f}(boldsymbol{x})) ) ( A cdot(-infty, 1] ) в. ( left[frac{3}{4}, inftyright) ) ( c cdot(-infty, infty) ) D. ( R ) |
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| 304 | If ( y=e^{x}+sin x-4 x^{3}, ) find ( frac{d y}{d x} ) | 12 |
| 305 | If f(x)= x(VxVx+1), then (1985 – 2 Marks) (a) f(x) is continuous but not differentiable at x=0 (b) f(x) is differentiable at x = 0 (c) f(x) is not differentiable at x=0 (d) none of these |
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| 306 | If ( boldsymbol{y}=sqrt{frac{1-sin 2 x}{1+sin 2 x}}, ) then ( left(frac{d y}{d x}right)_{x=0}= ) This question has multiple correct options A ( cdot frac{1}{2} ) B. c. -2 D. |
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| 307 | Let ( mathbf{f}: mathbf{R} rightarrow mathbf{R} ) be any function. Define ( mathbf{g}: mathbf{R} rightarrow mathbf{R} ) by ( boldsymbol{g}(boldsymbol{x})=|boldsymbol{f}(boldsymbol{x})| ) for all ( boldsymbol{x} ) Then ( g ) is A. ( g ) may be bounded even if ( f ) is unbounded B. one-one if fis one c. continuous if ( f ) is continuous D. differentiable if f is differentiable |
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| 308 | If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{p}|sin boldsymbol{x}|+boldsymbol{q} e^{|boldsymbol{x}|}+boldsymbol{r}|boldsymbol{x}|^{3} ) and if ( f(x) ) is differentiable at ( x=0, ) then A ( cdot p=q=r=0 ) B. ( p+q=0 ; r ) is any real number c. ( q+r=0 ; p ) is any real number D. ( r=0 ; p=0, q ) is any real number |
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| 309 | Differentiate the following function with respect to ( x ) ( boldsymbol{x}^{5}left(boldsymbol{3}-boldsymbol{6} boldsymbol{x}^{-boldsymbol{9}}right) ) A. ( 15 x^{-4}+24 x^{-5} ) B ( cdot 15 x^{5}+24 x^{-5} ) ( mathbf{c} cdot 15 x^{4}+24 x^{-4} ) D. ( 15 x^{4}+24 x^{-5} ) |
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| 310 | ( lim _{x rightarrow 0^{+}}left(left(x^{x^{x}}right)-x^{x}right) ) is A. Equal to 0 B. Equal to 1 c. Equal to – 1 D. Non existent |
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| 311 | Differentiate w.r.t. ( x ) ( boldsymbol{y}=(cos boldsymbol{x})left(1-sin ^{2} boldsymbol{x}right) ) |
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| 312 | The radius of a sphere is changing at the rate of ( 0.1 mathrm{cm} / ) sec. The rate of change of its surface area when the radius is ( 200 mathrm{cm}, ) is. A ( cdot 8 pi c m^{2} / ) sec В. ( 12 pi c m^{2} / )sec c. ( 160 pi c m^{2} / ) sec D. ( 200 pi c m^{2} / ) sec |
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| 313 | If ( y=sin ^{-1}left(x^{2}right) ) then find ( frac{d y}{d x} ) using first principle. A. ( frac{2 x}{sqrt{1-x^{4}}} ) в. ( frac{2}{sqrt{1-x^{2}}} ) c. ( frac{x}{sqrt{1-x^{4}}} ) D. ( -frac{1}{sqrt{1-x^{4}}} ) |
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| 314 | Let ( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x} ) and ( boldsymbol{g}(boldsymbol{x})=[boldsymbol{x}+mathbf{2}] ) where [.] denotes the greatest integer function. Then, ( (g o f)^{prime}left(frac{pi}{2}right) ) is? ( mathbf{A} cdot mathbf{1} ) B. c. -1 D. Does not exist |
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| 315 | Find the value of ( p ) if following function ( boldsymbol{f}(boldsymbol{x})= ) [ left{begin{array}{ll} frac{sqrt{1+p x}-sqrt{1-p x}}{x}, & text { if }-1 leq x< \ frac{2 x+2}{x-2}, & text { if } 0 leq x<1 end{array}right. ] is continuous at ( x=0 ) |
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| 316 | Differentiate the following function with respect to ( x ) ( 1+3 x ) |
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| 317 | If ( f(x)=frac{e^{x^{2}}-cos x}{x^{2}}, ) for ( x neq 0 ) is continuous at ( boldsymbol{x}=mathbf{0}, ) then value of ( boldsymbol{f}(mathbf{0}) ) is A ( cdot frac{2}{3} ) в. ( frac{5}{2} ) ( c cdot 1 ) D. |
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| 318 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( sin ^{-1}left(2 x^{2}-1right), 0<x<1 ) |
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| 319 | 8. If f(x)=va – (2) 1 (1) 1 , then f (a) = (co (d) a |
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| 320 | ( boldsymbol{y}=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}+boldsymbol{2} ) Find ( frac{d y}{d x} ) if the given function is continuous. |
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| 321 | Differentiate the following w.r.t. ( x: ) ( e^{sin ^{-1} x} ) | 12 |
| 322 | The value of ( frac{boldsymbol{f}(boldsymbol{t})}{boldsymbol{f}^{prime}(boldsymbol{t})} cdot frac{boldsymbol{f}^{prime prime}(-boldsymbol{t})}{boldsymbol{f}^{prime}(-boldsymbol{t})}- ) ( frac{f(-t)}{f^{prime}(-t)} cdot frac{f^{prime prime}(t)}{f^{prime}(t)} forall t epsilon R ) is equal to A . -2 B. 2 c. -4 ( D ) |
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| 323 | ( frac{boldsymbol{d}(sin boldsymbol{x})}{boldsymbol{d} boldsymbol{x}} ) ( A cdot cos x ) B. ( sec x ) ( c .-cos x ) D. – ( tan x ) |
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| 324 | Solve the following differential equation ( frac{d y}{d x}=3 x ) | 12 |
| 325 | What is the derivative of ( x^{3} ) with respect to ( x^{2} ? ) A ( cdot 3 x^{2} ) в. ( frac{3 x}{2} ) c. ( x ) D. ( frac{3}{2} ) |
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| 326 | If ( f(x) ) satisfies the conditions of Rolle’s theorem in [1,2] and ( f(x) ) is continuous in [1,2] then ( int_{1}^{2} f^{prime}(x) d x ) is equal to A . 3 B. 0 ( c .1 ) D. 2 |
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| 327 | If ( f(x)=(cos x+i sin x)(cos 3 x+ ) ( i sin 3 x) ldots(cos (2 n-1) x+i sin (2 n- ) 1) ( x ), then ( f^{prime prime}(x)= ) ( mathbf{A} cdot n^{2} f(x) ) B . ( -n^{4} f(x) ) ( mathbf{c} cdot-n^{2} f(x) ) D. ( n^{4} f(x) ) |
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| 328 | If the function ( f(x)=x^{3}-6 a x^{2}+5 x ) satisfies the conditions of Lagrange’s mean theorem for the interval [1,2] and the tangent to the curve ( y=f(x) ) at ( boldsymbol{x}=mathbf{7} / mathbf{4} ) is parallel to the chord joining the points of intersection of the curve with the ordinates ( x=1 ) and ( x=2 ) Then the value of a is? ( mathbf{A} cdot 35 / 16 ) B. ( 35 / 48 ) c. ( 7 / 16 ) D. ( 5 / 16 ) |
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| 329 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(tan ^{-1} boldsymbol{x}right) ) A ( cdot frac{1}{1+x^{2}} ) B ( cdot frac{-1}{1+x^{2}} ) c. ( frac{-1}{1-x^{2}} ) D. ( frac{1}{1-x^{2}} ) |
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| 330 | Solve: ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=cos (boldsymbol{x}+boldsymbol{y}) ) | 12 |
| 331 | The function ( f(x)=sin ^{-1}(cos x) ) is A. Discontinuous at ( x=0 ) B. continuous at ( x=0 ) C. differentiable at ( x=0 ) D. None of these |
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| 332 | ( y=tan ^{-1}left(frac{1}{x}right) ) find ( frac{d y}{d x} ) | 12 |
| 333 | If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{t}-sin boldsymbol{t}), boldsymbol{y}=boldsymbol{a}(1+cos boldsymbol{t}) ) then find ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} ) |
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| 334 | If ( y=sin left(log _{e} xright) ) prove that ( frac{d y}{d x}= ) ( frac{sqrt{1-y^{2}}}{x} ) |
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| 335 | If ( x^{2}+2 x y+y^{3}=42, ) find ( frac{d y}{d x} ) | 12 |
| 336 | Differentiate ( cos ^{-1} x ) A ( cdot frac{-1}{sqrt{left(1-x^{2}right)}} ) B. ( frac{1}{sqrt{left(1-x^{2}right)}} ) c. ( frac{-1}{sqrt{left(1+x^{2}right)}} ) D. ( frac{1}{sqrt{left(1+x^{2}right)}} ) |
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| 337 | Differentiate w.r.t. ( mathbf{x} ) ( f(x)=sqrt{sin (cos x)} ) |
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| 338 | Let ( boldsymbol{J}, boldsymbol{g}:lfloor-mathbf{1}, boldsymbol{2}rfloor rightarrow boldsymbol{K} ) be continuous functions which are twice differential on the interval ( (-1,2) . ) Let the values of ( f ) and ( g ) at the points -1,0 and 2 be as given in the following table: ( f(x)=left{begin{array}{l}3 ; x=-1 \ 6 ; x=0 quad text { and } g(x)= \ 0 ; x=2end{array}right. ) ( left{begin{array}{l}0 ; x=-1 \ 1 ; x=0 \ 0 ; x=-1end{array}right. ) In each of the intervals (-1,0) and ( (0,2), ) the function ( (f-3 g)^{prime prime} ) never vanishes. Then the correct statement(s) is (are) This question has multiple correct options A ( cdot f^{prime}(x)-3 g^{prime}(x)=0 ) has exactly three solutions in (-1,0)( cup(0,2) ) B. ( f^{prime}(x)-3 g^{prime}(x)=0 ) has exactly one solutions in (-1,0) C ( cdot f^{prime}(x)-3 g^{prime}(x)=0 ) has exactly one solutions in (0,2) D. ( f^{prime}(x)-3 g^{prime}(x)=0 ) has exactly two solutions in (-1,0) and exactly two solutions in (0,2) |
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| 339 | 2. is For a real number y, let [y] denotes the greatest integer less than or equal to y: Then the function f(x) = – tan(Te[x – 1) 1+[x]? (1981 – 2 Marks) (a) discontinuous at some x (b) continuous at all x, but the derivative f'(x) does not exist for somex c) f'(x) exists for all x, but the second derivative f'(x) does not exist for some x (d) f'(x) exists for all x |
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| 340 | From means value theorem ( boldsymbol{f}(boldsymbol{b})- ) ( boldsymbol{f}(boldsymbol{a})=(boldsymbol{b}-boldsymbol{a}) boldsymbol{f}^{prime}left(boldsymbol{x}_{1}right) ; mathbf{0}<boldsymbol{a}<boldsymbol{x}_{1}<boldsymbol{b} ) if ( f(x)=frac{1}{x}, ) then ( x_{1}= ) A ( cdot sqrt{a b} ) в. ( frac{a+b}{2} ) c. ( frac{2 a b}{a+b} ) D. ( frac{b-a}{b+a} ) |
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| 341 | (a) – (b) 1+2 (d) None of these |
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| 342 | Derivative of ( (x+3)^{2}(x+4)^{3}(x+5)^{4} ) ( boldsymbol{w} cdot boldsymbol{r} cdot operatorname{to} boldsymbol{x} ) is A ( cdot(x+3)(x+4)(x+5)^{2}left(9 x^{2}+70 x+133right) ) B cdot ( (x+3)(x+4)^{2}(x+5)^{3}left(9 x^{2}+70 x+133right) ) C ( cdot(x+3)(x+4)^{2}(x+5)left(9 x^{2}-70 x-133right) ) D. none of these |
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| 343 | ( f(x)=frac{1+e^{1 / x}}{1-e^{1 / x}}(x neq 0), f(0)=1, ) then ( f(x) ) is A. left coninuous at ( x=0 ) B. right continuous at ( x=0 ) c. continuous at ( x=0 ) D. none |
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| 344 | Differentiate the following w.r.t. ( x ) ( e^{x^{3}} ) | 12 |
| 345 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( (x-4)(x-5)(x-6)(x-7) ) then A ( cdot f^{prime}(x)=0 ) has four roots B. three roots of ( f^{prime}(x)=0 ) lie in (4,5)( cup(5,6) cup(6,7) ) C. the equation ( f^{prime}(x)=0 ) has only one root D. three roots of ( f^{prime}(x)=0 ) lie in (3,4)( cup(4,5) cup(5,6) ) |
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| 346 | If ( boldsymbol{y}=boldsymbol{e}^{2 x}(boldsymbol{a} boldsymbol{x}+boldsymbol{b}), ) show that ( boldsymbol{y}_{2} ) ( mathbf{4} boldsymbol{y}_{1}+mathbf{4} boldsymbol{y}=mathbf{0} ) |
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| 347 | A function ( f(x) ) defined as ( f(x)= ) ( left{begin{array}{ll}sin x, & x text { is rational } \ cos x, & x text { is irrational }end{array} ) is continuous right. at A ( cdot x=n pi+frac{pi}{4}, n in I ) В ( cdot x=n pi+frac{pi}{8}, n in I ) c. ( x=n pi+frac{pi}{6}, n in I ) D・ ( x=n pi+frac{pi}{3}, n in I ) |
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| 348 | The set of all points where ( boldsymbol{f}(boldsymbol{x})= ) ( sqrt[3]{x^{2}|x|}-|x|-1 ) is not differentiable is ( A cdot{0} ) B ( cdot{-1,0,1} ) ( mathbf{c} cdot{0,1} ) D. None of these |
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| 349 | Find number of terms in ( left(1+x+x^{4}right)^{12} ) | 12 |
| 350 | ( lim _{boldsymbol{x} rightarrow infty} sum_{boldsymbol{r}=1}^{boldsymbol{n}} tan ^{-1}left(frac{2 r}{1-boldsymbol{r}^{2}+boldsymbol{r}^{4}}right) ) is equal to ( mathbf{A} cdot pi / 4 ) B. ( pi / 2 ) ( c cdot frac{3 pi}{4} ) D. None of these |
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| 351 | If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{m} sin ^{-1} boldsymbol{x}} ) Then ( left(1-x^{2}right)left(frac{d y}{d x}right)^{2}=A y^{2}, ) then ( A= ) ? A . ( m ) B. ( -m ) ( mathrm{c} cdot m^{2} ) D. ( -m^{2} ) |
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| 352 | If ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) satisfies the property ( (boldsymbol{x}- ) ( boldsymbol{y}) boldsymbol{f}(boldsymbol{x}+boldsymbol{y})-(boldsymbol{x}+boldsymbol{y}) boldsymbol{f}(boldsymbol{x}-boldsymbol{y})= ) ( 4 x yleft(x^{2}-y^{2}right), f(1)=1, ) then the number of real roots of ( boldsymbol{f}(boldsymbol{x})=mathbf{4} ) will be A . 1 B . 2 ( c cdot 3 ) D. |
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| 353 | ( operatorname{Let} F(x)=f(x) g(x) h(x) ) for all real ( x ) where ( f(x), g(x) ) and ( h(x) ) are differentiable functions. At some point ( boldsymbol{x}_{0}, boldsymbol{F}^{prime}left(boldsymbol{x}_{0}right)=mathbf{2 1} boldsymbol{F}left(boldsymbol{x}_{0}right), boldsymbol{f}^{prime}left(boldsymbol{x}_{0}right)= ) ( 4 fleft(x_{0}right), g^{prime}left(x_{0}right)=-7 gleft(x_{0}right) ) and ( boldsymbol{h}^{prime}left(boldsymbol{x}_{0}right)=boldsymbol{k h}left(boldsymbol{x}_{0}right) . ) Then ( boldsymbol{k} ) is equal to |
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| 354 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( boldsymbol{x} sin 2 boldsymbol{x}+mathbf{5}^{boldsymbol{x}}+boldsymbol{k}^{boldsymbol{k}}+left(tan ^{2} boldsymbol{x}right)^{2} ) |
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| 355 | The graph of ( f(x) ) is given below. Based on this graph determine where the function is discontinuous. |
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| 356 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}2 a-x, & text { if }-a<x<a \ 3 x-2 a, & text { if } a leq xend{array} . ) Then right. which of the following is true? A. ( f(x) ) is discontinuous at ( x=a ) B. ( f(x) ) is not differentiable at ( x=a ) c. ( f(x) ) is differentiable at ( x geq a ) D. ( f(x) ) is continuous at all ( x<a ) |
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| 357 | If ( a x+b y^{2}=cos y, ) then find ( frac{d y}{d x} ) | 12 |
| 358 | Domain of the function is ( boldsymbol{y}= ) ( frac{sqrt{cos x-1 / 2}}{sqrt{6+35 x-6 x^{2}}} ) ( mathbf{A} cdot therefore D_{1} cap D_{2}=[0, pi / 6] cup[5 pi / 3,6] ) B ( ldots D_{1} cap D_{2}=[0, pi / 3] cup[5 pi / 3,6] ) ( mathrm{c} cdot therefore D_{1} cap D_{2}=[0, pi / 3] cup[5 pi / 6,6] ) D. ( therefore D_{1} cap D_{2}=[0, pi / 2] cup[5 pi / 3,6] ) |
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| 359 | Use Rolle’s theorem to prove that equation ( a x^{2}+b x=frac{a}{3}+frac{b}{2} ) has a root between 0 and 1 |
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| 360 | Find the derivative with respect to ( x ) of the function ( left(log _{cos x} sin xright)left(log _{sin x} cos xright)^{-1}+ ) ( sin ^{-1} frac{2 x}{1+x^{2}} ) at ( x=frac{pi}{4} ) A ( cdot gleft(frac{4}{pi^{2}+16}-frac{1}{log 2}right) ) B. ( -8left(frac{4}{(pi+4)^{2}}-frac{1}{log 2}right) ) ( ^{mathbf{c}} cdot_{8}left(frac{4}{pi^{2}+16}+frac{1}{log 2}right) ) D ( cdot gleft(frac{4}{(pi+4)^{2}}-frac{1}{log 2}right) ) |
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| 361 | if ( x_{1}, x_{2}, x_{3} dots x_{n} ) denote the values of ( x ) where ( f(x) ) vanishes such that ( x_{1}> ) ( x_{2}>x_{3} ldots x_{n}, ) then ( lim _{n rightarrow infty} sum_{r=1}^{n} frac{r}{x_{r}} ) is equal to A . -8 B. -4 ( c .-2 ) D. – |
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| 362 | will be 4. If y = sin(x2), then (a) 2t cos(x2) (c) 4×2 sin (+2) (b) 2 cos (+2) – 4t sin (12) (d) 2 cos (12) |
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| 363 | Let ( boldsymbol{f}(boldsymbol{x})=frac{2}{pi} operatorname{cosec}^{-1} frac{x+1}{2} ) Then ( mathbf{A} cdot lim _{x rightarrow 1^{+}} f(x)=0 ) ( mathbf{B} cdot lim _{x rightarrow 1} f(x)=frac{pi}{2} ) ( mathbf{C} cdot lim _{x rightarrow-3^{-}} f(x)=-1 ) ( mathbf{D} cdot lim _{x rightarrow-3} f(x)=1 ) E ( cdot lim _{x rightarrow-3} f(x)=-frac{pi}{2} ) |
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| 364 | ( f_{n}(x)=e^{f_{n-1}(x)} ) for all ( n epsilon N ) and ( f_{0}(x)=x, ) then ( frac{d}{d x}left{f_{n}(x)right} ) is This question has multiple correct options A ( cdot f_{n}(x) frac{d}{d x}left{f_{n-1}(x)right} ) В ( cdot f_{n}(x) f_{n-1}(x) ) ( mathbf{c} cdot f_{n}(x) f_{n-1}(x) cdots f_{2}(x) cdot f_{1}(x) ) D. none of these |
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| 365 | If ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y}+boldsymbol{z})=boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y}) cdot boldsymbol{f}(boldsymbol{z}) ) for all ( boldsymbol{x}, boldsymbol{y}, boldsymbol{z} ) and ( boldsymbol{f}(boldsymbol{2})=boldsymbol{4}, boldsymbol{f}^{prime}(boldsymbol{0})=boldsymbol{3}, ) then ( f^{prime}(2) ) equals A . 12 B. 9 c. 16 D. 6 |
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| 366 | If the Rolle’s theorem holds for the function ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{mathbf{3}}+boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x} ) in interval [-1,1] for the point ( c=frac{1}{2}, ) then find the value of ( 2 a+b ? ) A B. – ( c cdot 2 ) ( D cdot-2 ) |
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| 367 | ( operatorname{Let} f(x)=left(frac{tan left(frac{pi}{4}-xright)}{cot 2 x}right)left(x neq frac{pi}{4}right) ) The value which should be assigned to at ( frac{pi}{4} ) so that it is continuous everywhere, is A ( cdot frac{1}{2} ) B. ( c cdot 2 ) D. None of these |
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| 368 | ff ( y=tan ^{-1} frac{1}{x^{2}+x+1}+ ) ( tan ^{-1} frac{1}{x^{2}+3 x+3}+ ) ( tan ^{-1} frac{1}{x^{2}+5 x+7}+ldots . ) to ( n ) terms then A ( cdot frac{d y}{d x}=frac{1}{1+(x+n)^{2}}-frac{1}{1+x^{2}} ) B. ( frac{d y}{d x}=frac{1}{(x+n)^{2}}-frac{1}{1+x^{2}} ) c. ( frac{d y}{d x}=frac{1}{1+(x+n)^{2}}+frac{1}{1+x^{2}} ) D. None of these |
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| 369 | Let ( boldsymbol{f} ) be an increasing function on ( [boldsymbol{a}, boldsymbol{b}] ) and ( g ) be a decreasing function on ( [a, b] ) then on ( [a, b] ) This question has multiple correct options A. fog is a decreasing function. B. gof is an increasing function. c. ( f o g ) is an increasing function. D. None of these |
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| 370 | If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{3}}, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 371 | If ( f(x)=frac{1}{x-1}, ) then determine the number of points of discontinuity of ( boldsymbol{f}[boldsymbol{f}{boldsymbol{f}(boldsymbol{x})}] ) |
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| 372 | ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{sqrt{1+boldsymbol{p} boldsymbol{x}}-sqrt{mathbf{1}-boldsymbol{p} boldsymbol{x}}}{boldsymbol{x}} \ frac{mathbf{2} boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{2}}, quad mathbf{0} leq boldsymbol{x} leq mathbf{1}end{array} quad, quad-mathbf{1} leq boldsymbol{x}<right. ) 0 is continuous in the interval [-1,1] then ( p ) equals- A . -1 B. ( -frac{1}{2} ) ( c cdot frac{1}{2} ) ( D ) |
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| 373 | If ( y=sqrt{frac{sec x-1}{sec x+1}} ) then ( frac{d y}{d x}= ) A. ( frac{1}{2} sec ^{2} frac{x}{2} ) B. ( sec ^{2} frac{x}{2} ) c. ( frac{1}{2} tan frac{x}{2} ) D. ( tan frac{x}{2} ) |
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| 374 | ( y=6 x^{3}+3 x^{2}+4 x+5 ) Find the value of ( frac{d y}{d x} ? ) |
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| 375 | f ( boldsymbol{y}=log _{7}(log boldsymbol{x}) ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 376 | ( operatorname{Let} f(x)=left(2-frac{x}{a}right)^{tan left(frac{pi x}{2 a}right)}, x neq a ) The value which should be assigned to ( f ) at ( x=a ) so that it is continuous everywhere is A ( -frac{2}{pi} ) B . ( e^{-2 / pi} ) ( c cdot 2 ) ( mathbf{D} cdot e^{2 / pi} ) |
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| 377 | Let ( f ) be a function which is continuous and differentiable for all real ( x ). If ( boldsymbol{f}(mathbf{2})=-mathbf{4} ) and ( boldsymbol{f}^{prime}(boldsymbol{x}) geq mathbf{6} ) for all ( boldsymbol{x} in ) ( [2,4], ) then ( mathbf{A} cdot f(4)<8 ) B ( cdot f(4) geq 8 ) ( mathbf{c} cdot f(4) geq 12 ) D. none of these |
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| 378 | Find ( frac{d y}{d x}, ) if ( x+y=sin (x-y) ) A ( cdot frac{cos (x-y)-1}{cos (x-y)+1} ) B ( cdot frac{cos (x-y)+1}{cos (x-y)-1} ) C ( frac{cos (x+y)+1}{cos (x-y)-1} ) D. ( frac{cos (x+y)-1}{cos (x-y)+1} ) |
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| 379 | Illustration 2.35 If f(x) = x cos x, find f”(x). | 12 |
| 380 | If ( boldsymbol{y}=sec ^{-1}left(frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right)+sin ^{-1}left(frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}right) ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) |
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| 381 | Differentiate the following expression ( boldsymbol{w} cdot boldsymbol{r} cdot boldsymbol{t} cdot boldsymbol{x} ) ( boldsymbol{y}=csc ^{2}left(boldsymbol{x}^{2}right) ) |
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| 382 | Differentiate the given function w.r.t. ( x ) ( log (log x), x>1 ) |
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| 383 | ( operatorname{Let} g(x)=lim _{n rightarrow infty} frac{x^{n} f(x)+h(x)+1}{2 x^{n}+3 x+3}, x^{1} 1 ) and ( g(1)=lim _{x rightarrow 1} frac{sin ^{2}left(pi cdot 2^{x}right)}{ln left(sec left(pi cdot 2^{x}right)right)} ) be a continuous function at ( x=1, ) find the value of ( 4 g(1)+2 f(1)-h(1) . ) Assume that ( f(x) ) and ( h(x) ) are continuous at ( boldsymbol{x}=mathbf{1} ) |
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| 384 | 6. The function for The function In(1+ ax) – In(1-bx) is not defined (x)= x -0. The value which should be assigned tofat x that it is continuous at x=0, is (1983 – 1 Mark) (a) a-b (b) a + b (C) In a – In b (d) none of these |
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| 385 | If ( boldsymbol{x}^{m}+boldsymbol{y}^{m}=mathbf{1} ) such that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=-frac{boldsymbol{x}}{boldsymbol{y}} ) then what should be the value of ( m ? ) ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. None of the above |
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| 386 | If ( f(x)=sqrt{1-e^{-x^{2}}}, ) then at ( x=0, f(x) ) is A. differentiable as well as continuous B. continuous but not differentiable c. differentiable but not continuous D. neither differetiable nor continuous |
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| 387 | If ( f(x)=log left(frac{x^{2}+a b}{x(a+b)}right), ) then the value of ( ^{prime} C^{prime} ) for which ( f^{prime}(c)=0 ) in ( [a, b] ) ( mathbf{A} cdot C=pm sqrt{frac{a}{b}} ) B. ( C=pm sqrt{a b} ) ( ^{c} cdot c=pm sqrt{frac{b}{a}} ) D. none of these |
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| 388 | Find the derivatives of the following functions at the indicated points. ( boldsymbol{f}(boldsymbol{x})=sin 4 boldsymbol{x} cos mathbf{4} boldsymbol{x}, boldsymbol{f}^{prime}(boldsymbol{pi} / mathbf{3})=? ) |
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| 389 | ( boldsymbol{y}=tan ^{-1}left[frac{log left(frac{boldsymbol{e}}{boldsymbol{x}^{2}}right)}{log left(boldsymbol{e} boldsymbol{x}^{2}right)}right]+ ) ( tan ^{-1}left(frac{3+2 log x}{1-6 log x}right), ) then ( frac{d^{2} y}{d x^{2}}= ) ( A ) в. ( c cdot 0 ) D. – |
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| 390 | Examine the applicability of Mean Value Theorem for the following function. ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] ) for ( boldsymbol{x} boldsymbol{epsilon}[boldsymbol{2}, boldsymbol{2}] ) |
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| 391 | The value of ( c ) in the lagranges mean value theorem for ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}, boldsymbol{a}= ) ( 1, h=frac{1}{2} ) is A ( cdot frac{1}{3} ) в. ( sqrt{frac{19}{56}} ) c. ( sqrt{frac{19}{3}}+2 ) D. ( sqrt{frac{19}{3}}-2 ) |
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| 392 | Consider the functions defined implicitly by the equation ( y^{3}-3 y+ ) ( x=0 ) on various intervals in the real line. If ( boldsymbol{x} epsilon(-infty,-2) cup(2, infty), ) the equation implicitly defines a unique real valued differentiable function ( y= ) ( f(x) . ) If ( x epsilon(-2,-2) ) the equation implicitly defines a unique real valued differentiable function ( boldsymbol{y}=boldsymbol{g}(boldsymbol{x}) ) satisfying ( boldsymbol{g}=boldsymbol{g}(mathbf{0})=mathbf{0} ) If ( f(-10 sqrt{2})=2 sqrt{2} ) then ( f^{prime prime}(-10 sqrt{2})= ) A ( cdot frac{4 sqrt{2}}{7^{3} cdot 3^{2}} ) B. ( -frac{4 sqrt{2}}{7^{3} cdot 3^{2}} ) c. ( frac{4 sqrt{2}}{7^{3} cdot 3^{3}} ) D. ( frac{4 sqrt{2}}{7 cdot 3} ) |
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| 393 | In the function ( f(x)=a x^{3}+b x^{2}+ ) ( 11 x-6 ) satisfies condition of rolle’s therorem in [1,3] and ( f^{prime}left(2+frac{1}{3}right)=0 ) then value of ( a ) and ( b ) are respectively A. 1,-6 B. -1,6 c. -2,1 D. ( -1, frac{1}{2} ) |
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| 394 | differentiate ( e^{-2 tan ^{-1} x^{2}} ) | 12 |
| 395 | ff ( f(x)=e^{x} g(x) ) ( boldsymbol{g}(mathbf{0})=mathbf{1}, boldsymbol{g}^{prime}(mathbf{0})=mathbf{3}, ) then ( boldsymbol{f}^{prime}(mathbf{0}) ) is ( mathbf{A} cdot mathbf{0} ) B. 4 ( c cdot 3 ) D. |
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| 396 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}=sin boldsymbol{y} ) | 12 |
| 397 | Illustration 2.25 If y= 12++3]|2x* + 1], then find dy Illustration 2.25 If y = 31|2x + 1), then find |
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| 398 | Differentiate with respect to ( x ) : ( log left(cos x^{2}right) ) |
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| 399 | Find if limit of function exists as ( x ) tends to zero ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{lr}frac{mathbf{1}}{boldsymbol{x}}-mathbf{1} & boldsymbol{x} neq mathbf{0} \ boldsymbol{e} boldsymbol{x}+mathbf{1} \ mathbf{0} & boldsymbol{x}=mathbf{0}end{array}right. ) |
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| 400 | Let ( f(x) ) be a real valued function not identically zero, such that ( boldsymbol{f}left(boldsymbol{x}+boldsymbol{y}^{n}right)=boldsymbol{f}(boldsymbol{x})+(boldsymbol{f}(boldsymbol{y}))^{n} quad forall boldsymbol{x}, boldsymbol{y} in ) ( boldsymbol{R} ) where ( n in N(n neq 1) ) and ( f^{prime}(0) geq 0 . ) We may get an explicit form of the function ( boldsymbol{f}(boldsymbol{x}) ) The value of ( f(5) ) is : A . 6 B. 3 ( c cdot 5 n ) D. 5 |
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| 401 | 26. If y=x2 sin x, then will be … |
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| 402 | If ( boldsymbol{y}=tan boldsymbol{x}+cot boldsymbol{x} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 403 | Differentiate the following function with respect to ( x ) ( tan h^{-1}(3 x+1) ) |
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| 404 | Let ( f, g ) and ( h ) are differentiable functions. If ( boldsymbol{f}(mathbf{0})=mathbf{1} ; boldsymbol{g}(mathbf{0})=mathbf{2} ; boldsymbol{h}(mathbf{0})= ) 3 and the derivative of their pair wise products at ( x=0 ) are ( (f g)^{prime}(0)= ) ( mathbf{6} ;(boldsymbol{g} boldsymbol{h})^{prime}(mathbf{0})=mathbf{4} ) and ( (boldsymbol{h} boldsymbol{f})^{prime}(mathbf{0})=mathbf{5} ) then compute the value of ( (f g h)^{prime}(0) ) A . 12 B . 15 c. 16 D. None of these |
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| 405 | Examine if Mean value Theorem applies to ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+mathbf{3} boldsymbol{x}^{2}-mathbf{5} boldsymbol{x} ) in the interval [1,2]. If it does, then find the intermediate point whose existence is asserted by theorem. A. Mean Value theorem is applicable and intermediate points are ( c=-3.55,1.55 ) B. Mean Value theorem is not applicable c. Mean Value theorem is applicable and intermediate points are ( c=3.55,-1.55 ) D. none of these |
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| 406 | Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cl}boldsymbol{alpha}+frac{sin [boldsymbol{x}]}{boldsymbol{x}} & boldsymbol{i f} boldsymbol{x}>0 \ boldsymbol{2} & boldsymbol{i f} boldsymbol{x}=mathbf{0} \ boldsymbol{beta}+left[frac{sin boldsymbol{x}-boldsymbol{x}}{boldsymbol{x}^{3}}right] & boldsymbol{i} boldsymbol{f} boldsymbol{x}<0end{array}right. ) where ( [x] ) denotes the integral part of ( y ) If ( f ) is continuous at ( x=0, ) then ( beta-alpha= ) ( A ) B. 1 ( c cdot 0 ) ( D ) |
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| 407 | If ( boldsymbol{x}^{boldsymbol{y}}=boldsymbol{a}^{boldsymbol{x}}, ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x} log _{e} boldsymbol{a}-boldsymbol{y}}{boldsymbol{x} log _{e} boldsymbol{x}} ) | 12 |
| 408 | If ( boldsymbol{y}=sin ^{-1}left(frac{mathbf{5} boldsymbol{x}+mathbf{1 2} sqrt{mathbf{1}-boldsymbol{x}^{2}}}{mathbf{1 3}}right), ) then ( frac{d y}{d x} ) is equal to A. ( -frac{1}{sqrt{1-x^{2}}} ) B. ( frac{1}{sqrt{1-x^{2}}} ) c. ( frac{3}{sqrt{1-x^{2}}} ) D. ( -frac{x}{sqrt{1-x^{2}}} ) |
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| 409 | f ( boldsymbol{y}=log [boldsymbol{x}+sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}}], ) show that ( left(x^{2}+a^{2}right) frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}=0 ) |
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| 410 | If ( mathbf{a}+mathbf{b}+mathbf{c}=mathbf{0}, ) then the equation ( 3 a x^{2}+2 b x+c=0 ) has at least one root in This question has multiple correct options A ( .(1,2) ) в. (0,1) c. (-1,1) D. (2,3) |
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| 411 | If ( a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+ ) ( c=0 ) then ( frac{d y}{d x}= ) A. ( -left(frac{a x+h y+g}{h x+b y+f}right) ) в. ( -left(frac{a x+h y+g}{b x+h y+f}right) ) c. ( -left(frac{h x+b y+f}{a x+h y+g}right) ) D. ( -left(frac{h x+b y+f}{h x+a y+g}right) ) |
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| 412 | If the function ( f(x)=frac{log x-1}{x-e}, ) for ( x neq e ) is continuous at ( x=e, ) then find ( boldsymbol{f}(boldsymbol{e}) ) |
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| 413 | If ( boldsymbol{y}=|cos boldsymbol{x}|+|sin boldsymbol{x}|, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}= ) ( frac{2 pi}{3} ) is A ( cdot frac{1}{2}(sqrt{3}+1) ) B. ( 2(sqrt{3}-1) ) c. ( frac{1}{2}(sqrt{3}-1) ) D. none of these |
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| 414 | 7. „Osx<1 Let f(x) = (1983 – 2 Marks) *+2,15×52 Discuss the continuity of f,f' and f" on [0,2]. |
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| 415 | If ( f(x) ) is differentiable everywhere, then ( |boldsymbol{f}(boldsymbol{x})|^{2} ) is differentiable everywhere. Enter ( 1 text { if true or } 0 text { otherwise }) ) |
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| 416 | Dfferentiate w.r.t ( x ) : ( tan ^{2} 7 x ) |
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| 417 | Find whether the following function is differentiable at ( x=1,2: ) ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}boldsymbol{x}, & boldsymbol{x} leq mathbf{1} \ boldsymbol{2}-boldsymbol{x}, & boldsymbol{1} leq boldsymbol{x} leq mathbf{2} \ -boldsymbol{2}+boldsymbol{3} boldsymbol{x}-boldsymbol{x}^{2}, & boldsymbol{x}>boldsymbol{2}end{array}right. ) |
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| 418 | A derivable function ( boldsymbol{f}: boldsymbol{R}^{+} rightarrow boldsymbol{R} ) satisfies the condition ( f(x)-f(y) geq ) ( ln frac{x}{y}+x-y ; forall x, y in R^{+} . ) If ( g ) denotes the derivative of ( f ) then the value of the ( operatorname{sum} sum_{n=1}^{100} gleft(frac{1}{n}right) ) is ( 1030 k . ) Find the value of ( k ) |
12 |
| 419 | 4. st integer less than or equal to x. If (1986-2 Marks) Let [x] denote the greatest integer less than $x)=[r sin tx], then f(x) is a) continuous at r=0 (b) continuous in (c) differentiable at x=1 (d) differentiable in (1,1) (e) none of these |
12 |
| 420 | Verify Rolle’s Theorem for the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}-boldsymbol{8}, boldsymbol{x} in[-boldsymbol{4}, boldsymbol{2}] ) |
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| 421 | Differentiate the following function with respect to ( x ) ( left(1+x^{2}right) cos x ) |
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| 422 | If ( left(1+x^{2}right) y_{1}=x(1-y), y(0)=frac{4}{3} ) then ( y(sqrt{8})-frac{1}{9} ) is |
12 |
| 423 | ( f(x)=left{begin{array}{lr}frac{2^{x+2}-16}{4^{x}-16}, text { if } x neq 2 \ k, & text { if } x=2end{array}right. ) continuous at ( x=2, ) find ( k ) |
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| 424 | Differentiate the following function with respect to ( x ) ( x^{2} e^{x} ) log ( x ) A ( cdot x e^{x}(x log x+2 log x) ) B. ( x e^{x}(1+2 log x) ) C ( cdot x e^{x}(1+x log x) ) D. ( x e^{x}(1+x log x+2 log x) ) |
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| 425 | Suppose that ( boldsymbol{f} ) is a differentiable function with the property that ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})+boldsymbol{x} boldsymbol{y} ) and ( lim _{h rightarrow 0} frac{1}{h} f(h)=3 ) then A. ( f ) is a linear function B ( cdot f(x)=3 x+x^{2} ) c. ( f(x)=3 x+frac{x^{2}}{2} ) D. None of these |
12 |
| 426 | Let ( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}-mathbf{1}}+ ) ( sqrt{x+24-10 sqrt{x-1}} ; 1<x<26 ) be real valued function. Then ( f^{prime}(x) ) for ( 1< ) ( boldsymbol{x}<26 ) is ( A cdot 0 ) в. ( frac{1}{sqrt{x-1}} ) c. ( 2 sqrt{x-1}-5 ) D. none of these |
12 |
| 427 | Illustration 2.30 Find the derivative of y = sin(x+ – 4). a 2 1 1 |
12 |
| 428 | If ( f(x)=log _{x^{2}}left(log _{e} xright), ) then ( f^{prime}(x) ) at ( boldsymbol{x}=boldsymbol{e} ) is A . 1 B. c. ( frac{1}{2 e} ) D. |
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| 429 | Let ( f(x) ) be defined on ( [0, pi] ) by ( f(x)= ) ( left{begin{array}{ll}x+a sqrt{2} sin x & , 0 leq x leq pi / 4 \ 2 x cot x+b & , frac{pi}{4}<x leq frac{pi}{2} . text { If } f \ a cos 2 x-b sin x & , frac{pi}{2}<x<piend{array}right. ) is continuous on ( [0, pi] ) then This question has multiple correct options A ( a=frac{pi}{6} ) в. ( b=-frac{pi}{12} ) c. ( a=frac{pi}{6} ) and ( b=-frac{pi}{12} ) D・ ( a=frac{pi}{3} ) and ( b=-frac{pi}{12} ) |
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| 430 | Testify the mean value theorem in the interval ( [boldsymbol{a}, boldsymbol{b}], boldsymbol{f}(boldsymbol{x})=frac{1}{4 boldsymbol{x}-1} ) where ( a=1 ) and ( b=4 ) |
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| 431 | Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem: ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{2} boldsymbol{x}^{2}-boldsymbol{x}+boldsymbol{3} ) on ( [mathbf{0}, mathbf{1}] ) |
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| 432 | Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is a function which is defined by ( f(x)=max left{x, x^{3}right} . ) The set of all points on which ( f(x) ) is not differentiable is A ( cdot{-1,1} ) the B. {-1,0} ( c cdot(0,1) ) D. {-1,0,1} |
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| 433 | The value of ( c ) in Rolle’s theorem for the function ( boldsymbol{f}(boldsymbol{x})=cos frac{boldsymbol{x}}{2} ) on ( [boldsymbol{pi}, boldsymbol{3} boldsymbol{pi}] ) is ( A cdot 0 ) B. ( 2 pi ) c. ( frac{pi}{2} ) D. ( frac{3 pi}{2} ) |
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| 434 | The function ( f(x)=frac{cos x-sin x}{cos 2 x} ) is not defined at ( x=frac{pi}{4} . ) The value of ( fleft(frac{pi}{4}right) ) so that ( f(x) ) is continuous everywhere, is ( A ) B. ( c cdot sqrt{2} ) D. ( frac{1}{sqrt{2}} ) |
12 |
| 435 | Find the derivative of ( y=frac{2 x}{1-x^{2}} ) | 12 |
| 436 | Find the value of ( x ) for which the derivative of the function ( f(x)= ) ( 20 cos 3 x+12 cos 5 x-15 cos 4 x ) is equal to zero? |
12 |
| 437 | If ( f(x)=x^{2}-x+5, x>frac{1}{2}, ) and ( g(x) ) is its inverse function, then ( g^{prime}(7) ) equals: A. ( -frac{1}{3} ) в. ( frac{1}{13} ) c. ( frac{1}{3} ) D. ( -frac{1}{13} ) |
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| 438 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of the following implicit functions ( : boldsymbol{y}^{3}-boldsymbol{3} boldsymbol{y}^{2} boldsymbol{x}=boldsymbol{x}^{3}+boldsymbol{3} boldsymbol{x}^{2} boldsymbol{y} ) |
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| 439 | Draw the graph of function ( boldsymbol{f}(boldsymbol{x})= ) ( |x| / x . ) Is ( f(x) ) defined at ( x=0 ? ) Does the limit of ( f(x) ) exist when ( x rightarrow 0 ? ) |
12 |
| 440 | Evaluate ( : int(tan x-cot x)^{2} d x ) | 12 |
| 441 | If ( f^{1}(x)=sin (log x) ) and ( y= ) ( fleft(frac{2 x+3}{3-2 x}right), ) then ( frac{d y}{d x} ) equals A. ( frac{12}{(3-2 x)^{2}} ) B. ( sin left[log left(frac{2 x+3}{3-2 x}right)right] ) c. ( frac{12}{(3-2 x)^{2}} sin left[log left(frac{2 x+3}{3-2 x}right)right] ) D. ( frac{12}{(3-2 x)^{2}} cos left[log left(frac{2 x+3}{3-2 x}right)right] ) |
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| 442 | The value of ( c ) in Lagrange’s theorem for the function ( f(x)=|x| ) in the interval [-1,1] is A. 0 B. ( 1 / 2 ) c. ( -1 / 2 ) D. non-existent in the interval |
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| 443 | If ( s=sqrt{t^{2}+1}, ) then ( frac{d^{2} s}{d t^{2}} ) is equal to A ( cdot frac{1}{s} ) в. ( frac{1}{s^{2}} ) c. ( frac{1}{s^{3}} ) D. ( frac{1}{s^{4}} ) |
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| 444 | ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}3 x-8 & text { if } x leq 5 \ 2 k & text { if } x>5end{array}right. ) continuous, find ( k ) ( A cdot frac{2}{7} ) B. 3 ( overline{7} ) ( c cdot frac{4}{7} ) D. |
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| 445 | The function ( y=frac{2-x^{2}}{x^{4}} ) takes on equal values at the end-points of the interval ( [-1,1] . ) Is Rolle’s theorem valid in this interval? |
12 |
| 446 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( sin ^{-1}left{frac{sin x+cos x}{sqrt{2}}right}, frac{pi}{4}<x<frac{3 pi}{4} ) |
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| 447 | [ begin{array}{rlr} text { If } boldsymbol{f}(boldsymbol{x}) & =frac{boldsymbol{x}^{2}-mathbf{9}}{boldsymbol{x}-mathbf{3}}+boldsymbol{alpha}, text { for } boldsymbol{x}>mathbf{3} \ & =mathbf{5}, & text { for } boldsymbol{x}=mathbf{3} \ & =mathbf{2} boldsymbol{x}^{2}+mathbf{3} boldsymbol{x}+boldsymbol{beta}, & text { for } boldsymbol{x}<mathbf{3} end{array} ] is continuous at ( x=3, ) find ( alpha ) and ( beta ) |
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| 448 | If ( x^{4}+7 x^{2} y^{2}+9 y^{4}=24 x y^{3}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A ( cdot frac{x}{y} ) B. ( underline{y} ) c. ( -frac{x}{y} ) D. ( -frac{y}{x} ) |
12 |
| 449 | Differentiate ( boldsymbol{y}=sin ^{-1}left(frac{mathbf{2}^{x+1} mathbf{3}^{x}}{1+mathbf{3 6}^{x}}right) ) | 12 |
| 450 | If ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) forall boldsymbol{x}, boldsymbol{y} ) and ( boldsymbol{f}(mathbf{5})=mathbf{2}, boldsymbol{f}^{prime}(mathbf{0})=mathbf{3} ; ) then ( boldsymbol{f}^{prime}(mathbf{5}) ) is equal to- A .2 B. 4 ( c cdot 6 ) D. 8 |
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| 451 | ( operatorname{Let} f(x)=sin frac{1}{x}, x neq 0 . ) Then ( f(x) ) can be continuous at ( 4 x=0 ) A. If ( f(0)=1 ) B. If ( f(0)=0 ) c. If ( f(0)=-1 ) D. For no definite value of ( f(0) ) |
12 |
| 452 | If ( a x^{2}+2 x y+b y^{2}=0 ) then find ( frac{d y}{d x} ) | 12 |
| 453 | Find the value of ( k ) is continuous at ( x= ) where 2 ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}frac{boldsymbol{k} cos boldsymbol{x}}{boldsymbol{pi}-mathbf{2} boldsymbol{x}}, text { if } boldsymbol{x} neq frac{boldsymbol{pi}}{mathbf{2}} \ boldsymbol{3}, quad text { if } boldsymbol{x}=frac{boldsymbol{pi}}{2}end{array}right. ) |
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| 454 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{x} sin boldsymbol{y}+boldsymbol{y} sin boldsymbol{x}=mathbf{0} ) | 12 |
| 455 | Sketch the graph ( y=|x+3| . ) Evaluate ( int_{-6}^{0}|x+3| d x . ) What does this integral represent on the graph? | 12 |
| 456 | Differentiate the following function with respect to ( x ) ( boldsymbol{x}^{-4}left(boldsymbol{3}-boldsymbol{4} boldsymbol{x}^{-boldsymbol{5}}right) ) A . ( -12 x^{-6}+36 x^{-10} ) B. ( -12 x^{-5}+36 x^{-11} ) c. ( -12 x^{-5}+36 x^{-10} ) D. ( -12 x^{5}+36 x^{-10} ) |
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| 457 | If ( boldsymbol{f}(boldsymbol{x})=sqrt{1-sin 2 boldsymbol{x}}, ) then ( boldsymbol{f}^{prime}(boldsymbol{x}) ) is equal to: This question has multiple correct options A. ( -(cos x+sin x) ), for ( x in(pi / 4, pi / 2) ) B. ( (cos x+sin x) ), for ( x in(0, pi / 4) ) c. ( -(cos x+sin x) ), for ( x in(0, pi / 4) ) D. None of these |
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| 458 | Consider the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}2 boldsymbol{x}-mathbf{1}, & boldsymbol{0} leq boldsymbol{x}<mathbf{2} \ boldsymbol{x}+boldsymbol{a} & boldsymbol{2} leq boldsymbol{x} leq mathbf{4} \ boldsymbol{3} boldsymbol{x}+boldsymbol{b} & boldsymbol{4}<boldsymbol{x} leq mathbf{6}end{array}right. ) (i) Find ( f(2-) ) and ( f(2+) ) (ii) Find ( a ) if ( f ) is continuous at ( x=2 ) (iii) Find ( b ) if ( f ) is continuous on [0,6] |
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| 459 | If ( y=frac{x+c}{1+x^{2}}, ) then the value of ( x y ) where ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=mathbf{0} ) is A ( cdot frac{1}{2} ) B. ( frac{3}{4} ) ( c cdot frac{5}{4} ) D. None of these |
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| 460 | By Rolles theorem for ( f(x)=(x- ) ( a)^{m}(x-b)^{n} ) on ( [a, b] ; m, n ) being positive integer. Find the value of ( c ) which lies between ( a ) & b. A ( cdot c=frac{m b+n a}{m+n} ) в. ( c=frac{m b-n a}{m+n} ) c. ( _{c}=frac{n b+m a}{m+n} ) D. ( c=frac{n b-m a}{m+n} ) |
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| 461 | The function ( f ) is defined as ( f(x)= ) ( left{begin{array}{ll}x^{2}+a x+b, & text { if } 0 leq x<2 \ 3 x+2, & text { if } 2 leq x leq 4, text { If } f text { is } \ 2 a x+5 b, & text { if } 4<x leq 8end{array}right. ) continuous in [0,8] find the values of ( a ) and ( b ) |
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| 462 | Find the derivative of ( boldsymbol{y}=(boldsymbol{x}+ ) 1) ( (x+2)^{2} ) |
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| 463 | If ( sqrt{1-x^{2}}+sqrt{1-y^{2}}=a(x-y) ) prove that ( frac{d y}{d x}=sqrt{frac{1-y^{2}}{1-x^{2}}} ) |
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| 464 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: sin boldsymbol{x}+mathbf{7} boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{5} ) | 12 |
| 465 | If ( y=sqrt{x log _{e} x, text { then } frac{d y}{d x} text { at } x=e text { is }} ) ( A cdot frac{1}{e} ) B. ( frac{1}{sqrt{e}} ) ( c cdot sqrt{e} ) D. None of these |
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| 466 | Verify that ( y=a e^{-x} ) is a solution of ( frac{d^{2} y}{d x^{2}}=frac{1}{y}left(frac{d y}{d x}right)^{2} ) | 12 |
| 467 | Assertion Statement ( -1 f(x)=|x| cos x ) is not differentiable at ( mathbf{x}=mathbf{0} ) Reason Statement – 2 Every absolute value functions are not differentiable. A. Statement-1 is True, Statement-2 is True Statement- 2 is a correct explanation for Statement-1. B. Statement-1 is True, Statement-2 is True Statement-2 is NOT a correct explanation for Statement- c. Statement-1 is True, Statement-2 is False D. Statement-1 is False, Statement-2 is True |
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| 468 | Let ( boldsymbol{y}=sin ^{-1} boldsymbol{x}, ) then find ( left(mathbf{1}-boldsymbol{x}^{2}right) boldsymbol{y}_{2} ) ( boldsymbol{x} boldsymbol{y}_{1} ) Where ( y_{1} ) and ( y_{2} ) denote the first and second order derivatives respectively. ( mathbf{A} cdot mathbf{1} ) в. – 1 c. 0 D. |
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| 469 | Differentiate with respect to ( x ) : ( frac{x^{2}+2}{sqrt{cos x}} ) |
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| 470 | If ( y=e^{sqrt{x}}+e^{-sqrt{x}} ) then ( frac{d y}{d x} ) equals This question has multiple correct options A ( cdot frac{e^{sqrt{x}}-e^{-sqrt{x}}}{2 sqrt{x}} ) ( frac{e^{sqrt{x}}-e^{-sqrt{x}}}{2 x} ) c. ( frac{1}{2 sqrt{x}} sqrt{y^{2}-4} ) D. ( frac{1}{2 sqrt{x}} sqrt{y^{2}+4} ) |
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| 471 | If ( f(x)=|cos 2 x| ) then ( f^{prime}left(frac{pi}{4}+0right) ) is equal to A .2 B. c. -2 D. none of these |
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| 472 | If ( f ) be a continuous function on ( [mathbf{0}, mathbf{1}] ) differentiable in (0,1) such that ( f(1)=0, ) then there exists some ( c in ) (0,1) such that A ( cdot c f^{prime}(c)-f(c)=0 ) B. ( f^{prime}(c)+c f(c)=0 ) c. ( f^{prime}(c)-c f(c)=0 ) D. ( c f^{prime}(c)+f(c)=0 ) |
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| 473 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: ) ( boldsymbol{2} boldsymbol{y}^{2}+boldsymbol{6} boldsymbol{x}=mathbf{5} ) |
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| 474 | Find the slope of tangent to the curve ( y=3 x^{2}-6 ) at the point on it whose ( x ) coordinate is 2 |
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| 475 | If ( boldsymbol{u}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{h} boldsymbol{x} boldsymbol{y}+boldsymbol{b} boldsymbol{y}^{2} ) ( operatorname{then} x frac{partial^{2} u}{partial x partial y}+y frac{partial^{2} u}{partial y^{2}}=? ) ( mathbf{A} cdot 2(h x+b y) ) B. ( 2(h x-b y) ) c. ( 2(b x+h y) ) D. ( 2(b x-h y) ) |
12 |
| 476 | Differentiate ( boldsymbol{y}=log (log sqrt{boldsymbol{x}}) ) | 12 |
| 477 | Explain Mean Value Theorem | 12 |
| 478 | 15. A function f:R → R satisfies the equation f (x + y)=f(x)f) for all x, y in Randf(x) #0 for anyx in R. Let the function be differentiable at x=0 and f'()=2. Show that f'(x)=2f(x) for all x in R. Hence, determine f(x). (1990 – 4 Marks) |
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| 479 | ( mathbf{f} boldsymbol{y}=tan ^{-1}left[frac{sqrt{mathbf{1}+boldsymbol{x}^{2}}-mathbf{1}}{boldsymbol{x}}right], ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ? ) | 12 |
| 480 | If ( f(x)=x cdotleft(frac{a^{1 / x}-a^{-1 / x}}{a^{1 / x}+a^{-1 / x}}right), x neq ) ( mathbf{0}(boldsymbol{a}>mathbf{0},) boldsymbol{f}(mathbf{0})=mathbf{0} ) then A . fis differentiable at ( x=0 ) B. fis not differentiable at ( x=0 ) c. ( f ) is not continuous at ( x=0 ) D. None of these |
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| 481 | 26. Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f(x)= x cos(To(x+[x])) is discontinuous? (JEE Adv. 2017) (a) x=-1 (b) x=0 (c) X=1 (d) x=2 |
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| 482 | Differentiate with respect to ( x ) : ( frac{2^{x} cos x}{left(x^{2}+3right)^{2}} ) |
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| 483 | A function ( f ) is defined as follows: ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}1 & text { for }-infty< \ 1+sin x & text { for } 0 leq x<frac{pi}{2} \ 2+left(x-frac{pi}{2}right)^{2} & text { for } frac{pi}{2} leq x<+inftyend{array}right. ) Discuss the continunity and differentiability at ( boldsymbol{x}=mathbf{0} & boldsymbol{x}=boldsymbol{pi} / mathbf{2} ) This question has multiple correct options A. continuous but not differentiable at ( x=0 ) B. differentiable and continuous at ( x=pi / 2 ) c. neither continuous but nor differentiable at ( x=0 ) D. continuous but not differentiable at ( x=pi / 2 ) |
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| 484 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{cot ^{-1} frac{sqrt{1+boldsymbol{x}}-sqrt{1-boldsymbol{x}}}{sqrt{1+boldsymbol{x}}+sqrt{1-boldsymbol{x}}}right}= ) ( A cdot frac{1}{sqrt{1-x^{2}}} ) B ( cdot frac{-1}{2 sqrt{1-x^{2}}} ) ( mathbf{C} cdot frac{1}{1+x^{2}} ) D. ( frac{-1}{2left(1+x^{2}right)} ) |
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| 485 | A balloon which always remains spherical, has a variable diameter ( frac{3}{2}(2 x+3) . ) The rate of change of volume with respect to ( x ) will be A ( cdot frac{27 pi}{8}(2 x-3)^{2} ) в. ( frac{27 pi}{8}(2 x+3)^{2} ) c. ( frac{27 pi}{8}(3 x-2)^{2} ) D. ( frac{8}{27 pi}(2 x+3)^{2} ) |
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| 486 | Let ( mathbf{g}(mathbf{x})=log (mathbf{f}(mathbf{x})) ) where ( mathbf{f}(mathbf{x}) ) is a twice differentiable positive function on ( (0, infty) ) such that ( f(x+1)=x f(x) ) Then, for ( mathbf{N}=mathbf{1}, mathbf{2}, mathbf{3}, dots mathbf{g}^{prime prime}left(mathbf{N}+frac{mathbf{1}}{mathbf{2}}right)- ) ( mathrm{g}^{prime prime}left(frac{1}{2}right)= ) A ( cdot-4left(1+frac{1}{9}+frac{1}{25}+ldots+frac{1}{(2 mathrm{N}-1)^{2}}right) ) B. ( 4left(1+frac{1}{9}+frac{1}{25}+ldots+frac{1}{(2 mathrm{N}-1)^{2}}right) ) c. ( -4left(1+frac{1}{9}+frac{1}{25}+ldots+frac{1}{(2 mathrm{N}+1)^{2}}right) ) D ( 4left(1+frac{1}{9}+frac{1}{25}+ldots+frac{1}{(2 mathrm{N}+1)^{2}}right) ) |
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| 487 | Find ( frac{d y}{d x}, ) when ( y=x^{x}-2^{sin x} ) | 12 |
| 488 | If ( x^{2}-2 x^{2} y^{2}+5 x+y-5=0 ) and ( boldsymbol{y}(1)=1, ) then This question has multiple correct options A ( cdot y^{prime}(1)=1 ) в. ( y^{prime prime}(1)=-frac{4}{3} ) c. ( quad y^{prime prime}(1)=-frac{22}{3} ) D. ( y^{prime}(1)=frac{2}{3} ) |
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| 489 | Let ( boldsymbol{f}(boldsymbol{x} boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y}) ) for all ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) f ( f^{prime}(1)=2 ) and ( f(4)=4, ) then ( f^{prime}(4) ) equal to ( mathbf{A} cdot mathbf{4} ) B. c. ( frac{1}{2} ) D. 8 |
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| 490 | If ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{1+boldsymbol{x}^{2}+boldsymbol{x}^{4}}{1+boldsymbol{x}+boldsymbol{x}^{2}}right)=boldsymbol{a} boldsymbol{x}+ ) ( b, ) then ( (a, b)= ) A. (-1,2) B. (-2,1) c. (2,-1) D. (1,2) |
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| 491 | If the graphs of ( y=f(x) ) and ( y=g(x) ) intersect in coincident points the ( lambda ) can take values: This question has multiple correct options A . 3 B. 1 ( c cdot-1 ) D. |
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| 492 | If ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}+mathbf{1}|(|boldsymbol{x}|+|boldsymbol{x}-mathbf{1}|) ) then at what points the function is/are not differentiable at in the interval [-2,2] This question has multiple correct options A . -1 B. c. 1 D. ( 1 / 2 ) |
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| 493 | Differentiate: ( boldsymbol{x}^{mathbf{1 0 0}}+sin boldsymbol{x}-mathbf{1} ) A ( cdot 100 x^{99}-cos x ) ( x ) B. ( 100 x^{99}+cos x ) c. ( x^{99}+cos x ) D. ( 100 x^{99}+sin x ) |
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| 494 | If ( y=frac{x}{|n| c x mid} ) (where ( c ) is an arbitrary constant) is the general solution of the differential equation ( frac{d y}{d x}=frac{y}{x}+phileft(frac{x}{y}right) ) then the function ( phileft(frac{x}{y}right) ) A ( cdot frac{x^{2}}{y^{2}} ) в. ( -frac{x^{2}}{y^{2}} ) c. ( frac{y^{2}}{x^{2}} ) D. ( -frac{y^{2}}{x^{2}} ) |
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| 495 | Solve ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}-sin boldsymbol{theta}), boldsymbol{y}=boldsymbol{a}(mathbf{1}+cos boldsymbol{theta}) ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ? ) |
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| 496 | If ( mathbf{y}=mathbf{b} cos log left(frac{boldsymbol{x}}{boldsymbol{n}}right)^{boldsymbol{n}}, boldsymbol{t h e n} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A ( cdot frac{-n b sin log (x)^{n}}{x} ) B. ( n b sin log left(frac{x}{n}right)^{n} ) c. ( _{-n b sin log }left(frac{x}{n}right)^{n} ) D. None of these |
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| 497 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left(frac{cos boldsymbol{x}}{mathbf{1}+sin boldsymbol{x}}right) ) A. ( -frac{1}{2} ) B. ( -frac{1}{4} ) ( c cdot-frac{1}{8} ) D. 3 |
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| 498 | Let ( boldsymbol{f} ) be differential for all ( boldsymbol{x} ). If ( boldsymbol{f}(mathbf{1})= ) -2 and ( f^{prime}(x) geq 2 ) for ( x epsilon[1,6], ) then ( ? ) A. ( f(6)=5 ) В. ( f(6)<5 ) C. ( f(6)<8 ) D. ( f(6) geq 5 ) is ( 5(6) geq 5 ) |
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| 499 | Value of ( c ) of Lagranges mean theorem for [ boldsymbol{f}(boldsymbol{x})=mathbf{2}+boldsymbol{x}^{mathbf{3}} text { if } boldsymbol{x} leq mathbf{1} ] ( =3 x ) if ( x>1 ) on [-1,2] is ( A cdot pm frac{sqrt{5}}{3} ) B. ( pm frac{sqrt{3}}{2} ) ( c cdot pm frac{sqrt{2}}{5} ) ( D cdot pm frac{3}{sqrt{5}} ) |
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| 500 | Find the value of ( k ) so that ( boldsymbol{f}(boldsymbol{x})left{begin{array}{ll}boldsymbol{k} boldsymbol{x}+mathbf{1} & boldsymbol{i} boldsymbol{f} boldsymbol{x} leq boldsymbol{pi} \ cos boldsymbol{x} & boldsymbol{i} boldsymbol{f} boldsymbol{x}>piend{array}right. ) continuous at ( boldsymbol{x}=boldsymbol{pi} ) |
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| 501 | If ( f(x) ) is continuous and ( fleft(frac{9}{2}right)=frac{2}{9} ) ( operatorname{then} lim _{x rightarrow 0} fleft(frac{1-cos 3 x}{x^{2}}right) ) is equal to | 12 |
| 502 | If the p.d.f of a continuous random variable ( boldsymbol{x} ) is ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}k x^{2}(1-x) & , 0<x<1 \ 0 & , text { otherwise }end{array}right. ) then value of ( k ) is A . 12 B. 10 c. -12 D. |
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| 503 | Find the derivative of the following functions (it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers) ( sin (x+a) ) |
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| 504 | Let ( mathbf{f}=left{begin{array}{ll}mathbf{a x}^{2}+mathbf{1} & text { for } mathbf{x}>mathbf{1} \ mathbf{x}+mathbf{a} & text { for } mathbf{x} leq mathbf{1}end{array} text { then } mathbf{f}right. ) is derivable at ( x=1 ) if ( mathbf{A} cdot mathbf{a}=0 ) B. ( a=frac{1}{2} ) ( mathbf{c} cdot mathbf{a}=1 ) ( mathbf{D} cdot mathbf{a}=mathbf{2} ) |
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| 505 | State Rolle’s theorem. | 12 |
| 506 | Let ( y=2^{x}+x^{2}+2 ) then find ( frac{d y}{d x} ) | 12 |
| 507 | Differentiate the following w.r.t ( x: frac{e^{x}}{sin x} ) | 12 |
| 508 | Say true or false. Derivative of ( x^{n} ) is ( n x^{n-1} ) A. True B. False |
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| 509 | Is Rolle’s theorem valid for the function ( boldsymbol{y}=boldsymbol{x}^{3}+boldsymbol{4} boldsymbol{x}^{2}-boldsymbol{7} boldsymbol{x}-mathbf{1 0} ) in the interval [-1,2] |
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| 510 | Prove cos is continuous on R. | 12 |
| 511 | If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}+boldsymbol{e}^{-boldsymbol{x}}+log boldsymbol{x}^{2}, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 512 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} cot left(frac{x}{2}right)= ) | 12 |
| 513 | f ( boldsymbol{y}=tan ^{-1} frac{boldsymbol{x}}{mathbf{1}+sqrt{mathbf{1}-boldsymbol{x}^{2}}}+ ) ( sin left(2 tan ^{-1} sqrt{frac{1-x}{1+x}}right), ) then find ( frac{d y}{d x} ) for ( boldsymbol{x} in(-1,1) ) |
12 |
| 514 | ( left{begin{array}{cl}text { Evaluate } lim _{x rightarrow 2^{+}} f(x), text { where } f(x)= \ (x-[x], quad x2end{array}right. ) | 12 |
| 515 | If ( 5 f(x)+3 fleft(frac{1}{x}right)=x+2 ) and ( y= ) ( boldsymbol{x} boldsymbol{f}(boldsymbol{x}) ) ( operatorname{then} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{n}} ) at ( boldsymbol{x}=mathbf{1} ) A . 14 в. ( frac{7}{8} ) c. 1 D. |
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| 516 | ( f(x)=frac{a sin x-b x+c x^{2}+x^{3}}{2 x^{2} ell n(1+x)-2 x^{3}+x^{4}} ) when ( x neq 0 ) and ( f(x) ) is continuous at ( boldsymbol{x}=mathbf{0}, ) find value of ( mathbf{2 0 0} times boldsymbol{f}(mathbf{0}) ) |
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| 517 | Differentiate the functions with respect to ( x ) ( sec (tan (sqrt{x})) ) |
12 |
| 518 | Find ( frac{d y}{d x}=sin ^{-1} x ) | 12 |
| 519 | Discuss the continuity of the function defined by ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{5}| ) | 12 |
| 520 | Differentiate the following functions with respect to ( x ) ( sin ^{-1}left{frac{x+sqrt{1-x^{2}}}{sqrt{2}}right},-1<x<1 ) |
12 |
| 521 | Answer the following question in one word or one sentence or as per exact requirement of the question. If ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}|+|boldsymbol{x}-mathbf{1}|, ) write the value of ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(boldsymbol{f}(boldsymbol{x})) ) |
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| 522 | If ( sin ^{-1} x+sin ^{-1} y=frac{pi}{2}, ) then ( frac{d y}{d x} ) is equal to A ( cdot frac{x}{y} ) в. ( -frac{x}{y} ) c. ( frac{y}{x} ) D. ( -frac{y}{x} ) |
12 |
| 523 | Find derivative of ( (boldsymbol{a} boldsymbol{x}+boldsymbol{b})^{n}(boldsymbol{c} boldsymbol{x}+boldsymbol{d})^{boldsymbol{n}} ) | 12 |
| 524 | Differentiate the following with respect to ( x ) ( cos ^{-1} 2 x sqrt{1-x^{2}}, frac{1}{sqrt{2}}<x<1 ) |
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| 525 | For every twice differentiable function f:R 2,2W (JEE Adv. 2018) (0) +(f'(o))2 = 85, which of the following statements) is (are) TRUE? a) There exist r,seR, wherer (d) There exists a € (-4,4) such that f(a)+f”(a) = 0 and f'(a)+0 |
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| 526 | Let ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}]+[-boldsymbol{x}] . ) Then for any integer ( n ) and ( k in R-I ) This question has multiple correct options A ( cdot lim _{x rightarrow n} f(x) ) exists B. ( lim _{x rightarrow k} f(x) ) exists c. f is continuous at ( x=n ) D. f is continuous at ( x=k ) |
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| 527 | The function ( boldsymbol{f}(boldsymbol{x})left{begin{array}{l}frac{sin sqrt[3]{boldsymbol{x}} log (1+boldsymbol{3} boldsymbol{x})}{left(tan ^{-1} sqrt{boldsymbol{x}}right)^{2}left(boldsymbol{e}^{boldsymbol{5}} sqrt[3]{boldsymbol{x}}-mathbf{1}right)} quad, boldsymbol{x} neq mathbf{0} \ boldsymbol{a}, quad boldsymbol{x}=mathbf{0}end{array}right. ) is continuous at ( boldsymbol{x}=mathbf{0}, ) if ( mathbf{A} cdot a=0 ) B. ( a=frac{5}{3} ) ( mathbf{c} cdot a=2 ) D. ( a=frac{3}{5} ) |
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| 528 | ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{f}(boldsymbol{x})= ) ( frac{boldsymbol{x}left(boldsymbol{x}^{4}+mathbf{1}right)(boldsymbol{x}+mathbf{1})+boldsymbol{x}^{4}+mathbf{2}}{boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1}}, operatorname{then} boldsymbol{f}(boldsymbol{x}) ) is A. one-one ito B. many-one onto c. one-one onto D. many-one into |
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| 529 | If ( y=sqrt{frac{1-x}{1+x}} ) then ( frac{d y}{d x} ) equals- A ( cdot frac{y}{1-x^{2}} ) в. ( frac{y}{x^{2}-1} ) c. ( frac{y}{1+x^{2}} ) D. ( frac{y}{y^{2}-1} ) |
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| 530 | 14. The function f (x) = (x2 – 1) x2-3x+2 +cos ( x is NOT differentiable at (1999-2 Marks) (a) -1 (6) 0 (c) 1 (d) 2 |
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| 531 | Differentiate ( frac{x^{4}}{4}-frac{x^{-3}}{3}-frac{2}{x}+C ) |
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| 532 | The left hand derivative of ( f(x)= ) ( [x] sin pi x ) at ( x=k, k ) is an integer, is ( mathbf{A} cdot(-1)^{k}(k-1) pi ) B cdot ( (-1)^{k-1}(k-1) pi ) ( mathbf{c} cdot(-1)^{k} k pi ) D. ( (-1)^{k-1} k pi ) |
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| 533 | Let ( S ) be the set of all functions ( f: ) ( [0,1] rightarrow R, ) which are continuous on [0,1] and differentiable on ( (0,1) . ) Then for every ( f ) in ( S, ) there exists ( a c in(0,1) ) depending on ( f, ) such that ( ^{text {A } cdot frac{f(1)-f(c)}{1-c}=f^{prime}(c)} ) B cdot ( |f(c)-f(1)|<(1-c)left|f^{prime}(c)right| ) c. ( |f(c)+f(1)|<(1+c)left|f^{prime}(c)right| ) D ( cdot|f(c)-f(1)|<mid f^{prime}(c) ) |
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| 534 | If ( sin y=x sin (a+y), ) then show that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{sin ^{2}(boldsymbol{a}+boldsymbol{y})}{sin boldsymbol{a}} ) |
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| 535 | ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2})(boldsymbol{x}-boldsymbol{3}), boldsymbol{x} in[mathbf{0}, boldsymbol{4}] ) find ( ^{prime} c^{prime} ) if ( L M V T ) can be applied. |
12 |
| 536 | If ( boldsymbol{y}=tan ^{2}left(log boldsymbol{x}^{3}right), ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 537 | Show that ( f, ) given by ( f(x)= ) ( frac{boldsymbol{x}-|boldsymbol{x}|}{boldsymbol{x}}(boldsymbol{x} neq mathbf{0}), ) is continuous on ( mathrm{R}[mathbf{0}] ) | 12 |
| 538 | Find the derivative of ( sqrt{tan x} ) with respect to ( x ) using the first principle. A ( cdot frac{sec ^{2} x}{2 sqrt{tan x}} ) в. ( frac{s e c x}{2 sqrt{tan x}} ) ( ^{mathrm{c}} cdot frac{sec ^{2} x}{sqrt{tan x}} ) D. ( frac{sec ^{2} x}{2 sqrt{tan } x} ) |
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| 539 | ( operatorname{Let} f^{prime}(x)=e^{x^{2}} ) and ( f(0)=10 . ) If ( A< ) ( f(1)<B ) can be concluded from the mean value theorm, then the largest value of ( (boldsymbol{A}-boldsymbol{B}) ) equals ( A ) B. ( 1-e ) ( mathbf{c} cdot e-1 ) D. ( 1+e ) |
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| 540 | If ( x^{2}+y^{2}=t-frac{1}{t} ) and ( x^{4}+y^{4}=t^{2}+ ) ( frac{1}{t^{2}}, ) then prove that ( frac{d y}{d x}=frac{1}{x^{3} y} ) |
12 |
| 541 | Illustration 2.20 If y= – = (x)-1/2, then find dyldx. |
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| 542 | If ( mathbf{y}=mathbf{c e}^{x /(x-a)}, ) then ( frac{mathbf{d y}}{mathbf{d x}} ) equals ( A cdot a(x-a) ) B. ( -frac{text { ay }}{(x-a)^{2}} ) ( c cdot a^{2}(x-a)^{2} ) D. |
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| 543 | Let ( (a-b cos y)(a+b cos x)=a^{2}-b^{2} ) and ( frac{d y}{d x}=frac{sin x f(y))}{(a+b cos x)^{2}} cdot ) If ( a^{2}-b^{2}= ) ( 192, ) then ( f(pi / 2) ) |
12 |
| 544 | If ( f(x)=sec (3 x), ) then ( f^{prime}left(frac{3 pi}{4}right)= ) A. ( -3 sqrt{2} ) B. ( -frac{3 sqrt{2}}{2} ) ( c cdot frac{3}{2} ) D. ( frac{3 sqrt{2}}{2} ) E. ( 3 sqrt{2} ) |
12 |
| 545 | If ( f(x)=left|x^{2}-4 x+3right|, ) then ( f^{prime}(x) ) is A. ( 2 x-4 ) for ( 1<x<3 ) B. ( 4-2 x ) for ( 1<x<3 ) c. ( 2 x-4 ) for ( 1 leq x leq 3 ) D. ( 4-2 x ) |
12 |
| 546 | If ( y=log _{10}(sin x), ) then ( frac{d y}{d x} ) equals to: A ( cdot sin x log _{10} e ) B. ( cos x log _{10} e ) C. ( cot x log _{10} e ) D. ( cot x ) |
12 |
| 547 | Show that ( boldsymbol{f}(boldsymbol{x})= ) ( begin{array}{ll}frac{sin 3 x}{tan 2 x}, & text { if } x0end{array} ) |
12 |
| 548 | Find ( frac{d y}{d x}, x=aleft(cos t+log tan frac{t}{2}right), y= ) ( a sin t ) |
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| 549 | The derivative of the function ( f(x)= ) ( sqrt{x^{2}-2 x+1} ) in the interval [0,2] is A . -1 B. ( c cdot 0 ) D. does not exist |
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| 550 | Differentiate with respect to ( x e^{x} x^{5} ) A ( cdot 5 e^{x} x^{4}+e^{x} x^{5} ) B. ( 4 e^{x} x^{5}+e^{x} x^{5} ) ( mathbf{c} cdot 5 e^{x} x^{4}+e^{x} x^{4} ) D. ( 4 e^{x} x^{5}+e^{x} x^{4} ) |
12 |
| 551 | The derivative of ( f(tan x) ) with respect ( operatorname{tog}(sec x) ) at ( quad x=frac{pi}{4}, ) where ( f^{prime}(1)= ) ( mathbf{2} ; quad boldsymbol{g}^{prime}(sqrt{mathbf{2}})=mathbf{4} ) is A ( cdot frac{1}{sqrt{2}} ) B. ( sqrt{2} ) c. 1 D. |
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| 552 | ( x=t cos t, y=t+sin t . ) Then ( frac{d^{2} x}{d y^{2}} ) at ( t=frac{pi}{2} ) is A ( cdot frac{pi+4}{2} ) в. ( -frac{pi+4}{2} ) c. -2 D. none of these |
12 |
| 553 | Find: ( frac{d y}{d x}=sin (x+y)+cos (x+y) ) |
12 |
| 554 | If for all ( x, y ) the function ( f ) is defined by ( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})+boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y})=1 ) and ( boldsymbol{f}(boldsymbol{x})>0 . ) When ( boldsymbol{f}(boldsymbol{x}) ) is differentiable ( boldsymbol{f}^{prime}(boldsymbol{x})= ) A . -1 B. 1 ( c .0 ) D. cannot be determined |
12 |
| 555 | Let S (x) be defined in the interval [-2,2] such that f(x) =1-1-2sxso Xx-10<x2 and g(x)=f(xD) + f(x)! Test the differentiability of g(x) in (-2,2). (1986-5 Marks) |
12 |
| 556 | Differentiate w.r.t ( x, ) the following function: ( log sqrt{frac{1+cos x}{1-cos x}} ) |
12 |
| 557 | Consider the function ( y=|x-1|+ ) ( |x-2| ) in the interval [0,3] and discuss the continuity and differentiability of the function in this interval. This question has multiple correct options A. continuous everywhere B. differentiable everywhere except at ( x=1 ) and ( x=2 ) c. differentiable everywhere D. continuous everywhere except at ( x=1 ) and ( x=2 ) |
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| 558 | Find ( frac{d y}{d x} ) if (a) ( x^{3}+2 x^{2} y+3 x y^{2}+4 y^{3}=5 ) (b) ( x=2 cos ^{3} theta, y=2 sin ^{3} theta ) (c) ( y=sin ^{-1}(2 x sqrt{1-x^{2}}) ;-1 leq x leq 1 ) |
12 |
| 559 | Differentiate with respect to ( x ) : ( e^{tan ^{-1} sqrt{x}} ) |
12 |
| 560 | Differentiate the following functions with respect to ( x: ) ( tan ^{-1}left(frac{a+b x}{b-a x}right) ) |
12 |
| 561 | If ( f(x)=frac{1}{2} x-1, ) then on the interval ( [mathbf{0}, boldsymbol{pi}] ) A ( cdot tan (f(x)) ) and ( frac{1}{f(x)} ) are continuous в. ( tan (f(x)) ) and ( frac{1}{f(x)} ) are discontinuous c. ( tan (f(x)) ) is continuous but ( frac{1}{f(x)} ) is discontinuous D ( cdot tan (f(x)) ) is discontinuous but ( frac{1}{f(x)} ) is continuous |
12 |
| 562 | If ( boldsymbol{x}^{boldsymbol{y}}=boldsymbol{e}^{boldsymbol{x}-boldsymbol{y}} ) then ( frac{d y}{d x}=frac{log x}{(1-log x)^{2}} ) A . True B. False |
12 |
| 563 | Let ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) ) and ( boldsymbol{f}(boldsymbol{x})= ) ( 1+(sin 2 x) g(x) ) where ( g(x) ) is continuous, then ( f^{prime}(x) ) equals A ( . f(x) g(0) ) B ( .2 f(x) g(0) ) D. ( 2 f(0) ) |
12 |
| 564 | Let ( f(x) ) be a real valued function not identically zero, such that ( boldsymbol{f}left(boldsymbol{x}+boldsymbol{y}^{n}right)=boldsymbol{f}(boldsymbol{x})+(boldsymbol{f}(boldsymbol{y}))^{n} quad forall boldsymbol{x}, boldsymbol{y} in ) ( boldsymbol{R} ) where ( n in N(n neq 1) ) and ( f^{prime}(0) geq 0 . ) We may get an explicit form of the function ( boldsymbol{f}(boldsymbol{x}) ) The value of ( $ $ f^{prime}(0) $ $ f^{prime}(0) ) is : A. B. ( n ) c. ( n+1 ) D. 2 |
12 |
| 565 | COMO 10. Let f (x) be a continuous and g(x) be a discontinuous function. prove that f(x) + g(x) is a discontinuous function. (1987-2 Marks) |
12 |
| 566 | If ( x=frac{1-t^{2}}{1+t^{2}} ) and ( y=frac{2 t}{1+t^{2}} ) at then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? ) A ( cdot frac{-1}{x^{3}} ) в. ( frac{y}{x} ) c. ( frac{-x}{y} ) D. ( frac{x}{y} ) |
12 |
| 567 | What is the nature of the graph: ( boldsymbol{y}=boldsymbol{6} e^{-4 boldsymbol{x}} ) A. monotonically increasing B. monotonically decreasing c. Increasing then decreasing D. decreasing then increasing |
12 |
| 568 | Differentiate with respect to ( x ) ( boldsymbol{y}=log (1+sin boldsymbol{x}) ) |
12 |
| 569 | Find ‘c’ of Lagrange’s mean-value theorem for (i) ( f(x)=left(x^{3}-3 x^{2}+2 xright) ) on ( left[0, frac{1}{2}right] ) (ii) ( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{2 5}-boldsymbol{x}^{2}} ) on ( [mathbf{0}, mathbf{5}] ) (iii) ( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}+boldsymbol{2}} ) on ( [boldsymbol{4}, boldsymbol{6}] ) |
12 |
| 570 | Differentiate ( sin ^{2} 3 x cdot tan ^{3} 2 x ) | 12 |
| 571 | The function ( mathbf{f}(mathbf{x})=frac{cos boldsymbol{x}-sin boldsymbol{x}}{cos mathbf{2} boldsymbol{x}} ) is not defined at ( x=frac{pi}{4} ) The value of ( fleft(frac{pi}{4}right) ) so that ( mathbf{f}(mathbf{x}) ) is continuous at ( boldsymbol{x}=frac{boldsymbol{pi}}{mathbf{4}} ) is A ( cdot frac{1}{sqrt{2}} ) B. ( sqrt{2} ) ( c cdot-sqrt{2} ) D. |
12 |
| 572 | What is the derivative of ( |x-1| ) at ( x= ) ( mathbf{2} ? ) A . -1 B. 0 c. 1 D. Derivative does not exist |
12 |
| 573 | If ( y ) is a function of ( x, ) then ( frac{d^{2} y}{d x^{2}}+ ) ( y frac{d y}{d x}=0 . ) If ( x ) is a function of ( y, ) then the equation becomes A ( cdot frac{d^{2} x}{d y^{2}}+x frac{d x}{d y}=0 ) в. ( frac{d^{2} x}{d y^{2}}+yleft(frac{d x}{d y}right)^{3}=0 ) c. ( frac{d^{2} x}{d y^{2}}-yleft(frac{d x}{d y}right)^{2}=0 ) D ( cdot frac{d^{2} x}{d y^{2}}-xleft(frac{d x}{d y}right)^{2}=0 ) |
12 |
| 574 | Differentiate ( 2 x^{3 / 2}+2 x^{5 / 2}+C ) A ( cdot frac{d y}{d x}=sqrt{x}(3+5 x) ) B. ( frac{d y}{d x}=sqrt{x}(3-5 x) ) c. ( frac{d y}{d x}=-sqrt{x}(3+5 x) ) D. None of these |
12 |
| 575 | Find the derivative of the following functions (it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers) ( (a x+b)^{n}(c x+d)^{m} ) |
12 |
| 576 | If ( e^{y}(x+1)=1 ) show that ( frac{d y}{d x}=-e^{y} ) | 12 |
| 577 | If ( boldsymbol{x}+boldsymbol{y}=tan ^{-1} boldsymbol{y} ) and ( boldsymbol{y}^{prime prime}=boldsymbol{f}(boldsymbol{y}) boldsymbol{y}^{prime} ) then ( boldsymbol{f}(boldsymbol{y})= ) A ( cdot frac{1}{yleft(1+y^{2}right)} ) В. ( frac{3}{yleft(1+y^{2}right)} ) c. ( frac{2}{yleft(1+y^{2}right)} ) D. ( frac{-2}{yleft(1+y^{2}right)} ) |
12 |
| 578 | If ( boldsymbol{f}(boldsymbol{x})=min left(|boldsymbol{x}|^{2}-mathbf{5}|boldsymbol{x}|, mathbf{1}right) ) then ( boldsymbol{f}(boldsymbol{x}) ) is non differentiable at ( lambda ) points, then ( lambda+13 ) equals |
12 |
| 579 | f ( boldsymbol{y}=boldsymbol{a} cos (log boldsymbol{x})-boldsymbol{b} sin (log boldsymbol{x}), ) then the value of ( x^{2} frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}+y ) is ( mathbf{A} cdot mathbf{0} ) B. 1 c. 2 D. 3 |
12 |
| 580 | If ( x=A cos 4 t+B sin 4 t, ) then ( frac{d^{2} x}{d t^{2}}= ) ( A ) B . ( -16 x ) ( c .15 x ) D. ( 16 x ) E . ( -15 x ) |
12 |
| 581 | If ( y=log _{x^{2}+4}left(7 x^{2}-5 x+1right), ) then ( frac{d y}{d x}= ) A ( cdot frac{1}{log _{e}left(x^{2}+4right)}left(frac{14 x-5}{7 x^{2}-5 x+1}-frac{2 x y}{x^{2}+4}right) ) В ( cdot frac{1}{log _{e}left(x^{2}+4right)}left(frac{14 x-5}{7 x^{2}-5 x+1}+frac{2 x y}{x^{2}+4}right) ) c. ( -frac{1}{log _{e}left(x^{2}+4right)}left(frac{14 x-5}{7 x^{2}-5 x+1}-frac{2 x y}{x^{2}+4}right) ) D. None of these |
12 |
| 582 | A value of c which the conclusion of Mean Value Theorem holds for the function ( boldsymbol{f}(boldsymbol{x})=log _{e} boldsymbol{x} ) on the interval [1,3] is ( mathbf{A} cdot 2 log _{3} e ) B – ( frac{1}{2} log _{e} 3 ) ( c cdot log _{3} ) ( mathrm{D} cdot log _{e} 3 ) |
12 |
| 583 | If ( x=sec theta-cos theta ) and ( y=sec ^{3} theta- ) ( sec ^{3} theta-cos ^{3} theta, ) then the value of ( left(frac{d y}{d x}right)^{2} ) at ( x=0 ) A . 0 B. 2 ( c cdot 4 ) D. |
12 |
| 584 | If ( f(x)=frac{1-tan x}{1-sqrt{2} sin x}, ) for ( x neq ) ( frac{pi}{4} ) is continous ( boldsymbol{a t} quad boldsymbol{x}= ) ( frac{pi}{4}, quad ) find ( quad fleft(frac{pi}{4}right) ) |
12 |
| 585 | Find the value of ( k ) for which ( f(x)= ) ( left{begin{array}{l}frac{1-cos 4 x}{8 x^{2}}, text { when } x neq 0 \ k, quad text { when } x=0end{array}right. ) continuous at ( boldsymbol{x}=mathbf{0} ) |
12 |
| 586 | ( operatorname{Let} g(x)=frac{f(x)}{x+1} ) where ( f(x) ) is differentiable on [0,5] such that ( boldsymbol{f}(mathbf{0})=mathbf{4}, boldsymbol{f}(mathbf{5})=-1 . ) There exists ( boldsymbol{c} in ) (0,5) such that ( g^{prime}(c) ) is ? ( A cdot-frac{1}{6} ) B. ( frac{1}{6} ) c. ( -frac{5}{6} ) D. – |
12 |
| 587 | ( frac{boldsymbol{d}}{d x}left(e^{tan x}right) ) ( mathbf{A} cdot e^{tan x} cdot sec ^{2} x ) B ( cdot e^{cot x} cdot sec ^{2} x ) ( mathbf{C} cdot e^{cos x} cdot sec ^{2} x ) D ( cdot e^{sin x} cdot sec ^{2} x ) |
12 |
| 588 | Find ( frac{d y}{d x} ) of ( x y+y^{2}=tan x+y ) | 12 |
| 589 | ff ( y=e^{x}(sin x+cos x), ) then prove that ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}-boldsymbol{2} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{2} boldsymbol{y}=mathbf{0} ) |
12 |
| 590 | If ( f(x) ) is continuous in [0,1] and ( fleft(frac{1}{3}right)=1 ) then ( lim _{n rightarrow infty} fleft(frac{n}{sqrt{9 n^{2}+1}}right) ) is ( mathbf{A} cdot mathbf{1} ) B. c. ( frac{1}{3} ) D. none of these |
12 |
| 591 | Assertion Derivative of ( (boldsymbol{p} boldsymbol{x}+boldsymbol{q})left(frac{boldsymbol{r}}{boldsymbol{x}}+boldsymbol{s}right) ) is ( boldsymbol{p s}+ ) ( frac{boldsymbol{q} boldsymbol{r}}{boldsymbol{x}^{2}} ) Reason ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(boldsymbol{u} boldsymbol{v})=boldsymbol{u}^{prime} boldsymbol{v}+boldsymbol{u} boldsymbol{v}^{prime} ) where ( u ) and ( v ) are two distinct functions. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct |
12 |
| 592 | (a) Differentiate ( boldsymbol{y}=cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right) ) with respect to ( boldsymbol{x}, mathbf{0}<boldsymbol{x}<mathbf{1} ) (b) Differentiate ( x^{x}-2^{sin x} ) with respect to ( x ) |
12 |
| 593 | If ( x^{y}-y^{x}=1, ) then the value of ( frac{d y}{d x} ) is : | 12 |
| 594 | If ( boldsymbol{x}^{p} cdot boldsymbol{y}^{boldsymbol{q}}=(boldsymbol{x}+boldsymbol{y})^{boldsymbol{p}+boldsymbol{q}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? ) A. ( frac{y}{x} ) B. ( -frac{y}{x} frac{y}{x} ) c. ( frac{x}{y} ) D. ( -frac{x}{y} ) |
12 |
| 595 | If ( x^{3}-y^{3}+3 x y^{2}-3 x^{2} y+1=0, ) then ( operatorname{at}(0,1) frac{d y}{d x} ) equals ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot 2 ) D. |
12 |
| 596 | ff ( x=a t^{4}, y=b t^{3}, ) then find ( frac{d y}{d x} ) | 12 |
| 597 | If ( f(x)=cos ^{-1}left{frac{1-left(log _{e} xright)^{2}}{1+left(log _{e} xright)^{2}}right} ), then ( boldsymbol{f}^{prime}(boldsymbol{e}) ) A. Does not exist B. c. D. Is equal to 1 |
12 |
| 598 | ( * boldsymbol{f}(boldsymbol{x})= ) ( frac{1-sin x}{(pi-2 x)^{2}} cdot frac{log sin x}{log left(1+pi^{2}-4 pi x+4 x^{2}right)}, x ) ( pi / 2 ) The assigned to function at ( x= ) ( pi / 2 ) in order that it may be continuous at ( x=pi / 2 ) is ( -frac{1}{m} . ) Find ( m ) |
12 |
| 599 | ( operatorname{Let} f(x)=left{begin{array}{ll}2 x+3 & ,-3 leq x<-2 \ x+1, & -2 leq x<0 \ x+2, & 0 leq x leq 1end{array}right. ) At what points the function is/are not differentiable in the interval (-3,1) This question has multiple correct options A . -2 B. ( c .1 ) D. ( 1 / 2 ) |
12 |
| 600 | f ( f(x)=sqrt{1+cos ^{2}left(x^{2}right)}, ) then ( f^{prime}(x) ) is? | 12 |
| 601 | Find ( frac{d y}{d x} ) of ( a x+b y^{2}=cos y ) | 12 |
| 602 | Differentiate w.r.t ( x ) ( boldsymbol{y}=log left(boldsymbol{4} boldsymbol{e}^{boldsymbol{3} boldsymbol{x}}right) ) |
12 |
| 603 | ( boldsymbol{y}=sin ^{1}left[frac{boldsymbol{2} boldsymbol{x}}{mathbf{1}+boldsymbol{x}^{2}}right], boldsymbol{t h e n} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 604 | If ( x sin y=sin (y+a) ) and ( frac{d y}{d x}= ) ( frac{A}{1+x^{2}-2 x cos a} ) then the value of ( A ) is ( A cdot 2 ) B. ( cos a ) ( c .-sin a ) D. – – |
12 |
| 605 | [ operatorname{Let} f(x)=x+frac{1}{2 x+frac{1}{2 x+frac{1}{2 x+ldots . . infty}}} ] Compute the value of ( boldsymbol{f}(mathbf{1 0 0}) cdot boldsymbol{f}^{prime}(mathbf{1 0 0}) ) |
12 |
| 606 | Solve the following differential equation ( frac{d y}{d x}=x^{2} ) | 12 |
| 607 | If ( y=sqrt{sin x+y} ) then ( frac{d y}{d x} ) equals to A ( cdot frac{cos x}{2 y-1} ) B. ( frac{cos }{1-2 y} ) c. ( frac{sin x}{1-2 y} ) D. ( frac{sin x}{2 y-1} ) |
12 |
| 608 | Find the derivation of ( sqrt{tan x} ) with respect to x using first principle. | 12 |
| 609 | Differentiate the following functions with respect to ( x: ) ( mathbf{f} boldsymbol{y}=mathbf{s e c}^{-1}left(frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right)+ ) ( sin ^{-1}left(frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}right), boldsymbol{x}>mathbf{0} . ) Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) |
12 |
| 610 | ( f(x)=frac{sin 2 x+1}{sin x-cos x} ) is discontinuous at ( boldsymbol{x}= ) A ( cdot frac{pi}{4} ) в. ( frac{pi}{3} ) c. D. |
12 |
| 611 | ( mathbf{f} boldsymbol{y}=sin left{tan ^{-1} sqrt{left(frac{1-x}{1+x}right)}right} ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{- x}}{sqrt{mathbf{1 – x}^{2}}} ) |
12 |
| 612 | If ( f ) is a real-valued differentiable function satisfying ( |boldsymbol{f}(boldsymbol{x})-boldsymbol{f}(boldsymbol{y})| leq ) ( (x-y)^{2}, quad x, y in R ) and ( f(0)=0, ) then ( boldsymbol{f}(mathbf{1}) ) equals ( mathbf{A} cdot mathbf{1} ) B. 2 c. 0 D. – |
12 |
| 613 | Differentiate the following function with respect to ( x ) ( x^{3} sin x ) |
12 |
| 614 | Verify lagrange’s mean value theorem for the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{3} ) where ( boldsymbol{x} in[boldsymbol{4}, boldsymbol{6}] ) |
12 |
| 615 | If ( y=x tan y, ) then ( frac{d y}{d x} ) is equal to A. ( frac{tan y}{x-x^{2}-y^{2}} ) в. ( frac{y}{x-x^{2}-y^{2}} ) c. ( frac{tan y}{y-x} ) D. ( frac{tan x}{x-y^{2}} ) |
12 |
| 616 | Differentiate w.r.t ( boldsymbol{x} ) ( e^{operatorname{cosec}^{2} x} ) |
12 |
| 617 | 25. If|cs and f(x) is a differentiable function at x = 0 given + x bsin -1 <x<0 by f(x) = { x=0 2x 12 – 1 0<x< 1 / 2 Find the value of a' and prove that 64 b2=4-c2 (2004 – 4 Marks) |
12 |
| 618 | The value of ( c ) in Lagrange mean value theorem for ( f(x)=log (sin x) ) in ( left[frac{pi}{6}, frac{5 pi}{6}right] ) is This question has multiple correct options A ( cdot frac{pi}{4} ) в. ( frac{pi}{2} ) c. ( frac{2 pi}{3} ) D. ( frac{3 pi}{4} ) |
12 |
| 619 | ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{r}frac{boldsymbol{K} cos boldsymbol{x}}{boldsymbol{pi}-mathbf{2} boldsymbol{x}} ; boldsymbol{x} neq frac{boldsymbol{pi}}{mathbf{2}} \ boldsymbol{5} ; boldsymbol{x}=frac{boldsymbol{pi}}{2}end{array}right. ) Find the value of ( K ) so that the function is continuous at the point ( boldsymbol{x}=frac{boldsymbol{pi}}{mathbf{2}} ) |
12 |
| 620 | Left hand derivative and right hand derivative of a function ( f(x) ) at a point ( x=a ) are defined as ( f^{prime}left(a^{-}right)=lim _{h rightarrow 0^{+}} frac{f(a)-f(a-h)}{h}= ) ( lim _{h rightarrow 0^{-}} frac{f(a)-f(a-h)}{h}= ) ( lim _{x rightarrow a^{+}} frac{f(a)-f(x)}{a-x} ) respectively Let ( f ) be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function. If ( f ) is even, which of the following is Right hand derivative of ( boldsymbol{f}^{prime} ) at ( boldsymbol{x}=boldsymbol{a} ) A. ( lim _{h rightarrow 0^{-}} frac{f^{prime}(a)+f^{prime}(-a+h)}{h} ) B. ( lim _{h rightarrow+} frac{f^{prime}(a)+f^{prime}(-a-h)}{h} ) c. ( lim _{h rightarrow 0} frac{-f^{prime}(a)+f^{prime}(-a+h)}{-h} ) D. ( lim _{h rightarrow 0^{+}} frac{f^{prime}(a)+f^{prime}(-a+h)}{h} ) |
12 |
| 621 | 1. Differentiation of sin(x?) w.r.t. x is (a) cos(x2) (b) 2x cos(x2) (c) x2 cos(x2) (d) -cos(2x) dy |
12 |
| 622 | Differentiate the following functions with respect to ( boldsymbol{x} ) [ mathbf{f} boldsymbol{y}=cot ^{-1}left{frac{sqrt{mathbf{1}+sin boldsymbol{x}}+sqrt{mathbf{1}-sin boldsymbol{x}}}{sqrt{mathbf{1}+sin boldsymbol{x}}-sqrt{mathbf{1}-sin boldsymbol{x}}}right} ] ( 0<x<frac{pi}{2}, ) show that ( frac{d y}{d x} ) is independent of ( boldsymbol{x} ) |
12 |
| 623 | Find the derivative of the following functions: ( 3 cot x+5 cos e c x ) | 12 |
| 624 | Find the derivative of the following function. ( log (log x) ) |
12 |
| 625 | Let ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] ) and ( boldsymbol{g}(boldsymbol{x})= ) ( left{begin{array}{cc}mathbf{0}, & boldsymbol{x} in boldsymbol{Z} \ boldsymbol{x}^{2}, & boldsymbol{x} in boldsymbol{R}-boldsymbol{Z}end{array} . ) Then which of the right. following is not true ([.] represents the greatest integer function)? This question has multiple correct options A ( cdot lim _{x rightarrow 1} g(x) ) exists but ( g(x) ) is not continuous at ( x=1 ) B. ( lim _{x rightarrow 1} f(x) ) does not exist and ( f(x) ) is not continuous at [ x=1 ] c. ( g o f ) is a continuous function. D. gof is a discontinuous function. |
12 |
| 626 | If ( y^{2}+b^{2}=2 x y, ) then ( frac{d y}{d x} ) equals This question has multiple correct options A ( cdot frac{1}{x y-b^{2}} ) в. ( frac{y}{y-x} ) c. ( frac{x y-b^{2}}{(y-x)^{2}} ) ( frac{x y-b^{2}}{y} ) |
12 |
| 627 | Discuss the continuity of the following function at the indicated point(s): [ boldsymbol{f}(boldsymbol{x})=left{begin{array}{l} frac{boldsymbol{e}^{boldsymbol{x}}-mathbf{1}}{log (mathbf{1}+mathbf{2} boldsymbol{x})}, boldsymbol{i} boldsymbol{f} boldsymbol{x} neq mathbf{0} \ boldsymbol{7}, boldsymbol{i} boldsymbol{f} quad boldsymbol{x}=mathbf{0} end{array}right. ] ( boldsymbol{x}=mathbf{0} ) |
12 |
| 628 | If ( x^{2}+2 x y+2 y^{2}=1, ) then ( frac{d y}{d x} ) at the point where ( y=1 ) is equal to: A . B. 2 ( c cdot-1 ) D. – ( E ) |
12 |
| 629 | Verify Rolle’s theorem for the following function: ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{x}+mathbf{1 0} ) on ( [mathbf{0}, boldsymbol{4}] ) | 12 |
| 630 | Let ( f(x) ) satisfy the requirements of Lagrange’s mean value theorem in [0,1] ( boldsymbol{f}(mathbf{0})=mathbf{0} ) and ( boldsymbol{f}^{prime}(boldsymbol{x}) leq mathbf{1}-boldsymbol{x}, forall boldsymbol{x} boldsymbol{epsilon}(mathbf{0}, mathbf{1}) ) then A. ( f(x) geq x ) в. ( |f(x)| geq 1 ) C ( . f(x) leq x(1-x) ) D. ( f(x) leq frac{1}{4} ) |
12 |
| 631 | d? dv2 equals (2007-3 marks) (d |
12 |
| 632 | Find ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}, ) where ( boldsymbol{y}=log left(frac{boldsymbol{x}^{2}}{boldsymbol{e}^{2}}right) ) | 12 |
| 633 | Let ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{mathbf{3}}+mathbf{3} boldsymbol{x} forall boldsymbol{x} in boldsymbol{R}, ) then equation of tangent for ( y=f^{-1}(x) ) at ( boldsymbol{x}=mathbf{5} ) will be A ( cdot 9 y-x=4 ) В. ( 9 y-4 x=-19 ) c. ( 49 y-9 x=4 ) D. ( 9 y-2 x=-1 ) |
12 |
| 634 | 9. The following functions are continuous on (0,7). (1991 – 2 Marks) (a) tan x o 0<**** V x sin x, sin(+*), <x<* |
12 |
| 635 | Solve: ( lim _{x rightarrow 3} frac{left(x^{frac{1}{3}}+3 sqrt{3}right)left(x^{frac{1}{3}}-3 sqrt{3}right)}{x-3} ) |
12 |
| 636 | For instantaneous speed, the distance traveled by the object and the time taken are both equal to zero. A. True B. False |
12 |
| 637 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{x}^{boldsymbol{y}}=boldsymbol{e}^{boldsymbol{x}-boldsymbol{y}} ) | 12 |
| 638 | ( boldsymbol{x}^{frac{1}{2}}+mathbf{1}=boldsymbol{t} ) differentiate w.r.t. ( mathbf{x} ) ( mathbf{A} cdot frac{d t}{d x}=frac{1}{2 sqrt{x}} ) B. ( frac{d t}{d x}=frac{1}{4 sqrt{x}} ) ( mathbf{C} cdot frac{d t}{d x}=frac{1}{8 sqrt{x}} ) D. ( frac{d t}{d x}=frac{1}{16 sqrt{x}} ) |
12 |
| 639 | If ( x sqrt{1+y}+y sqrt{1+x}=0, ) for ( -1< ) ( x<1, ) prove that ( frac{d y}{d x}=frac{1}{(1+x)^{2}} ) |
12 |
| 640 | Differentiate w.r.t ( mathbf{x} ) ( boldsymbol{y}=boldsymbol{x}^{2} ln (sqrt{frac{boldsymbol{x}^{2}+mathbf{9}}{boldsymbol{x}^{2}+boldsymbol{4}}} ) | 12 |
| 641 | If ( boldsymbol{y}=sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}+sqrt{boldsymbol{x}+sqrt{boldsymbol{y}+ldots infty}}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A ( cdot frac{y^{2}-x}{2 y^{3}-2 x y-1} ) В. ( frac{x^{2}-x}{2 x^{3}-2 x y-1} ) C. ( frac{x^{2}-x}{2 x^{3}-2 x y^{2}-1} ) D. None of these |
12 |
| 642 | Find ( k, ) if the given function is continuous at ( x=2 ) ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}mathbf{3} boldsymbol{x}-mathbf{4} text { for } mathbf{0} leq boldsymbol{x} leq mathbf{2} \ mathbf{2} boldsymbol{x}+boldsymbol{k} text { for } mathbf{2} leq boldsymbol{x} leq mathbf{4}end{array}right. ) |
12 |
| 643 | If ( f(x) ) is continuous at ( x=c ) and ( g(x) ) is continuous at ( x=f(c) ) then which of the following is/are continuous at ( x=c ) ( ? ) A ( cdot(f(x)-g(x)) * f(x) ) в. ( f(g(x)) ) c. ( f(f(x)) ) D. None |
12 |
| 644 | If ( y^{2}=a x^{2}+b x+c, ) then ( y^{3} frac{d^{2} y}{d x^{2}} ) is A . a constant B. a function of x only c. a function of y only D. a function of ( x ) and ( y ) |
12 |
| 645 | Show that ( 3+frac{1}{1+} frac{1}{6+1+} frac{1}{6+} dots= ) ( 3left(1+frac{1}{3+2+3+2+cdots}right) ) |
12 |
| 646 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=log left(frac{1-boldsymbol{x}^{2}}{1+boldsymbol{x}^{2}}right) ) | 12 |
| 647 | Let ( f ) be differentiable for all ( mathbf{x} . ) If ( boldsymbol{f}(mathbf{1})= ) -2 and ( f^{prime}(x) geq 2 ) for all ( x epsilon[1,6] ) then A ( cdot f(6)<8 ) B. ( f(6) geq 8 ) ( mathbf{c} cdot f(6) geq 5 ) D. ( f(6) leq 5 ) |
12 |
| 648 | Is every continuous function differentiable? |
12 |
| 649 | If ( x^{2}+y^{2}=4, ) then ( y frac{d y}{d x}+x ) is equal to ( A cdot 4 ) B. 0 c. 1 D. – |
12 |
| 650 | Examine the continuity of the following function at the point ( boldsymbol{x}=-frac{1}{2} ) ( f(x)=left{begin{array}{ll}frac{4 x^{2}-1}{2 x+1} & x neq-frac{1}{2} \ -2, & x=-frac{1}{2}end{array}right. ) | 12 |
| 651 | Differentiate w.r.t. ( boldsymbol{x} ) ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{3} boldsymbol{x}} sin boldsymbol{4} boldsymbol{x} ) |
12 |
| 652 | If ( g(X)=frac{x}{[X]} ) for ( X>2 ) then ( lim _{x rightarrow 2^{+}} ) ( frac{boldsymbol{g}(boldsymbol{X})-boldsymbol{g}(boldsymbol{2})}{boldsymbol{X}-boldsymbol{2}}= ) A . – B. c. ( frac{1}{2} ) D. |
12 |
| 653 | Find the derivative of the following functions: (i) ( tan x cos x ) (ii) ( sec x ) |
12 |
| 654 | ( y=e^{x^{2}} ) the value of ( frac{d y}{d x} ) is ( m x e^{x^{2}} . ) Find ( m ) |
12 |
| 655 | Form the differential equation from the following primitive, where constant is arbitrary. ( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} ) A ( cdot frac{d y}{d x}=0 ) B. ( frac{d^{2} y}{d x^{2}}=0 ) c. ( frac{d^{3} y}{d x^{3}}=0 ) D. None of these |
12 |
| 656 | Check the continuity of the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}frac{|boldsymbol{x}|}{boldsymbol{x}}, & boldsymbol{x} neq mathbf{0} \ mathbf{0}, & boldsymbol{x}=mathbf{0}end{array} quad mathbf{a t} boldsymbol{x}=mathbf{0}right. ) | 12 |
| 657 | Is the function defined by ( f(x)=|x|, ) a continuous function? |
12 |
| 658 | If ( f(x)= ) ( left{begin{array}{cc}(mathbf{1}+|sin boldsymbol{x}|)^{frac{a}{|sin |}} & ;-frac{pi}{6}<boldsymbol{x}<mathbf{0} \ boldsymbol{b} & ; boldsymbol{x}=mathbf{0} \ boldsymbol{e}^{left(frac{tan 2 x}{tan 3 x}right)} & ; mathbf{0}<boldsymbol{x}<frac{pi}{mathbf{6}}end{array}right. ) is a continuous function on ( left(-frac{pi}{6}, frac{pi}{6}right) ) then A ( cdot a=frac{2}{3}, b=e^{2} ) в. ( a=frac{1}{3}, b=e^{1 / 3} ) c. ( _{a=frac{2}{3}, b=e^{2 / 3}} ) D. ( a=e^{2 / 3}, b=frac{2}{3} ) |
12 |
| 659 | If ( 2^{x}-2^{y}=2^{x+y} ) then ( frac{d y}{d x}= ) ( mathbf{A} cdot 2^{y-x} ) B . ( 2^{y / x} ) ( mathbf{c} cdot-2^{y / x} ) D. ( 2^{x / y} ) |
12 |
| 660 | Differentiate the following function with respect to ( x ) ( frac{boldsymbol{x}^{n}}{sin boldsymbol{x}} ) |
12 |
| 661 | Determine the values of ( a, b, c ) for which [ left{begin{array}{ll} f(x)=frac{sin (a+1) x+sin x}{x} & text { for } x end{array}right. ] is continuous at ( boldsymbol{x}=mathbf{0} ) |
12 |
| 662 | Differentiate the following w.r.t.x: ( 5^{x} cdot sec ^{-1} 2 x ) |
12 |
| 663 | The function ( boldsymbol{f}:(boldsymbol{R}-mathbf{0}) rightarrow mathbf{R} ) given by ( f(x)=frac{1}{x}-frac{2}{e^{2 x}-1} ) can be made continuous at ( x=0 ) by defining ( f(0) ) as A . 2 B. – – c. 0 ( D ) |
12 |
| 664 | ffunction ( f(x) ) is continuous in interval ( [-2,2], ) find the value of ( (a+b) ) where ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{sin a x}{x}-2, & text { for }-2 leq x<0 \ 2 x+1, & text { for } 0 leq x leq 1 \ 2 b sqrt{x^{2}+3}-1, & text { for } 1<x leq 2end{array}right. ) |
12 |
| 665 | If ( f(x) ) is differentiable in the interval ( [2,5], ) where ( f(2)=frac{1}{5} ) and ( f(5)=frac{1}{2} ) then there exists a number ( c, 2<c< ) 5 for which ( f^{prime}(c)= ) A ( cdot frac{1}{2} ) B. ( frac{1}{5} ) ( c cdot frac{1}{10} ) D. None |
12 |
| 666 | Find the value ( f(0) ) so that the function ( boldsymbol{f}(boldsymbol{x})=frac{1}{x}-frac{2}{e^{2 x}-1}, boldsymbol{x} neq 0 ) is continuous at ( x=0 & ) examine the differentiability of ( f(x) ) at ( x=0 ) A ( cdot f(0)=0, ) differentiable at ( x=0 ) B. ( f(0)=0, ) not differentiable at ( x=0 ) ( mathrm{c} . f(0)=1, ) differentiable at ( x=0 ) D. ( f(0)=1, ) not differentiable at ( x=0 ) |
12 |
| 667 | ( frac{d}{d x}left(tan ^{-1} frac{cos x-sin x}{cos x+sin x}right) ) ( A ) B. – ( c cdot 1 ) D. |
12 |
| 668 | Find the derivative of ( frac{tan ^{-1} x}{1+tan ^{-1} x} ) w.r.t. ( tan ^{-1} x ) A ( cdot frac{1}{sec ^{-1} x} ) в. ( frac{1}{left(tan ^{-1} xright)^{2}} ) c. ( frac{1}{1+tan ^{2} x} ) D. ( frac{1}{left(1+tan ^{-1} xright)^{2}} ) |
12 |
| 669 | If ( int frac{sin x}{sin (x-alpha)} d x=A x+ ) ( B log sin (x-alpha)+c ) then find the value of ( (boldsymbol{A}, boldsymbol{B}) ) |
12 |
| 670 | The function ( f(x)=[x], ) at ( x=5 ) is: A. Ieft continuous B. right continuous c. continuous D. cannot be determined |
12 |
| 671 | Differentiate ( tan ^{-1}left(frac{sin x}{1+cos x}right) ) w.r.t. | 12 |
| 672 | If the function ( mathbf{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{2^{x+2}-16}{4^{x}-16} & text { for } x neq 2 \ mathbf{A} & x=2end{array} ) is continuous right. at ( boldsymbol{x}=mathbf{2}, ) then ( mathbf{A}= ) ( A cdot 2 ) B. ( c cdot frac{1}{4} ) D. |
12 |
| 673 | Derivative of an odd function. A. May be even or may be odd B. Is always odd C. Is always even D. None of these |
12 |
| 674 | The function ( f(x)=sin ^{-1}(cos x) ) is : A. Discontinuous at ( x=0 ) B. Continuous at ( x=0 ) c. Differentiable at ( x=0 ) D. None of these |
12 |
| 675 | Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function such that ( boldsymbol{f}left(frac{boldsymbol{x}+boldsymbol{y}}{mathbf{2}}right)=frac{boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})}{2} ) for all ( mathbf{x}, mathbf{y} ) and ( f(0)=3 ) and ( f^{prime}(0)=3 . ) Then A . ( f(x) / x ) is continuous on ( mathbb{R} ) B. ( f(x) ) is continuous on ( R ) c. ( f(x) ) is bounded on ( R ) D. none of these |
12 |
| 676 | Find the derivative of ( f(x)=left(x^{2}-5right)left(x^{3}-2 x+3right) ) |
12 |
| 677 | If ( y=a^{frac{1}{2} log _{a} cos x} . ) Find ( frac{d x}{d} ) | 12 |
| 678 | If ( y=x^{2} tan x, ) find ( frac{d y}{d x} ) | 12 |
| 679 | If ( y=e^{x} sin x, ) then find ( frac{d y}{d x} ) A ( cdot e^{x}(sin x+cos x) ) B . ( e^{x}(sin x-cos x) ) ( mathbf{c} cdot e^{x} sin x ) D. None of these |
12 |
| 680 | Discuss the continuity of the function ( f ) defined by ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x} neq mathbf{0} ) | 12 |
| 681 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[sin ^{2}left(cot ^{-1} sqrt{frac{boldsymbol{1}+boldsymbol{x}}{1-boldsymbol{x}}}right)right]=? ) | 12 |
| 682 | Find ( frac{d y}{d x}, ) if ( y=log left(sqrt{x}-frac{1}{sqrt{x}}right) ) | 12 |
| 683 | If ( y=log (log x)+2 sin x, ) find ( frac{d y}{d x} ) | 12 |
| 684 | Show that ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{2}|+|boldsymbol{x}-mathbf{3}| ) is not differentiable at ( x=2 ) |
12 |
| 685 | 20. If y = 2 sin x, then dyldt will be… | 12 |
| 686 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[log _{e}left{left(boldsymbol{e}^{boldsymbol{x}}+boldsymbol{2}right)+right.right. ) ( sqrt{boldsymbol{e}^{2 boldsymbol{x}}+boldsymbol{4} boldsymbol{e}^{boldsymbol{x}}+boldsymbol{5}}}]= ) A. ( frac{1}{sqrt{e^{2 x}+4 e^{x}+5}} ) C. ( frac{e^{x}}{sqrt{e^{2 x}+4 e^{x}+5}} ) D. ( frac{e^{x}}{sqrt{e^{2 x}+4 e^{x}+3}} ) ( sqrt{e^{2 x}+4 e^{x}+3} ) |
12 |
| 687 | If ( g ) is inverse function of ( f ) where ( f(x)=int_{0}^{pi} frac{1}{sqrt{1+t^{2}}} d t quad ) and ( int gleft(g^{prime}(x)right)^{2} d x=frac{left[1+(g(x))^{alpha}right]^{beta}}{gamma}+c ) Then the value of ( alpha beta gamma ) is equal to [where ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} in boldsymbol{R}] ) A . 9 B. 6 ( c .3 ) D. |
12 |
| 688 | Differentiate: ( y=sin (2 x+3) ) w.r.t ( x ) | 12 |
| 689 | Suppose that ( f ) is differentiable for all ( boldsymbol{x} in boldsymbol{R} ) and that ( boldsymbol{f}^{prime}(boldsymbol{x}) leq 2 ) for all ( boldsymbol{x} . ) If ( f(1)=2 ) and ( f(4)=8, ) then ( f(2) ) has the value equal to A . 3 B. 4 ( c cdot 6 ) D. 8 |
12 |
| 690 | The value of ( c ) in Lagrange’s theorm for the function ( f(x)=log sin x ) in the interval ( [boldsymbol{pi} / mathbf{6}, mathbf{5} boldsymbol{pi} / mathbf{6}] ) is ( mathbf{A} cdot pi / 4 ) в. ( pi / 2 ) ( mathrm{c} cdot 2 pi / 3 ) D. none of these |
12 |
| 691 | Assertion Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function defined by ( boldsymbol{f}(boldsymbol{x})=max left{boldsymbol{x}, boldsymbol{x}^{3}right} . ) Then, ( boldsymbol{f}(boldsymbol{x}) ) is not differentiable at ( boldsymbol{x}=-mathbf{1}, mathbf{0}, mathbf{1} ) Reason ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}boldsymbol{x}, boldsymbol{x} leq-mathbf{1} \ boldsymbol{x}^{3},-mathbf{1}<boldsymbol{x} leq mathbf{0} \ boldsymbol{x}, mathbf{0}mathbf{1}end{array}right. ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
12 |
| 692 | Let ( mathbf{f}:(-mathbf{1}, mathbf{1}) rightarrow mathbf{R} ) be a differentiable function with ( mathbf{f}(mathbf{0})=-mathbf{1} ) and ( mathbf{f}^{prime}(mathbf{0})=mathbf{1} ) Let ( mathbf{g}(mathbf{x})=[mathbf{f}(mathbf{2 f}(mathbf{x})+mathbf{2})]^{2} . ) Then ( mathbf{g}^{prime}(mathbf{0})= ) A . -4 B. ( c cdot-2 ) D. 4 |
12 |
| 693 | In [0,1] Lagrange’s Mean Value theorem is NOT applicable to A ( cdot(mathrm{x})=left{begin{array}{ll}frac{1}{2}-mathrm{x}, & mathrm{x}<frac{1}{2} \ left(frac{1}{2}-mathrm{x}right)^{2} & mathrm{x} geq frac{1}{2}end{array}right. ) B. ( f(x)=left{begin{array}{ll}frac{sin x}{x}, & x neq 0 \ 1, & x=0end{array}right. ) c. ( f(x)=x|x| ) D. ( f(x)=|x| ) |
12 |
| 694 | If ( f(x) ) is a polynomial in ( x, ) then the second derivative of ( fleft(e^{x}right) ) at ( x=1 ) is A ( cdot e f^{prime prime}(e)+f^{prime}(e) ) B . ( left(f^{prime prime}(e)+f^{prime}(e)right) e^{2} ) c. ( e^{2} f^{prime prime}(e) ) D. ( left(f^{prime prime}(e) e+f^{prime}(e)right) e ) |
12 |
| 695 | The law of the mean can also be put in the form ( mathbf{A} cdot f(a+h)=f(a)-h f^{prime}(a+q h) 0<q<1 ) B ( cdot f(a+h)=f(a)+h f^{prime}(a+q h) 0<q<1 ) ( mathbf{c} cdot f(a+h)=f(a)+h f^{prime}(a-q h) 0<q<1 ) D. ( f(a+h)=f(a)-h f^{prime}(a-q h) 0<q<1 ) |
12 |
| 696 | If ( f(x), phi(x), varphi(x) ) are continuous on ( [a, b] ) and differentiable on ( (a, b) exists c epsilon(a, b), ) then ( left|begin{array}{lll}boldsymbol{f}(boldsymbol{a}) & boldsymbol{phi}(boldsymbol{a}) & boldsymbol{varphi}(boldsymbol{a}) \ boldsymbol{f}(boldsymbol{b}) & boldsymbol{phi}(boldsymbol{b}) & boldsymbol{varphi}(boldsymbol{b}) \ boldsymbol{f}^{prime}(boldsymbol{c}) & boldsymbol{phi}^{prime}(boldsymbol{c}) & boldsymbol{varphi}^{prime}(boldsymbol{c})end{array}right|= ) ( A cdot f^{prime}(c) ) B. ( phi^{prime}(c) ) c. ( varphi^{prime}(c) ) D. |
12 |
| 697 | If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(boldsymbol{x}-boldsymbol{2})(boldsymbol{x}-boldsymbol{4}), mathbf{1} leq boldsymbol{x} leq mathbf{4} ) then a number satisfying the conditions of the mean value theorem is ( mathbf{A} cdot mathbf{1} ) B. ( frac{5}{2} ) ( c .3 ) D. ( frac{7}{2} ) |
12 |
| 698 | If ( f(x)=x^{3} ) and ( g(x)=x^{3}-4 x ) in ( -2 leq x leq 2, ) then consider the statements: (a) ( f(x) ) and ( g(x) ) satisfy mean value theorem. (b) ( f(x) ) and ( g(x) ) both satisfy Rolle’s theorem. (c) Only ( g(x) ) satisfies Rolle’s theorem. Of these statements A . (a) alone is correct. B. (a) and (c) are correct c. (a) and (b) are correct D. None is correct |
12 |
| 699 | Examine the continuity of the function [ boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll} frac{log boldsymbol{x}-log mathbf{7}}{boldsymbol{x}-mathbf{7}} & text { for } boldsymbol{x} neq mathbf{7} \ mathbf{7}, & text { for } boldsymbol{x}=mathbf{7} end{array}right. ] at ( x=7 ) |
12 |
| 700 | Applying Lagranges’s MeanValue Theorem for a suitable function ( f(x) ) in ( [0, h], ) we have ( f(h)+h f^{prime}(theta h), 0<theta< ) 1. Then for ( f(x)=cos x, ) the value of ( lim _{h rightarrow 0^{+}} theta ) is A . B. 0 ( c cdot frac{1}{2} ) D. |
12 |
| 701 | Suppose that a function ( f ) satisfies the following conditions for all real values of ( x ) and ( y ) ( (i) f(x+y)=f(x) cdot f(y) ) ( (i i) f(x)=1+x g(x), ) where ( lim _{x rightarrow 0} g(x)=1 . ) The value of ( log f(8) ) is |
12 |
| 702 | The function ( frac{|boldsymbol{x}|}{boldsymbol{x}^{2}+mathbf{2} boldsymbol{x}}, boldsymbol{x} neq mathbf{0} ) and ( boldsymbol{f}(mathbf{0})=mathbf{0} ) is not continuous at ( boldsymbol{x}=mathbf{0} ) because- A ( cdot lim _{x rightarrow 0} f(x) neq f(0) ) B. ( lim _{x rightarrow 0^{+}} f(x) ) does not exist c. ( lim _{x rightarrow 0^{-}} f(x) ) does not exist D. ( lim _{x rightarrow 0} f(x) ) does not exist |
12 |
| 703 | If ( f(x) ) is a differentiable function in the interval ( (0, infty) ) such that ( f(1)= ) 1 ( operatorname{and} lim _{t rightarrow x} frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1, ) for eacch ( boldsymbol{x}>0, ) then ( boldsymbol{f}(mathbf{3} / 2) ) is equal to: ( mathbf{A} cdot frac{13}{6} ) B . ( frac{23}{18} ) c. ( frac{25}{9} ) D. ( frac{31}{18} ) |
12 |
| 704 | Discuss the applicability of Lagrange’s mean value theorem for the function ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) on [-1,1] |
12 |
| 705 | if ( y=5 x^{2}+8 x ) find ( frac{d y}{d x} ) A. ( 10 x+8 ) B. ( 5 x+8 ) c. ( 10 x^{2}+8 x ) D. none of these |
12 |
| 706 | Solve ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} sin ^{boldsymbol{n}} boldsymbol{x} ) | 12 |
| 707 | ( frac{dleft(x^{n}right)}{d x}=? ) A ( cdot n x^{n-1} ) В . ( n x^{n} ) c. ( (n-1) x^{n-1} ) D. ( (n-1) x^{n} ) |
12 |
| 708 | The function ( f: R / 0 rightarrow R ) given by ( f(x)= ) ( frac{1}{x}-frac{2}{e^{2 x}-1} ) can be made continuous at ( x=0 ) by defining ( f(0) ) as A . B. ( c cdot 2 ) D. – |
12 |
| 709 | The number of continuous functions on R which satisfy ( (f(x))^{2}=x^{2} ) for all ( x in ) ( boldsymbol{R} ) is ( mathbf{A} cdot mathbf{1} ) B . 2 ( c cdot 4 ) D. 8 |
12 |
| 710 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ccc}x & text { if } & x text { is rational } \ 2-x & text { if } & x text { is irrational }end{array} ) Then right. fof ( (x) ) is continuous A. everywhere B. no where c. at all irrational ( x ) D. at all rational ( x ) |
12 |
| 711 | If ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{20^{x}+3^{x}-6^{x}-10^{x}}{1-cos 8 x} ; text { for } x neq 0 \ left(frac{k}{16}right) log left(frac{10}{3}right) cdot log 2 ; text { for } x=0end{array}right. ) continous at ( x=0, ) then the value of ( k ) is ( A cdot sin ^{2} 30^{circ} ) B. ( log _{3}left(frac{1}{2}right) ) [ 3 ] ( c cdot sqrt[3]{1} ) D. ( frac{log 2^{2}}{3} ) |
12 |
| 712 | Differentiate each of the functions with respect to ( ^{prime} boldsymbol{x}^{prime} ) ( frac{a x+b}{c x+d} ) |
12 |
| 713 | Verify Rolle’s theoremlquad for the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{-boldsymbol{x}} sin boldsymbol{x}, boldsymbol{x} boldsymbol{epsilon}[mathbf{0}, boldsymbol{pi}] ) |
12 |
| 714 | If ( boldsymbol{f}(boldsymbol{x})= ) ( lim _{p rightarrow infty} frac{boldsymbol{x}^{p} boldsymbol{g}(boldsymbol{x})+boldsymbol{h}(boldsymbol{x})+mathbf{7}}{mathbf{7} boldsymbol{x}^{boldsymbol{p}}+mathbf{3} boldsymbol{x}+mathbf{1}} ; boldsymbol{x} neq mathbf{1} ) and ( boldsymbol{f}(mathbf{1})=mathbf{7}, boldsymbol{f}(boldsymbol{x}), boldsymbol{g}(boldsymbol{x}) ) and ( boldsymbol{h}(boldsymbol{x}) ) are all continuous functions at ( x=1 . ) Then which of the following statement(s) is/are correct This question has multiple correct options A. ( g(1)+h(1)=70 ) B. ( g(1)-h(1)=28 ) D. ( g(1)-h(1)=-28 ) |
12 |
| 715 | Illustration 2.19 If y = – = x-10, then find dyldx. |
12 |
| 716 | Answer the following question in one word or one sentence or as per exact requirement of the question. f ( frac{pi}{2}<x<pi, ) then find ( frac{d}{d x}(sqrt{frac{1+cos 2 x}{2}}) ) |
12 |
| 717 | Show that the function ( |x| ) is not differentiable at ( x=0 ) |
12 |
| 718 | Identify the graph of the polynomial function ( boldsymbol{f} ) ( f(x)=x^{3}-2 x^{2}-x+2 ) A. graph a B. graph b c. graph c D. graph d |
12 |
| 719 | If ( x=t^{3}+t+5 & y=sin t ) then ( frac{d^{2} y}{d x^{2}}= ) A. ( frac{left(3 t^{2}+1right) sin t+6 t cos t}{left(3 t^{2}+1right)^{3}} ) B. ( frac{left(3 t^{2}+1right) sin t+6 t cos t}{left(3 t^{2}+1right)^{2}} ) c. ( frac{left(3 t^{2}+1right) sin t+6 t cos t}{left(3 t^{2}+1right)^{2}} ) D. ( frac{c o s t}{3 t^{2}+1} ) |
12 |
| 720 | The function ( f(x)=sin ^{-1}(cos x) ) is? This question has multiple correct options A. Discontinuous at ( x=0 ) B. Continuous at ( x=0 ) C. Differentiable at ( x=0 ) D. None of these |
12 |
| 721 | Show that ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(cos boldsymbol{h}^{-1} boldsymbol{x}right)=frac{mathbf{1}}{sqrt{boldsymbol{x}^{2}-mathbf{1}}} ) | 12 |
| 722 | If ( x=y sqrt{1-y^{2}}, ) then ( frac{d y}{d x} ) equals to? | 12 |
| 723 | If ( y=cos ^{3} x, ) then ( frac{d y}{d x}= ) | 12 |
| 724 | Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem: ( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{2 5}-boldsymbol{x}^{2}} ) on ( [-mathbf{3}, boldsymbol{4}] ) |
12 |
| 725 | Find the derivative of ( frac{2}{x+1}-frac{x^{2}}{3 x-1} ) |
12 |
| 726 | 9. Letf:R Letfir → Rbe a differentiable function and (1) able function and f(1) =4. Then f( 2t the value of lim (1990 – 2 Marks) dt is X-1 X1 (a) 8 f'(l) (b) 4 f'(1) (c) 2 f'(1) (d) f'(1) |
12 |
| 727 | The set of points where f(x) = x is differentiable is (a) (-0,0) (0,00) (b) (-00,-1) (-1,00) (c) (-00,00) (d) (0,00) [2006] |
12 |
| 728 | Let ( lim _{x rightarrow a} f(x) ) exists but it is not equal to ( f(a) . ) Then ( f(x) ) is discontinuous at ( x=a ) and a is called a removable discontinuity. If ( lim _{x rightarrow a^{-}} f(x)= ) land ( lim _{x rightarrow a^{+}} f(x)=m ) exist but ( l neq ) ( m . ) Then a is called a jump discontinuity. If one of the limits (left hand limit or right hand limit ) does not exist, then a is called an infinite discontinuity. ( operatorname{Let} f(x)=left{begin{array}{cl}x^{2}+|x|, & x-5end{array} . ) Then right. ( boldsymbol{x}=-mathbf{5} ) is A. a point of discontinuity B. a jump discontinuity c. a removable discontinuity D. an infinite discontinuity |
12 |
| 729 | Given that ( prod_{n=1}^{n} cos frac{x}{2^{n}}=frac{sin x}{2^{n} sin left(frac{x}{2^{n}}right)} ) and ( boldsymbol{f}(boldsymbol{x})= ) [ left{begin{array}{c} lim _{n rightarrow infty} sum_{n=1}^{n} frac{1}{2^{n}} tan left(frac{x}{2^{n}}right), quad x in(0, \ frac{2}{pi} end{array}right. ] Then which one of the following is true? A. ( f(x) ) has non-removable discontinuity of finite type at [ x=frac{pi}{2} ] B. ( f(x) ) has removable discontinuity at ( x=frac{pi}{2} ) C ( quad f(x) ) is continuous at ( x=frac{pi}{2} ) D. ( f(x) ) has non-removable discontinuity of infinite type at ( x=frac{pi}{2} ) |
12 |
| 730 | The value of ( p ) for which the function [ left{begin{array}{ccc} boldsymbol{f}(boldsymbol{x})= & & \ & left(boldsymbol{4}^{boldsymbol{x}}-mathbf{1}right)^{mathbf{3}} & \ hline multirow{2}{*} {sin frac{boldsymbol{x}}{boldsymbol{p}} log left(1+frac{boldsymbol{x}^{2}}{mathbf{3}}right)} & ; & boldsymbol{x} neq mathbf{0} \ & mathbf{1 2}(log mathbf{4})^{mathbf{3}} & ; boldsymbol{x}=mathbf{0} end{array}right. ] continuous at ( boldsymbol{x}=mathbf{0}, ) is A . 4 B. 2 ( c .3 ) ( D ) |
12 |
| 731 | The function ( mathbf{f}(boldsymbol{x})=frac{|boldsymbol{x}|}{boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{0} ) is A. Ieft continuous B. right continuous c. continuous D. Discontinuous |
12 |
| 732 | T applicable (2003) In [0,1] Lagranges Mean Value theorem is NOT appli to V – – X X (a) f(x)= (1 – x xz 2 ΛΙ sin x , X0 x=0 (b) f(x) = { x T 1, (C) f(x) = x/x/ (d) f(x) = |x| |
12 |
| 733 | Let ( boldsymbol{f}:[mathbf{2}, mathbf{7}] rightarrow[mathbf{0}, infty] ) be a continuous and differentiable function. Then, the value of ( (boldsymbol{f}(mathbf{7})- ) ( f(2)) frac{(f(7))^{2}+(f(2))^{2}+f(2) cdot f(7)}{3}, ) is (where ( c epsilon(2,7)) ) ( mathbf{A} cdot 3 f^{2}(c) f^{prime}(c) ) B. ( 5 f^{2}(c) . f(c) ) c. ( 5 f^{2}(c) cdot f^{prime}(c) ) D. none of these |
12 |
| 734 | Find the values of ( a ) and ( b ) such that the function defined by ( f(x)=left{begin{array}{l}5, text { if } x leq 2 \ a x+b, text { if } 2<x<10 text { is a } \ 21, text { if } x geq 10end{array}right. ) continuous function. |
12 |
| 735 | If ( y=tan ^{-1}left(frac{2^{x}}{1+2^{2 x+1}}right), ) then ( frac{d y}{d x} ) at ( boldsymbol{x}=mathbf{0} ) is ( ? ) ( A cdot-frac{1}{5} ) B. 2 ( c cdot epsilon 2 ) D. none of these |
12 |
| 736 | Solve- ( cos x^{3} cdot sin ^{2}left(x^{5}right) ) |
12 |
| 737 | Prove that ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(cot boldsymbol{h}^{-1} boldsymbol{x}right)=frac{-mathbf{1}}{left(boldsymbol{x}^{2}-mathbf{1}right)} ) |
12 |
| 738 | Let ( f ) be a function defined for all ( boldsymbol{x} boldsymbol{epsilon} boldsymbol{R} ) If ( f ) is differentiable and ( fleft(x^{3}right)=x^{5} ) for all ( boldsymbol{x} boldsymbol{epsilon} boldsymbol{R}(boldsymbol{x} neq boldsymbol{0}) ) then the value of ( boldsymbol{f}^{prime}(mathbf{2 7}) ) is A . 15 B. 45 c. 0 D. None |
12 |
| 739 | ( left(3 x^{4}-x^{3}+4right)^{5 / 2} ) differentiate w.r.t ( x ) | 12 |
| 740 | Show that between any two roots of the equation ( e^{x} cos x=1 ) there exists atleast one root of ( e^{x} sin x-1=0 ) by continuity and differentiability. |
12 |
| 741 | If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{a} boldsymbol{x}^{2}-boldsymbol{b}, & boldsymbol{i} boldsymbol{f} quad|boldsymbol{x}|<-1 \ -frac{1}{|boldsymbol{x}|} boldsymbol{i} boldsymbol{f} & |boldsymbol{x}| geq-1end{array}right. ) differential at ( x=1 . ) Find the values of ( a ) and ( b ) A ( . a=1 / 2 ; b=3 / 2 ) В. ( a=1 / 2 ; b=-3 / 2 ) c. ( a=-1 / 2 ; b=3 / 2 ) |
12 |
| 742 | f ( y=x^{x}+(sin x)^{cot x} ). find ( frac{d y}{d x} ) | 12 |
| 743 | Differentiate the following functions w.r.t. ( x ) ( e^{log (log x)} ) |
12 |
| 744 | Find the derivative of the following functions from first principle: ( sin (x+1) ) |
12 |
| 745 | Differentiate the given function w.r.t. ( x ) ( frac{cos x}{log x}, x>0 ) | 12 |
| 746 | ( frac{1+tan ^{2} x}{1-tan ^{2} x} d x ) is equal to A ( cdot log frac{1-tan x}{1+tan x}+c ) B. ( log frac{1+tan x}{1-tan x}+c ) c. ( frac{1}{2} log frac{1-tan x}{1+tan x}+c ) D. ( frac{1}{2} log frac{1+tan x}{1-tan x}+c ) |
12 |
| 747 | Let ( f(x)=1+|sin x| . ) Then This question has multiple correct options ( mathbf{A} cdot f(x) ) is continuous nowhere B . ( f(x) ) is continuous everywhere ( mathbf{C} cdot f(x) ) is differentiable nowhere D. ( f^{prime} ) (0) does not exist |
12 |
| 748 | Verify Rolle’s Theorem for the function ( f(x)=x(x-1)^{2} ) in the interval [0,1] |
12 |
| 749 | Evaluate : ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}+mathbf{1}}{boldsymbol{2} boldsymbol{x}+boldsymbol{2} boldsymbol{y}+boldsymbol{3}} ) |
12 |
| 750 | Assertion Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be any function. Define ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) by ( boldsymbol{g}(boldsymbol{x})=|boldsymbol{f}(boldsymbol{x})| ) for all ( boldsymbol{x} ) Then, ( g ) is continuous is ( boldsymbol{f} ) is continuous. Reason Composition of two continuous functions is a continuous function A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
12 |
| 751 | ( boldsymbol{g}(boldsymbol{x})=left{begin{array}{ll}mathbf{1} & boldsymbol{x} leq-boldsymbol{2} \ frac{mathbf{1}}{mathbf{2}} boldsymbol{x} & -boldsymbol{2}<boldsymbol{x}<mathbf{4} text { .then } \ sqrt{boldsymbol{x}} & , boldsymbol{x} geq mathbf{4}end{array}right. ) A. ( g ) is a continuous function B. all the discontinuities are removable discontinuities c. all the discontinuities are jump D. all the discontinuities are infinitt |
12 |
| 752 | ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{4} ) find ( boldsymbol{y} ) in terms of ( boldsymbol{x} ) | 12 |
| 753 | If ( boldsymbol{f}(boldsymbol{x})= ) ( frac{1 . cos x+5 cos 3 x+cos 5 x}{cos 6 x+6 cos 4 x+15 cos 2 x+10} ) then ( boldsymbol{f}(mathbf{0})+boldsymbol{f}^{prime}(mathbf{0})+boldsymbol{f}^{prime prime}(mathbf{0})= ) A ( cdot frac{1}{2} ) B. ( c cdot-frac{1}{2} ) ( D ) |
12 |
| 754 | If ( y=sqrt{frac{1-x}{1+x}}, ) then ( frac{d y}{d x} ) equals A ( cdot frac{y}{1-x^{2}} ) в. ( frac{y}{x^{2}-1} ) c. ( frac{y}{1+x^{2}} ) D. ( frac{y}{y^{2}-1} ) |
12 |
| 755 | f ( x=tan left(frac{1}{a} log yright), ) prove that ( (1+ ) ( left.x^{2}right) frac{d^{2} y}{d x^{2}}+2 x frac{d y}{d x}-a 0 ) |
12 |
| 756 | ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cl}frac{tan [x]-[x] tan 1}{x} & ; boldsymbol{x} neq 0 \ boldsymbol{0} & ; boldsymbol{x}=0end{array}, ) then right. ( boldsymbol{f}^{prime}left(mathbf{0}^{-}right) ) is [where ( [boldsymbol{x}] ) denotes integer part of ( boldsymbol{x} ) A . 0 B. ( c cdot-1 ) D. Does not exist |
12 |
| 757 | If ( boldsymbol{y}= ) ( tan ^{-1}left(cot left(frac{pi}{2}-xright)right) ) then ( frac{d y}{d x}= ) ( A ) B. – – ( c cdot 0 ) D. |
12 |
| 758 | If ( boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}=tan boldsymbol{x}+boldsymbol{y}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is equa to A ( frac{sec ^{2} x-y}{(x+2 y-1)} ) B. ( frac{cos ^{2} x+y}{(x+2 y-1)} ) C ( frac{sec ^{2} x-y}{(2 x+y-1)} ) D. ( frac{cos ^{2} x+y}{(2 x+2 y-1)} ) |
12 |
| 759 | If ( x^{y}=e^{x-y}, ) then show that ( frac{d y}{d x}= ) ( frac{log boldsymbol{x}}{(1+log boldsymbol{x})^{2}} ) | 12 |
| 760 | If ( y=left(tan ^{-1} xright)^{2}, ) show that ( left(x^{2}+right. ) 1) ( ^{2} y_{2}+2 xleft(x^{2}+1right) y_{1}=2 ) |
12 |
| 761 | ( boldsymbol{f}(boldsymbol{x})=mathbf{1} /left(mathbf{1}-boldsymbol{e}^{-mathbf{1} / boldsymbol{x}}right), boldsymbol{x} neq mathbf{0}, boldsymbol{f}(mathbf{0})= ) 0 at ( x=0 ) Is function continuous at ( x=0 ? ) |
12 |
| 762 | Differentiate with respect to ( x ) : ( log _{7}(2 x-3) ) |
12 |
| 763 | Let ( y=e^{frac{1}{x}} ) then find ( frac{d^{2} y}{d x^{2}} ) | 12 |
| 764 | ( operatorname{Let} int_{0}^{x}left(frac{b t cos 4 t-a sin 4 t}{t^{2}}right) d t= ) ( frac{a sin 4 x}{x} ) then ( a ) and ( b ) are given by A ( cdot a=frac{1}{4}, b=1 ) В. ( a=2, b=2 ) c. ( a=-1, b=4 ) D. ( a=2, b=4 ) |
12 |
| 765 | If ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{sqrt{1+p x}-sqrt{1-p x}}{x} & -1 leq x<0 \ frac{2 x+1}{x-2} & 0 leq x leq 1end{array}right. ) continuous in the interval ( [-1,1], ) then ( boldsymbol{p}= ) A . – B. ( frac{-1}{2} ) ( c cdot frac{1}{2} ) ( D ) |
12 |
| 766 | The function ( f(x)=|x| ) at ( x=0 ) is: A. continuous but non-differentiable B. discontinuous and differentiable c. discontinuous and non-differentiable D. continuous and differentiable |
12 |
| 767 | Examine the continuity of the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}|boldsymbol{x}| cos frac{1}{x} & , text { if } boldsymbol{x} neq mathbf{0} \ boldsymbol{0} & , text { if } boldsymbol{x}=mathbf{0}end{array} text { at } boldsymbol{x}=right. ) ( mathbf{n} ) |
12 |
| 768 | If the function ( left{begin{array}{cc}frac{k cos x}{(pi-2 x)}, & text { when } x neq frac{pi}{2} \ 3, & x=frac{pi}{2}end{array}right. ) be continue at ( x=frac{pi}{2}, ) then the value of ( k ) is A . 3 в. -3 c. -5 D. 6 |
12 |
| 769 | If ( boldsymbol{y}=|cos boldsymbol{x}|+|sin boldsymbol{x}|, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}= ) ( frac{2 pi}{3} ) is A ( cdot frac{1-sqrt{3}}{2} ) B. 0 c. ( frac{1}{2}(sqrt{3}-1) ) D. None of these |
12 |
| 770 | Find the slope of the tangent to the curve ( y=x^{3}-x ) at ( x=2 ) | 12 |
| 771 | If ( y=tan ^{-1} frac{cos x+sin x}{cos x-sin x}, ) then find ( frac{d y}{d x} ) | 12 |
| 772 | If ( y=frac{sin x+cos x}{sin x-cos x} ) find ( frac{d y}{d x} ) at ( x=frac{pi}{4} ) | 12 |
| 773 | If ( x=a(cos theta+sin theta) ) and ( y= ) ( a(sin theta-cos theta), ) then find ( frac{d^{2} y}{d x^{2}} ) |
12 |
| 774 | Differentiate the following functions with respect to ( boldsymbol{x} ) ff ( y=sin ^{-1}left(frac{2 x}{1+x^{2}}right)+ ) ( sec ^{-1}left(frac{1+x^{2}}{1-x^{2}}right), 0<x<1, ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{4}}{boldsymbol{1}+boldsymbol{x}^{2}} ) |
12 |
| 775 | If ( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{pi}(|boldsymbol{x}|+[boldsymbol{x}]), ) then ( boldsymbol{f}(boldsymbol{x}) ) is/are (where [.] denotes greatest integer function) This question has multiple correct options A . continuous at ( x=frac{1}{2} ) B. continuous at ( x=0 ) c. Differentiable in (2,4) D. Differentiable in (0,1) |
12 |
| 776 | If ( y=frac{1}{2}left(sin ^{-1} xright)^{2}, ) then find ( (1- ) ( left.boldsymbol{x}^{2}right) boldsymbol{y}_{2}-boldsymbol{x} boldsymbol{y}_{1} ) Where ( y_{1} ) and ( y_{2} ) denote first and second derivatives of ( y ) respectively. A . -1 B. 0 c. 1 D. 2 |
12 |
| 777 | If ( boldsymbol{y}=tan ^{-1} sqrt{frac{1-sin x}{1+sin x}}, ) then the value of ( frac{d y}{d x} ) at ( x=frac{pi}{6} ) is A. ( -frac{1}{2} ) B. c. 1 D. – |
12 |
| 778 | If ( f(x)=frac{x^{2}-10 x+25}{x^{2}-7 x+10} ) for ( x neq 5 ) and ( f ) is continuous ( a t x=5, ) then ( f(5) ) has the value equal to- A . B. 5 c. 10 D. 25 |
12 |
| 779 | Differentiate w.r.t. ( x ) ( boldsymbol{y}=sin left(5 x^{3}+2 xright) ) |
12 |
| 780 | If ( boldsymbol{x} sqrt{boldsymbol{y}}+boldsymbol{y} sqrt{boldsymbol{x}}=1, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) equals ( mathbf{A} cdot-frac{y+2 sqrt{x y}}{x+2 sqrt{x y}} ) B. ( -sqrt{frac{x}{y}}left(frac{y+2 sqrt{x y}}{x+2 sqrt{x y}}right) ) ( mathbf{c} cdot-sqrt{frac{y}{x}}left(frac{y+2 sqrt{x y}}{x+2 sqrt{x y}}right) ) D. None of these |
12 |
| 781 | The differential coefficient of ( f(sin x) ) with respect to ( x ) where ( f(x)=log x ) is: A . ( tan x ) B. ( cot x ) c. ( f(cos x) ) D. |
12 |
| 782 | Find the derivative ( : cot x ) | 12 |
| 783 | The points where the function ( f(x)= ) ( [boldsymbol{x}]+|mathbf{1}-boldsymbol{x}|,-mathbf{1} leq boldsymbol{x} leq mathbf{3}, ) where [ denotes the greatest integer function, is not differentiable are A. ( x=-1,0,1,2,3 ) B. ( x=-1,0,2 ) c. ( x=0,1,2,3 ) D. ( x=-1,0,1,2 ) |
12 |
| 784 | f ( y=cos 2 x cos 3 x, ) then ( y_{n} ) is equal to Where, ( y_{n} ) denotes the ( n^{t h} ) derivative of ( boldsymbol{y} ) A ( cdot 6^{n} cos left(2 x+frac{n pi}{2}right) cos left(3 x+frac{n pi}{2}right) ) B. ( frac{1}{2}left[5^{text {n }} cos left(frac{text { n } pi}{2}+5 xright)+cos left(frac{text { n } pi}{2}+xright)right] ) c. ( frac{1}{2}left[5^{mathrm{n}} sin left(5 x+frac{mathrm{n} pi}{2}right)+sin left(x+frac{pi}{2}right)right] ) D. |
12 |
| 785 | If ( boldsymbol{f}(boldsymbol{x}) ) defined by ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{left|x^{2}-xright|}{x^{2}-x}, x neq 0,1 \ 1, quad x=0 quad text { then } f(x) text { is } \ -1, quad x=1end{array}right. ) continuous for all ( A ) B. ( x ) except at ( x=0 ) c. ( x ) except at ( x=1 ) D. ( x ) except at ( x=0 ) and ( x=1 ) |
12 |
| 786 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=left(sin 20 x+a^{2 x}+10right) ) |
12 |
| 787 | Differentiate the following function with respect to ( x ) ( frac{1}{sin x} ) |
12 |
| 788 | Differentiate with respect to ( x ) : ( sin ^{-1}left(frac{2^{x+1}}{1+4^{x}}right) ) |
12 |
| 789 | Let ( f(x) ) be a continuous function whose range is ( [2,6.5] . ) If ( h(x)= ) ( left[frac{cos x+f(x)}{lambda}right], lambda in N, ) be continuous where [.] denotes the greatest integer function, then the least value of ( lambda ) is ( mathbf{A} cdot mathbf{6} ) B. 7 ( c cdot 8 ) D. None of these |
12 |
| 790 | Examine whether the given function ( f(x) ) is continuous at ( x=3 ) [ begin{array}{c} boldsymbol{f}(boldsymbol{x})= \ frac{boldsymbol{x}^{4}-boldsymbol{8} boldsymbol{x}}{sqrt{boldsymbol{x}^{2}+mathbf{5}}-mathbf{3}}, quad text { for } quad boldsymbol{x} neq boldsymbol{3} \ boldsymbol{3} quad text { for } quad boldsymbol{x}=mathbf{3} end{array} ] [ boldsymbol{f}(boldsymbol{x})=mathbf{3} ] |
12 |
| 791 | Find the diffrentiation of ( x sin x ) | 12 |
| 792 | 1. Let f: R → R, g: R → R and h: R → R be differentiable functions such that f(x)=x3 + 3x +2, g(f(x))= x and h (g(g(x)))=x for all x e R. Then (JEE Adv. 2016) (a) f(2)= 15 (6) h'(1)=666 (d) h(g(3))=36 (c) h(0)=16 |
12 |
| 793 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{1-sin ^{3} x}{3 cos ^{2} x}, quad text { if } quad xfrac{pi}{2}end{array}end{array}right. ) continuous at ( x=frac{pi}{2}, ) find ( a ) and ( b ) |
12 |
| 794 | If ( boldsymbol{f}(boldsymbol{x})=sqrt{1}+sqrt{boldsymbol{x}}, boldsymbol{x}>0, ) then ( boldsymbol{f}(boldsymbol{x}) ) ( f^{prime}(x) ) is equal to A ( cdot frac{1}{2 sqrt{x}} ) B. ( frac{1}{2} ) c. ( frac{1}{4 sqrt{x}} ) D. ( frac{2 sqrt{x}+1}{4 sqrt{x}} ) |
12 |
| 795 | ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be such that ( |boldsymbol{f}(boldsymbol{x})-boldsymbol{f}(boldsymbol{y})|^{2} leq|boldsymbol{x}-boldsymbol{y}|^{3} ) for all ( boldsymbol{x}, boldsymbol{y} in ) ( R ) then the value of ( f^{prime}(x) ) is A ( cdot f(x) ) B. constant possibly different from zero c. ( (f(x))^{2} ) D. |
12 |
| 796 | If ( tan (x+y)+tan (x-y)=1 ), then find ( frac{d y}{d x} ) |
12 |
| 797 | Trace the curve ( boldsymbol{y}=boldsymbol{x}^{3} ) | 12 |
| 798 | ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}+boldsymbol{a} & ; quad boldsymbol{x}<0 \ |boldsymbol{x}-1| & ; quad boldsymbol{x} geq 0end{array}right. ) ( boldsymbol{g}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}+mathbf{1} & ; text { if } boldsymbol{x}0), ) then A ( . a=2, b=0 ) В. ( a=2, b=1 ) c. ( a=1, b=0 ) D. ( a=1, b=1 ) |
12 |
| 799 | The function [ boldsymbol{f}(boldsymbol{x})=left{begin{array}{r} boldsymbol{x}^{2}-boldsymbol{a} boldsymbol{x}+mathbf{3}, boldsymbol{x} text { is ratio } \ boldsymbol{2}-boldsymbol{x}, quad boldsymbol{x} text { is irratio } end{array}right. ] is continuous at exactly two points then the possible values of ‘ ( a ) ‘ are ( A cdot(2, infty) ) В ( cdot(-infty, 3) ) c. ( (-infty,-1) cup(3, infty) ) D. ( R ) |
12 |
| 800 | If ( x^{m} cdot y^{n}=(x+y)^{m+n}, ) then ( frac{d y}{d x} ) is ( ? ) A. ( frac{y}{x} ) в. ( frac{x+y}{x y} ) c. ( x y ) D. ( frac{x}{y} ) |
12 |
| 801 | Statement I: The function ( f(x) ) in the figure is differentiable at ( x=a ) Statement II: The function ( f(x) ) continuous at ( x=a ) A. Both Statement I and Statement II are true and the Statement II is the correct explanation of the Statement I B. Both Statement I and Statement II are true and the Statement II is not the correct explanation of the Statement I c. Statement l is true but Statement II is false Statement I is false but Statement II is true |
12 |
| 802 | If ( f(x)=(x-1)(x-2) ) and interval given was ( (0,4), ) find ‘c’ using Langrange’s mean value theorem. A ( cdot 2+frac{2}{sqrt{3}} ) B. ( 2-frac{2}{sqrt{3}} ) c. 2 D. Both A and B |
12 |
| 803 | is derivable and has a continuous derivative at ( boldsymbol{x}=mathbf{0} ) A ( . m in(1, infty) ) B. ( m in[2, infty) ) c. ( m in(2, infty) ) D. ( m in(-infty, 2) ) |
12 |
| 804 | ff ( f(x)=1+x+x^{2}+ldots ldots+x^{1000}, ) then ( boldsymbol{f}^{prime}(-1)=ldots . . ) A . -50 B . -500 c. -100 D. 500500 |
12 |
| 805 | If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{t}-sin boldsymbol{t}), boldsymbol{y}=boldsymbol{a}(boldsymbol{1}-cos boldsymbol{t}) ) find ( boldsymbol{d} boldsymbol{y} / boldsymbol{d} boldsymbol{x} ). at ( boldsymbol{t}=boldsymbol{pi} ) |
12 |
| 806 | If ( boldsymbol{y}=sec sqrt{boldsymbol{a}+boldsymbol{b} boldsymbol{x}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) equals- A ( cdot frac{b}{b sqrt{a+b x}} sec sqrt{a+b x} tan sqrt{a+b x} ) в. ( frac{b}{2 sqrt{a+b x}} sec sqrt{a+b x} tan sqrt{a+b x} ) c. ( 2 b sqrt{a+b x} sec sqrt{a+b x} tan sqrt{a+b x} ) D. None of these |
12 |
| 807 | If ( f(x)=frac{tan x}{sqrt{1+tan ^{2} x}}, lim _{x rightarrow(pi / 2)^{-}} f(x)= ) ( boldsymbol{a} ) and ( lim _{boldsymbol{x} rightarrow(boldsymbol{pi} / mathbf{2})^{+}} boldsymbol{f}(boldsymbol{x})=boldsymbol{b} ) then ( mathbf{A} cdot a=b ) B . ( a=1+b ) ( mathbf{c} cdot a+b=0 ) D. ( a+b=2 ) |
12 |
| 808 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{x}^{10}+mathbf{1 0}^{boldsymbol{x}}+mathbf{1 0} boldsymbol{x}+mathbf{1 0} ) | 12 |
| 809 | Check the continuity of ( f ) given by ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ccc}left(x^{2}-9right) /left(x^{2}-2 x-3right) & text { if } & 0<x< \ 1.5 & text { if } & x=varepsilonend{array}right. ) and ( x neq 3 ) at the point 3 |
12 |
| 810 | If ( y=ln left(x^{e^{x}}right) ) find ( frac{d y}{d x} ) | 12 |
| 811 | Differentiate the given function w.r.t. ( x ) ( boldsymbol{y}=log left(cos e^{x}right) ) |
12 |
| 812 | The solution of differential equation ( boldsymbol{y} boldsymbol{d} boldsymbol{x}+left(boldsymbol{x}-boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{y}=mathbf{0} ) ( mathbf{A} cdot e^{frac{y}{x}}=sin x+c ) B. ( y=c x log x ) c. ( x=frac{y^{2}}{3}+frac{c}{y} ) D. ( cos left(frac{y-2}{x}right)=a ) |
12 |
| 813 | If Rolle’s theorem is applicable to the function ( f(x)=frac{ln x}{x},(x>0) ) over the interval ( [a, b] ) where ( a epsilon I, b epsilon I, ) then the value of ( a^{2}+b^{2} ) can be A . 20 B . 25 c. 45 D. 10 |
12 |
| 814 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left[frac{sqrt{mathbf{1}+sin boldsymbol{x}}-sqrt{mathbf{1}-sin boldsymbol{x}}}{sqrt{mathbf{1}+sin boldsymbol{x}}-sqrt{mathbf{1}-sin boldsymbol{x}}}right]= ) ( A ) B. ( -frac{1}{2} ) ( c cdot frac{1}{2} ) ( D ) |
12 |
| 815 | Draw a graph of the function y=[x] +1-, -18×3. Determine the points, if any, where this function is not differentiable. (1989- 4 Marks) |
12 |
| 816 | Find the derivative of the following function from first principle: ( -x ) |
12 |
| 817 | The set of all points of differentiability of the function ( mathbf{f}(mathbf{x})=frac{sqrt{mathbf{x}+mathbf{1}}-mathbf{x}}{sqrt{mathbf{x}}} ) for ( mathbf{x} ) ( neq 0 ) and ( mathrm{f}(0)=0 ) is ( mathbf{A} cdot(-infty, infty) ) B. ( [0, infty) ) ( c cdot(0, infty) ) D. ( (-infty, infty) sim{0} ) |
12 |
| 818 | For the function ( f(x)=frac{x^{100}}{100}+frac{x^{99}}{99}+ ) ( +frac{x^{2}}{2}+x+1, f^{prime}(1)= ) ( mathbf{A} cdot x^{100} ) в. 100 ( c .10 ) D. None of these |
12 |
| 819 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for the following i) ( y=tan ^{-1}left(frac{3 x-x^{3}}{1-3 x^{2}}right),-frac{1}{sqrt{3}}<x< ) ( frac{1}{sqrt{3}} ) ii) ( y=sin ^{-1}left(frac{1-x}{1+x}right), 0<x<1 ) |
12 |
| 820 | f ( y=log sqrt{frac{1+tan x}{1-tan x}}, ) prove that ( frac{d y}{d x}= ) ( sec 2 x ) |
12 |
| 821 | If ( lim _{x rightarrow c} frac{f(x)-f(c)}{x-c} ) exists finitely, then ( mathbf{A} cdot lim _{x rightarrow c} f(x)=f(c) ) B. ( lim _{x rightarrow c} f^{prime}(x)=0 ) ( mathbf{C} cdot lim _{x rightarrow c} f(x) ) does not exist D. ( lim _{x rightarrow c} f(x) ) may or may not exist |
12 |
| 822 | Find derivative of ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} cos boldsymbol{x} ) | 12 |
| 823 | If ( f(x) ) be such that ( f(x)=max ) ( left{|2-x|, 2-x^{3}right}, ) then This question has multiple correct options A continuous ( forall x in R ) [ f(x) text { is continuous } x in R ] B. ( f(x) ) is differentiable ( forall x in R ) c. ( f(x) ) is non-differentiable at one point only D. ( f(x) ) is non-differentiable at 4 points only |
12 |
| 824 | Find the differential of ( mathbf{y}=left(sin ^{-1} mathbf{x}right)^{2}+mathbf{A} cos ^{-1} mathbf{x}+mathbf{B} ) where ( A, B ) are arbitrary constants |
12 |
| 825 | If ( x y=tan ^{-1}(x y)+cot ^{-1}(x y), ) then ( frac{d y}{d x} ) is equal to A ( cdot frac{y}{x} ) B. ( -frac{y}{x} ) c. ( frac{x}{y} ) D. ( -frac{x}{y} ) |
12 |
| 826 | If ( boldsymbol{x}=operatorname{cost} ) and ( boldsymbol{y}=sin 4 boldsymbol{t} ) then ( (1- ) ( left.boldsymbol{x}^{2}right) boldsymbol{y}_{2}-boldsymbol{x} boldsymbol{y}_{1}= ) A . ( 4 y ) в. ( -4 y ) c. ( 16 y ) D. ( -16 y ) |
12 |
| 827 | Differentiate with respect to ( x ) : ( log left{cot left(frac{pi}{4}+frac{x}{2}right)right} ) |
12 |
| 828 | Differentiate ( frac{x}{sin x} ) with respect to ( x ) A ( frac{sin x+x cos x}{sin ^{2} x} ) B. ( frac{sin x-x cos x}{sin ^{2} x} ) ( frac{cos x+x cos x}{sin x} ) D. ( frac{cos x-x sin x}{sin x} ) |
12 |
| 829 | For every pair of continuous functions ( boldsymbol{f}, boldsymbol{g}:[mathbf{0}, mathbf{1}] rightarrow boldsymbol{R} ) such that max ( {boldsymbol{f}(boldsymbol{x}): boldsymbol{x} in[mathbf{0}, mathbf{1}]}= ) ( max {g(x): x in[0,1]}, ) the correct statement (s) is (are) A ( cdot(f(c))^{2}+3 f(c)=(g(c))^{2}+3 g(c) ) for some ( c in[0,1] ) B ( cdot(f(c))^{2}+f(c)=(g(c))^{2}+3 g(c) ) for some ( c in[0,1] ) ( (f(c))^{2}+3 f(c)=(g(c))^{2}+g(c) ) for some ( c in[0,1] ) D ( cdot(f(c))^{2}=(g(c))^{2} ) for some ( c in[0,1] ) |
12 |
| 830 | Find derivative of ( tan ^{-1} frac{cos x-sin x}{cos x+sin x} ) w.r.t. ( boldsymbol{x} ) A . -1 B. c. 1 D. |
12 |
| 831 | Differential coefficient of ( log (sin x) ) with respect to ( x ) is: A . ( cot x ) B. ( operatorname{cosec} x ) ( c cdot tan x ) D. sec ( x ) |
12 |
| 832 | If ( y=sqrt{frac{1+sin x}{1-sin x}} ) then ( frac{d y}{d x}=? ) A ( cdot frac{1}{2} sec ^{2}left(frac{pi}{4}-frac{x}{2}right) ) B ( cdot frac{1}{2} csc ^{2}left(frac{pi}{4}-frac{x}{2}right) ) ( ^{mathrm{c}} cdot frac{1}{2} csc left(frac{pi}{4}-frac{x}{2}right) cot left(frac{pi}{4}-frac{x}{2}right) ) D. none of these |
12 |
| 833 | f ( boldsymbol{y}=log (log sin boldsymbol{x}), ) then evaluate ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 834 | If ( lim _{n rightarrow infty} frac{1^{a}+2^{a}+3^{a}+dots+n^{a}}{n^{a+1}}=frac{1}{5} ) ( (w h e r e a>-1) ) then the value of ( a ) is A .2 B. 3 ( c cdot 4 ) D. 5 |
12 |
| 835 | Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem: ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-mathbf{1} ) on ( [mathbf{2}, boldsymbol{3}] ) |
12 |
| 836 | If ( y=tan ^{-1} sqrt{frac{1-cos x}{1+cos x}}, ) then for ( 0< ) ( boldsymbol{x}<frac{boldsymbol{pi}}{mathbf{2}}, frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( mathbf{A} cdot 2 sec ^{2}(x / 2) ) B ( cdot frac{1}{2} sec ^{2}(x / 2) ) ( c cdot frac{1}{2} ) D. ( -frac{1}{2} sec ^{2}(x / 2) ) |
12 |
| 837 | Examine the applicability of Mean Value Theorem for the following function. ( f(x)=x^{2}-1 ) for ( x epsilon[1,2] ) | 12 |
| 838 | If ( f(x y)=f(x) . f(y) forall x, y in R ) If the function is continuous at one point ( boldsymbol{x}= ) ( a, ) then ( f(x) ) is: A. continuous for all ( x in R-{0} ) B. continuous forall ( x in R ) c. discontinuous on D. continuous at ( x=0 ) |
12 |
| 839 | Differentiate with respective to ( x ) ( log (sec x+tan x) ) | 12 |
| 840 | If ( y ) and ( z ) are the functions of ( x ) and if ( boldsymbol{y}^{2}+boldsymbol{z}^{2}=boldsymbol{lambda}^{2}, ) then ( boldsymbol{y} frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{boldsymbol{y}}{boldsymbol{lambda}}right)+ ) ( frac{d}{d x}left(frac{z^{2}}{lambda}right) ) is equal to A ( cdot frac{z}{lambda} frac{d z}{d x} ) B. ( frac{z}{lambda} frac{d x}{d z} ) c. ( frac{lambda}{z} frac{d z}{d x} ) D. None of these |
12 |
| 841 | For ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{2} cdot sin boldsymbol{x}} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 842 | If the function ( boldsymbol{f}(boldsymbol{x})= ) ( frac{boldsymbol{x}^{2}-(boldsymbol{A}+mathbf{2}) boldsymbol{x}+boldsymbol{A}}{boldsymbol{x}-mathbf{2}}, ) for ( boldsymbol{x} neq mathbf{2} ) and ( f(2)=2, ) is continuous at ( x=2, ) then find the value of ( boldsymbol{A} ) ? |
12 |
| 843 | If ( f ) and ( g ) are differentiable functions then ( D *(f g) ) is equal to A. ( f D * g+g D * f ) В. ( D * f D * g ) C ( cdot f^{2} D * g+g^{2} D * f ) D ( cdot f(D * g)^{2}+g(D * f)^{2} ) |
12 |
| 844 | Solve: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(operatorname{cosec} boldsymbol{x})=? ) | 12 |
| 845 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{4}+boldsymbol{3} boldsymbol{x}+1,[-boldsymbol{2},-1] ) then ( A cdot f ) has exactly two zeros in [-2,-1] B. f has exactly one zero in [-2,-1 C ( cdot ) f has at least one zero in [-2,-1] D. f has no zero in [-2,-1] |
12 |
| 846 | ( operatorname{Let} f(x)=left{begin{array}{cc}1 /|x| & text { for }|x| geq 1 \ a x^{2}+b & text { for }|x|<1end{array} ) The right. coefficients a and b so that fis continuous and differentiable at any point, are equal to A. ( a=-1 / 2, b=3 / 2 ) в. ( a=1 / 2, b=-3 / 2 ) c. ( a=1, b=-1 ) D. none of these |
12 |
| 847 | For some constants ( a ) and ( b ) find the derivative of ( left(a x^{2}+bright)^{2} ) |
12 |
| 848 | Find the derivative of ( sec ^{-1}left(frac{x+1}{x-1}right)+ ) ( sin ^{-1}left(frac{x-1}{x+1}right) ) A . B. ( c cdot-1 ) D. ( frac{x+1}{x-1} ) |
12 |
| 849 | If ( boldsymbol{x}^{boldsymbol{y}}=boldsymbol{e}^{boldsymbol{x}-boldsymbol{y}}, ) then find ( frac{d boldsymbol{y}}{d boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{1} ) | 12 |
| 850 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{e}^{sin sqrt{tan boldsymbol{x}}} ) | 12 |
| 851 | Differentiate w.r.t. ( boldsymbol{x} ) ( frac{boldsymbol{x}}{boldsymbol{y}^{3}}=mathbf{1} ) |
12 |
| 852 | Find the second order derivatives of ( tan ^{-1} x ) |
12 |
| 853 | Find ( frac{d y}{d x} ) if ( 3 x+4 y=9 ) | 12 |
| 854 | Find the values of ( a ) and ( b ) such that the function defined by ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ccc}5, & text { if } & x leq 2 \ a x+b, & text { if } & 2<x<10 text { is } \ 21, & text { if } & x geq 10end{array}right. ) continuous function. |
12 |
| 855 | Show that the function ( boldsymbol{f}(boldsymbol{x})= ) ( left[cos left(x^{10}+1right)right]^{1 / 3}, x in R ) is a continuous function. |
12 |
| 856 | A non zero polynomial with real coefficients has the property that ( boldsymbol{g}(boldsymbol{x})=boldsymbol{g}^{prime}(boldsymbol{x}) cdot boldsymbol{g}^{prime prime}(boldsymbol{x}) cdot ) Let the leading coefficient of ( g(x) ) be ( a ). Then ( 36 a= ) ( A cdot 6 ) B. 4 ( c .3 ) D. |
12 |
| 857 | If ( int_{0}^{y} frac{1}{sqrt{1+9 u^{2}}} d u=u, ) then ( frac{d^{2} y}{d u^{2}} ) is ( mathbf{A} cdot sqrt{1+9 y^{2}} ) В. ( frac{1}{sqrt{1+9 y^{2}}} ) c. ( 9 y ) D. ( 9 y^{2} ) |
12 |
| 858 | Differentiate the function with respect to ( x ) ( 2 sqrt{cot left(x^{2}right)} ) |
12 |
| 859 | If ( y=ln sqrt{frac{1-sin x}{1+sin x}} ) then ( frac{d y}{d x} ) equals- ( A cdot sec x ) B. – sec ( x ) ( c cdot csc x ) ( mathbf{D} cdot sec x tan x ) |
12 |
| 860 | If ( boldsymbol{f}(boldsymbol{x})=sqrt{1+cos ^{2}left(boldsymbol{x}^{2}right)}, ) then ( f^{prime}left(frac{sqrt{pi}}{2}right) ) equal to A ( cdot frac{sqrt{pi}}{6} ) в. ( -sqrt{frac{pi}{6}} ) c. ( frac{1}{sqrt{6}} ) D. ( frac{pi}{sqrt{6}} ) |
12 |
| 861 | f ( x sin (a+y)=sin y, ) then ( y^{prime}=? ) | 12 |
| 862 | Solve: ( lim _{x rightarrow 0} frac{sin ^{3} x^{2}}{x^{6}} ) ( mathbf{A} cdot mathbf{1} ) B. ( c cdot-1 ) D. ( infty ) |
12 |
| 863 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=log left(frac{1+boldsymbol{x}}{1-boldsymbol{x}}right) ) | 12 |
| 864 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}= ) ( sin ^{-1}left(frac{boldsymbol{6} boldsymbol{x}-boldsymbol{4} sqrt{boldsymbol{1}-boldsymbol{4} boldsymbol{x}^{2}}}{mathbf{5}}right) ) |
12 |
| 865 | If ( y=frac{sin ^{-1} x}{sqrt{1-x^{2}}}, ) prove that ( (1- ) ( left.x^{2}right) frac{d y}{d x}=(x y+1) ) |
12 |
| 866 | Assertion ( f(x)=sin ^{2} x+sin ^{2}left(x+frac{pi}{3}right)+ ) ( cos x cos left(x+frac{pi}{3}right) ) then ( f^{prime}(x)=0 ) Reason Derivative of a constant function is always zero A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
12 |
| 867 | Differentiate ( frac{x^{2}-1}{x} ) w.r.t ( x ) | 12 |
| 868 | ( frac{boldsymbol{d}^{2} boldsymbol{x}}{boldsymbol{d} boldsymbol{y}^{2}} ) equals ( mathbf{A} cdotleft(frac{d^{2} y}{d x^{2}}right)^{-1} ) B ( cdot-left(frac{d^{2} y}{d x^{2}}right)^{-1}left(frac{d y}{d x}right)^{-3} ) C ( cdotleft(frac{d^{2} y}{d x^{2}}right)^{-1}left(frac{d y}{d x}right)^{-2} ) D ( cdotleft(-frac{d^{2} y}{d x^{2}}right)^{-1}left(frac{d y}{d x}right)^{-2} ) |
12 |
| 869 | If ( y=x sin y, ) then prove that ( frac{d y}{d x}= ) ( frac{boldsymbol{y}}{boldsymbol{x}(1-boldsymbol{x} cos boldsymbol{y})} ) | 12 |
| 870 | If ( f^{prime}(3)=2 ) then ( lim _{h rightarrow 0} frac{fleft(3+h^{2}right)-fleft(3-h^{2}right)}{2 h^{2}} ) is ( A ) B. 2 c. 37 ( D cdot frac{1}{1} ) |
12 |
| 871 | The function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}2 x^{2}-1, & text { if } 1 leq x leq 4 \ 151-30 x, & text { if } 4<x leq 5end{array} ) is not right. suitable to apply Rolle's theorem, since B. ( f(1) neq f(5) ) c. ( f(x) ) is continuous only at ( x=4 ) D. ( f(x) ) is not differentiable in (4,5) E. ( f(x) ) is not differentiable at ( x=4 ) |
12 |
| 872 | ( boldsymbol{f}(boldsymbol{x})=sqrt{log _{1 / 2}left(frac{mathbf{5} boldsymbol{x}-boldsymbol{x}^{2}}{mathbf{4}}right)} ) | 12 |
| 873 | (d) For any illu Letf:(0,7) → be a twice differentiable function such tha: f(x) sint-f(t)sinx = sina x for all x € (0,7). lim I-r then which of the following statement(s) 12 (JEE Adv. 2018) is (are) TRUE? (6) f(x) <* _ xfor allx e (0,7) © There exists a e (0, Tt) such that f'(a)=0 |
12 |
| 874 | Differentiation gives us the instantaneous rate of change of one variable with respect to another A. True B. False |
12 |
| 875 | If ( f(x)=(x)^{frac{1}{x-1}} ) for ( x neq 1 ) and ( f ) is continuous at ( mathbf{x}=1 ) then ( mathbf{f}(mathbf{1})= ) ( A ) B . e- ( c cdot e^{-2} ) D. e ( ^{2} ) |
12 |
| 876 | The value of ( c ) in Lagrange’s theorem for the function ( |x| ) in the interval [-1,1] is ( mathbf{A} cdot mathbf{0} ) в. ( 1 / 2 ) c. ( -1 / 2 ) D. non existent in the interval |
12 |
| 877 | The function ( boldsymbol{f}(boldsymbol{x})= ) ( frac{log (1+a x)-log (1-b x)}{x} ) is not defined at ( x=0 . ) The value which should be assigned to ( f ) at ( x=0 ) so that it is continuous there, is A ( . a-b ) B. ( a+b ) ( mathbf{c} cdot log a+log b ) D. none of these |
12 |
| 878 | Verify Rolle’s theorem for ( boldsymbol{f}(boldsymbol{x})= ) ( x sqrt{a^{2}-x^{2}} ) in ( [0, a] ) | 12 |
| 879 | Diffrentiate w.r.t ( x: ) ( boldsymbol{y}=cos ^{-1}left[frac{1-x}{1+x}right] ) | 12 |
| 880 | For ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1} ; boldsymbol{x} in[-1,1], ) the constant of Meanvalue theorem is ( mathbf{A} cdot mathbf{0} ) B. ( frac{1}{2} ) ( c cdot-frac{1}{2} ) D. |
12 |
| 881 | 28. Let f:R →R and g: R R be two non-constant differentiable functions. If f'(x)=(e ^)-(x)))g'(x) for all x eR, and f(1) = g(2)=1, then which of the following statement (s) is (are) TRUE? (JEE Adv. 2018) (a) f(2)1-loge 2 (c) g(1) >1-loge 2 (d) g(1) <1-loge 2 |
12 |
| 882 | If ( boldsymbol{y}=sin ^{-1}left(3^{-x}right), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A ( cdot frac{-log 3}{sqrt{3^{2 x}-1}} ) в. ( frac{3^{x} log 3}{sqrt{3^{2 x}-1}} ) c. ( frac{-3^{-x} log 3}{sqrt{3^{2 x}-1}} ) D. ( frac{log 3}{3^{x} sqrt{3^{2 x}-1}} ) |
12 |
| 883 | If ( f(x)=[x sin pi x], ) then which of the following is incorrect? A. ( f(x) ) is continuous at ( x=0 ) B. ( f(x) ) is continuous in (-1,0) c. ( f(x) ) is differentiable at ( x=1 ) D. ( f(x) ) is differentiable in (-1,1) |
12 |
| 884 | Value of ( c ) of Rolles theorem for ( boldsymbol{f}(boldsymbol{x})= ) ( sin x-sin 2 x ) on ( [0, pi] ) ( ^{mathbf{A}} cdot cos ^{-1}left(frac{1+sqrt{33}}{8}right) ) B. ( cos ^{-1}left(frac{1+sqrt{35}}{8}right) ) ( ^{mathbf{c}} cdot cos ^{-1}left(frac{1-sqrt{38}}{5}right) ) D. does not exist |
12 |
| 885 | If ( f(x)=left{begin{array}{l}frac{1-sqrt{2} sin x}{pi-4 x} x neq frac{pi}{4} frac{pi}{4} \ a, x=frac{pi}{4}end{array}right. ) is continuous at ( x=frac{pi}{4} ) then ( a= ) ( mathbf{A} cdot mathbf{4} ) B. 2 c. 1 D. |
12 |
| 886 | If ( f ) is a real valued function defined by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+mathbf{4} boldsymbol{x}+mathbf{3}, ) then find ( boldsymbol{f}^{prime}(mathbf{1}) ) and ( boldsymbol{f}^{prime}(boldsymbol{3}) ) |
12 |
| 887 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( frac{e^{2 x}+e^{-2 x}}{e^{2 x}-e^{-2 x}} ) |
12 |
| 888 | Solve: ( frac{d y}{d x}=x^{2}(x-2), ) given ( y=2 ) where ( boldsymbol{x}=mathbf{0} ) |
12 |
| 889 | ff ( y=e^{4 x}+2 e^{-x} ) satisfies the equation ( boldsymbol{y}_{3}+boldsymbol{A} boldsymbol{y}_{1}+boldsymbol{B} boldsymbol{y}=mathbf{0} ) then the value of ( A B ) is |
12 |
| 890 | Evaluate ( lim _{x rightarrow 4} frac{3-sqrt{5+x}}{1-sqrt{5-x}} ) A ( cdot frac{1}{3} ) B. ( -frac{1}{3} ) ( c cdot frac{2}{3} ) D. ( -frac{2}{3} ) |
12 |
| 891 | If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{t}-sin t) ) and ( boldsymbol{y}=boldsymbol{a}(boldsymbol{1}+cos boldsymbol{t}) ) then the value of ( y_{2} ) at ( t=frac{pi}{2} ) is ( A ) B . a ( ^{2} ) ( c cdot frac{1}{a} ) D. ( frac{1}{a^{2}} ) |
12 |
| 892 | ( y=left(frac{2^{x+1}}{1+4^{x}}right) ) Find ( frac{d y}{d x} ) | 12 |
| 893 | ( f(x)=left{begin{array}{ll}frac{x^{2}-4}{x-2} & x neq 2 \ 4 & x=2end{array} ) discus right. continuity at ( boldsymbol{f}(boldsymbol{2}) ) |
12 |
| 894 | If ( boldsymbol{y}=boldsymbol{x}^{2}+mathbf{5} boldsymbol{x} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 895 | State True or False, Differentiating the equation of the curve at a point gives the slope of the tangent to the curve at that point. A . True B. False |
12 |
| 896 | The value of ( boldsymbol{f}(mathbf{0}) ) so that ( boldsymbol{f}(boldsymbol{x})= ) ( frac{sqrt{1+x}-sqrt[3]{1+x}}{x} ) is continuous is A ( cdot frac{1}{6} ) B. ( frac{1}{4} ) c. ( frac{1}{3} ) D. – |
12 |
| 897 | if ( y^{2}=a x+b x+c, ) then ( y^{3} frac{d^{2} y}{d x^{2}} ) is A . a constant B. a function of ( x ) only c. a function of ( y ) only D. a function of ( x ) and ( y ) |
12 |
| 898 | If a function ( f ) satisfy ( fleft(frac{x+y}{3}right)= ) ( frac{mathbf{2}+boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})}{mathbf{3}} ) for real ( mathbf{x} ) and ( mathbf{y} ) ( boldsymbol{f}^{prime}(2)=3 ) then ( boldsymbol{f}(boldsymbol{x}) ) is equal to A ( cdot-frac{1}{12} x^{3}+x^{2} ) B . ( 24 log (3 x+2) ) ( mathbf{c} cdot(3 x+2) ) D. ( frac{3}{4} x^{2}+2 ) |
12 |
| 899 | Find the value of the constant ( k ) so that the function given below is continuous at ( boldsymbol{x}=mathbf{0} ) ( boldsymbol{f}(boldsymbol{x})= ) ( left{frac{1-cos 2 x}{2 x^{2}}, x neq 0 quad k x=0right} ) |
12 |
| 900 | 14. A value of c for which conclusion of Mean Value Theorem holds for the function f(x) = log, x on the interval [1, 3] is [2007] (a) logze (b) log3 (C) 2 logze (d) = log;e |
12 |
| 901 | Let ( g ) is the inverse function of ( f ) and ( boldsymbol{f}^{prime}(boldsymbol{x})=frac{boldsymbol{x}^{mathbf{1 0}}}{left(mathbf{1}+boldsymbol{x}^{2}right)} cdot ) If ( boldsymbol{g}(boldsymbol{2})=boldsymbol{a} ) then ( boldsymbol{g}^{prime}(boldsymbol{2}) ) is equal to A ( cdot frac{5}{2^{10}} ) в. ( frac{1+a^{2}}{a^{10}} ) c. ( frac{a^{10}}{1+a^{2}} ) D. ( frac{1+a^{10}}{a^{2}} ) |
12 |
| 902 | Find the derivatives of the following functions. ( log _{2}left(2 x^{2}-3 x+1right) ) |
12 |
| 903 | If ( f(x)=frac{sin 4 x}{5 x}+a, quad ) for ( x>0 ) ( =boldsymbol{x}+mathbf{4}-boldsymbol{b} quad ) for ( boldsymbol{x}<mathbf{0} ) ( =1 ) for ( x=0 ) is continuous ( a t x=0, ) find ( a ) and ( b ) |
12 |
| 904 | For some constants ( a ) and ( b ) find the derivative of ( frac{x-a}{x-b} ) |
12 |
| 905 | ( f(x)=left{begin{array}{cl}m x+1, & x leq frac{pi}{2} \ sin x+n, & x>frac{pi}{2}end{array} ) is right. continuous at ( x=frac{pi}{2}, ) then A ( . m=1, n=0 ) B. ( m=frac{n pi}{2}+1 ) c. ( n=frac{m}{2} ) D. ( m=n=frac{n pi}{2} ) |
12 |
| 906 | Find the derivative of tan ( x ) using first principle of derivatives | 12 |
| 907 | If ( e^{x}+e^{y}=e^{x+y}, ) show that ( frac{d y}{d x}= ) ( -e^{y-x} ) |
12 |
| 908 | By using ( L M V T ), prove that ( frac{beta-alpha}{1+beta^{2}}<tan ^{-1} beta-tan ^{-1} alpha< ) ( frac{beta-alpha}{1+alpha^{2}}, beta-alpha<0 ) |
12 |
| 909 | If ( (cos x)^{y}=(cos y)^{x} ) then find ( frac{d y}{d x} ) | 12 |
| 910 | Write an example of a function which is everywhere continuous but fails to be differentiable exactly at five points. | 12 |
| 911 | Let ( y=x^{3}-8 x+7 ) and ( x=f(t) . ) If ( frac{d y}{d t}=2 ) and ( x=3 ) at ( t=0, ) then ( frac{d x}{d t} ) at ( t=0 ) is given by ( mathbf{A} cdot mathbf{1} ) B. ( frac{19}{2} ) ( c cdot frac{2}{19} ) D. None of these |
12 |
| 912 | If ( sqrt{1-x^{6}}+sqrt{1-y^{6}}=aleft(x^{3}-y^{3}right) ) and ( frac{d y}{d x}=f(x, y) sqrt{frac{1-y^{6}}{1-x^{6}}} ) then ( begin{array}{ll}text { A } cdot f(x, y)=frac{y}{x} & text { B. } f(x, y)=frac{x^{2}}{y^{2}} \ text { c. } f(x, y)=2 frac{y^{2}}{x^{2}} & \ text { D. } f(x, y)=frac{y^{2}}{x^{2}} & end{array} ) |
12 |
| 913 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{mathbf{3} boldsymbol{x}+boldsymbol{4}}{mathbf{2} boldsymbol{x}-mathbf{3}}right) ) A ( cdot frac{17}{(2 x-3)^{2}} ) B ( cdot frac{1}{(2 x-3)^{2}} ) c. ( frac{-1}{(2 x-3)^{2}} ) D. ( frac{-17}{(2 x-3)^{2}} ) |
12 |
| 914 | If ( boldsymbol{y}=sin ^{-1}left[operatorname{atan}^{-1} sqrt{frac{1-x}{1+x}}right] ) then find ( frac{d boldsymbol{y}}{d x} ) |
12 |
| 915 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}=sin boldsymbol{y} ) | 12 |
| 916 | f ( y(n)=e^{x} e^{x^{2}} ldots e^{x^{n}}, 0<x<1 ) then ( lim _{n rightarrow infty} frac{boldsymbol{d} boldsymbol{y}(boldsymbol{n})}{boldsymbol{d} boldsymbol{x}} ) at ( frac{1}{2} ) is ( A ) B. ( 4 e ) ( c cdot 2 e ) D. ( 3 e ) |
12 |
| 917 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: ) ( sin boldsymbol{x}-boldsymbol{3} boldsymbol{x}=mathbf{5} boldsymbol{y} ) |
12 |
| 918 | If ( 2^{x}+2^{y}=2^{x+y}, ) then ( frac{d y}{d x} ) has the value equal to This question has multiple correct options A ( cdot-frac{2^{y}}{2^{x}} ) B. ( frac{1}{1-2^{x}} ) ( mathbf{C} cdot 1-2^{y} ) D ( cdot frac{2^{x}left(1-2^{y}right)}{2^{y}left(2^{x}-1right)} ) |
12 |
| 919 | Show that ( f(x)=(x-1) e^{x}+1 ) is an increasing function for all ( x>0 ) |
12 |
| 920 | The value of ‘c’ in Rolle’s theorem for the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{c}boldsymbol{x}^{2} cos left(frac{1}{boldsymbol{x}}right), boldsymbol{x} neq mathbf{0} \ mathbf{0}, boldsymbol{x}=mathbf{0}end{array}right. ) the interval [-1,1] is? A ( cdot frac{-1}{2} ) B. ( frac{1}{4} ) ( c cdot 0 ) D. Non-existent in the interval |
12 |
| 921 | Number of points where ( boldsymbol{f}(boldsymbol{x})=(1- ) ( x)left|x-x^{2}right|+x ) is not differentiable is A . B. ( c cdot 2 ) D. |
12 |
| 922 | If ( boldsymbol{f}(boldsymbol{x})=left(frac{boldsymbol{x}^{a}}{boldsymbol{x}^{b}}right)^{boldsymbol{a}+boldsymbol{b}} cdotleft(frac{boldsymbol{x}^{b}}{boldsymbol{x}^{c}}right)^{b+c} cdotleft(frac{boldsymbol{x}^{c}}{boldsymbol{x}^{a}}right)^{c+a} ) then ( f^{prime}(x) ) is equal to ( mathbf{A} cdot mathbf{1} ) B. ( mathbf{c} cdot x^{a+b+c} ) D. None of these |
12 |
| 923 | If ( boldsymbol{y}=mathbf{2}^{2^{x}}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A ( cdot y(log 2)^{2} times 2^{x} ) В . ( y times 2(log 2) 2^{x} ) C ( cdot y times 2(log 2)^{2} times 2^{x} ) D. ( -y(log 2) times 2^{x} ) |
12 |
| 924 | If ( y=log [tan x], ) find ( frac{d y}{d x} ) | 12 |
| 925 | If ( y^{2}=a x^{2}+b x+c, ) where ( a, b, c ) are constants, then ( y^{3} frac{d^{2} y}{d x^{2}} ) is equal to. A ( cdot frac{-1}{4} ) в. ( frac{-1}{6} ) c. ( frac{-3}{4} ) D. ( frac{-1}{8} ) |
12 |
| 926 | Let ( [x] ) denote the greatest integer less than or equal to ( x ). If ( f(x)=[x sin pi x] ) ( operatorname{then} f(x) ) is : This question has multiple correct options A. continuous at ( x=0 ) B. continuous in (-1,0) c. differentiable at ( x=1 ) D. differentiable in (-1,1) |
12 |
| 927 | Let ( f(x) ) be a continuous function which satisfies ( boldsymbol{f}left(boldsymbol{x}^{2}+mathbf{1}right)=frac{mathbf{2}}{boldsymbol{f}left(mathbf{2}^{x}right)-mathbf{1}} boldsymbol{&} ) ( boldsymbol{f}(boldsymbol{x})>mathbf{0} forall boldsymbol{x} varepsilon boldsymbol{R} ) Then ( lim _{boldsymbol{x} rightarrow mathbf{1}} boldsymbol{f}(boldsymbol{x}) ) is ( A cdot 4 ) B. 2 ( c ) D. does not exist |
12 |
| 928 | Verify the Rolle’s theorem for the following functions: ( f(x)=x^{4}-1 ) on the interval [-1,1] A. True B. False |
12 |
| 929 | Let ( x y=x+y ) then prove that ( frac{d y}{d x}+ ) ( frac{1}{(x-1)^{2}}=0 ) | 12 |
| 930 | ( y=(sin x)^{cos x}+(cos x)^{sin x}, ) find ( frac{d y}{d x} ) | 12 |
| 931 | Examine the functions for continuity: ( f x=left{begin{array}{ll}frac{sin 2 x}{sin 3 x}, & text { when } x neq 0 \ 2, & text { when } x=0end{array} text { at } x=0right. ) |
12 |
| 932 | If ( boldsymbol{y}=tan ^{-1}left(cot left(frac{pi}{2}-xright)right), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( A ) B. -1 c. 0 D. |
12 |
| 933 | Find: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[frac{sin (boldsymbol{x}+boldsymbol{a})}{cos boldsymbol{x}}right] ) | 12 |
| 934 | ( fleft(x^{2}+y^{2}right)^{2}=x y, ) then ( left(frac{d y}{d x}right) ) is | 12 |
| 935 | For a real number ( boldsymbol{y},[boldsymbol{y}] ) denotes the greatest integer less than or equal to ( y ) then ( f(x)=frac{tan (pi[x-pi])}{1+[x]^{2}} ) is A. discontinuous at some ( x ) B. continuous at all ( x ), but ( f^{prime}(x) ) does not exist for same ( x ) ( mathbf{C} cdot f^{prime}(x) ) exists for all ( x ) but ( f^{prime}(x) ) does not exist D. ( f^{prime}(x) ) exists for all ( x ) |
12 |
| 936 | The equation of a curve is ( y=frac{e^{2 x}}{4 x+1} ) and the point ( P ) on the curve has ( y ) coordinate 10. Find the gradient of the curve at ( P ) |
12 |
| 937 | 35. If the function f: [0,4] → Ris differentiable then show that (1) For a, b € (0,4), (S(4))2 – (0))2=8f'(a)fb) (ii) [ f(t)dt = 2[af (a?)+BF (B2)]0<a,ß < 2 |
12 |
| 938 | If ( boldsymbol{f}(boldsymbol{a})=boldsymbol{a}^{2}, boldsymbol{phi}(boldsymbol{a})=boldsymbol{b}^{2} ) and ( boldsymbol{f}^{prime}(boldsymbol{a})= ) ( mathbf{3} phi^{prime}(boldsymbol{a}) ) then ( lim _{x rightarrow 0} frac{sqrt{boldsymbol{f}(boldsymbol{x})}-boldsymbol{a}}{sqrt{boldsymbol{phi}(boldsymbol{x})}-boldsymbol{b}} ) is ( mathbf{A} cdot b^{2} / a^{2} ) в. ( b / a ) c. ( 2 b / a ) D. None of these |
12 |
| 939 | Range of ( boldsymbol{f}(boldsymbol{x}) ) is ( ? ) A. ( R ) B . ( R-{0} ) ( c cdot R^{+} ) D. ( (0, e) ) |
12 |
| 940 | If ( y=frac{sin ^{-1} x}{sqrt{1-x^{2}}}, ) prove that ( left(1-x^{2}right) frac{d y}{d x}-x y=1 ) |
12 |
| 941 | Show that ( f(x)=x^{9}+4 x^{7}+11 ) is an increasing function for all ( boldsymbol{x} in boldsymbol{R} ) |
12 |
| 942 | ff ( y=x-x^{2} ), then the derivative of ( boldsymbol{y}^{2} boldsymbol{w} cdot boldsymbol{r} cdot boldsymbol{t} cdot boldsymbol{x}^{2} ) is A ( cdot 2 x^{2}+3 x-1 ) B . ( 2 x^{2}-3 x+1 ) c. ( 2 x^{2}+3 x+1 ) D. None of these |
12 |
| 943 | Derivate ( e^{sqrt{2 x+1}} ) where ( x=12 ) w.r.t. ( x ) | 12 |
| 944 | Compute the value of ( theta ) in the first mean value theorem ( boldsymbol{f}(boldsymbol{x}+boldsymbol{h})=boldsymbol{f}(boldsymbol{x})+ ) ( boldsymbol{h} boldsymbol{f}^{prime}(boldsymbol{x}+boldsymbol{theta h}) ) if ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} ) A ( cdot frac{1}{2} ) в. ( frac{1}{3} ) ( c cdot frac{1}{4} ) D. ( frac{1}{5} ) |
12 |
| 945 | an D >00 24. f’O)= lim wy(9) and $0) = 0. Using this find diem (n + 1}{cos- ” () “).cs-4 ICOS nOC (2004 – 2 Marks) |
12 |
| 946 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ccc}x & text { if } & x text { is rational } \ 2-x & text { if } & x text { is irrational }end{array} ) Then right. fof ( (x) ) is continuous A. everywhere B. no where c. at all irrational ( x ) D. at all rational ( x ) |
12 |
| 947 | Let ( lim _{x rightarrow a} f(x) ) exists but it is not equal to ( f(a) ). Then ( f(x) ) is discontinuous at ( x=a ) and a is called a removable discontinuity. If ( lim _{x rightarrow a^{-}} f(x)= ) ( l ) and ( lim _{x rightarrow a^{+}} f(x)=m ) exist but ( l neq ) ( m . ) Then a is called a jump discontinuity. If one of the limits (left hand limit or right hand limit ) does not exist, then a is called an infinite discontinuity. ( operatorname{Let} f(x)left{begin{array}{cc}2|x|, & x leq-1 \ 2 x, & -1 leq x leq 0 \ x+1, & 01end{array} ) Then right. ( f(x) ) at This question has multiple correct options A. ( x=-1 ) is a removable discontinuity B. ( x=0 ) is a jump discontinuity c. ( x=1 ) is a removable discontinuity D. ( x=-1 ) is a jump discontinuity |
12 |
| 948 | Differentiate: ( boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}=tan boldsymbol{x}+boldsymbol{y} ? ) |
12 |
| 949 | ( f(x)=left{begin{array}{ll}frac{1-cos x}{x^{2}}, & text { when } x neq 0 \ 1, & text { when } x=0end{array}right. ) then show that ( f(x) ) is discontinuous at ( boldsymbol{x}=mathbf{0} ) |
12 |
| 950 | Solve the different equation:- ( left(tan ^{-1} y-xright) d y=left(1+y^{2}right) d x ) |
12 |
| 951 | If ( f(x)=frac{e^{1 / x}-1}{e^{1 / x}+1}, x neq 0 ) and ( f(0)=0 ) then ( f(x) ) is A. Continuous at 0 B. Right continuous at 0 c. Discontinuous at 0 D. Left continuous at 0 |
12 |
| 952 | Express a in terms of b if the function defined by ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{a} boldsymbol{x}+mathbf{1} & , boldsymbol{x} leq mathbf{3} \ boldsymbol{b} boldsymbol{x}+mathbf{3} & boldsymbol{x}>mathbf{3}end{array}right} ) is continuous at ( x=3 ) |
12 |
| 953 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: ) ( boldsymbol{x}+boldsymbol{y}^{2}=log boldsymbol{y}+boldsymbol{x}^{2} ) |
12 |
| 954 | Let ( boldsymbol{f}(boldsymbol{x}) ) be defined in the interval [-2,2] such that ( f(x)= ) ( left{begin{array}{ll}-1, & -2 leq x leq 0 \ x-1, & 0<x leq 2end{array} text { and } g(x)=right. ) ( boldsymbol{f}(|boldsymbol{x}|)+|boldsymbol{f}(boldsymbol{x})| ) Test the differentiablity of ( g(x) ) in (-2,2) A. not derivable at ( x=0 ) and ( x=1 ) B. derivable at all points c. not derivable at ( x=0 ) D. not derivable at ( x=1 ) |
12 |
| 955 | If ( e^{x y}=y+sin ^{2} x, ) then at ( x=0, d y / d x ) is equal to |
12 |
| 956 | Find derivative of ( tan ^{-1} frac{cos x}{1+sin x} ) A ( cdot frac{1}{2} ) B. ( -frac{1}{2} ) ( c cdot frac{3}{2} ) D. ( -frac{3}{2} ) |
12 |
| 957 | Let ( boldsymbol{f}(boldsymbol{x}) ) be a polynomial in ( mathbf{x} . ) The second derivative of ( fleft(e^{x}right) ) at ( x=1 ) is ( mathbf{A} cdot e f^{prime prime}(e)+f^{prime}(e) ) В ( cdotleft(f^{prime prime}(e)+f^{prime}(e)right) e^{2} ) c. ( e^{2} f^{prime prime}(e) ) D. ( left(f^{prime prime}(e) e+f^{prime}(e)right) e ) |
12 |
| 958 | ( f(x)=left{begin{array}{ll}frac{1-sin ^{3} x}{3 cos ^{2} x}, & text { if } quad xfrac{pi}{2}end{array}right. ) so that ( f(x) ) is continuous at ( x=frac{pi}{2} ) then This question has multiple correct options A ( a=frac{1}{2} ) в. ( b=4 ) ( c cdot a=1 ) D. ( b=-4 ) |
12 |
| 959 | Let ( f ) be a twice differentiable such that ( boldsymbol{f}^{prime prime}(boldsymbol{x})=-boldsymbol{f}(boldsymbol{x}) ) and ( boldsymbol{f}^{prime}(boldsymbol{x})=boldsymbol{g}(boldsymbol{x}) . ) If ( boldsymbol{h}(boldsymbol{x})={boldsymbol{f}(boldsymbol{x})}^{2}+{boldsymbol{g}(boldsymbol{x})}^{2}, ) where ( h(5)=11 . ) Find ( h(10) ) ( mathbf{A} cdot mathbf{1} ) B . 10 c. 11 D. 100 |
12 |
| 960 | If ( f(x)=|x|^{3} . ) show that ( f^{prime prime}(x) ) exists for all real ( x ) and find it. |
12 |
| 961 | 4. If y = cos(sin x2), then then at x = 12 (a) -2 (b) 2 |
12 |
| 962 | f ( boldsymbol{y}=sin boldsymbol{x} ) and ( boldsymbol{x} ) changes from ( boldsymbol{pi} / mathbf{2} ) to ( 22 / 14, ) what is the approximate change in ( boldsymbol{y} ? ) |
12 |
| 963 | ff ( y=log _{sin x}(tan x), ) then ( left(frac{d y}{d x}right)_{pi / 4} ) is equal to A ( cdot frac{4}{log 2} ) B. ( -4 log 2 ) c. ( frac{-4}{log 2} ) D. None of these |
12 |
| 964 | State true or false: The differential coefficient of ( boldsymbol{f}(log boldsymbol{x}) ) w.r.t. ( log x ) where ( f(x)=log x ) is ( frac{1}{log x} ) A. True B. False |
12 |
| 965 | Find ( boldsymbol{f}^{prime}(mathbf{0}) ) for ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}}+ ) ( boldsymbol{a}^{2} log (boldsymbol{x}+sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}}) ) A . – 2a B. 2a ( c cdot-a ) ( D ) |
12 |
| 966 | If ( boldsymbol{x}=boldsymbol{a} cos ^{3} boldsymbol{theta} ) and ( boldsymbol{y}=boldsymbol{a} sin ^{3} boldsymbol{theta}, ) then find the value of ( frac{d^{2} y}{d x^{2}} ) at ( theta=frac{n}{6} ) |
12 |
| 967 | Check the continuity of the function ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{2}|+boldsymbol{x} ) | 12 |
| 968 | Let ( f(x) ) be a function such that ( lim _{x rightarrow 0} frac{f(x)}{x}=1 . ) If [ lim _{x rightarrow 0} frac{x(1+a cos x)-b sin x}{{f(x)}^{3}}=1 ] then ( |a+b|= ) |
12 |
| 969 | If the function ( f(x)=2 x^{2}+3 x+5 ) satisfies LMVT at ( x=2 ) on the closed interval ( [1, a] ) then the value of ‘ ( a^{prime} ) is equal to A . 3 B. 4 ( c cdot 6 ) D. |
12 |
| 970 | ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x} ) and ( boldsymbol{f}^{prime}(boldsymbol{pi}) ) A . -1 B. c. 1 D. None of these |
12 |
| 971 | ff ( y=frac{sin 4 x}{x^{2}+16}, ) then find ( frac{d y}{d x} ) | 12 |
| 972 | f ( boldsymbol{y}=cot ^{-1}left[frac{sqrt{1+x^{2}}+1}{x}right], ) then find the value of ( frac{d y}{d x} ) |
12 |
| 973 | ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=cos boldsymbol{x}+sin boldsymbol{x} ) | 12 |
| 974 | If ( 4 a+2 b+c=0 ) then the equation ( 3 a x^{2}+2 b x+c=0 ) has at least one real root lying between A. 0 and 1 B. 1 and 2 c. 0 and 2 D. none of these |
12 |
| 975 | If ( boldsymbol{y}=boldsymbol{e}^{sin ^{2} boldsymbol{x}+sin ^{4} boldsymbol{x}+sin ^{6} boldsymbol{x}+ldots infty}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A ( cdot e^{tan ^{2} x} ) B. ( e^{tan ^{2} x} sec ^{2} x ) C ( cdot 2 e^{tan ^{2} x} tan x cdot sec ^{2} x ) D. 1 |
12 |
| 976 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}int_{0}^{x}{1+|1-t|} d t & text { if } x>2 \ mathbf{5 x – 7} & text { if } x leq 2end{array} ) then right. A. ( f ) is not continuous at ( x=2 ) B. ( f ) is continuous but not differentiable at ( x=2 ) c. ( f ) is differentiable everywhere D ( cdot f^{prime}(2+) ) doesn’t exist |
12 |
| 977 | Match the columns | 12 |
| 978 | Differentiate: ( boldsymbol{y}=boldsymbol{c}^{2}+frac{boldsymbol{c}}{boldsymbol{x}} ) |
12 |
| 979 | 47. f(x) is a differentiable function and g(x) is a double function such that f(x) <1 and f'(x) = g(x). Iff-(0)73 Prove that there exists some CE-3, 3) such on and g(x) is a double differentiable 1 and f '(x)=g(x). Iff2(0)+7(0)=9. some ce(-3, 3) such that gc.g"C) <0. (2005 – 6 Marks) |
12 |
| 980 | Find the value of ( c ) of Rolle’s theorem for ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) in [-1,1] ( mathbf{A} cdot mathbf{0} ) B. ( c cdot-1 ) D. does not exist |
12 |
| 981 | ( x=frac{(n+1)^{n}}{(n+2)} ) ( frac{d x}{d n}=? ) |
12 |
| 982 | The value of ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} int_{2}^{boldsymbol{x}^{2}}(boldsymbol{t}-mathbf{1}) boldsymbol{d} boldsymbol{t} ) A ( cdotleft(x^{2}-1right) ) В. ( xleft(x^{2}-1right) ) c. ( 2 xleft(x^{2}-1right) ) D. none of these |
12 |
| 983 | Derivative of which function is ( boldsymbol{f}^{prime}(boldsymbol{x})= ) ( x sin x ? ) This question has multiple correct options A. ( x sin x+cos x ) B. ( x cos x+sin x ) c. ( x sin left(frac{pi}{2}-xright)+cos left(frac{pi}{2}-xright) ) D. ( x cos left(frac{pi}{2}-xright)+sin left(frac{pi}{2}-xright) ) |
12 |
| 984 | Given ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{[{|boldsymbol{x}|}] boldsymbol{e}^{boldsymbol{x}^{2}}{[boldsymbol{x}+{boldsymbol{x}}]}}_{left(frac{1}{boldsymbol{e}^{x^{2}}-1}right)} boldsymbol{s} boldsymbol{g n}(sin boldsymbol{x}) \ boldsymbol{0} quad text { for } quad boldsymbol{x}=mathbf{0}end{array}right. ) Where ( {x} ) is the fractional part function; ( [x] ) is the step up function and ( operatorname{sgn}(x) ) is the signum function of ( x ) then, ( boldsymbol{f}(boldsymbol{x}) ) A. Is continuous at ( x=0 ) B. Is discontinuous at ( x=0 ) C. Has a removable discontinuity at ( x=0 ) D. Has in irremovable disconitnuity at ( x=0 ) |
12 |
| 985 | If ( boldsymbol{y}=mathbf{s e c}^{-1}left[frac{sqrt{boldsymbol{x}}+mathbf{1}}{sqrt{boldsymbol{x}}-mathbf{1}}right]+ ) ( sin ^{-1}left[frac{sqrt{boldsymbol{x}}-1}{sqrt{boldsymbol{x}}+1}right] ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( A ) в. c. D. |
12 |
| 986 | Show that the Lagrange’s mean value theorem is not applicable to the function ( boldsymbol{f}(boldsymbol{x})=frac{1}{boldsymbol{x}} ) on [-1,1] | 12 |
| 987 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} sec boldsymbol{x}= ) ( mathbf{A} cdot sec x tan x ) ( mathbf{B} cdot cos x tan x ) ( c cdot sin x tan x ) ( mathbf{D} cdot sec x cot x ) |
12 |
| 988 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) ( boldsymbol{y}=-boldsymbol{x}^{mathbf{3}}+boldsymbol{x} ) |
12 |
| 989 | ( operatorname{Let} g(x)= ) ( left{begin{array}{ll}2(x+1), & -infty<x leq-1 \ sqrt{1-x^{2}}, & -1<x<1 \ |x+1|, & 1 leq x<inftyend{array} ) then right. A ( cdot g(x) ) is discontinuous at exactly three points B . ( g(x) ) is continuous in ( (-infty, 1] ) C ( cdot g(x) ) is continuous in ( [1, infty) ) D. ( g(x) ) has finite type of discontinuity at ( x=1, ) but continuous at ( x=-1 ) |
12 |
| 990 | Which of the following given statements is/are not correct? This question has multiple correct options A ( cdot frac{d}{d x}(operatorname{cosec} x)=operatorname{cosec} x cdot cot x ) B. ( frac{d}{d x}(sec x)=sec x . tan x ) C ( cdot frac{d}{d x}(3 cot x)=-3 operatorname{cosec}^{2} x ) D ( cdot frac{d}{d x}(2 tan x)=-2 sec ^{2} x ) |
12 |
| 991 | Differentiate the following function w.r.t.x. ( frac{1}{left(x^{2}+3^{2}right)} ) |
12 |
| 992 | Illustration 2.26 Find the derivative of y=- . +2 +1 |
12 |
| 993 | Assertion If ( boldsymbol{y}= ) ( (1+x)left(1+x^{2}right)left(1+x^{4}right) ldotsleft(1+x^{2^{n}}right) ) then ( frac{d y}{d x} ) at ( x=0 ) is 1 Reason ( y=frac{1-x^{2^{n+1}}}{1-x} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
12 |
| 994 | ff ( y=tan ^{-1}(3 x), ) then find ( frac{d^{2} y}{d x^{2}} ) | 12 |
| 995 | Left hand derivative and right hand derivative of a function ( f(x) ) at a point ( x=a ) are defined as ( f^{prime}left(a^{-}right)=lim _{h rightarrow 0^{+}} frac{f(a)-f(a-h)}{h}= ) ( lim _{h rightarrow 0^{-}} frac{f(a)-f(a-h)}{h}= ) ( lim _{x rightarrow a^{+}} frac{f(a)-f(x)}{a-x} ) respectively Let ( f ) be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function. If ( f ) is odd, which of the following is Lefthand derivative of ( f ) at ( x=a ) A ( cdot lim _{h rightarrow 0^{-}} frac{f(a-h)-f(a)}{-h} ) в. ( lim _{h rightarrow 0^{-}} frac{f(a-h)-f(a)}{h} ) c. ( lim _{h rightarrow 0^{+}} frac{f(a)+-f(a-h)}{-h} ) D. ( lim _{h rightarrow 0^{-}} frac{f(-a)-f(-a-h)}{-h} ) |
12 |
| 996 | ( operatorname{Let} boldsymbol{F}(boldsymbol{x})=left|begin{array}{ccc}sin boldsymbol{x} & cos boldsymbol{x} & sin boldsymbol{x} \ cos boldsymbol{x} & -sin boldsymbol{x} & cos boldsymbol{x} \ boldsymbol{x} & boldsymbol{1} & boldsymbol{1}end{array}right| ) Which of the following statement hold true? This question has multiple correct options A ( cdot ) Range of ( F(x) ) is ( (-infty, infty) ) B ( cdot F^{prime}left(frac{pi}{2}right)= ) c. ( F(x) ) is bounded D. ( F(x) ) is continuous and differentiable every where in its domain |
12 |
| 997 | If ( boldsymbol{y}=boldsymbol{x}+boldsymbol{e}^{boldsymbol{x}}, ) then what will be the value of ( frac{d^{2} x}{d y^{2}} ? ) |
12 |
| 998 | (x² + 2x) Illustration 2.28 If y= , then find (3x – 4) dx |
12 |
| 999 | 18. Differentiation of 2×2 + 3x w.r.t. x is… …fondra |
12 |
| 1000 | If ( boldsymbol{y}=sec ^{-1}left(frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right)+sin ^{-1}left(frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}right) ) then ( frac{d y}{d x} ) is equal to A . B. ( x+1 ) ( c cdot 1 ) D. – |
12 |
| 1001 | Using the ahove approximation, the value ( sqrt{104} ) is ( mathbf{A} cdot 10.18 ) B. 10.49 c. 10.2 D. 10.28 |
12 |
| 1002 | Assertion(A): Let ( boldsymbol{f}(boldsymbol{x}) ) be twice differentiable function such that ( boldsymbol{f}^{prime prime}(boldsymbol{x})=-boldsymbol{f}(boldsymbol{x}) ) and ( boldsymbol{f}^{prime}(boldsymbol{x})=boldsymbol{g}(boldsymbol{x}) . ) If ( boldsymbol{h}(boldsymbol{x})=[boldsymbol{f}(boldsymbol{x})]^{2}+[boldsymbol{g}(boldsymbol{x})]^{2} ) and ( boldsymbol{h}(mathbf{1})=mathbf{8} ) ( operatorname{then} h(2)=8 ) Reason (R): Derivative of a constant function is zero. A. Both A and R are true R is correct reason of A B. Both A and R are true R is not correct reason of A c. A is true but R is false D. A is false but R is true |
12 |
| 1003 | Let ( [t] ) denote the greatest integer ( leq t ) and ( lim _{x rightarrow 0} xleft[frac{4}{x}right]=A . ) Then the function, ( f(x)=left[x^{2}right] sin (pi x) ) is discontinuous, when ( x ) is equal to: ( A cdot sqrt{A} ) B. ( sqrt{A+1} ) c. ( sqrt{A+5} ) D. ( sqrt{A+21} ) |
12 |
| 1004 | f ( boldsymbol{y}=boldsymbol{f}left(boldsymbol{a}^{boldsymbol{x}}right) ) and ( boldsymbol{f}^{prime}(sin boldsymbol{x})=log _{e} boldsymbol{x}, ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if it exists, where ( frac{boldsymbol{pi}}{boldsymbol{2}}<boldsymbol{x}<boldsymbol{pi} ) |
12 |
| 1005 | Differentiate the given function w.r.t. ( x ) ( frac{cos ^{-1} frac{x}{2}}{sqrt{2 x+7}},-2<x<2 ) | 12 |
| 1006 | ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}mathbf{3} boldsymbol{x}-mathbf{4}, & mathbf{0} leq boldsymbol{x} leq mathbf{2} \ mathbf{2} boldsymbol{x}+boldsymbol{lambda}, & mathbf{2}<boldsymbol{x} leq mathbf{3}end{array} . text { If } boldsymbol{f} ) is right. continuous at ( x=2, ) then ( lambda ) is ( A ) B. c. -2 ( D ) |
12 |
| 1007 | Ify=(x + V1+x? “, then (1+x) 2 x is 12002] (b) – n’y (d) 2xy |
12 |
| 1008 | 5. f(x) = x2 – 3x, then the points at which f(x) = f(x) are (a) 1,3 (b) 1, -3 (c)-1,3 (d) None of these |
12 |
| 1009 | Left hand derivative and right hand derivative of a function ( f(x) ) at a point ( boldsymbol{x}=boldsymbol{a} ) are defined as ( f^{prime}left(a^{-}right)=lim _{h rightarrow 0^{+}} frac{f(a)-f(a-h)}{h}= ) ( lim _{h rightarrow 0^{-}} frac{f(a)-f(a-h)}{h}= ) ( lim _{x rightarrow a^{+}} frac{f(a)-f(x)}{a-x} ) respectively Let ( f ) be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function The statement ( lim _{h rightarrow 0} frac{f(-x)-f(-x-h)}{h}= ) ( lim _{h rightarrow 0} frac{f(x)-f(x-h)}{-h} ) implies that for all ( mathbf{x} boldsymbol{epsilon} boldsymbol{R} ) ( A . f ) is odd B. ( f ) is even c. ( f ) is neither odd nor ever D. nothing can be concluded |
12 |
| 1010 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(sqrt{boldsymbol{3}} sin left(boldsymbol{2} boldsymbol{x}+frac{boldsymbol{pi}}{boldsymbol{3}}right)+cos left(boldsymbol{2} boldsymbol{x}+frac{boldsymbol{pi}}{boldsymbol{3}}right)right) ) A. ( 4 cos 2 x ) B . ( -4 sin 2 x ) ( c .4 sin 2 x ) D. ( -4 cos 2 x ) |
12 |
| 1011 | If ( y=2 sin x-3 x^{4}+8, ) then ( frac{d y}{d x} ) is B . ( 2 cos x-12 x^{3} ) ( mathbf{c} cdot 2 cos x+12 x^{3} ) D. ( 2 sin x+12 x^{3} ) |
12 |
| 1012 | ( boldsymbol{x}=sin boldsymbol{t} quad boldsymbol{y}=cos boldsymbol{m} boldsymbol{t} ) Prove ( :left(boldsymbol{1}-boldsymbol{x}^{2}right) boldsymbol{y}_{n+2}-(boldsymbol{2} boldsymbol{n}+boldsymbol{1}) boldsymbol{y}_{n+1}- ) ( left(boldsymbol{n}^{2}-boldsymbol{m}^{2}right) boldsymbol{y}_{n}=boldsymbol{0} ) |
12 |
| 1013 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[frac{boldsymbol{operatorname} boldsymbol{operatorname { a n } boldsymbol { x }}-boldsymbol{operatorname { c o t } boldsymbol { x }}}{boldsymbol{operatorname { t a n } boldsymbol { x }}+boldsymbol{operatorname { c o t } boldsymbol { x }}}right]= ) ( A cdot 2 sin 2 x ) B. – -2 ( sin 2 x ) ( c cdot 2 cos 2 x ) D. ( -2 cos 2 x ) |
12 |
| 1014 | If ( e^{y}+x y=e ) then at ( x=0, frac{d^{2} y}{d x^{2}}=e^{-lambda} ) then numerical quantity ( -lambda ) should be equal to A .2 B. 3 ( c cdot 4 ) D. 5 |
12 |
| 1015 | The differential coefficient of ( boldsymbol{f}left(log _{e} boldsymbol{x}right) ) with respect to ( x, ) where ( f(x)=log _{e} x ) is A ( cdot frac{x}{log _{e} x} ) в. ( frac{1}{x} log _{e} x ) c. ( frac{1}{x log _{e} x} ) D. none of these |
12 |
| 1016 | Find ( boldsymbol{f}^{prime}(boldsymbol{3}) ) if ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+mathbf{5} boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{x}+mathbf{5} ) A . 28 B. 54 ( c .32 ) D. None |
12 |
| 1017 | ( frac{boldsymbol{d}}{d x}left(e^{tan x}right) ) ( mathbf{A} cdot e^{tan x} cdot sec ^{2} x ) B ( cdot e^{cot x} cdot sec ^{2} x ) ( mathbf{C} cdot e^{cos x} cdot sec ^{2} x ) D ( cdot e^{sin x} cdot sec ^{2} x ) |
12 |
| 1018 | Find the derivative of ( boldsymbol{y}= ) ( n sqrt{frac{1-sin x}{1+sin x}} ) |
12 |
| 1019 | If ( x=sin ^{-1} t ) and ( y=log left(1-t^{2}right), ) then ( frac{d^{2} y}{d x^{2}} ) at ( t=1 / 2 ) is A ( cdot frac{-8}{3} ) в. ( frac{8}{3} ) ( c cdot frac{3}{4} ) D. ( frac{-3}{4} ) |
12 |
| 1020 | Find the derivative of the following functions (it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers) ( sec x-1 ) ( sec x+1 ) |
12 |
| 1021 | Find the derivative of ( x^{2} ) with respect to ( log boldsymbol{x} ) |
12 |
| 1022 | Show that ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(tan boldsymbol{h}^{-1} boldsymbol{x}right)=frac{mathbf{1}}{mathbf{1 – x}^{mathbf{2}}} ) | 12 |
| 1023 | If ( y=ln sqrt{tan x} ) then the value of ( frac{d y}{d x} ) at ( x=frac{pi}{4} ) is ( A cdot infty ) B. c. D. |
12 |
| 1024 | Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem: ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}-boldsymbol{x}^{2} ) on ( [mathbf{0}, mathbf{1}] ) |
12 |
| 1025 | A metal sphere with radius of ( 10 mathrm{cm} ) is to be covered with a ( 0.02 mathrm{cm} ) coating of silver approximately silver required is ( left(operatorname{in} c m^{3}right) ) A ( .2 pi ) в. ( 10 pi ) ( c .6 pi ) D. ( 8 pi ) |
12 |
| 1026 | The number of real solutions of the equation ( e^{x}=x ) is ( mathbf{A} cdot mathbf{1} ) B. 2 c. 0 D. none of these |
12 |
| 1027 | If ( S_{n} ) denotes the sum of ( n ) terms of ( g . p ) whose common ratio is ( r, ) then ( (r-1) frac{d S_{n}}{d r} ) is equal to A ( cdot(n-1) S_{n}+n S_{n-1} ) B . ( (n-1) S_{n}-n S_{n-1} ) ( mathbf{c} cdot(n-1) S_{n} ) D. None of these |
12 |
| 1028 | Differentiate the following w.r.t. ( x ) ( left(2 x^{2}+9right)^{3} ) A ( cdot 4left(2 x^{2}+9right)^{2} x ) B. ( 12left(2 x^{2}+9right)^{2} x ) c. ( 12left(2 x^{2}+9right)^{3} x ) D. ( 6left(2 x^{2}+9right)^{2} x ) |
12 |
| 1029 | If ( y=left(5 x^{3}-4 x^{2}-8 xright)^{9}, ) find ( frac{d y}{d x} ) | 12 |
| 1030 | ( boldsymbol{y}=cos ^{-1}left{frac{2 boldsymbol{x}-boldsymbol{3} sqrt{1-boldsymbol{x}^{2}}}{sqrt{mathbf{1 3}}}right}, ) find ( frac{boldsymbol{d} boldsymbol{2}}{boldsymbol{d}} ) | 12 |
| 1031 | Verify Rolle’s theorem for the function ( boldsymbol{y}=boldsymbol{x}^{2}+mathbf{2} . boldsymbol{x} boldsymbol{epsilon}|-mathbf{2}, mathbf{2}| ) |
12 |
| 1032 | ( operatorname{Let} y=sqrt{x}+2 x^{frac{3}{4}}+3 x^{frac{5}{6}}(x>0) . ) Find the derivative of ( y ) with respect to ( x ) |
12 |
| 1033 | Solve ( : I_{n}=int_{0}^{frac{pi}{2}} e^{-x} sin ^{n} x d x ) | 12 |
| 1034 | Find the derivative ( x^{3} ) | 12 |
| 1035 | Find ( frac{d y}{d x}, ) when ( boldsymbol{x}=boldsymbol{a}(mathbf{1}-cos boldsymbol{theta}) ) and ( boldsymbol{y}=boldsymbol{a}(boldsymbol{theta}+sin boldsymbol{theta}) ) ( operatorname{at} theta=frac{pi}{2} ) |
12 |
| 1036 | If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{c}boldsymbol{a}^{2} cos ^{2} boldsymbol{x}+boldsymbol{b}^{2} sin ^{2} boldsymbol{x}, boldsymbol{x} leq mathbf{0} \ boldsymbol{e}^{boldsymbol{a} boldsymbol{x}+boldsymbol{b}}, boldsymbol{x}>mathbf{0}end{array}right. ) ( f(x) ) is continuous at ( x=0 ) then ( mathbf{A} cdot 2 log |a|=b ) B. ( 2 log |b|=e ) c. ( log a=2 log mid b ) ( mathbf{D} cdot a=b ) |
12 |
| 1037 | For the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{6} boldsymbol{x}^{2}+ ) ( boldsymbol{a} boldsymbol{x}+boldsymbol{b} . ) If Rolle’s theorem holds in ( [mathbf{1}, boldsymbol{3}] ) with ( c=2+frac{1}{sqrt{3}} ) then ( (a, b) ) A ( cdot(11,12) ) B . (11,11) c. ( (11, text { any real value }) ) D. (any real value, 0 ) |
12 |
| 1038 | Consider the function: ( f(-infty, infty) rightarrow ) ( (-infty, infty) ) defined by ( boldsymbol{f}(boldsymbol{x})= ) ( frac{x^{2}-a x+1}{x^{2}+a x+1}, 0<a<2 ) Which of the following is true? A. ( f(x) ) is decreasing on (-1,1) and has a local minimum at ( x=1 ) B. ( f(x) ) is increasing on (-1,1) and has a local maximum at ( x=1 ) C. ( f(x) ) is increasing on (-1,1) and has neither a local maximum nor a local minimum at ( x=-1 ) D. ( f(x) ) is decreasing on (-1,1) and has neither a local maximum nor a local minimum at ( x=1 ) |
12 |
| 1039 | The function ( f(x)=sin ^{-1}(tan x) ) is not differentiable at- A ( . x=0 ) В. ( x=-pi / 6 ) c. ( x=pi / 6 ) D. ( x=pi / 4 ) |
12 |
| 1040 | Check continuity of the function ( x^{2}|x| ) at the origin. | 12 |
| 1041 | Find ( k, ) if ( f ) is continuous at ( x=0 ) wher ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}frac{16^{x}-2^{x}}{k^{x}-1} text { when } x neq 0 \ 3 text { when } x=0end{array}right. ) |
12 |
| 1042 | Differentiate: ( sin ^{2} 3 x cdot tan ^{3} 2 x ) | 12 |
| 1043 | The ordered pair (a,b) such that ( f(x)= ) ( left{begin{array}{ll}frac{b e^{x}-cos x-x}{x} & , x>0 \ a & , x=0 \ frac{2 tan ^{-1}left(e^{x}right)-frac{pi}{4}}{x} & , x<0end{array}right. ) continuous at ( x-0 ) is |
12 |
| 1044 | If ( f(x)=left{begin{array}{ll}frac{1-cos x}{x^{2}}, & x<0 \ frac{1}{2} e^{x}, quad x geq 0end{array} ) then at right. ( boldsymbol{x}=mathbf{0}, boldsymbol{f} ) is A. continuous B. not continuous ( c . ) differentiable D. none of these |
12 |
| 1045 | Find the derivative of the following functions from first principle ( frac{1}{x^{2}} ) | 12 |
| 1046 | If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}-sin theta) ) and ( boldsymbol{y}=boldsymbol{a}(boldsymbol{1}+cos boldsymbol{theta}) ) then ( frac{d y}{d x} ) is A . ( cot theta ) B. ( cot frac{theta}{2} ) ( c cdot-cot frac{theta}{2} ) D. none of these |
12 |
| 1047 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) when ( boldsymbol{y}=boldsymbol{x}^{boldsymbol{x} cos boldsymbol{x}}+left(frac{boldsymbol{x}^{2}+mathbf{1}}{boldsymbol{x}^{2}-mathbf{1}}right) ) | 12 |
| 1048 | [ f(x)=left{begin{array}{ll} frac{1-sin ^{2} x}{3 cos ^{2} x}, & xfrac{pi}{2} end{array}right. ] then ( f(x) ) is continuous at ( x=frac{pi}{2} ) |
12 |
| 1049 | ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}+mathbf{1}: & boldsymbol{x} leq mathbf{1} \ boldsymbol{3}-boldsymbol{a} boldsymbol{x}^{2}: & boldsymbol{x}>1end{array}right. ) Find the value of ( a ) if ( f ) is continuous at ( boldsymbol{x}=mathbf{1} ) |
12 |
| 1050 | Consider the following statements in respect of the function ( boldsymbol{f}(boldsymbol{x})=sin left(frac{mathbf{1}}{boldsymbol{x}}right) ) for ( boldsymbol{x} neq mathbf{0} ) and ( boldsymbol{f}(mathbf{0})=mathbf{0} ) 1. ( lim _{x rightarrow 0} f(x) ) exists 2. ( f(x) ) is continuous at ( x=0 ) Which of the above statements is/are correct? A. 1 only B. 2 only c. Both 1 and 2 D. Neither 1 nor 2 |
12 |
| 1051 | Let ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y}), forall boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) suppose that ( boldsymbol{f}(mathbf{3})=mathbf{3}, boldsymbol{f}^{prime}(mathbf{0})=mathbf{1 1}, ) then ( f^{prime}(3) ) is given by A .22 B. 44 c. 28 D. 33 |
12 |
| 1052 | f ( mathbf{y}=boldsymbol{a} cos (log mathbf{x})+mathbf{b} sin (log mathbf{x}), ) then ( boldsymbol{x}^{2} boldsymbol{y}^{prime prime}+boldsymbol{x} boldsymbol{y}^{prime}= ) A. ( -y ) B. ( c cdot y ) D. ( y^{2} ) |
12 |
| 1053 | If ( boldsymbol{y}= ) ( (1+x)left(1+x^{2}right)left(1+x^{4}right) dotsleft(1+x^{2^{n}}right) ) then ( frac{d y}{d x} ) at ( x=0 ) is ( mathbf{A} cdot mathbf{1} ) B. – – c. 0 D. none of these |
12 |
| 1054 | Let ( boldsymbol{f}=boldsymbol{R} rightarrow boldsymbol{R} ) be a continuous function defined by ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{boldsymbol{e}^{boldsymbol{x}}+mathbf{2} boldsymbol{e}^{-boldsymbol{x}}} ) Statement ( 1: f(c)=frac{1}{3}, ) for some ( c epsilon R ) Statement ( 2: 0<f(x) leq frac{1}{2 sqrt{2}}, ) for al ( boldsymbol{x} epsilon boldsymbol{R} ) |
12 |
| 1055 | ( operatorname{Let} mathbf{f}(boldsymbol{x})=left{begin{array}{ll}frac{3 x+4 tan x}{x} & text { for } x neq 0 \ 7 & text { for } x=0end{array}right. ) ( operatorname{then} f(x) ) is A. continuous at ( x=0 ) B. not continuous at ( x=0 ) c. not determined at ( x=0 ) D. ( L t_{x rightarrow 0} f(x)=8 ) |
12 |
| 1056 | If ( f(x) ) is continuous on ( [a, b] ) and ( boldsymbol{f}(boldsymbol{a}) neq boldsymbol{f}(boldsymbol{b}), ) then for any value ( c ) belongs ( (f(a), f(b)), ) there is at least one number ( boldsymbol{x}_{o} ) in ( (boldsymbol{a}, boldsymbol{b}) ) for which ( boldsymbol{f}left(boldsymbol{x}_{boldsymbol{o}}right)=boldsymbol{c} ) A. ( fleft(x_{o}right)=b ) В. ( fleft(x_{o}right)=c ) c. ( fleft(x_{o}right)=f(c) ) D. ( fleft(x_{o}right)=0 ) |
12 |
| 1057 | For ( boldsymbol{x} in boldsymbol{R}, boldsymbol{f}(boldsymbol{x})=|log 2-sin boldsymbol{x}| ) and ( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(boldsymbol{f}(boldsymbol{x})), ) then: A. g is not differential at ( X=0 ) B ( cdot g^{prime}(0)=cos (log 2) ) C・ ( g^{prime}(0)=-cos (log 2) ) D. g is differentiable at ( x=0 ) and ( g^{prime}(0)=-sin (log 2) ) |
12 |
| 1058 | Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a continuous function such that ( boldsymbol{f}(boldsymbol{x})-mathbf{2} boldsymbol{f}left(frac{boldsymbol{x}}{mathbf{2}}right)+boldsymbol{f}left(frac{boldsymbol{x}}{mathbf{4}}right)=boldsymbol{x}^{mathbf{2}} ) The equation ( f(x)-x-f(0)=0 ) have exactly: A. no solution B. one solution c. two solutions D. Infinite solutions |
12 |
| 1059 | Consider the functions, ( boldsymbol{f}(boldsymbol{x})=mid boldsymbol{x} ) ( mathbf{2}|+| boldsymbol{x}-mathbf{5} mid, boldsymbol{x} in boldsymbol{R} ) Statement 1: ( boldsymbol{f}^{prime}(mathbf{4})=mathbf{0} ) Statement ( 2: f ) is continuous in [2,5] differentiable in (2,5) and ( f(2)=f(5) ) A. Statement 1 is false, Statement 2 is true B. Statement 1 is true, Statement 2 is true; Statement 2 is correct explanation for Statement c. statement 1 is true, statement 2 is true; Statement 2 is not a correct explanation for Statement 1 D. Statement 1 is true, Statement 2 is false |
12 |
| 1060 | The function ( boldsymbol{f}(boldsymbol{x})=frac{1+sin boldsymbol{x}-cos boldsymbol{x}}{1-sin boldsymbol{x}-cos boldsymbol{x}} ) is not defined at ( x=0 . ) The value of ( f(0) ) so that ( f(x) ) is continuous at ( x= ) ( 0, ) is ( A ) B. – ( c cdot 0 ) D. none of these |
12 |
| 1061 | Find the derivative of ( frac{x}{(x-1)} ) | 12 |
| 1062 | Find the derivative of ( 99 x ) at ( x=100 ) | 12 |
| 1063 | 31. Consider the function, f(x)=bx-2+bx-51, XER. Statement-1: f ‘(4)=0 Statement-2 :fis continuous in [2,5), differentiable in (2,5) and f(2)=f(5). [2012] (a) Statement-1 is false, Statement-2 is true. (b) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1. Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1. ) Statement-1 is true, statement-2 is false. |
12 |
| 1064 | Illustration 2.31 Find the derivative of y= – 21 Since |
12 |
| 1065 | ff ( boldsymbol{y}=boldsymbol{x}+frac{1}{boldsymbol{x}+frac{1}{boldsymbol{x}+frac{1}{boldsymbol{x}+cdots}}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A ( cdot frac{d y}{d x}=frac{x}{2 x-y} ) в. ( frac{d y}{d x}=frac{y}{2 y-x} ) c. ( frac{d y}{d x}=frac{2 y}{2 x-y} ) D. None of these |
12 |
| 1066 | 0 000 Iffis a real valued differentiable function satisi If(x)-) < (x – y)2, x,y e Randf(0) = 0, then f (1) equals [2005] (a) -1 (6) 0 (0) 2 (d) 1. |
12 |
| 1067 | ( y=frac{2(x-sin x)^{frac{3}{2}}}{sqrt{x}}, ) find ( frac{d y}{d x} ) ( mathbf{A} cdot frac{d y}{d x}=yleft{frac{3}{2} cdot frac{1-cos x}{x-sin x}-frac{1}{2 x}right} ) B. ( frac{d y}{d x}=yleft{frac{3}{4} cdot frac{1-sin x}{x-cos x}+frac{1}{2 x}right} ) ( mathbf{c} cdot frac{d y}{d x}=yleft{frac{3}{4} cdot frac{1-cos x}{x-sin x}-frac{1}{2 x}right} ) D. ( frac{d y}{d x}=yleft{frac{3}{2} cdot frac{1-sin x}{x-cos x}-frac{1}{2 x}right} ) |
12 |
| 1068 | f ( boldsymbol{y}=log sin boldsymbol{x}+boldsymbol{e}^{boldsymbol{x}} tan boldsymbol{x}, ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 1069 | If ( y=sqrt{x}+frac{1}{sqrt{x}}, ) show that ( 2 x frac{d y}{d x}+ ) ( boldsymbol{y}=mathbf{2} sqrt{boldsymbol{x}} ) |
12 |
| 1070 | If ( y^{2}=4 a x, ) then ( frac{left(1+y_{1}^{2}right)^{3 / 2}}{y_{2}} ) at ( x=a ) is A. ( -4 sqrt{2} a ) an ( a sqrt{2} ) an ( sqrt{2} ) B. ( 4 sqrt{2} a ) ( c cdot frac{4 sqrt{2}}{a} ) D. ( -4 a ) |
12 |
| 1071 | Show that the function ( f ) fiven below by: ( left{begin{array}{ll}frac{e^{1 / x}-1}{e^{1 / x}+1} & , text { if } x neq 0 \ -1 & , text { if } x=0end{array} ) is discontinuous at right. ( boldsymbol{x}=mathbf{0} ) |
12 |
| 1072 | If the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{1-cos 4 x}{x^{2}} & text { if } x0end{array}right. ) is continuous at ( x=0 ) then ( a= ) ( mathbf{A} cdot mathbf{8} ) B. ( c .-8 ) ( D ) |
12 |
| 1073 | Find the value of ( ^{prime} a^{prime} ) for which the function ( f ) defined by ( f(x)=left{begin{array}{l}a sin frac{pi}{2}(x+1), x leq 0 \ frac{tan x-sin x}{x^{3}}, x>0end{array}right. ) continuous at ( boldsymbol{x}=mathbf{0} ) |
12 |
| 1074 | Consider the function for ( boldsymbol{x}=[-mathbf{2}, mathbf{3}] ) ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}boldsymbol{x}^{mathbf{3}}-mathbf{2} boldsymbol{x}^{mathbf{2}}-mathbf{5} boldsymbol{x}+mathbf{6} & boldsymbol{i} boldsymbol{f} boldsymbol{x} neq mathbf{1} \ boldsymbol{x}-mathbf{1} & boldsymbol{i} boldsymbol{f} boldsymbol{x}=mathbf{1}end{array}right. ) then A. ( f ) is discontinuous at ( x=1 Rightarrow ) Rolles theorem is not applicable in [-2,3] B. ( f(-2) neq f(3) Rightarrow ) Rolles theorem is not applicable in [-2,3] C. ( f ) is not derivable in (-2,3)( Rightarrow ) Rolles theorem is not applicable D. Rolles theorem is applicable as ( f ) satisfies all the conditions and ( c ) of Rolles theorem is |
12 |
| 1075 | Find the value of ( k ) if ( f(x)= ) ( left{begin{array}{ll}frac{1-cos 2 x}{1+cos 2 x}, & x neq 0 \ k, & x=0end{array} ) is continuous at right. ( boldsymbol{x}=mathbf{0} ) |
12 |
| 1076 | Differentiate: ( e^{sin ^{-1} x} ) | 12 |
| 1077 | 10. Let f(x) = 0, *<0 then for all x , x20 (a) f' is differentiable (b) fis differentiable (c) f'is continuous (d) fis continuous |
12 |
| 1078 | If ( boldsymbol{x}=cos ^{n} boldsymbol{theta}, boldsymbol{y}=sin ^{n} boldsymbol{theta} operatorname{then} frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}= ) A ( cdot frac{n}{n-1} cdot frac{cos ^{2 n-1} theta}{sin ^{n-3} theta} ) B. ( frac{n-2}{n} cdot frac{tan ^{n-3} theta}{cos ^{n+1} theta sin theta} ) c. ( n-1 . tan ^{n-2} theta cdot sec ^{2} theta ) D. ( quad n frac{sin ^{n-1} theta}{cos ^{n-2} theta} ) |
12 |
| 1079 | If ( x sqrt{1+y}+y sqrt{1+x}=0, ) then ( frac{d y}{d x} ) is equal to ( ^{A} cdot frac{1}{(1+x)^{2}} ) B. ( frac{-1}{(1+x)^{2}} ) c. ( frac{1}{(1-x)^{2}} ) D. None of these |
12 |
| 1080 | Differentiate the following from first principle. ( f(x)=cos left(x-frac{pi}{8}right) ) | 12 |
| 1081 | The value of ( c ) in lagrange’s theorem for the function ( f(x)=log sin x ) in the interval ( left[frac{pi}{6}, frac{5 pi}{6}right] ) is A ( cdot frac{pi}{4} ) в. c. ( frac{2 pi}{3} ) D. None of these |
12 |
| 1082 | If, is continuous at ( x=2, ) find the value of ( k ) ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}frac{boldsymbol{x}^{3}+boldsymbol{x}^{2}-mathbf{1 6} boldsymbol{x}+mathbf{2 0}}{(boldsymbol{x}-mathbf{2})^{2}}, & boldsymbol{x} neq mathbf{2} \ boldsymbol{k}, & boldsymbol{x}=mathbf{2}end{array}right. ) |
12 |
| 1083 | If ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ccc}x^{alpha} cos left(frac{1}{x}right), & text { if } & x neq 0 \ 0 & , text { if } & x=0end{array}right. ) continuous at ( x=0 ) then ( mathbf{A} cdot alpha0 ) c. ( alpha=0 ) D. ( alpha geq 0 ) |
12 |
| 1084 | If ( u ) and ( v ) are differentiable functions of ( x ) and if ( y=u+v ) then ( frac{d y}{d x}=frac{d u}{d x}+frac{d v}{d x} ) |
12 |
| 1085 | Discuss the continuity of the following functions: (a) ( f(x)=sin x+cos x ) (b) ( f(x)=sin x-cos x ) |
12 |
| 1086 | Which of the following is(can be) continuous at each point of its domain- This question has multiple correct options A ( . f(x) ) в. ( g(x) ) c. ( k(x) ) D. all three ( f, g, k ) |
12 |
| 1087 | Let ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}]^{2}+sqrt{{boldsymbol{x}}} ) where ( mathbb{D} boldsymbol{&} ) Orespectively denotes the greatest integer and fractional part functions, then which of the following is correct? A. ( f(x) ) is continuous at all integral points B. ( f(x) ) is not differentiable ( forall x in I ) c. ( f(x) ) is discontinuous as ( x in I-{1} ) ( f(x) ) is continuous ( & ) differentiable at ( x=0 ) 0 |
12 |
| 1088 | 37. Using Rolle’s theorem, prove that there is at least one root in (451/100, 46) of the polynomial P(x) = 51×101 – 2323(x)100 – 45x + 1035. (2004 – 2 Marks) |
12 |
| 1089 | Discuss the applicability of Rolle’s theorem for the following function on the indicated interval: ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2 / 3} ) on [-1,1] |
12 |
| 1090 | In which of the following functions is Rolle’s theorem applicable A ( , quad f(x)=left{begin{array}{l}x, 0 leq x<1 \ 0, x=1end{array} text { on }[0,1]right. ) в. ( quad f(x)=left{begin{array}{l}frac{sin x}{x},-pi leq x<0 \ 0, x=0end{array} text { on }[-pi, 0]right. ) c. ( f(x)=frac{x^{2}-x-6}{x-1} ) on [-2,3] D ( quad f(x)=left{begin{array}{l}frac{x^{3}-2 x^{2}-5 x+6}{x-1}, text { if } x neq 1 \ -6, text { if } x=1end{array}right. ) |
12 |
| 1091 | If ( y=a^{frac{1}{2} log _{a} cos x} . ) Find ( frac{d x}{d} ) | 12 |
| 1092 | Identify the graph of the polynomial function ( boldsymbol{f} ) ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{boldsymbol{4}}+mathbf{1} ) begin{tabular}{|l|l|l|l|l|} hline 1 & & & & & \ hline & & & & & \ hline & & & & & \ hline & & & & & \ hline & & & & ( mathrm{d} ) & & \ hline & & & & & & \ hline end{tabular} A. graph a B. graph b C. graph c D. graph d |
12 |
| 1093 | Differentiate the following functions with respect to ( x ) ( frac{sqrt{x^{2}+1}+sqrt{x^{2}-1}}{sqrt{x^{2}+1}-sqrt{x^{2}-1}} ) |
12 |
| 1094 | The differential coefficient of ( a^{log _{10}left(operatorname{cosec}^{-1} xright)} ) is A ( cdot frac{a^{log _{10}left(operatorname{cosec}^{-1} xright)}}{[operatorname{cosec}]^{-1} mathrm{x}} frac{1}{mathrm{x} sqrt{mathrm{x}^{2}-1}} log _{10} mathrm{a} ) ( ^{mathrm{B}}-frac{a^{log _{10}left(operatorname{cosec}^{-1} xright)}}{[operatorname{cosec}]^{-1} mathrm{x}} frac{1}{|x| sqrt{mathrm{x}^{2}-1}} log _{10} mathrm{a} ) ( ^{mathbf{C}} cdot frac{a^{log _{10}left(operatorname{cosec}^{-1} xright)}}{[operatorname{cosec}]^{-1} mathbf{x}} frac{1}{|x| sqrt{mathrm{x}^{2}-1}} log _{mathrm{a}} 10 ) D. ( frac{a^{log _{10}left(operatorname{cosec}^{-1} xright)}}{[operatorname{cosec}]^{-1} mathrm{x}} frac{1}{mathrm{x} sqrt{mathrm{x}^{2}-1}} log _{mathrm{a}} 10 ) |
12 |
| 1095 | If ( boldsymbol{y}=boldsymbol{x}^{3} log left(frac{1}{x}right) . ) Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 1096 | ( operatorname{Let} f(x)=left{begin{array}{cl}min left{x, x^{2}right}, & x geq 0 \ max left{2 x, x^{2}-1right}, & x<0end{array}right. ) Then which of the following is not true? A. ( f(x) ) is continuous at ( x=0 ) B. ( f(x) ) is not differentiable at ( x=1 ) ( mathrm{c} . f(x) ) is not differentiable at exactly three points D. none of these |
12 |
| 1097 | Differentiate ( e^{sin x} ) ( mathbf{A} cdot e^{sin x} cos x ) B. ( -e^{sin x} cos x ) ( mathbf{c} cdot e^{-sin x} cos x ) ( mathbf{D} cdot e^{cos x} sin x ) |
12 |
| 1098 | Find ( lim _{x rightarrow 0} f(x), ) where ( f(x)= ) ( left{begin{array}{ll}frac{x}{|x|}, & x neq 0 \ 0, & x=0end{array}right. ) |
12 |
| 1099 | Let ( boldsymbol{f}(boldsymbol{x}) ) be a continuous function whose range is ( [2,6,5] . ) If ( h(x)= ) ( left[frac{cos x+f(x)}{lambda}right], lambda in N ) be continuous where [.] denotes the greatest integer function, then the least value of ( lambda ) is ( A cdot 6 ) B. 7 c. 8 D. None of these |
12 |
| 1100 | For what triplets of real numbers ( (a, b, ) c) with ( a neq 0 ) the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}boldsymbol{x} & boldsymbol{x} leq mathbf{1} \ boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} & text { otherwise }end{array}right. ) differentiable for all real x? A ( cdot{(a, 1-2 a, a) / a in R, a neq 0} ) в. ( {(a, 1-2 a, c) / a, c in R, a neq 0} ) c. ( {(a, b, c) / a, b, c in R, a+b+c=1} ) D. ( {(a, 1=2 a, 0) / a in R, a neq 0} ) |
12 |
| 1101 | Discuss the continuity of the following function at ( x=0 . ) If the function has a removable discontinuity, redefine the function so as to remove the discontinuity ( f(x)=left{begin{array}{ll}frac{4^{x}-e^{x}}{6^{x}}-1 & text { for } x neq 0 \ log left(frac{2}{3}right) & text { for } x=0end{array}right. ) |
12 |
| 1102 | Examine the continuity of the function: ( f(x)=frac{log 100+log (0.01+x)}{3 x}, ) for ( boldsymbol{x} neq mathbf{0} ) ( =frac{mathbf{1 0 0}}{mathbf{3}} quad ) for ( boldsymbol{x}= ) ( mathbf{0} ; boldsymbol{a} boldsymbol{t} boldsymbol{x}=mathbf{0} ) |
12 |
| 1103 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}x^{3}-x^{2}+10 x-5 quad, x leq 1 \ -2 x+log _{2}left(b^{2}-2right), x>1end{array} ) the set of right. values of ( b ) for which ( f(x) ) has greatest value at ( x=1 ) is given by: A. ( 1 leq b leq 2 ) В . ( b={1,2} ) c. ( b in(-infty,-1) ) D ( cdot b in[-sqrt{130},-sqrt{2}] cup[sqrt{2}, sqrt{130}] ) |
12 |
| 1104 | Which of the following functions is every where continuous- A . ( x+|x| ) B . ( x-|x| ) c. ( x|x| ) D. All of the above |
12 |
| 1105 | A value of ( C ) for which the conclusion of Mean Value Theorem holds for the function ( f(x)=log _{e} x ) on the interval [1,3] is ( mathbf{A} cdot 2 log _{3} e ) B. ( frac{1}{2} log _{3} e ) ( mathbf{c} cdot log _{3} e ) D. ( 2 log _{e} 3 ) |
12 |
| 1106 | If ( f(x)=operatorname{sgn}left(x^{5}right) ) then which of the following is/are false (where sgn denotes signum function) This question has multiple correct options ( mathbf{A} cdot f^{prime}left(0^{+}right)=1 ) B. ( f^{prime}left(0^{-}right)=1 ) C. ( f ) is continuous but not differentiable at ( x=0 ) D. fis discontinuous at ( x=0 ) |
12 |
| 1107 | Assertion For the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{3} boldsymbol{x}+boldsymbol{2} ) ( boldsymbol{L} M boldsymbol{V} boldsymbol{T} ) is applicable in ( [mathbf{1}, boldsymbol{2}] ) and the value of ( c ) is ( frac{3}{2} ) Reason If ( L M V T ) is known to be applicable for any quadratic polynomial in ( [a, b], ) then ( c ) of ( L M V T ) is ( frac{(a+b)}{2} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
12 |
| 1108 | Let ( y=log (log (x)) ) then find ( frac{d y}{d x} ) | 12 |
| 1109 | ( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b}, ) a,b being constants. A ( cdot x frac{d^{2} y}{d x^{2}}-frac{d y}{d x}=0 ) в. ( x frac{d^{2} y}{d x^{2}}+frac{d y}{d x}=0 ) c. ( 2 x frac{d^{2} y}{d x^{2}}-frac{d y}{d x}=0 ) D. ( x frac{d^{2} y}{d x^{2}}-2 frac{d y}{d x}=0 ) |
12 |
| 1110 | ff ( y=tan ^{-1}left[frac{sqrt{1+x^{2}}-sqrt{1-x^{2}}}{sqrt{1+x^{2}}+sqrt{1-x^{2}}}right] ) what would be ( frac{d y}{d x} ) A ( cdot frac{-x}{sqrt{1-x^{4}}} ) в. ( frac{1}{sqrt{1-x^{4}}} ) c. ( frac{x}{sqrt{1-x^{4}}} ) D. none of thes |
12 |
| 1111 | Find the value of ( frac{d y}{d x} quad ) a ( theta=frac{n}{4}, ) if ( x= ) ( boldsymbol{a} e^{theta}(sin theta-cos theta) ) and ( y=a e^{theta}(sin theta- ) ( cos theta) ) |
12 |
| 1112 | Obtain the differential equation whose solution is ( boldsymbol{y}=boldsymbol{x} sin (boldsymbol{x}+boldsymbol{A}), mathbf{A} ) being constant A ( cdotleft(x y_{1}-yright)^{2}+x^{2} y^{2}=x^{4} ) B. ( left(x y_{1}-yright)^{2}-x^{2} y^{2}=x^{4} ) c. ( left(x y_{1}-yright)^{2}+x^{2} y^{2}=x^{2} ) D. ( left(x y_{1}-yright)^{2}-x^{2} y^{2}=x^{2} ) |
12 |
| 1113 | If ( boldsymbol{y}(boldsymbol{n})=boldsymbol{e}^{boldsymbol{x}} boldsymbol{e}^{boldsymbol{x}^{2}} ldots boldsymbol{e}^{boldsymbol{x}^{n}}, boldsymbol{0}<boldsymbol{x}<1 . ) Then ( lim _{n rightarrow infty} frac{boldsymbol{d} boldsymbol{y}(boldsymbol{n})}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=frac{mathbf{1}}{mathbf{2}} ) is ( A cdot e ) в. ( 4 € ) ( c cdot 2 e ) D. ( 3 e ) |
12 |
| 1114 | Solve: ( lim _{x rightarrow 3} frac{x^{2}-9}{x-3} ) |
12 |
| 1115 | Let a function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) satisfy the equation ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y}) ) for all ( x, y . ) If the function ( f(x) ) is continuous at ( x=0, ) then A. ( f(x)=0 ) continuous for all ( x ) B. ( f(x) ) is continuous for all positive real ( x ) c. ( f(x) ) is continuous for all ( x ) D. None of these |
12 |
| 1116 | Let ( mathbf{f}(mathbf{x}) ) be differentiable on the interval ( (0, infty) ) such that ( f(1)=1, ) and ( lim _{t rightarrow x} frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1 ) for each ( mathbf{x}>0 . ) Then ( mathbf{f}(mathbf{x}) ) is A ( frac{1}{3 mathrm{x}}+frac{2 mathrm{x}^{2}}{3} ) B. ( -frac{1}{3 x}+frac{4 x^{2}}{3} ) c. ( -frac{1}{x}+frac{2}{x^{2}} ) ( D cdot underline{1} ) |
12 |
| 1117 | Differentiate ( log left(cos e^{x}right) ) w.r.t to ( x ) | 12 |
| 1118 | ( boldsymbol{f}(boldsymbol{x})=([boldsymbol{x}]-[-boldsymbol{x}]) boldsymbol{s i n}^{-1}|boldsymbol{x}-mathbf{1}| ) Which of the following statements is/are correct? (Note : [.] denotes the greatest integer function) This question has multiple correct options A. ( f(x) ) is continuous at ( x=1 ) B. ( f(x) ) is differentiable at ( x=1 ) c. ( f(x) ) is not differentiable at ( x=1 ) D. ( f(x) ) is discontinuous at ( x=1 ) |
12 |
| 1119 | Find ( frac{d y}{d x} ) of ( 2 x+3 y=sin x ) | 12 |
| 1120 | Let ( boldsymbol{f}(boldsymbol{x}) ) be a polynomial function of degree 2 and ( f(x)>0 ) for all ( x in R ) If ( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}^{prime}(boldsymbol{x})+boldsymbol{f}^{prime prime}(boldsymbol{x}), ) then for any ( boldsymbol{x} ) A ( . g(x)0 ) ( mathbf{c} cdot g(x)=0 ) D. ( g(x) geq 0 ) |
12 |
| 1121 | Suppose that ( f(x)=x^{3}-3 x^{2}-4 x+12 ) ( operatorname{and} h(x)=left{begin{array}{ll}frac{f(x)}{x-3} & x neq 3 \ K & x=3end{array}, ) then right. find the value of ( mathrm{K} ) that makes h’ continuous at ( x=3 ) |
12 |
| 1122 | The set onto which the derivative of the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(log boldsymbol{x}-1) ) maps the ray ( [1, infty) ) is ( ? ) A. ( [1, infty) ) (n) в. ( (10, infty) ) ( c cdot[0, infty) ) D. (0,0) |
12 |
| 1123 | Let ( f ) and ( g ) be differential functions satisfying ( boldsymbol{g}^{prime}(boldsymbol{a})=mathbf{2}, boldsymbol{g}(boldsymbol{a}) boldsymbol{b} ) and ( boldsymbol{f} boldsymbol{o} boldsymbol{g}=boldsymbol{I} ) (identify function) then ( boldsymbol{f}^{prime}(boldsymbol{b})= ) A ( cdot 1 / 2 ) B . 2 ( c cdot 2 / 3 ) D. None of these |
12 |
| 1124 | Find the derivative of the following functions (it is to be understood that ( a, b, c, d, p, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers) ( frac{boldsymbol{a}}{boldsymbol{x}^{4}}-frac{boldsymbol{b}}{boldsymbol{x}^{2}}+cos boldsymbol{x} ) |
12 |
| 1125 | Prove that the difference of the infinite continued fractions ( frac{1}{a+b+c} frac{1}{c+} dots, frac{1}{b+a+c+} dots ., ) is equal to ( frac{a-b}{1+a b} ) |
12 |
| 1126 | If ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}frac{xleft(3 e^{1 / x}+4right)}{2-e^{1 / x}}, & x neq 0 text { then } f(x) \ 0 & , x=0end{array}right. ) A. ( f ) is not continuous B. ( f ) is continuous but not differentiable at ( x=0 ) C ( cdot f^{prime prime}(0) ) exist ( mathbf{D} cdot f^{prime}(0+)=2 ) |
12 |
| 1127 | If ( boldsymbol{f}(boldsymbol{x})= ) ( frac{sin 3 x+A sin 2 x+B sin x}{x^{5}}, x neq 0 ) is continous at ( x=0 ) then This question has multiple correct options A. ( A=-4 ) B. ( B=5 ) c. ( f(0)=1 ) D. ( A=-3 ) |
12 |
| 1128 | ( operatorname{Let} f(x)=left{begin{array}{ll}g(x) cdot cos frac{1}{x} & text { if } x neq 0 \ 0 & text { if } x=0end{array}right. ) where ( g(x) ) is an even function differentiable at ( x=0, ) passing through the origin. Then ( boldsymbol{f}^{prime}(mathbf{0}) ) A. is equal to 1 B. is equal to 0 c. is equal to 2 D. does not exist |
12 |
| 1129 | If ( y=frac{x^{2}}{2}+frac{1}{2} times sqrt{x^{2}+1}, ) then ( 2 y= ) ( x y^{prime}, ) where ( y^{prime} ) denotes the derivative of ( y ) w.r.t. ( boldsymbol{x} ) A. True B. False |
12 |
| 1130 | Verify Rolle’s Theorem for the function ( f(x)=e^{x}(sin x-cos x) ) on ( left[frac{pi}{4}, frac{5 pi}{4}right] ) | 12 |
| 1131 | Differentiate: ( left(sin ^{-1} x+frac{1}{2} log frac{1+x}{1-x}right) ) | 12 |
| 1132 | If ( y=tan ^{-1}left(frac{4 x}{1+5 x^{2}}right)+ ) ( tan ^{-1}left(frac{2+3 x}{2-3 x}right), ) then ( frac{d y}{d x} ) is ( frac{1}{1+4 x^{2}} ) – s. ( frac{3}{1+4 x^{2}} ) ( frac{5}{1+25 x^{2}} ) ( frac{5}{left(1+25 x^{2}right)}-frac{1}{left(1+x^{2}right)}-frac{1.5}{left(1+2.25 x^{2}right)} ) |
12 |
| 1133 | Suppose, ( A=frac{d y}{d x} ) of ( x^{2}+y^{2}=4 ) at ( (sqrt{2}, sqrt{2}), B=frac{d y}{d x} ) of ( sin y+sin x= ) ( sin x cdot sin y operatorname{at}(pi, pi) ) and ( C=frac{d y}{d x} ) of ( 2 e^{x y}+e^{x} e^{y}-e^{x}=e^{x y+1} ) at ( (1,1), ) then ( (A-B-C) ) has the value equal to A. ( frac{1}{2} ) в. ( frac{1}{3} ) c. 1 D. 2 |
12 |
| 1134 | If ( y=log (log x) ) then ( frac{d^{2} y}{d x^{2}} ) is equal to A ( cdot frac{-(1+log x)}{x^{2} log x} ) B. ( frac{(1+log x)}{x^{2} log x} ) c. ( frac{-(1+log x)}{(x log x)^{2}} ) D. ( frac{(1+log x)}{left(x^{2} log xright)^{2}} ) |
12 |
| 1135 | ( lim _{x rightarrow 5} frac{2 x^{2}+9 x-5}{x+5} ) | 12 |
| 1136 | If the function ( f ) defined on ( left(frac{pi}{6}, frac{pi}{3}right) ) b ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cl}frac{sqrt{mathbf{2}} cos boldsymbol{x}-mathbf{1}}{cot boldsymbol{x}-mathbf{1}}, & boldsymbol{x} neq frac{boldsymbol{pi}}{boldsymbol{4}} \ boldsymbol{k}, & boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{4}}end{array}right. ) continuous, then ( k ) is equal to? ( A cdot frac{1}{2} ) B. ( c cdot frac{1}{sqrt{2}} ) ( D ) |
12 |
| 1137 | ( mathbf{f} boldsymbol{y}=sin ^{-1}(mathbf{3} boldsymbol{x})+mathbf{s} mathbf{e} mathbf{c}^{-1}left(frac{mathbf{1}}{mathbf{3} boldsymbol{x}}right), ) find ( frac{boldsymbol{d} mathbf{2}}{boldsymbol{d}} ) | 12 |
| 1138 | Differentiate the following function with respect to ( boldsymbol{x} ) ( sin h^{-1}(sqrt{x}) ) |
12 |
| 1139 | Find the values of ( k ) so that the function ( f ) is continuous at the indicated point: [ boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll} boldsymbol{k} boldsymbol{x}^{2}, & text { if } boldsymbol{x} leq boldsymbol{pi} \ cos boldsymbol{x}, & text { if } boldsymbol{x}>pi end{array}right. ] at ( boldsymbol{x}=boldsymbol{pi} ) |
12 |
| 1140 | Find the second order derivative of the following function: ( x^{3}+tan x ) |
12 |
| 1141 | If ( boldsymbol{f}:[-boldsymbol{2}, boldsymbol{2}] rightarrow boldsymbol{R} ) is defined by ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{sqrt{1+e x}-sqrt{1-e x}}{x} & text { for }-2 leq x< \ frac{x+3}{x+1} & text { for } 0 leq x leqend{array}right. ) is continuous on ( [-2,2], ) then ( e= ) A ( cdot frac{2}{sqrt{3}} ) B. 3 ( c cdot frac{3}{2} ) D. ( frac{3}{sqrt{2}} ) |
12 |
| 1142 | The value of ( f(0) ) so that the function ( f(x)=frac{2 x-sin ^{-1} x}{2 x+tan ^{-1} x} ) is continuous at each point in its domain, is equal to A . 2 в. ( frac{1}{3} ) c. ( frac{2}{3} ) D. ( frac{-1}{3} ) |
12 |
| 1143 | Consider the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cl}frac{tan k x}{x}, & x<0 \ 3 x+2 k^{2}, & x geq 0end{array} . ) What is the non- right. zero value of k for which the function is continuous at ( boldsymbol{x}=mathbf{0} ? ) A . ( 1 / 4 ) B. ( 1 / 2 ) c. 1 D. |
12 |
| 1144 | If ( y=left(x^{2}+1right) sin x ) then ( left(frac{pi}{2}right)^{2}- ) ( y_{20}left(frac{pi}{2}right) ) is equal to |
12 |
| 1145 | Differentiate ( sin boldsymbol{h}^{-1}left(frac{mathbf{1}}{boldsymbol{x}}right) ) with respect to ( boldsymbol{x}(boldsymbol{x}>mathbf{0}) ) |
12 |
| 1146 | If ( f(x)=frac{sin 3 x}{sin x}, x neq 0 ) is continuous [ =boldsymbol{K}, boldsymbol{x}=mathbf{0} ] function, then ( boldsymbol{K}= ) A . B. 3 ( c cdot frac{1}{3} ) D. |
12 |
| 1147 | The constant ( c ) of Rolle’s theorem for the function ( boldsymbol{f}(boldsymbol{x})=log frac{boldsymbol{x}^{2}+boldsymbol{a} boldsymbol{b}}{(boldsymbol{a}-boldsymbol{b}) boldsymbol{x}} ) in ( [boldsymbol{a}, boldsymbol{b}] ) where ( mathbf{0} notin[boldsymbol{a}, boldsymbol{b}] ) is A. ( sqrt{a b} ) в. ( frac{a+b}{2} ) c. ( frac{a-b}{2} ) D. ( frac{b-a}{2} ) |
12 |
| 1148 | Find ( mathrm{k} ) so that the function [ begin{array}{cc} boldsymbol{f}(boldsymbol{x})=left{1-cos 2 boldsymbol{x} / 2 boldsymbol{x}^{2}right. & boldsymbol{x} neq mathbf{0} \ {boldsymbol{k} & boldsymbol{x}=mathbf{0} end{array} ] is coutinous at ( x=0 ) |
12 |
| 1149 | Evaluate: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(boldsymbol{x}^{100}+boldsymbol{x}^{99}+boldsymbol{x}^{98}+ldots+boldsymbol{x}^{2}+boldsymbol{x}+right. ) ( mathbf{1} ) |
12 |
| 1150 | Among the following, the continuous function is ? A. ( tan x ) B. ( sec x ) ( c cdot sin 1 / x ) D. None of these |
12 |
| 1151 | ( f(x)=left{begin{array}{ll}cos x ; & x geq 0 \ x+k & ; x<0end{array} ) find the right. value of ( k ) if ( f(x) ) is continuous at ( x=0 ) |
12 |
| 1152 | Let ( f(x) ) be a continuous function defined for ( 1 leq x leq 3 ). If ( f(x) ) takes rational values for all ( x ) and ( f(2)=10 ) then the value of ( boldsymbol{f}(mathbf{1} . mathbf{5}) ) is A . 7.5 B. 10 ( c .5 ) D. none of these |
12 |
| 1153 | If ( f(x) ) is continuous for ( 0 leq x<infty ) then the most suitable values of ( a ) and ( b ) are A ( . a=1, b=-1 ) B. ( a=-1, b=1+sqrt{2} ) c. ( a=-1, b=1 ) D. none of these |
12 |
| 1154 | Differentiate with respect to ( x ) : ( log (csc x-cot x) ) |
12 |
| 1155 | f ( y=tan ^{-1}left(frac{1}{1+x+x^{2}}right)+ ) ( tan ^{-1}left(frac{1}{x^{2}+3 x+3}right)+ ) ( tan ^{-1}left(frac{1}{x^{2}+5 x+7}right)+—+ ) upto n terms, then ( boldsymbol{y}^{prime}(mathbf{0})= ) A ( cdot frac{-1}{1+n^{2}} ) в. ( frac{-n^{2}}{1+n^{2}} ) c. ( frac{n^{2}}{1+n^{2}} ) D. |
12 |
| 1156 | Assertion The function ( y=f(x), ) defined parametrically as ( boldsymbol{y}=boldsymbol{t}^{2}+boldsymbol{t}|boldsymbol{t}|, boldsymbol{x}= ) ( mathbf{2} t-|boldsymbol{t}|, boldsymbol{t} in boldsymbol{R}, ) is continuous for all rea ( boldsymbol{x} ) Reason ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}2 boldsymbol{x}^{2}, & boldsymbol{x} geq mathbf{0} \ mathbf{0}, & boldsymbol{x}<mathbf{0}end{array}right. ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
12 |
| 1157 | The derivative of ( boldsymbol{y}=(1-boldsymbol{x})(2- ) ( x) ldots(n-x) ) at ( x=1 ) is equal to A . 0 в. (-1)( (n-1) ! ) c. ( n !-1 ) D ( cdot(-1)^{n-1}(n-1) ! ) E ( cdot(-1)^{n}(n-1) ! ) |
12 |
| 1158 | Evaluate: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{tan ^{-1} frac{boldsymbol{x}}{mathbf{1}+boldsymbol{x}^{2}}+tan ^{-1} frac{mathbf{1}+boldsymbol{x}^{2}}{boldsymbol{x}}right} ) ( A ) в. ( c cdot frac{1}{2} ) ( D ) |
12 |
| 1159 | Suppose that ( f ) is differentiable function with the property ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})= ) ( f(x)+f(y)+x^{2} y^{2} ) and ( lim _{x rightarrow 0} frac{f(x)}{x}=10 ) then ( f^{prime}(0) ) is equal to |
12 |
| 1160 | A point on the curve ( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}^{2}-boldsymbol{4}} ) defined in [2,4] where the tangent is parallel to the chord joining two points on the curve A ( cdot(sqrt{2}, sqrt{6}) ) B . ( (sqrt{6}, sqrt{2}) ) C ( cdot(2,6) ) D ( cdot(6,2) ) |
12 |
| 1161 | Let ( f(x)=left{begin{array}{cc}-2, & -3 leq x leq 0 \ x-2, & x<x leq 3end{array} ) and right. ( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(|boldsymbol{x}|)+mid boldsymbol{f}(boldsymbol{x}) ) Which of the following statements are correct? 1. ( g(x) ) is continuous at ( x=0 ) 2. ( g(x) ) is continuous at ( x=2 ) 3. ( g(x) ) is continuous at ( x=-1 ) Select the correct answer using the code given below A. 1 and 2 only B. 2 and 3 only c. 1 and 3 only D. 1,2 and 3 |
12 |
| 1162 | If ( boldsymbol{y}=4 boldsymbol{x}^{4}+boldsymbol{2} boldsymbol{x}^{3}+frac{mathbf{5}}{boldsymbol{x}}+boldsymbol{9}, ) then find ( boldsymbol{d} boldsymbol{y} / boldsymbol{d} boldsymbol{x} ) |
12 |
| 1163 | If ( f(x)=frac{log left(e^{x^{2}}+2 sqrt{x}right)}{tan sqrt{x}}, x neq 0, ) then the value of ( f(0) ) so that ( f ) is continuous at ( x=0 ) is A ( cdot frac{1}{2} ) B. ( sqrt{2} ) ( c cdot 2 ) D. ( frac{1}{sqrt{2}} ) |
12 |
| 1164 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} sqrt{cos boldsymbol{x}}= ) | 12 |
| 1165 | Differentiate : ( e^{e^{x}} ) |
12 |
| 1166 | Differentiate ( boldsymbol{x}^{3} ) w.r.t ( boldsymbol{x} ) | 12 |
| 1167 | The function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{-|boldsymbol{x}|} ) is A. continuous everywhere but not differentiable at ( x=0 ) B. continuous and differentiable everywhere C. not continuous at ( x=0 ) D. None of the above |
12 |
| 1168 | For the discontinuous function given below, find the value of ( boldsymbol{f}(-mathbf{3}) ) ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}^{2}+mathbf{1}, & boldsymbol{i f} quad boldsymbol{x}<mathbf{0} \ frac{boldsymbol{2} boldsymbol{x}}{mathbf{3}}-mathbf{1}, & boldsymbol{i f} quad boldsymbol{0}<boldsymbol{x}3end{array}right. ) ( A cdot-3 ) B. ( c cdot 10 ) D. – 3, 7, and 10 |
12 |
| 1169 | f ( boldsymbol{x}=sin t, boldsymbol{y}=sin boldsymbol{k} boldsymbol{t} ) satisfies ( left(1-x^{2}right) y_{2}-x y_{1}+A y=0 ) then ( A ) is Equal to ( A cdot k ) B. ( mathbf{c} cdot k^{2} ) D. ( 1+k ) |
12 |
| 1170 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[(boldsymbol{x}+mathbf{1})left(boldsymbol{x}^{2}+mathbf{1}right)left(boldsymbol{x}^{4}+mathbf{1}right)left(boldsymbol{x}^{boldsymbol{8}}+right.right. ) 1) ( ] ) ( =frac{left(15 x^{p}-16 x^{q}+1right)}{(x-1)^{2}} Rightarrow(p, q)= ) A . (12,11) B. (15, 14) ( c cdot(16,14) ) D. (16, 15) |
12 |
| 1171 | If ( y=A sin (omega t-k x), ) then the value of ( frac{d y}{d x} ) is A. ( A cos (omega t-k x) ) B. ( -A omega cos (omega t-k x) ) c. ( A k cos (omega t-k x) ) D. ( -A k cos (omega t-k x) ) |
12 |
| 1172 | ( fleft(x^{2}+y^{2}right)^{2}=x y, ) find ( frac{d y}{d x} ) | 12 |
| 1173 | If ( y=left[log log sin x^{circ}right]^{7}, ) find ( frac{d y}{d x} ) A ( cdot frac{6 pi}{180^{circ}}left[log left(log sin x^{circ}right)right]^{7} cdot frac{tan x^{circ}}{log sin x^{circ}} ) в. ( frac{7 pi}{180^{circ}}left[log left(log sin x^{circ}right)right]^{6} cdot frac{cot x^{circ}}{log sin x^{circ}} ) c. ( frac{pi}{30^{circ}}left[log left(log sin x^{circ}right)right]^{7} frac{cot x^{circ}}{log sin x^{circ}} ) D. none of these |
12 |
| 1174 | Differentiate ( sqrt{e^{sqrt{x}}}, x>0 ) | 12 |
| 1175 | ax 14. If y = a sin x + b cos x, then y2 + (a) Function of x (b) Function of y (c) Function of x and y (d) Constant |
12 |
| 1176 | The width of each of five continuous classes in a frequency distribution is 5 and the lower class limit of the lowest class is 10 The upper class limit of the highest class is A . 25 B. 30 ( c .35 ) D. 50 |
12 |
| 1177 | If ( 27 a+9 b+3 c+d=0, ) then the equation ( 4 a x^{3}+3 b x^{2}+2 c x+d=0 ) has atleast one real root lying between A. 0 and 1 B. 1 and 3 c. 0 and 3 D. None |
12 |
| 1178 | If ( y=frac{5 x}{(1-x)^{2 / 3}}+cos ^{2}(2 x+1), ) find ( frac{d y}{d x} ) A. ( frac{5}{3(1-x)^{5 / 9}}(3-x)-2 sin (4 x+2) ) в. ( frac{-5}{3(1-x)^{5 / 9}}(3-x)-2 sin (4 x+2) ) c. ( frac{4}{3(1-x)^{5 / 9}}(3-x)-2 sin (4 x+2) ) D. ( frac{5}{3(1-x)^{5 / 9}}(3-x)-sin (4 x+2) ) |
12 |
| 1179 | Find the derivative of the following functions (it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers: ( (x+a) ) |
12 |
| 1180 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}[boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{g}(boldsymbol{x})]=boldsymbol{f}(boldsymbol{x}) frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} boldsymbol{g}(boldsymbol{x})+ ) ( g(x) frac{d}{d x} f(x) ) is known as ( _{–}—r u l e ) A. Product B. Sum c. Multiplication D. None of these |
12 |
| 1181 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{y}=log left(boldsymbol{e}^{boldsymbol{x}} sin ^{boldsymbol{5}} boldsymbol{x}right) ) | 12 |
| 1182 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[sin ^{-1}left(frac{sqrt{mathbf{1}+boldsymbol{x}}+sqrt{mathbf{1}-boldsymbol{x}}}{mathbf{2}}right)right] mathbf{w} cdot mathbf{r} ) to ( x ) equals A ( cdot frac{1}{2 sqrt{1-x^{2}}} ) в. ( frac{-2}{sqrt{1-x^{2}}} ) c. ( frac{-1}{2 sqrt{1-x^{2}}} ) D. None of these |
12 |
| 1183 | Let ( [x] ) denote the integral part of ( x in ) ( boldsymbol{R}, boldsymbol{g}(boldsymbol{x})=boldsymbol{x}-[boldsymbol{x}] . ) Let ( boldsymbol{f}(boldsymbol{x}) ) be any continuous function with ( boldsymbol{f}(mathbf{0})=boldsymbol{f}(mathbf{1}) ) then the function ( h(x)=f(g(x)) ) A. has finitely many discontinuities B. is discontinuous at some ( x=c, c in I ) c. is continuous on ( R ) D. is a constant function |
12 |
| 1184 | Differentiate the function with respect to ( x ) ( cos x^{3} cdot sin ^{2}left(x^{5}right) ) |
12 |
| 1185 | Illustration 2.22 If y = 4x*+ 2x + + 9, then find dyldx. |
12 |
| 1186 | Show that the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cll}3-x, & text { if } & x1end{array}right. ) ( boldsymbol{x}=mathbf{1} ) |
12 |
| 1187 | Differentiate the following ( (x+2)^{3} ) |
12 |
| 1188 | If ( y=sin ^{-1} x operatorname{th} e nleft(1-x^{2}right) frac{d^{2} y}{d x^{2}}= ) A ( cdot-x frac{d y}{d x} ) B. c. ( _{x} frac{d y}{d x} ) D. ( xleft(frac{d y}{d x}right)^{2} ) |
12 |
| 1189 | ( lim _{x rightarrow 0} frac{sqrt{frac{1}{2}(1-cos x)}}{x}= ) ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot 0 ) D. does not exist |
12 |
| 1190 | Find the value of ( f(0) ) so that the function ( boldsymbol{f}(boldsymbol{x})= ) ( frac{mathbf{9 6}left[log left(1+frac{boldsymbol{x}}{mathbf{1 2}}right)-log left(mathbf{1}-frac{boldsymbol{x}}{mathbf{8}}right)right]}{boldsymbol{x}}, boldsymbol{x} neq ) 0 is continuous on [0,8] |
12 |
| 1191 | Differentiate the following w.r.t. ( x ) ( sin ^{2} sqrt{x} ) A ( cdot frac{1}{2 sqrt{x}} sin (3 sqrt{x}) ) B. ( frac{1}{sqrt{x}} sin (2 sqrt{x}) ) c. ( frac{1}{2 sqrt{x}} sin (2 sqrt{x}) ) D. ( frac{1}{2 sqrt{x}} sin (4 sqrt{x}) ) |
12 |
| 1192 | The length ( x ) of a rectangle is decreasing at a rate of ( 3 mathrm{cm} / mathrm{min} ) and width ( y ) is increasing at a rate of ( 2 c m / m i n . ) When ( x=10 c m ) and ( y= ) ( 6 c m, ) find the rates of change of (i) the perimeter (ii) the area of the rectangle. |
12 |
| 1193 | Show that the function is continuous at [ begin{array}{l} x=0, text { if } f(x)=frac{sin 3 x}{tan 2 x}, x0 \ =frac{3}{2}, x=0 end{array} ] |
12 |
| 1194 | ( operatorname{Let} f(x)=2 tan ^{-1} x+sin ^{-1}left(frac{2 x}{1+x^{2}}right) ) Then A ( cdot f^{prime}(2)=f^{prime}(3) ) B . ( f^{prime}(2)=0 ) ( mathbf{c} cdot f^{prime}(1 / 2)=16 / 5 ) D. All of these |
12 |
| 1195 | If ( f(x)= ) [ frac{sin 3 x+A sin 2 x+B sin x}{x^{5}} quad(x neq 0) ] continuous at ( x=0, ) then find ( A+B ) |
12 |
| 1196 | For the function ( f(x)=e^{cos x}, ) Rolle’s theorem is A ( cdot ) applicable, when ( frac{pi}{2} leq x leq frac{3 pi}{2} ) B. applicable, when ( 0 leq x leq frac{pi}{2} ) C . applicable, when ( 0 leq x leq pi ) D. applicable, when ( frac{pi}{4} leq x leq frac{pi}{2} ) |
12 |
| 1197 | If ( frac{1}{2}left(e^{y}-e^{-y}right)=x, ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{1}{sqrt{boldsymbol{x}^{2}+mathbf{1}}} ) |
12 |
| 1198 | If ( f(x) ) and ( g(x) ) are both continuous at ( x=c ) then which of the following is/are always continuous at ( x=c ? ) This question has multiple correct options A. ( f(x)+g(x) ) B. ( (f(x)-g(x)) times f(x) ) c. ( g(x) times f(x) ) D. ( frac{f(x)-g(x)}{g(x)} ) |
12 |
| 1199 | Let ( boldsymbol{f}_{boldsymbol{p}}(boldsymbol{a})=boldsymbol{e}^{frac{i a}{p^{2}}} cdot boldsymbol{e}^{frac{2 i a}{p^{2}}} cdot boldsymbol{e}^{frac{3 i a}{p^{2}}} ldots cdot boldsymbol{e}^{frac{i a}{p}} ) (Where ( i=sqrt{-1} ) and ( p in ) ( N) ) then ( lim _{n rightarrow infty} f_{n}(pi) ) ( mathbf{A} cdot mathbf{1} ) B. c. -1 D. ( -i ) |
12 |
| 1200 | Differentiate from first principle: ( y=x^{2} ) ( mathbf{A} cdot 2 x ) B . ( (x-1)^{2} ) ( mathbf{c} cdot x^{3} ) D. ( frac{1}{sqrt{x}} ) |
12 |
| 1201 | Differentiate with respect to ( x ) : ( e^{sin ^{-1} 2 x} ) |
12 |
| 1202 | Verify Rolle’s theorem for the function ( f(x)=sin x+cos x-1 ) in the interval ( left[0, frac{pi}{2}right. ) |
12 |
| 1203 | If ( f(x)=sin 2 x-cos 2 x, ) find ( f^{prime}left(frac{pi}{6}right) ) | 12 |
| 1204 | Consider the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}a x-2 & text { for }-2<x<-1 \ -1 & text { for }-1 leq x leq 1 \ a+2(x-1)^{2} & text { for } quad 1<x<2end{array}right. ) What is the value of a for which ( f(x) ) is continuous at ( x=-1 ) and ( x=1 ? ) ( A cdot-1 ) B. ( c cdot 0 ) ( D ) |
12 |
| 1205 | If ( boldsymbol{x}=cos t ) and ( boldsymbol{y}=ln t ; ) then at ( boldsymbol{t}=frac{boldsymbol{pi}}{2} ) ( left(frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}right) ) is equal to A . 0 B. – – ( c .1 ) ( D ) |
12 |
| 1206 | If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a}, boldsymbol{a} in boldsymbol{R}, ) then ( mathbf{A} cdot nabla f(x)=0 ) B ( cdot nabla f(x)=a ) ( mathbf{c} cdot nabla f(x)=2 a ) D. ( nabla f(x)=a^{2} ) |
12 |
| 1207 | If ( y=tan ^{-1}left(frac{2 x}{1-x^{2}}right)+ ) ( tan ^{-1}left(frac{3 x-x^{3}}{1-3 x^{2}}right) ) ( tan ^{1}left(frac{4 x-4 x^{3}}{1-6 x^{2}+x^{4}}right) ) then Show that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{mathbf{1}}{mathbf{1}+boldsymbol{x}^{2}} ) |
12 |
| 1208 | If ( y=frac{x}{a+frac{x}{b+y}}, ) then ( frac{d y}{d x} ) is ( mathbf{A} cdot frac{a}{a b+2 a y} ) B. ( frac{b}{a b+2 b y} ) c. ( frac{a}{a b+2 b y} ) D. ( frac{b}{a b+2 a y} ) |
12 |
| 1209 | The number of points at which the function ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{0 . 5}|+|boldsymbol{x}-mathbf{1}|+ ) ( tan x ) does not have a derivative in the interval (0,2) is/are? ( mathbf{A} cdot mathbf{1} ) B . 2 ( c .3 ) D. |
12 |
| 1210 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{g}^{prime}(1)+boldsymbol{g}^{prime prime}(2) ) and ( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(1) cdot boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{f}^{prime}(boldsymbol{x})+boldsymbol{f}^{prime prime}(boldsymbol{x}) ) then This question has multiple correct options A ( cdot f^{prime}(1)+f^{prime}(2)=0 ) B . ( g^{prime}(2)=g^{prime}(1) ) c. ( g^{prime prime}(2)+f^{prime prime}(3)=6 ) D. none of these |
12 |
| 1211 | Differentiate with respect to ( x ) : ( (log sin x)^{2} ) |
12 |
| 1212 | llustration 2.29 If y Illustration 2.29 sin x If y = – , then find x + cos x sin x then finden dx |
12 |
| 1213 | Find the value of ( k ) if ( f(x) ) is continuous at ( boldsymbol{x}=boldsymbol{pi} / 2, ) where ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{k cos x}{pi-2 x}, x neq pi / 2 \ 3, quad x=pi / 2end{array}right. ) |
12 |
| 1214 | If ( boldsymbol{y}=mathbf{1}+boldsymbol{x}+frac{boldsymbol{x}^{2}}{mathbf{2 !}}+frac{boldsymbol{x}^{mathbf{3}}}{mathbf{3 !}}+ldots+frac{boldsymbol{x}^{boldsymbol{n}}}{boldsymbol{n} !} ) then ( frac{d y}{d x} ) is equal to ( mathbf{A} cdot underline{y} ) B. ( y+frac{x^{n}}{n !} ) c. ( y-frac{x^{n}}{n !} ) D. ( y-1-frac{x^{n}}{n !} ) |
12 |
| 1215 | Differentiate ( -3 x^{2} cdotleft(sin 2 x^{3}right)left{cos left[cos ^{2}left(x^{3}right)right]right} ) |
12 |
| 1216 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[csc ^{-1}left(frac{sqrt{boldsymbol{2}}}{boldsymbol{x}-sqrt{mathbf{1}-boldsymbol{x}^{2}}}right)right] ) | 12 |
| 1217 | Find the inverse function of ( f(x)= ) ( 2 x-3 ) |
12 |
| 1218 | Differentiate with respect to ( x ) : ( sin ^{2}(log (2 x+3)) ) |
12 |
| 1219 | A particle moves along a straight line such that its displacement ( s ) at any time ( t ) is given by ( s=t^{3}-6 t^{2}+3 t+ ) ( 4 m, t ) being is seconds. Find the velocity of the particle when the acceleration is zero. |
12 |
| 1220 | Differentiate the following function with respect to ( x ) ( left(2 x^{2}-3right) sin x ) A ( cdot 4 x sin x+left(2 x^{2}+3right) cos x ) B. ( 4 x sin x+left(2 x^{2}-3right) sin x ) c. ( 4 x sin x+left(2 x^{2}-3right) cos x ) D. ( 4 x cos x+left(2 x^{2}-3right) cos x ) |
12 |
| 1221 | Differentiate the following function w.r.t. ( x: cos ^{-1}left(1-2 sin ^{2} xright) ) | 12 |
| 1222 | If the transformation ( z=log tan left(frac{x}{2}right) ) reduces the differential equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+cos boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+mathbf{4} boldsymbol{y} operatorname{cosec}^{2} boldsymbol{x}=mathbf{0} ) int ( frac{d^{2} y}{d x^{2}}+A y=0 ) then the value of ( A ) is |
12 |
| 1223 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[cos ^{2}left(tan ^{-1}left(sin left(cot ^{-1} xright)right)right)right]= ) A. ( frac{2}{left(x^{2}+2right)^{2}} ) B. ( frac{2 x}{left(x^{2}+2right)^{2}} ) c. ( frac{left(x^{2}+1right)}{left(x^{2}+2right)} ) D. ( frac{-2 x}{left(x^{2}-1right)^{2}} ) |
12 |
| 1224 | Find ( frac{d y}{d x}, ) if ( x=a(theta-sin theta) ) and ( y= ) ( boldsymbol{a}(mathbf{1}-cos boldsymbol{theta}) ) |
12 |
| 1225 | ( frac{d y}{d x} ) for ( y=tan ^{-1}{sqrt{frac{1+cos x}{1-cos x}}}, ) where ( mathbf{0}<boldsymbol{x}<boldsymbol{pi}, ) is? ( A cdot frac{-1}{2} ) B. ( c cdot 1 ) D. |
12 |
| 1226 | Which one of the following is correct in respect of the function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}^{2}}{|boldsymbol{x}|} ) for ( boldsymbol{x} neq mathbf{0} ) and ( boldsymbol{f}(mathbf{0})=mathbf{0} ? ) A. ( f(x) ) is discontinuous everywhere B. ( f(x) ) is continuous everywhere c. ( f(x) ) is continues at ( x=0 ) only D. ( f(x) ) is discontinuous at ( x=0 ) only |
12 |
| 1227 | Find ( frac{d y}{d x} ) if ( y=frac{x^{2}+x}{2} ) | 12 |
| 1228 | Using LMV Theorem, find a point on the curve ( y=(x-3)^{2} ), where the tangent is parallel to the chord joining (3,0) and ( (mathbf{5}, mathbf{4}) ) |
12 |
| 1229 | Answer the following question in one word or one sentence or as per exact requirement of the question. Write the value of ( lim _{x rightarrow a} frac{x f(a)-a f(x)}{x-a} ) |
12 |
| 1230 | If ( f(x)=frac{x-e^{x}+cos 2 x}{x^{2}}, x neq 0 ) is continuous at ( x=0, ) then A ( quad f(0)=frac{-5}{2} ) B . ( [f(0)]=-2 ) c. ( f(0)=-0.5 ) D. ( [f(0)] . f(0)=-1.5 ) |
12 |
| 1231 | Find the continuity of ( f(x) ), If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ccc}boldsymbol{x}+mathbf{1} & boldsymbol{i f} & boldsymbol{x} geq mathbf{1} \ boldsymbol{x}^{2}+mathbf{1} & boldsymbol{i f} & boldsymbol{x}<mathbf{1}end{array}right} ) | 12 |
| 1232 | ff ( y=4 x-5 ) is a tangent to the curve ( boldsymbol{y}^{2}=boldsymbol{p} boldsymbol{x}^{3}+boldsymbol{q} ) at ( (boldsymbol{2}, boldsymbol{3}), ) then ( (boldsymbol{p}+boldsymbol{q}) ) is equal to A . -5 B. 5 ( c .-9 ) D. E . |
12 |
| 1233 | If ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ccc}sqrt{1+k x}-sqrt{1-k x} & text { for } & -1 leq x<0 \ x & text { for } 0 leq x<1end{array}right. ) is continuous at ( boldsymbol{x}=mathbf{0} ) then ( boldsymbol{k}= ) |
12 |
| 1234 | If ( y=cos left(m cos ^{-1} xright), ) show that ( left(1-x^{2}right) frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}+m^{2} y=0 ) |
12 |
| 1235 | If ( f(1)=3 ) and ( f^{prime}(1)=-frac{1}{3} ) then the derivative of ( left(x^{11}+f(x)right)^{-2} ) at ( x=1 ) is A. ( -frac{1}{2} ) B. – – c. 1 D. ( f^{prime}(1) ) |
12 |
| 1236 | If ( x e^{x y}+y e^{-x y}=sin ^{2} x, ) then ( frac{d y}{d x} ) at ( boldsymbol{x}=mathbf{0} ) is A ( cdot 2 y^{2}-1 ) в. ( 2 y ) c. ( y^{2}-y ) D. ( y^{2}+1 ) E ( cdot y^{2}-1 ) |
12 |
| 1237 | NI-86 9. Let g(x) = log f(x) where f(x) is twice differ function on (0, 0) such that sex N=1,2,3, te() is twice differentible positive hat (x+1)=xf(x). Then, for (2008) 1 431+ 1 1 (h) + 9 +….. + 25′ (2N 4 1 + -+- +….. + 0 25 |
12 |
| 1238 | The value of the derivative of ( |boldsymbol{x}-mathbf{1}|+ ) ( |x-3| ) at ( x=2 ) is: ( A cdot 2 ) B. ( c .0 ) D. – 2 |
12 |
| 1239 | Verify Lagrange’s Mean Value Theorem for the function ( f(x)=x^{2}+x-1 ) in the interval ( [mathbf{0}, mathbf{4}] ) |
12 |
| 1240 | The velocity ( v ) of a particle is given by the equation ( v=6 t^{2}-6 t^{3}, ) where ( v ) is in the ( m s^{-1}, t ) is the instant of time in seconds while 6 and 6 are suitable dimensional constants. At what values of ( t ) will the velocity be maximum and minimum? Determine these maximum and minimum values of the velocity. |
12 |
| 1241 | The value of ( f(0) ) so that the function ( f(x)=frac{sqrt{1+x}-(1+x)^{1 / 3}}{x} ) becomes continuous is equal to ( A cdot frac{1}{6} ) B. c. 2 D. |
12 |
| 1242 | If ( boldsymbol{y}=cot ^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right) ) then ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} ) is equal to A . B. 1/2 c. ( frac{1}{1+sin x} ) D. ( frac{1}{sqrt{1+sin x}}+frac{1}{sqrt{1-sin x}} ) |
12 |
| 1243 | ( f(x)=[x] ) is a greatest integer function,then it is continuous at? ( A cdot R ) B. z ( c . ) D. R-z |
12 |
| 1244 | ff ( y=sec ^{-1}left(frac{sqrt{x}+1}{sqrt{x}-1}right)+sin ^{-1}left(frac{sqrt{x}-1}{sqrt{x}+1}right) ) then find ( frac{d y}{d x} ) |
12 |
| 1245 | Differentiate: ( boldsymbol{y}=8 sin boldsymbol{x} cos boldsymbol{x} ) w.r.t ( boldsymbol{x} ) | 12 |
| 1246 | Differentiate the following function with respect to ( x ) ( boldsymbol{x}^{-4}left(boldsymbol{3}-boldsymbol{4} boldsymbol{x}^{-boldsymbol{5}}right) ) |
12 |
| 1247 | If ( f(x)=sqrt{2} x+frac{4}{sqrt{2 x^{prime}}} ) then ( f^{prime}(2) ) is equal to ( mathbf{A} cdot mathbf{0} ) B. – c. 1 D. 2 |
12 |
| 1248 | If ( y=tan ^{-1}left(frac{a cos x-b sin x}{b cos x+a sin x}right) ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? ) ( A cdot frac{a}{b} ) B. ( frac{-b}{a} ) c. D. – |
12 |
| 1249 | The derivative of ( sin ^{-1} x ) with respect to ( cos ^{-1} sqrt{1-x^{2}} ) is? A ( cdot frac{1}{sqrt{1-x^{2}}} ) B. ( cos ^{-1} x ) c. 1 D. 0 |
12 |
| 1250 | What is derivative of ( left[frac{1}{x}right]^{x} ) | 12 |
| 1251 | If ( boldsymbol{y}=|cos boldsymbol{x}|+|sin boldsymbol{x}|, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}= ) ( frac{2 pi}{3} ) is A ( cdot frac{1-sqrt{3}}{2} ) B. c. ( frac{sqrt{3}-1}{2} ) D. None of these |
12 |
| 1252 | Solve for ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{r}^{2} ) | 12 |
| 1253 | The value of ( boldsymbol{f}^{prime}(boldsymbol{3}) ) is ( A cdot 8 ) B. 10 c. 12 D. 18 |
12 |
| 1254 | The displacement of ( S ) of a particle at time ( t(O<t<pi) ) is given by ( S= ) ( sin 2 t-6 cos t . ) Then the acceleration for the value of ( t ) for which its velocity is zero is A .0 unit/ ( sec ^{2} ) B. 3 unit/ sec ( ^{2} ) c. 2 unit ( mid sec ^{2} ) D. 4 unit/ ( sec ^{2} ) |
12 |
| 1255 | Differentiate the following function with respect to ( x ) ( boldsymbol{x}^{-3}(mathbf{5}+mathbf{3} boldsymbol{x}) ) |
12 |
| 1256 | Find the value of ( f(0) ) so that the function ( boldsymbol{f}(boldsymbol{x})= ) ( 8left(frac{1-cos ^{2} x+sin ^{2} x}{x^{2}}right), x neq 0 ) is continuous |
12 |
| 1257 | 10. If x= 1-2 1+72 and y=- 1+ 2, the |
12 |
| 1258 | ( operatorname{Let} mathbf{f}(mathbf{x})=frac{sin mathbf{4} boldsymbol{pi}[mathbf{x}]}{mathbf{1}+[mathbf{x}]^{2}}, mathbf{w h e r e}[x] ) is the greatest integer less than or equal to ( x ) then A ( cdot f(x) ) is not differentiable at some points B. ( f(x) ) exists but is different from zero ( c cdot f(x)=0 ) for all ( x ) D. ( f^{prime}(x)=0 ) but ( f ) is not a constant function |
12 |
| 1259 | f ( boldsymbol{y}=mathbf{1}+boldsymbol{x} cdot boldsymbol{e}^{boldsymbol{y}}, ) show that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{e}^{boldsymbol{y}}}{boldsymbol{2}-boldsymbol{y}} ) | 12 |
| 1260 | Find ( frac{d y}{d x} ) for ( y=log _{e}(x+sqrt{x^{2}-a^{2}}) ) | 12 |
| 1261 | Let ( f ) be a continuous function on ( mathrm{R} ) satisfying ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) ) for all ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) and ( boldsymbol{f}(1)=boldsymbol{4} ) then ( boldsymbol{f}(boldsymbol{3}) ) is equal to |
12 |
| 1262 | If ( boldsymbol{y}=(mathbf{1}+boldsymbol{x})left(mathbf{1}+boldsymbol{x}^{2}right)left(mathbf{1}+boldsymbol{x}^{4}right) dots . .(mathbf{1}+ ) ( left.x^{2^{n}}right), ) find ( frac{d y}{d x} ) at ( x=0 ) ( mathbf{A} cdot 2^{n} ) B. ( c .1 ) D. ( 2 n ) |
12 |
| 1263 | The set of points where ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{mathbf{1}+|boldsymbol{x}|} ) is differentiable, is ( mathbf{A} cdot(-infty, 0) cup(0, infty) ) B ( cdot(-infty,-1) cup(-1, infty) ) ( c cdot(-infty, infty) ) D・ ( (0, infty) ) |
12 |
| 1264 | Let ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}^{2} & text { if } boldsymbol{x} leq boldsymbol{x}_{0} \ boldsymbol{a} boldsymbol{x}+boldsymbol{b} & text { if } boldsymbol{x}>boldsymbol{x}_{0}end{array}right. ) The values of the coefficients a and b for which the function is continuous and has a derivative at ( x_{0} ). are A ( cdot a=x_{0}, b=-x_{0} ) B . ( a=2 x_{0}, b=-x_{0}^{2} ) C ( . a=x_{0}^{2}, b=-x_{0} ) D. ( a=x_{0}, b=-x_{0}^{2} ) |
12 |
| 1265 | If ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{sqrt{boldsymbol{x}+1}-sqrt{boldsymbol{x}}} ) be a real values function then ( A cdot f(x) ) is continuous, but ( f^{prime}(0) ) does not exist B. ( f(x) ) is differentiable at ( x=0 ) ( mathrm{c} cdot mathrm{f}(mathrm{x}) ) is not continuous at ( x=0 ) D. ( f(x) ) is not differentiable at ( x=0 ) |
12 |
| 1266 | If ( boldsymbol{y}=tan (boldsymbol{2} boldsymbol{x}+boldsymbol{3}) cdot ) Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 1267 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{cos boldsymbol{x}^{0}right}=? ) | 12 |
| 1268 | If Rolles theorem holds for the function ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{3}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{c} boldsymbol{x}, boldsymbol{x} in[-1,1] ) at the point ( x=frac{1}{2} ) then ( 2 b+c ) equals ( A ) B. ( c cdot 2 ) ( D cdot-3 ) |
12 |
| 1269 | n figure a square ( O A B C ) is inscribed in a quadrant ( O P B Q ) of a circle. If ( O A=21 c m . ) find the area of the shaded region |
12 |
| 1270 | Let ( boldsymbol{f} cdot boldsymbol{R} rightarrow boldsymbol{R} ) be defined as ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cl}0, & x text { is rational } \ sin |x|, & x text { is rational }end{array}right. ) Then which of the following is true? A. ( f ) is discontinuous for all ( x ) B. ( f ) is continuous for all ( x ) c. ( f ) is discontinuous at ( x=k pi ), where ( k ) is an integer D. ( f ) is continuous at ( x=k pi, ) where ( k ) is an integer |
12 |
| 1271 | Illustration 2.34 If x = ał”, y = bt”, then find ay dx This is nolladimnlinit differentina: |
12 |
| 1272 | If ( sin (x y)+frac{y}{x}=x^{2}-y^{2}, ) find ( frac{d y}{d x} ) | 12 |
| 1273 | If ( f(x)=b e^{a x}+a e^{b x}, ) then ( f^{prime prime}(0)= ) ( mathbf{A} cdot mathbf{0} ) в. ( 2 a ) ( mathbf{c} cdot a b(a+b) ) ( mathbf{D} cdot a b ) |
12 |
| 1274 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( log {x+2+sqrt{x^{2}+4 x+1}} ) |
12 |
| 1275 | Find the derivative of ( csc ^{2} x, ) by using first principle of derivatives. |
12 |
| 1276 | Differentiate ( frac{x^{2} sin x}{1-x} ) w.r.t ( x ) | 12 |
| 1277 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{c} boldsymbol{x}+boldsymbol{d} ) and ( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x}-boldsymbol{2} ) If ( lim _{x rightarrow 1} frac{f(x)}{g(x)}=1 ) and ( lim _{x rightarrow 2} frac{f(x)}{g(x)}=4, ) then find the value of ( frac{c^{2}+d^{2}}{a^{2}+b^{2}} ) |
12 |
| 1278 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( tan ^{-1}left{frac{5 x}{1-6 x^{2}}right},-frac{1}{sqrt{6}}<x<frac{1}{sqrt{6}} ) |
12 |
| 1279 | Say true or false. If ( y=2 sec x, ) then ( frac{d y}{d x} ) is ( 2 sec x tan x ) A. True B. False |
12 |
| 1280 | ( sum_{n=0}^{infty}(-1)^{n} frac{x^{2 n+1}}{2 n+1} ) is equal to ( (-1<x<1) ) A ( cdot tan ^{-1} x-x+c ) B ( cdot log (1+x) ) c. ( frac{1}{1-x}+frac{1}{1+x} ) ( mathbf{D} cdot sin ^{-1} x ) |
12 |
| 1281 | Solution of ( boldsymbol{y}^{2} boldsymbol{x}+boldsymbol{y}-boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=mathbf{0} ) is A ( cdot frac{y^{2}}{5}+frac{y^{2}}{4 x^{2}}=c ) в. ( frac{x^{2}}{2}+frac{x}{y}=C^{prime} ) c. ( frac{y^{2}}{4}+frac{y^{5}}{5 x^{4}}=c ) D. ( frac{x^{2}}{5}+frac{x^{4}}{4 y^{4}}=c ) |
12 |
| 1282 | Assertion(A): ( f(x)= ) ( left{begin{array}{ll}x^{2} sin left(frac{1}{x}right), & x neq 0 \ 0, & x=0end{array} ) is continuous at right. ( boldsymbol{x}=mathbf{0} ) Reason(R): Both ( h(x)=x^{2}, g(x)= ) ( left{begin{array}{ll}sin left(frac{1}{x}right), & x neq 0 \ 0, & x=0end{array} ) are continuous at right. ( boldsymbol{x}=mathbf{0} ) A. Both A and R are true and R is the correct explanation of B. Both A and R are true and R is not the correct explanation of c. ( A ) is true but ( R ) is false D. R is true but A is false |
12 |
| 1283 | Let ( f(x) ) be differentiable on the interval ( (0, infty) ) such that ( f(1)=1, ) and ( lim _{t rightarrow x} frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1 ) for ( operatorname{each} x>0 ) Then ( f(x) ) is A ( frac{1}{3 x}+frac{2 x^{2}}{3} ) в. ( -frac{1}{3 x}+frac{4 x^{2}}{3} ) c. ( -frac{1}{x}+frac{2}{x^{2}} ) D. |
12 |
| 1284 | Differentiate the following w.r.t. ( x ) ( left(3 x^{2}+2right)left(4 x-3 x^{3}right) ) ( mathbf{A} cdot 45 x^{4}+18 x^{2}+8 ) B. ( -45 x^{4}+18 x^{2}+8 ) c. ( -45 x^{4}+15 x^{2}+8 ) D. ( -45 x^{4}+18 x^{2}+18 ) |
12 |
| 1285 | 16-1) sinifx+1 25. Let (8) – )(x-1)sinif x+1 [2008) x-1 0 if x=1 Then which one of the following is true? (a) fis neither differentiable at x=0 nor at x=1 (b) fis differentiable at x=0 and at x=1 (C) fis differentiable at x =0 but not atx=1 (d) fis differentiable at x = 1 but not at x=0 i function with |
12 |
| 1286 | Discuss the continuity and differentiability of ( boldsymbol{f}(boldsymbol{x})=|log | boldsymbol{x} | ) | 12 |
| 1287 | If ( y=sin (sin x), ) then prove that ( frac{d^{2} y}{d x^{2}}+ ) ( tan x frac{d y}{d x}+y cos ^{2} x=0 ) |
12 |
| 1288 | 41. For a twice differentiable function f (x), g(x) is defined as g(x) = (f ‘(x)2 + f'(x)) f(x) on [a, e]. If for a<b<c<d<e, f(a) = 0, f (b) = 2, f (C) = -1, f (d) = 2, f(e)=0 then find the minimum number of zeros of g(x). (2006 – 6M) |
12 |
| 1289 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( cos ^{-1}left{frac{cos x+sin x}{sqrt{2}}right},-frac{pi}{4}<x<frac{pi}{4} ) |
12 |
| 1290 | The function ( mathbf{f}(mathbf{x})=frac{cos mathbf{3} boldsymbol{x}-cos mathbf{4} boldsymbol{x}}{boldsymbol{x} sin mathbf{2} boldsymbol{x}} ) for ( neq mathbf{0}, mathbf{f}(mathbf{0})=frac{mathbf{7}}{mathbf{4}} mathbf{a t} boldsymbol{x}=mathbf{0}, ) is This question has multiple correct options A. Continuous B. discontinuous c. left continuous D. right continuous |
12 |
| 1291 | If ( y=tan ^{-1}left(frac{2^{x}}{1+2^{2 x+1}}right), ) then ( frac{d y}{d x} ) at ( boldsymbol{x}=mathbf{0} ) is? A ( cdot frac{1}{10} log 2 ) B. ( frac{1}{5} log 2 ) c. ( -frac{1}{10} log 2 ) D. ( log 2 ) |
12 |
| 1292 | ( operatorname{Let} f(x)=left{begin{array}{c}sin x, quad text { for } x geq 0 \ 1-cos x, & text { for } x<0end{array} ) and right. ( g(x)=e^{x} ). Then ( (g o f)^{prime}(0) ) is |
12 |
| 1293 | ( f(x)=left{begin{array}{cc}1-cos x & x neq 0 \ x & \ k & x=0end{array}right} ) continuous at ( x=0, ) then the value of ( k ) is: A . 0 ( B cdot frac{1}{2} ) ( c cdot frac{1}{4} ) ( D cdot-frac{1}{2} ) |
12 |
| 1294 | Verify Lagrange’s Mean Value Theorem for the following function: ( boldsymbol{f}(boldsymbol{x})=2 sin boldsymbol{x}+sin 2 boldsymbol{x} ) on ( [mathbf{0}, boldsymbol{pi}] ) |
12 |
| 1295 | Differentiate the following function with respect to ( x ) ( frac{4 x+5 sin x}{3 x+7 cos x} ) |
12 |
| 1296 | If ( boldsymbol{y}=cot ^{-1}left(frac{1-x}{1+x}right) ) then ( frac{d y}{d x}=? ) A ( cdot frac{-1}{left(1+x^{2}right)} ) B. ( frac{1}{left(1+x^{2}right)} ) c. ( frac{1}{left(1+x^{2}right)^{3 / 2}} ) D. none of these |
12 |
| 1297 | ( f(x) ) is defined as under: ( f(x)= ) ( left{begin{array}{cc}a x(x-1)+b, & x3end{array}right. ) ( f^{prime}(x) ) is discontinuous at ( x=3 . ) Then ( boldsymbol{a} neq boldsymbol{k}, boldsymbol{b}=boldsymbol{m}, boldsymbol{c}=frac{mathbf{1}}{boldsymbol{h}}, boldsymbol{d}=-boldsymbol{p} . ) Find ( boldsymbol{k}+ ) ( boldsymbol{m}+boldsymbol{h}+boldsymbol{p} ? ) |
12 |
| 1298 | ( y=sin left(2 sin ^{-1} xright), frac{d y}{d x}= ) A ( cdot sqrt{left(frac{1-y^{2}}{1-x^{2}}right)} ) в. ( sqrt[2]{left(frac{1+y^{2}}{1-x^{2}}right)} ) ( ^{mathrm{c}} cdot sqrt{left(frac{1-y^{2}}{1+x^{2}}right)} ) D. ( sqrt{left(frac{1+y^{2}}{1+x^{2}}right)} ) |
12 |
| 1299 | Identify a possible graph for function ( boldsymbol{f} ) given by ( boldsymbol{f}(boldsymbol{x})=-sqrt{(boldsymbol{x}-mathbf{1})}-mathbf{1} ) A. graph a B. graph b c. graph c D. graph d |
12 |
| 1300 | Differentiate ( -frac{4 x+5 sin x}{3 x+7 cos x} ) w.r.t ( x ) | 12 |
| 1301 | Differentiate: ( frac{d}{d x}left(tan ^{-1} xright) ) |
12 |
| 1302 | The value of ( c ) in Lagrange’s theorem for the function in the interval [-1,1] is [ f(x)=left{begin{array}{cl} x cos left(frac{1}{x}right), & x neq 0 \ 0, & x=0 end{array}right. ] ( mathbf{A} cdot mathbf{0} ) B. ( frac{1}{2} ) ( c ) [ -frac{1}{2} ] D. Non existent in the interva |
12 |
| 1303 | If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{3}}, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 1304 | If for a continuous function ( boldsymbol{f}, boldsymbol{f}(mathbf{0})= ) ( boldsymbol{f}(1)=mathbf{0}, boldsymbol{f}^{prime}(1)=mathbf{2} ) and ( boldsymbol{g}(boldsymbol{x})= ) ( fleft(e^{x}right) e^{f(x)}, ) then ( g^{prime}(0) ) is equal to A . 1 B. 2 ( c cdot 0 ) D. None of these |
12 |
| 1305 | Given, ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cl}tan 4 x times cos 3 x & x neq 0 \ x & x=0end{array} . text { If } f ) is right. continuous at ( boldsymbol{x}=mathbf{0}, ) then ( boldsymbol{k}= ) ( A cdot 0 ) B. 4 c. ( frac{4}{3} ) D. |
12 |
| 1306 | Differentiate sec ( x ) by first principle. | 12 |
| 1307 | TOPIL-WIJL DU 12. Letf:(-1, 1) Rbe a differentiable function with f0=-1 and f” (0)=1. Let g(x)=[(2f(x) + 2)]2. Then g’O)= 120101 (a) 4 (a) (6) o (c) 2 (d) 4 4 |
12 |
| 1308 | Solve the differential equation ( cos left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)=boldsymbol{a},(boldsymbol{a} in boldsymbol{R}) ) | 12 |
| 1309 | If ( |y|=5 x-2 y, ) then which of the following is incorrect? A. ( y(x) ) is discontinuous at ( x=0 ) B. ( y(x) ) is continuous ( forall x in R ) c. ( y(x) ) is strictly increasing ( forall x epsilon R ) D. domain of ( y(x) ) is set of all real values |
12 |
| 1310 | If ( sqrt{boldsymbol{y}+boldsymbol{x}}+sqrt{boldsymbol{y}-boldsymbol{x}}=boldsymbol{c} text { (where } boldsymbol{c} neq mathbf{0}) ) then ( frac{d y}{d x} ) has the value equal to This question has multiple correct options A ( cdot frac{2 x}{c^{2}} ) в. ( frac{x}{y+sqrt{y^{2}-x^{2}}} ) c. ( frac{y-sqrt{y^{2}-x^{2}}}{x} ) D. ( frac{c^{2}}{2 y} ) |
12 |
| 1311 | 36. Forx I R, f(x) = log2 – sinx and g(x)=f(f(x)), then: JEEM 2016 (a) g'(0)=-cos(log2) (b) gis differentiable at x=0 and g’O)=-sin(log2) (c) g is not differentiable at x=0 (d) g’o= cos(log2) |
12 |
| 1312 | Differentiate with respect to ( times frac{(1+x)}{e^{x}} ) A ( .-x e^{-x} ) B . ( x e^{-x} ) ( mathrm{c} cdot-x e^{-2 x} ) D. ( x^{2} e^{-x} ) |
12 |
| 1313 | Assertion ( operatorname{Let} f(x)=left{begin{array}{ll}1 & text { if } 2 leq x leq 3 \ 3 & text { if } 3 leq x leq 5end{array}right. ) The mean value of ( boldsymbol{f} ) is attained Reason ( f ) is a bounded function but not continuous on ( [mathbf{2}, mathbf{5}] ) A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion C. Assertion is true but Reason is false D. Assertion is false but Reason is true |
12 |
| 1314 | f ( e^{y}(x+1)=1, ) show that ( y_{2}=y_{1}^{2} ) | 12 |
| 1315 | Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be given by ( boldsymbol{f}(boldsymbol{x})=mathbf{5} boldsymbol{x}, ) if ( boldsymbol{x} in boldsymbol{Q} ) and ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{6} ) if ( boldsymbol{x} in boldsymbol{R}-boldsymbol{Q} ) then A. f is continuous at ( x=1 ) and ( x=2 ) B. fis not continuous at ( x=1 ) and ( x=2 ) c. ( f ) is continuous at ( x=1 ) but not at ( x=2 ) D. fis continuous at ( x=2 ) but not at ( x=1 ) |
12 |
| 1316 | ( f(x)=frac{1-cos (1-cos x)}{x^{4}} ) is continuous at ( x=0, ) then ( f(0)= ) A ( cdot frac{1}{2} ) B. ( c cdot frac{1}{6} ) D. |
12 |
| 1317 | Using the definition, show that the function. ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sin (1 / boldsymbol{x}) ) if ( boldsymbol{x} neq mathbf{0}, boldsymbol{0} ) if ( boldsymbol{x}=mathbf{0} ) is continuous at the point ( boldsymbol{x}=mathbf{0} ) |
12 |
| 1318 | Find the derivative of the following functions from first principle: ( cos left(x-frac{pi}{8}right) ) |
12 |
| 1319 | If ( boldsymbol{y}=frac{boldsymbol{x}^{2}}{mathbf{2}}+frac{mathbf{1}}{mathbf{2}} boldsymbol{x} sqrt{boldsymbol{x}^{2}+mathbf{1}}+ ) ( ln sqrt{x+sqrt{x^{2}+1}} ), then the value of ( boldsymbol{x} boldsymbol{y}^{prime}+log boldsymbol{y}^{prime} ) is ( mathbf{A} cdot underline{y} ) B. ( 2 y ) c. 0 D. ( -2 y ) |
12 |
| 1320 | Verify L.M.V for the function? ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}2+x^{3} & x leq 1 \ 3 x & x>1end{array}right} o n[-1,2] ) | 12 |
| 1321 | Find the derivative of ( sin left(2 sin ^{-1} xright) ) A ( cdot frac{2 cos left(2 sin ^{-1} xright)}{sqrt{1-x^{2}}} ) B. ( frac{cos left(2 sin ^{-1} xright)}{sqrt{1-x^{2}}} ) c. ( frac{2 cos left(2 cos ^{-1} xright)}{sqrt{1-x^{2}}} ) D. ( -frac{cos left(2 cos ^{-1} xright)}{sqrt{1-x^{2}}} ) |
12 |
| 1322 | Verify Rolle’s theorem for ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(boldsymbol{x}+ ) ( mathbf{3}) e^{-boldsymbol{x} / 2} ) in ( (-mathbf{3}, mathbf{0}) ) A. Yes Rolle’s theorem is applicable and the stationary point is ( x=-2 ) B. Yes Rolle’s theorem is applicable and the stationary point is ( x=-1 ) C. No Rolle’s theorem is not applicable in the given interval D. Both A and B |
12 |
| 1323 | Consider the function ( g(x)= ) ( left{begin{array}{ll}frac{1+a^{x}+x a^{x} ln a}{a^{x} x^{2}} & x0end{array}right. ) where ( a>0 ) Find the value of ( a & g(0) ) so that the function ( g(x) ) is continuous at ( x=0 ) ( ^{A} cdot a=frac{1}{sqrt{2}}, g(0)=frac{(ln 2)^{2}}{8} ) B. ( a=-frac{1}{sqrt{2}}, g(0)=frac{(l n 2)^{2}}{8} ) c. ( a=-frac{1}{sqrt{2}}, g(0)=frac{-}{(l n 2)^{2}} 8 ) D. ( a=frac{1}{sqrt{2}}, g(0)=-frac{(l n 2)^{2}}{8} ) |
12 |
| 1324 | Differentiate the following functions with respect to ( boldsymbol{x} ) ( tan ^{-1}left(frac{sqrt{boldsymbol{x}}+sqrt{boldsymbol{a}}}{1-sqrt{boldsymbol{x} boldsymbol{a}}}right) ) |
12 |
| 1325 | The value of ( f(0), ) so that the function ( boldsymbol{f}(boldsymbol{x})= ) ( frac{sqrt{a^{2}-a x+x^{2}}-sqrt{a^{2}+a x+x^{2}}}{sqrt{a+x}-sqrt{a-x}} ) becomes continuous for all ( x, ) is given by ( mathbf{A} cdot a^{3 / 2} ) B. ( a^{1 / 2} ) c. ( -a^{1 / 2} ) D. ( -a^{3 / 2} ) |
12 |
| 1326 | ff ( y=y(x) ) and it follows the relation ( mathbf{2} e^{x y^{2}}+boldsymbol{y} cos left(x^{2}right)=4, ) then ( left|boldsymbol{y}^{prime}(mathbf{0})right| ) is equal to |
12 |
| 1327 | Assertion If ( f(x)=0 ) has two distinct positive real roots then number of non- differentiable points of ( boldsymbol{y}=|boldsymbol{f}(-|boldsymbol{x}|)| ) is ( mathbf{1} ) Reason Graph of ( boldsymbol{y}=boldsymbol{f}(|boldsymbol{x}|) ) is symmetrical about y-axis A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct |
12 |
| 1328 | ( x cos (a+y)=cos y ) then prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{cos ^{2}(boldsymbol{a}+boldsymbol{y})}{sin _{boldsymbol{a}}} ) |
12 |
| 1329 | Differentiate from first principles. (i) ( 3 x ) |
12 |
| 1330 | PICOU 23. If y=x then is |
12 |
| 1331 | Identify the graph of the polynomial function ( boldsymbol{f} ) ( f(x)=x^{4}+x^{3}-2 x^{2} ) begin{tabular}{|l|l|l|l|l|} hline 1 & 1 & & & \ hline & & & & \ hline & & & & \ hline & & & & \ hline & & & & \ hline & & & & \ hline end{tabular} A. graph a B. graph b c. graph c D. graph d |
12 |
| 1332 | Find the derivative of the following functions from the first principals w.r.t to ( boldsymbol{x} ) ( tan 2 x ) |
12 |
| 1333 | If ( y=x^{-frac{1}{2}}+log _{5} x+frac{sin x}{cos x}+2^{x} ), then find ( frac{d y}{d x} ) A. ( -frac{1}{2} x^{-3 / 2}+frac{1}{x log _{e} 5}+sec ^{2} x+2^{x} log 2 ) B. ( frac{1}{2} x^{-3 / 2}+frac{1}{x log _{e} 5}+sec ^{2} x+2^{x} log 2 ) c. ( -frac{3}{2} x^{-3 / 2}+frac{1}{x log _{e} 5}+sec ^{2} x+2^{x} log 2 ) D. ( -frac{1}{2} x^{-3 / 2}+frac{1}{x log _{e} 5}+cos ^{2} x+2^{x} log 2 ) |
12 |
| 1334 | Is the function ( f ) defined by ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}boldsymbol{x}, boldsymbol{i} boldsymbol{f} boldsymbol{x} leq 1 \ boldsymbol{5}, boldsymbol{i f} boldsymbol{x}>1end{array}right. ) continuous at ( boldsymbol{x}=mathbf{0} ? ) At ( boldsymbol{x}=mathbf{1} ) ? At ( boldsymbol{x}= ) ( mathbf{2} ? ) |
12 |
| 1335 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{g}^{prime}(1)+boldsymbol{g}^{prime prime}(2) ) and ( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{f}^{prime}(boldsymbol{2})+boldsymbol{f}^{prime prime}(boldsymbol{3}) . ) Then A ( cdot f^{prime}(1)=4+f^{prime}(2) ) B ( cdot g^{prime}(2)=8+g^{prime}(1) ) C ( cdot g^{prime prime}(2)+f^{prime prime}(3)=4 ) D. all of these |
12 |
| 1336 | Derivative of ( 2 tan x-7 sec x ) with respect to ( x ) is: ( mathbf{A} cdot 2 sec x+7 tan x ) B. ( sec x(2 sec x+tan x) ) c. ( 2 sec ^{2} x+sec x ). tan ( x ) D. ( sec x(2 sec x-7 tan x) ) |
12 |
| 1337 | ( cos ^{-1} x=log (y)^{1 / m} ) Evaluate ( frac{d y}{d x} ) |
12 |
| 1338 | Let ( f(x) ) be a real value function not identically zero satisifes the equation, ( fleft(x+y^{n}right)=f(x)+f(y)^{n} ) for all real ( x, y ) and ( f^{prime}(0) geq 0 ) where ( n(>1) ) is an odd natural number. ( boldsymbol{f}(mathbf{1 0})=mathbf{1 0 k . F i n d} ) ( boldsymbol{k} ) value |
12 |
| 1339 | 17. Let (9= 17. Let f(x) = {1+ sin x |3a|sin xl <x<0 b ; x = 0 etan 2x/tan 3x ; 0<x<* (1994 – 4 Marks) Determine a and b such that f(x) is continuous at x=0 |
12 |
| 1340 | x20, then show that f(x)=0 for all x 20. (2001 – 5 Marks) 21. Let a e R. Prove that a function f: R R is differentiable at a if and only if there is a function g: R R which is continuous at a and satisfies f(x)-f(a)=g(x) (x – a) for all XER. (2001 – 5 Marks) |
12 |
| 1341 | Differentiate the following functions with respect to ( x ) : ( tan ^{-1}left{frac{x^{1 / 3}+a^{1 / 3}}{1-(a x)^{1 / 3}}right} ) |
12 |
| 1342 | Assertion If both functions ( f(t) ) and ( g(t) ) are continuous on the closed interval ( [boldsymbol{a}, boldsymbol{b}] ) differentiable on the open interval ( (a, b) ) and ( g^{prime}(t) ) is not zero on that open interval, then there exists some ( c ) in ( (a, b) ) such that ( frac{f^{prime}(c)}{g^{prime}(c)}=frac{f(b)-f(a)}{g(b)-g(a)} ) Reason If ( f(t) ) and ( g(t) ) are continuous and differntiable in ( [a, b], ) then there exists some ( c ) in ( (a, b) ) such that ( f^{prime}(c)= ) ( frac{boldsymbol{f}(boldsymbol{b})-boldsymbol{f}(boldsymbol{a})}{boldsymbol{b}-boldsymbol{a}} ) and ( boldsymbol{g}^{prime}(boldsymbol{c})=frac{boldsymbol{g}(boldsymbol{b})-boldsymbol{g}(boldsymbol{a})}{boldsymbol{b}-boldsymbol{a}} ) from Lagrange’s mean value theorm. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
12 |
| 1343 | Check whether Lagrange’s mean value theorem is applicable on ( f(x)=sin x+ ) ( cos x ) interval ( left[0, frac{pi}{2}right] ) |
12 |
| 1344 | The function ( f(x)=sin ^{-1}(cos x) ) is ( :- ) A. discontinuous at ( x=0 ) B. continuous at ( x=0 ) C. differentiable at ( x=0 ) D. none of these |
12 |
| 1345 | Find the derivative of the following (it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m text { and } n text { are integers }): frac{sin x+cos x}{sin x-cos x} ) |
12 |
| 1346 | The point(s) on the curve ( y^{3}+3 x^{2}= ) ( 12 y ) where the tangent is vertical, is ( (operatorname{are}) ) A ( cdotleft(pm frac{4}{sqrt{3}},-2right) ) B ( cdotleft(pm frac{sqrt{11}}{3}, 1right) ) ( mathbf{c} cdot(0,0) ) D. ( left(pm frac{4}{sqrt{3}}, 2right) ) |
12 |
| 1347 | Find ( frac{d y}{d x} ) for ( 2 x^{2}+5 x y+3 y^{2}=1 ) | 12 |
| 1348 | Differentiate ( boldsymbol{y}=mathbf{1 0}^{boldsymbol{x}}+mathbf{1 0}^{tan boldsymbol{x}} ) | 12 |
| 1349 | Find from first principles the differential coefficient of ( 2 x^{2}+3 x ) |
12 |
| 1350 | The function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}x^{2} / a, & 0 leq x<1 \ a, & 1 leq x<sqrt{2} \ frac{2 b^{2}-4 b}{x^{2}}, & sqrt{2} leq x<inftyend{array}right. ) continuous for ( 0 leq x<infty, ) then the most suitable values of ( a ) and ( b ) are B. ( a=-1, b=1+sqrt{2} ) c. ( a=-1, b=1 ) D. None of these |
12 |
| 1351 | If ( f ) and ( g ) are differentiable functions in ( [mathbf{0}, mathbf{1}] ) satisfying ( boldsymbol{f}(mathbf{0})=mathbf{2}= ) ( g(1), g(0)=0 ) and ( f(1)=6, ) then for some ( boldsymbol{c} in[mathbf{0}, mathbf{1}] ) A ( cdot 2 f^{prime}(c)=g^{prime}(c) ) B ( cdot 2 f^{prime}(c)=3 g^{prime}(c) ) c. ( f^{prime}(c)=g^{prime}(c) ) D. ( f^{prime}(c)=2 g^{prime}(c) ) |
12 |
| 1352 | The range of the function ( Delta=f(|x|) ) is- A ( cdot[0,1] ) в. [0,1) c. (0,1] D. None of these |
12 |
| 1353 | If ( y=e^{a sin ^{-1} x} ) then prove that ( left(1-x^{2}right) y_{2}-x y_{1}-a^{2} y=0, ) where ( y_{1} ) and ( y_{2} ) are first and second order derivatives of ( y ) respectively. |
12 |
| 1354 | Sketch the graph ( y=|x-5| . ) Evaluate ( int_{0}^{1}|x-5| d x . ) What does this value of the integral represent on the graph? | 12 |
| 1355 | Differentiate ( log sqrt{frac{1+cos x}{1-cos x}} w . r . t . x ) | 12 |
| 1356 | ( boldsymbol{y}=sin left(boldsymbol{pi} / boldsymbol{6} e^{x y}right) ) putting ( boldsymbol{x}=mathbf{0} operatorname{than} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 1357 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(sec ^{2} boldsymbol{x}+operatorname{cosec}^{2} boldsymbol{x}right)= ) ( mathbf{A} cdot-4 sec x cdot tan x cdot cos e c x cdot cot x ) B. ( 4 sec x cdot cos ) ec ( x ) c. ( 2 sec x cdot tan x-2 cos e c x cdot cot x ) D. ( 2 sec ^{2} cdot tan x-2 operatorname{cosec}^{2} x cdot cot x ) |
12 |
| 1358 | The function ( f(x)=sin ^{-1}(cos x) ) is : A. discontinuous at ( =0 ) B. continuous at ( =0 ) c. differentiable ( =0 ) D. none of these |
12 |
| 1359 | If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{a} boldsymbol{x} ) satisfies the conditions of Rolles theorem on [1,3] with ( c=2+frac{1}{sqrt{3}} ) then ( (a, b)= ) ( mathbf{A} cdot(11,6) ) В ( cdot(11,-6) ) c. (-6,11) D. (6,11) |
12 |
| 1360 | ( operatorname{Let} f(x)=cos 2 x cdot cot left(frac{pi}{4}-xright) ) If ( f ) is continuous at ( x=frac{pi}{4} ) then the value of ( fleft(frac{pi}{4}right) ) is equal to A . 2 B. – c. ( frac{-1}{2} ) D. |
12 |
| 1361 | If ( boldsymbol{f}(boldsymbol{x})= ) ( left{x sin frac{1}{x}, x neq 0 quad k quad, x=0right} ) is continuous at ( x=0, ) then the value of ( k ) is ( A cdot 1 ) B. – 1 c. D. 2 |
12 |
| 1362 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) ( boldsymbol{y}=frac{1}{boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x}+boldsymbol{3}} ) |
12 |
| 1363 | If ( x=phi(t), y=psi(t), ) then ( frac{d^{2} y}{d x^{2}} ) is A ( cdot frac{phi^{prime} psi^{prime prime}-psi^{prime} phi^{prime prime}}{left(phi^{prime}right)^{2}} ) B. ( frac{phi^{prime} psi^{prime prime}-psi^{prime} phi^{prime prime}}{left(phi^{prime}right)^{3}} ) c. ( frac{phi^{prime prime}}{psi^{prime prime}} ) D. ( frac{psi^{prime prime}}{phi^{prime prime}} ) |
12 |
| 1364 | Differentiate with respect to ( x ) : ( e^{sqrt{cot x}} ) |
12 |
| 1365 | Find the minimum and maximum values of the function ( y=x^{3}-3 x^{2}+6 ) Also find the values of ( x ) at which these occur. |
12 |
| 1366 | If the function ( f(x)=x^{3}-6 x^{2}+a x+ ) ( b ) satisfies Rolle’s theorem in the interval [1,3] and ( f^{prime}left(frac{2 sqrt{3}+1}{sqrt{3}}right)=0, ) then A ( . a=-11 ) B. ( b=-6 ) ( mathbf{c} cdot a=6 ) D. ( a=11 ) |
12 |
| 1367 | For a curve at which the tangent lines at two distinct points coincide, then the curve cannot be A. a cubic curve B. a quadratic curve c. a curve of 4th power D. none of these |
12 |
| 1368 | 11. The function f(x)=[x]cos 2 The function f(x)=[x]cos 2x-1 , [.] denotes the greatest integer function, is discontinuous at (1995) (a) Allx (b) All integer points (C) Nox (d) x which is not an integer |
12 |
| 1369 | Sketch the graph ( y=|x-5| . ) Evaluate ( int_{0}^{1}|x-5| d x . ) What does this value of the integral represent on the graph? | 12 |
| 1370 | Show that ( left(frac{1}{a+} frac{1}{b+} frac{1}{c+} dotsright)left(c+frac{1}{b+} frac{1}{a+} frac{1}{c_{1}} dots .right)= ) ( frac{1+b c}{1+a b} ) |
12 |
| 1371 | The value of ( f(0) ) so that the function ( f(x) frac{log left(1+frac{x}{a}right)-log left(1-frac{x}{b}right)}{x},(x neq 0) ) is continuous at ( x=0 ) is A ( cdot frac{a+b}{a b b b b b} ) B . ( frac{a-b}{a b b b b} ) c. ( frac{a b}{a+b b} ) D. ( frac{a b}{a-b} ) |
12 |
| 1372 | If ( y=frac{1}{sqrt{a^{2}-x^{2}}}, ) find ( frac{d y}{d x} ) | 12 |
| 1373 | 2. If y = 2 sin? 0 + tan 0 then dy will be do (a) 4 sin cos 0 + sec 0 tan O (b) 2 sin 20+ seca e (c) 4 sin + seca e (d) 2 cos2 0 + sec2 e dy |
12 |
| 1374 | Which of the following limits vanishes? A ( cdot lim _{x rightarrow infty} x^{frac{1}{4}} sin frac{1}{sqrt{x}} ) B. ( lim _{x rightarrow pi^{2}}(1-sin x) cdot tan x ) C ( lim _{x rightarrow infty} frac{2 x^{2}+3}{x^{2}+x-5} cdot ) sgn D. ( lim _{x rightarrow 3^{circ}} frac{[x]^{2}-9}{x^{2}-9} ) where [] denotes greatest integer function |
12 |
| 1375 | 45. Let f(x) = 15-x-10 (a) © {5, 10, 153 {5, 10, 15,20} “(X)=15-x-101: X R. Then the set of all values of x, at the function, g(x)=f(f(x)) is not differentiable, is: JEEM 2019-9 April (M) (b) {10, 15) (d) {10} |
12 |
| 1376 | Find the derivative of the following functions(it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m text { and } n text { are integers }) ) ( frac{1}{a x^{2}+b x+c} ) |
12 |
| 1377 | If ( f(x)=frac{log (1+a x)-log (1-b x)}{x} ) for ( boldsymbol{x} neq mathbf{0} ) and ( boldsymbol{f}(mathbf{0})=boldsymbol{k} ) and ( boldsymbol{f}(boldsymbol{x}) ) is continuous at ( x=0, ) then ( k ) is equal to: A ( cdot a+b ) в. ( a-b ) ( c ) D. |
12 |
| 1378 | The number of points where the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}mathbf{1}+left[cos frac{pi x}{2}right], & mathbf{1}<boldsymbol{x} leq mathbf{2} \ mathbf{1}-{boldsymbol{x}}, & mathbf{0} leq boldsymbol{x}<mathbf{1} \ |boldsymbol{s i n} boldsymbol{pi} boldsymbol{x}|, & -mathbf{1} leq boldsymbol{x}<mathbf{0}end{array} ) and right. ( f(1)=0 ) is continuous but nondifferentiable is/are (where [.] and { represent greatest integer and fractional part functions, respectively) ( A cdot O ) в. ( c cdot 2 ) D. none of these |
12 |
| 1379 | ( operatorname{Let} f(x)=left{begin{array}{cc}x^{2} / 2 & 0<x leq 1 \ 2 x^{2}-3 x+3, & 1<x<2end{array}right. ) then which is incorrect This question has multiple correct options A. ( f ) is continuous in (0,2) B. ( f ) is not continuous at all points in (0,2) c. ( f ) is differentiable in (0,2) D. ( L t_{x rightarrow 1^{+}} f(x)=L t_{x rightarrow 1^{-}} f(x) ) |
12 |
| 1380 | Prove that ( e^{x}-x>1, ) if ( x>0 ) | 12 |
| 1381 | If ( x^{3} y^{5}=(x+y)^{8}, ) then show that ( frac{d y}{d x}=frac{y}{x} ) |
12 |
| 1382 | Differentiate ( frac{1}{3} tan ^{3} x-tan x+x ) w.r.t ( x ) | 12 |
| 1383 | If ( left(x^{2}+y^{2}right)^{2}=x y, ) find ( frac{d y}{d x} ) | 12 |
| 1384 | The value of ( c ) of mean value theorem when ( f(x)=x^{3}-3 x-2 ) in [-2,3] is A ( cdot sqrt{frac{7}{3}} ) B. ( sqrt{frac{3}{7}} ) c. ( frac{sqrt{7}}{3} ) D. ( frac{sqrt{3}}{7} ) |
12 |
| 1385 | From mean value theorem, ( boldsymbol{f}(boldsymbol{b})- ) ( boldsymbol{f}(boldsymbol{a})=(boldsymbol{b}-boldsymbol{a}) boldsymbol{f}^{1}left(boldsymbol{x}_{1}right) ; boldsymbol{a}<boldsymbol{x}_{1}< ) bif ( f(x)=frac{1}{x} ) then ( x_{1}= ) ( A cdot sqrt{a b} ) B. ( frac{a+b}{2} ) c. ( frac{a b}{a+b} ) D・ ( frac{a-b}{b-a-a-b} ) |
12 |
| 1386 | If ( f(x) ) is a continuous function on [0,1] differentiable in (0,1) such that ( f(1)=0, ) then there exists some ( c epsilon(0,1) ) such that A ( cdot c f^{prime}(c)-f(c)=0 ) B. ( f^{prime}(c)+c f(c)=0 ) c. ( f^{prime}(c)-c f(c)=0 ) D. ( c f^{prime}(c)+f(c)=0 ) |
12 |
| 1387 | At the point ( x=1, ) the function ( f(x)= ) ( left{begin{array}{ll}x^{3}-1, & 1<x<infty \ x-1, & -infty<x leq 1end{array}right. ) A. continuous and differentiable B. Continuous and not differentiable c. Discontinuous and differentiable D. Discontinuous and not differentiable |
12 |
| 1388 | If ( boldsymbol{y}=cos ^{-1}left(frac{5 cos x-12 sin x}{13}right), x in ) ( left(0, frac{pi}{2}right), ) then find the value of ( d y / d x ) |
12 |
| 1389 | 3. If y = sin x & x = 3t then will be (a) 3 cos (x) (c) 3 cos (x) (b) cos x (d) -cos x |
12 |
| 1390 | The set of all points of differentiability of the function ( f(x)=frac{sqrt{x+1}-1}{sqrt{x}} ) for ( x neq 0 ) and ( f(0)=0 ) is ( mathbf{A} cdot(-infty, infty) ) B. ( [0, infty) ) ( c cdot(0, infty) ) ( mathbf{D} cdot(-infty, infty) sim{0} ) |
12 |
| 1391 | Find the derivative of ( frac{x+cos x}{tan x} ) with respect to ( x ) |
12 |
| 1392 | ( boldsymbol{y}=frac{boldsymbol{x}}{2 sqrt{2}} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 1393 | If ( y=tan ^{-1}(cot x)+cot ^{-1}(tan x) ) then ( frac{d y}{d x} ) is equal to- ( A cdot 1 ) B. 0 c. -1 D. – 2 |
12 |
| 1394 | Find the derivative of ( y=log ^{3}left(x^{2}right) ) | 12 |
| 1395 | Let ( f ) be differentiable for all ( x ). If ( boldsymbol{f}(1)=-2 ) and ( boldsymbol{f}^{prime}(boldsymbol{x}) geq 2 ) for ( boldsymbol{x} in[mathbf{1}, boldsymbol{6}] ) This question has multiple correct options A. ( f(6)<8 ) B. ( f(6) geq 8 ) D. ( f(6) leq 5 ) |
12 |
| 1396 | f ( p(x)=51 x^{101}-2323 x^{100}-45 x+ ) 1035, then using Rolle’s Theorem. prove that atleast one foot lies between ( left(45^{1 / 100}, 46right) ) |
12 |
| 1397 | Differentiate ( e^{x}+e^{-x} ) with respect to ( x ) | 12 |
| 1398 | ff ( f(x)=left{begin{array}{lll}frac{e^{3 x}-1}{4 x} & text { for } & x neq 0 \ frac{k+x}{4} & text { for } & x=0end{array}right. ) continuous at ( x=0, ) then ( k= ) ( A cdot ) B. 3 ( c cdot 2 ) ( D ) |
12 |
| 1399 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(boldsymbol{x}^{2} boldsymbol{e}^{boldsymbol{a} boldsymbol{x}}right) ) A ( cdot e^{a x}left(a x^{2}+2 xright) ) B ( cdot e^{a x}left(2 a x^{2}+2 xright) ) C ( cdot e^{a x}left(a x^{2}+2 a xright) ) D. ( e^{a x}left(a x^{2}-2 a xright) ) |
12 |
| 1400 | 3. Given y= 5x 31(1 – x)2 + cos2 (2x+1) ; Find dy. (1980) |
12 |
| 1401 | If ( x+y=sin (x-y) ) then ( frac{d y}{d x} ) is equal to A ( cdot frac{1}{2} ) B. 0 c. -1 D. |
12 |
| 1402 | If ( boldsymbol{y}=cos ^{-1}(cos boldsymbol{x}), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, boldsymbol{a} boldsymbol{t} boldsymbol{x}= ) ( frac{5 pi}{4} ) is equal to ( mathbf{A} cdot mathbf{1} ) B . – c. ( frac{1}{sqrt{2}} ) D. ( -frac{1}{sqrt{2}} ) |
12 |
| 1403 | 3. If y = x sin x, then (a) I dy 1 = – + cot x = – + cotx y dx x (c) 1 dy-1 – cotx y dx x (d) None of these |
12 |
| 1404 | Differentiate w.r.t ( boldsymbol{x} ) ( sin ^{-1}left(frac{a cos x+b sin x}{sqrt{a^{2}+b^{2}}}right) ) | 12 |
| 1405 | On the interval ( boldsymbol{I}=[-2,2] ),for the function ( left{begin{array}{ll}(x+1) e^{-left[frac{1}{[x]}+frac{1}{x}right]} & (x neq 0) \ 0 & (x=0)end{array}right. ) (where []( text { is } G I F) ) which one of the following hold good? This question has multiple correct options A. is continuous for all values of ( x in I ) B. is continuous for all values of ( x in I-(0) ) c. assumes all intermediate values from ( f(-2) ) & ( f(2) ) D. has a maximum value equal to |
12 |
| 1406 | What are the value of ( c ) for which Rolle’s theorem for the function ( f(x)=x^{3}- ) ( 3 x^{2}+2 x ) in the interval [0,2] is verified? A ( . c=pm 1 ) B. ( _{c=1 pm frac{1}{sqrt{3}}} ) c. ( c=pm 2 ) D. None of these |
12 |
| 1407 | The function ( f(x)= ) ( frac{log (1+a x)-log (1-b x)}{x} ) is not defined at ( x=0 . ) The value of which should be assigned to ( f ) at ( x=0 ), is ( mathbf{A} cdot a-b ) B. ( a+b ) ( mathbf{c} cdot log a+log b ) D. None of these |
12 |
| 1408 | If ( f(x)=left{begin{array}{ll}frac{x}{1+e^{1 / x}} & x neq 0 \ 0 & x=0end{array} ), then the right. function ( f(x) ) is differentiable for: ( mathbf{A} cdot x in R^{+} ) B. ( x in R ) c. ( x in R-{0} ) D. ( x in R-{0,1} ) |
12 |
| 1409 | Let ( f(x) ) be a continous and differentiable function on ( [0,1], ) such that ( f(0) neq 0 ) and ( f(1)=0 . ) We can conclude that there exists ( c in(0,1) ) such that A ( cdot c . f^{prime}(c)-f(c)=0 ) B. ( f^{prime}(c)+c . f(c)=0 ) c. ( f^{prime}(c)-c . f(c)=0 ) D. ( c . f^{prime}(c)+f(c)=0 ) |
12 |
| 1410 | If ( f(x)=frac{a cos x-cos b x}{x^{2}}, x neq 0 ) and ( boldsymbol{f}(mathbf{0})=mathbf{4} ) continuous at ( boldsymbol{x}=mathbf{0}, ) then the ordered pair ( (a, b) ) is ( A cdot(neq 1,3) ) в. ( (1, neq 3) ) c. (-1,-3) D. (1,±3) |
12 |
| 1411 | ( operatorname{Let} boldsymbol{f}left(frac{boldsymbol{x}+boldsymbol{y}}{mathbf{2}}right)=frac{mathbf{1}}{mathbf{2}}[boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})] ) for real ( x ) and ( y . ) If ( f^{prime}(0) ) exists and equals -1 and ( f(0)=1 ) then the value of ( f(2) ) is A . 1 B. – c. ( 1 / 2 ) D. 2 |
12 |
| 1412 | The value of ( frac{d^{2} y}{d x^{2}} ) at the point where ( t= ) 0 is A . 1 B. 2 ( c cdot-2 ) D. 3 |
12 |
| 1413 | Differentiate the following function from first principle: ( sin ^{-1}(2 x+3) ) |
12 |
| 1414 | If ( f(x) ) is a polynomial of degree ( n(>2) ) and ( f(x)=f(k-x), ) (where ( k ) is a fixed real number), then degree of ( boldsymbol{f}^{prime}(boldsymbol{x}) ) is ( A ) B. ( n-1 ) ( mathbf{c} cdot n-2 ) D. None of these |
12 |
| 1415 | Rolle’s theorem cannot be applicable for: A ( cdot f(x)=cos x-1 ) in ( (0,2 pi) ) B – ( f(x)=x(x-2)^{2} ) in (0,2) c. ( f(x)=3+(x-1)^{frac{3}{5}} ) in (0,3) D. ( f(x)=sin ^{2} x ) in ( (0, pi) ) |
12 |
| 1416 | For what value of ( k ) the function ( f(x)= ) ( left{begin{array}{l}frac{sin 5 x}{3 x}, i f, x neq 0 \ k, text { if } x=0end{array} ) is continuous at right. ( boldsymbol{x}=mathbf{0} ) |
12 |
| 1417 | If the function ( left{begin{array}{l}boldsymbol{x}, quad text { if } quad boldsymbol{x} leq mathbf{1} \ boldsymbol{c x}+boldsymbol{k}, quad text { if } quad mathbf{1}<boldsymbol{x}<mathbf{4} \ -mathbf{2} boldsymbol{x}, quad text { if } quad boldsymbol{x} geq mathbf{4}end{array}right. ) is contionus everywhere, then the value of ( c ) and ( k ) are respectively: A. -3,-5 в. -3,5 c. -3,-4 D. -3,4 E . -3,3 |
12 |
| 1418 | ( frac{d(tan x .)}{d x} ) ( mathbf{A} cdot sec ^{2} x ) B. ( cot ^{2} x ) ( mathbf{c} cdot cos ^{2} x ) ( D cdot sin ^{2} x ) |
12 |
| 1419 | The function ( f(x)=left{begin{array}{cc}frac{e^{1 / x}-1}{e^{1 / x}+1} & x neq 0 \ 0, & x=0end{array}right. ) is A. continuous at ( x=0 ) B. discontinuous at ( x=0 ) c. discontinuous at ( x=0 ) but can be made continuous at ( x=0 ) D. None of these |
12 |
| 1420 | 23 | 12 |
| 1421 | ff ( y=operatorname{en}left{frac{x+sqrt{left(a^{2}+x^{2}right)}}{a}right}, ) then the value of ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is A ( cdot sqrt{a^{2}-x^{2}} ) B. ( a sqrt{a^{2}+x^{2}} ) c. ( frac{1}{sqrt{a^{2}+x^{2}}} ) D. ( x sqrt{a^{2}+x^{2}} ) |
12 |
| 1422 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[(boldsymbol{x}+boldsymbol{a})left(boldsymbol{x}^{2}+boldsymbol{a}^{2}right)left(boldsymbol{x}^{4}+boldsymbol{a}^{4}right)right]=? ) ( ^{mathbf{A} cdot frac{7 x^{8}+aleft(8 x^{7}-a^{7}right)}{(x-a)^{2}}} ) B. ( frac{7 x^{8}-aleft(8 x^{7}-a^{7}right)}{(x-a)^{2}} ) c. ( frac{7 x^{8}-aleft(8 x^{7}+a^{7}right)}{(x-a)^{2}} ) D. ( x^{4}+a^{4} ) |
12 |
| 1423 | Differentiate ( boldsymbol{x}=boldsymbol{y}+ ) | 12 |
| 1424 | Using the fact that ( sin (boldsymbol{A}+boldsymbol{B})= ) ( sin A cos B+cos A sin B ) and the differentiation, obtain the sum formula for cosines. |
12 |
| 1425 | If ( f(x)=frac{x-1}{x+2}, ) then ( frac{d f^{-1}(x)}{d x} ) is equal to A ( cdot frac{1}{(1-x)^{2}} ) B. ( frac{-3}{(1-x)^{2}} ) c. ( frac{3}{(1-x)^{2}} ) D. ( frac{-1}{(1-x)^{2}} ) |
12 |
| 1426 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of function ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{3}}+frac{mathbf{1}}{mathbf{2}} log boldsymbol{x} ) A ( cdot 2 . e^{x^{3}} x^{2}+frac{1}{2 x} ) B. ( e^{x^{3}} x^{2}+frac{1}{2 x} ) c. ( 3 . e^{x^{3}} x^{2}+frac{1}{2 x} ) D. ( 3 . e^{x^{3}} x^{2}+frac{1}{x} ) |
12 |
| 1427 | If ( boldsymbol{y}=boldsymbol{e}^{log left(1+boldsymbol{x}+boldsymbol{x}^{2}+boldsymbol{x}^{3}+cdotsright)}, ) where ( |boldsymbol{x}|<mathbf{1} ) then ( frac{d y}{d x} ) is equal to A ( cdot frac{-1}{(1-x)^{2}} ) B. ( frac{1}{(1-x)^{2}} ) c. ( frac{1}{(1+x)^{2}} ) D. None of these |
12 |
| 1428 | [ begin{aligned} boldsymbol{f}(boldsymbol{x})=& boldsymbol{x}, boldsymbol{i} boldsymbol{f} boldsymbol{x} leq 1 \ boldsymbol{5}, & boldsymbol{i} boldsymbol{f} boldsymbol{x} geq 1 end{aligned} ] Check whether ( f(x) ) is continuous at [ boldsymbol{x}=mathbf{0} ? boldsymbol{x}=mathbf{1} ? boldsymbol{x}=mathbf{2} ? ] |
12 |
| 1429 | ( f(x)=x^{2} ) in ( 2 leq x leq 3 ) Is Rolle’s theorem applicable? |
12 |
| 1430 | Differentiate the following functions with respect to ( x ) : If ( y=tan ^{-1}left(frac{2 x}{1-x^{2}}right), x>0, ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{4}}{boldsymbol{1}+boldsymbol{x}^{2}} ) |
12 |
| 1431 | Find ( frac{d y}{d x} ) where ( boldsymbol{y}=sqrt{frac{boldsymbol{x}^{2}+mathbf{1}}{boldsymbol{x}^{2}+mathbf{2}}} ) | 12 |
| 1432 | If f(x) and g(x) are differentiable function for 0 < x <1 such that f(0) = 2, g(0)=0, f(1) = 6; g(1)= 2, then show that there exist c satisfying 0<c<1 and f'(c)=2g'(c). |
12 |
| 1433 | Differentiate ( f(x)=4 x^{2}-5 x ) ( A cdot 8 x ) B. ( 8 x-5 ) ( c .5 ) D. None of these |
12 |
| 1434 | ( y=sin x cos x ) find ( frac{d y}{d x} ) | 12 |
| 1435 | The displacement ( x ) of a particle along the ( x ) -axis at time ( t ) is given by ( x= ) ( frac{a_{1}}{2} t+frac{a_{2}}{3} t^{2} . ) Find the acceleration of the particle. |
12 |
| 1436 | If ( y=x^{2} sin x, ) then ( frac{d y}{d x} ) will be A. ( x^{2} cos x+2 x sin x ) B. ( 2 x sin x ) C ( cdot x^{2} cos x ) D. ( 2 x cos x ) |
12 |
| 1437 | Let ( f(x) ) be a real-valued differentiable function not identically zero such that ( boldsymbol{f}left(boldsymbol{x}+boldsymbol{y}^{2 n+1}right)=boldsymbol{f}(boldsymbol{x})+ ) ( {f(y)}^{2 n+1}, n epsilon N ) and ( x, y ) are any real numbers and ( f^{prime}(0) geq 0 . ) Find the value of ( f(5) ) A . 0 B. 1 c. 2 D. 5 |
12 |
| 1438 | The function ( f(x)=x(x+3) e^{-(1 / 2) x} ) satisfies the condition of Rolle’s theorem in ( [-3,0] . ) The value of ( c ) is ( mathbf{A} cdot mathbf{0} ) B. – c. -2 D. – 3 |
12 |
| 1439 | Given ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}3-left[cot ^{-1}left(frac{2 x^{3}-3}{x^{2}}right)right] & text { for } x>0 \ left{x^{2}right} cos left(e^{1 / x}right) & text { for } x<0end{array}right. ) where {}( &[] ) denotes the fractional part and the integral part functions respectively, then which of the following statement does not hold good – This question has multiple correct options ( mathbf{A} cdot fleft(0^{-}right)=0 ) B . ( fleft(0^{+}right)=3 ) C ( . f(0)=0 Rightarrow ) continuity of ( f ) at ( x=0 ) D. irremovable discontinuity of ( f ) at ( x=0 ) |
12 |
| 1440 | Discuss the continuity of the following function ( : f(x)=sin x . cos x ) | 12 |
| 1441 | Differentiate ( boldsymbol{f}(boldsymbol{x}) ) with respect to ( boldsymbol{g}(boldsymbol{x}) ) for the following. ( boldsymbol{f}(boldsymbol{x})=log _{e} boldsymbol{x}, boldsymbol{g}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} ) |
12 |
| 1442 | If ( S_{n} ) denotes the sum of ( n ) terms of a G.P. whose common ratio is ( r, ) then ( (r-1) frac{d S_{n}}{d r} ) is equal to A ( cdot(n-1) S_{n}+n S_{n-1} ) B . ( (n-1) S_{n}-n S_{n-1} ) ( mathbf{c} cdot(n-1) S_{n} ) D. None of these |
12 |
| 1443 | If ( boldsymbol{x}+boldsymbol{y}=boldsymbol{x}^{boldsymbol{y}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} boldsymbol{e q u a l s -} ) A ( frac{y x^{y-1}-1}{1-x^{y} log x} ) В. ( frac{y x^{y-1}-1}{x^{y} log x-1} ) c. ( frac{y x^{y-1}+1}{x^{y} log x+1} ) D. None of these |
12 |
| 1444 | If ( sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}}=10, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{y}=boldsymbol{4} ) ( A cdot 4 ) в. -3 c. -4 D. 3 |
12 |
| 1445 | Differentiate: ( boldsymbol{y}=sin ^{-1}left(frac{1-boldsymbol{x}^{2}}{mathbf{1}+boldsymbol{x}^{2}}right), mathbf{0}< ) ( boldsymbol{x}<mathbf{1} ) |
12 |
| 1446 | 15. If f(x) = rex) x+0 then f(x) is [2003] „x=0 (a) discontinuous every where (b) continuous as well as differentialble for all x (c) continuous for all x but not differentiable at x=0 (d) neither differentiable nor continuous at x=0 |
12 |
| 1447 | Test the continuity of the function ( f ) at ( mathbf{x}=mathbf{0}, ) where [ begin{array}{l} boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2} sin left(frac{1}{x}right) text { for } boldsymbol{x} neq mathbf{0} \ quad=mathbf{1} text { for } boldsymbol{x}=mathbf{0} end{array} ] |
12 |
| 1448 | If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-mathbf{5} boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{x}, ) verify conditions of the mean value theorem satisfied for ( boldsymbol{a}=mathbf{1}, boldsymbol{b}=mathbf{3} . ) Find ( boldsymbol{c} boldsymbol{epsilon}(mathbf{1}, boldsymbol{3}) ) such that ( boldsymbol{f}^{prime}(boldsymbol{c})=frac{boldsymbol{f}(boldsymbol{3})-boldsymbol{f}(1)}{boldsymbol{3}-1} ) ( A cdot 2 ) B. ( c .3 ) D. |
12 |
| 1449 | ( operatorname{Let} f(x)=frac{ln (1+x tan x)}{4 x}, x neq 0 ) is continuous at ( boldsymbol{x}=mathbf{0}, ) then ( boldsymbol{f}(mathbf{0}) ) must be equal to A . 1 B. 0 ( c .3 ) D. |
12 |
| 1450 | Differentiate the following function w.r.t. ( boldsymbol{x} ) ( sqrt[3]{left(2 x^{2}-7 x-4right)^{5}} ) |
12 |
| 1451 | Find ‘c’ of the mean value theorem,if ( f(x)=x(x-1)(x-2) ) ( boldsymbol{a}=mathbf{0}, boldsymbol{b}=frac{mathbf{1}}{mathbf{2}} ) ( ^{mathrm{A}} cdot_{C}=1-frac{sqrt{21}}{5} ) в. ( quad C=1-frac{sqrt{21}}{6} ) ( ^{mathrm{c}} cdot_{C}=2-frac{sqrt{21}}{6} ) D. ( c=1+frac{sqrt{21}}{6} ) |
12 |
| 1452 | 17. Let $(x) = 1-tan x X ,X E f(x) is continuous 4x-tet [2004] (a) 165 (1) 1 |
12 |
| 1453 | If ( z=f ) of ( (x) ) where ( f(x)=x^{2}, ) then what is ( frac{d z}{d x} ) equal to? ( mathbf{A} cdot x^{3} ) В. ( 2 x^{3} ) ( c cdot 4 x^{3} ) D. ( 4 x^{2} ) |
12 |
| 1454 | If ( cos ^{-1}left(frac{x^{2}-y^{2}}{x^{2}+y^{2}}right)=k ) (a constant) then ( frac{d y}{d x}= ) A. ( frac{y}{x} ) в. ( frac{x}{y} ) c. ( frac{x^{2}}{y^{2}} ) D. ( frac{y^{2}}{x^{2}} ) |
12 |
| 1455 | Extend the definition of the following by continuity. ( f(x)=frac{1-cos 7(x-pi)}{5(x-pi)^{2}} ) at the point ( boldsymbol{x}=boldsymbol{pi} ) |
12 |
| 1456 | ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) in the interval [-1,1] Is Rolle’s Theorem applicable? |
12 |
| 1457 | The Rolle’s theorem is applicable in the interval ( -1 leq x leq 1 ) for the function ( mathbf{A} cdot f(x)=x ) ( mathbf{B} cdot f(x)=x^{2} ) ( mathbf{c} cdot f(x)=2 x^{3}+3 ) ( mathbf{D} cdot f(x)=|x| ) |
12 |
| 1458 | If ( y=e^{m sin ^{-1} x},-1 leq x leq 1 ), show that ( left(1-x^{2}right) frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}-a^{2} y=0 ) |
12 |
| 1459 | Differentiate the following w.r.t ( x ) ( sin left(x^{2}+5right) ) |
12 |
| 1460 | Find the second order derivatives of ( sin (log x) ) |
12 |
| 1461 | ff ( y=tan left(frac{5}{2} pi t+frac{pi}{6}right) ) then find the value of ( frac{boldsymbol{a} boldsymbol{y}}{boldsymbol{d} boldsymbol{t}} ) at ( boldsymbol{t}=mathbf{0} ) |
12 |
| 1462 | If ( y=cos ^{-1}left(frac{2 x}{1+x^{2}}right), ) then ( frac{d y}{d x} ) is equal to A ( cdot-frac{2}{1+x^{2}} ) of all ( |x|1 ) c. ( frac{2}{1+x^{2}} ) of all ( |x|<1 ) D. None of the above |
12 |
| 1463 | Find the derivative of the following functions: ( 5 sec x+4 cos x ) |
12 |
| 1464 | Given that ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x} boldsymbol{g}(boldsymbol{x})}{|boldsymbol{x}|}, boldsymbol{g}(mathbf{0})= ) ( boldsymbol{g}^{prime}(mathbf{0})=mathbf{0} ) and ( boldsymbol{f} ) is continuous at ( boldsymbol{x}=mathbf{0} ) the value of ( f^{prime}(0) ) is |
12 |
| 1465 | Using Rolle’s theorem, find points on the curve ( boldsymbol{y}=mathbf{1 6}-boldsymbol{x}^{2}, boldsymbol{x} in[-mathbf{1}, mathbf{1}], ) where tangent is parallel to ( x- ) axis. |
12 |
| 1466 | Differentiate the following function with respect to ( x ) ( frac{x+cos x}{tan x} ) |
12 |
| 1467 | Solve ( lim _{x rightarrow-2} frac{left(frac{1}{x}+frac{1}{2}right)}{x+2} ) | 12 |
| 1468 | Discuss the applicability of Rolle’s theorem for the following function on the indicated interval: ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}-mathbf{4} boldsymbol{x}+mathbf{5}, mathbf{0} leq boldsymbol{x} leq mathbf{1} \ mathbf{2} boldsymbol{x}-mathbf{3}, quad mathbf{1}<boldsymbol{x} leq mathbf{2}end{array}right. ) |
12 |
| 1469 | If ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}], ) where ( [.] ) is the greatest integer function, and ( boldsymbol{g}(boldsymbol{x})= ) ( xleft(1-x^{2}right)left(4-x^{2}right), ) then ( g[f(x)] ) is A. discontinuous at all integer B. continuous at all integer c. continuous at ( x=0,pm 1,pm 2 ) D. discontinuous at ( x=0,pm 1,pm 2 ) |
12 |
| 1470 | Find ( frac{d y}{d x}, ) when ( y=frac{x cos ^{-1} x}{sqrt{1-x^{2}}} ) | 12 |
| 1471 | ( frac{mathbf{d}^{2} mathbf{x}}{mathbf{d} mathbf{y}^{2}} ) equals A ( cdotleft(frac{mathrm{d}^{2} mathbf{y}}{mathrm{d} mathbf{x}^{2}}right)^{-1} ) B ( cdot quad-left(frac{mathrm{d}^{2} mathbf{y}}{mathrm{d} mathbf{x}^{2}}right)^{-1}left(frac{mathrm{d} mathbf{y}}{mathrm{d} mathbf{x}}right)^{-3} ) C ( cdotleft(frac{mathrm{d}^{2} mathbf{y}}{mathrm{d} mathbf{x}^{2}}right)left(frac{mathrm{d} mathbf{y}}{mathrm{d} mathbf{x}}right)^{-2} ) D ( cdot ) ( -left(frac{mathrm{d}^{2} mathbf{y}}{mathrm{d} mathbf{x}^{2}}right)left(frac{mathrm{d} mathbf{y}}{mathrm{d} mathbf{x}}right)^{-3} ) |
12 |
| 1472 | If ( sqrt{boldsymbol{y}+boldsymbol{x}}+sqrt{boldsymbol{y}-boldsymbol{x}}=boldsymbol{c}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is equal to This question has multiple correct options A ( cdot frac{2 x}{c^{2}} ) B. ( frac{x}{y+sqrt{y^{2}-x^{2}}} ) c. ( frac{y-sqrt{y^{2}-x^{2}}}{x} ) D. ( frac{c^{2}}{2 y} ) |
12 |
| 1473 | 21. Let f : [a, b] → [1,00) be a continuous function and let g: R → R be defined as (JEE Adv. 2014) if x <a, t, if a < x b. (a) g(x) is continuous but not differentiable at a (b) g(x) is differentiable on R c) g(x) is continuous but not differentiable at b (d) g(x) is continuous and differentiable at either (a) or (b) but not both For everunoir ofation…. |
12 |
| 1474 | 2 ( 10 полапс спасен 24. The function f :R/{0} → R given by [2007 f(x) = 1 2 r e2x – 1 can be made continuous at x =0 by defining f (0) as (a) o (6) 1 (c) 2 (d) -1 w |
12 |
| 1475 | The graph of the function ( f(x)=x^{3}+1 ) after translation 4 units to the right and 2 units up, resulted in a new graph ( l(x) ) What is the value of ( l(3.7) ? ) A. 0.973 B. 1.784 c. 1.973 D. 2.027 E . 2.973 |
12 |
| 1476 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(boldsymbol{3} cos left(frac{boldsymbol{pi}}{boldsymbol{6}}+boldsymbol{x}^{0}right)-boldsymbol{4} cos ^{3}left(frac{boldsymbol{pi}}{boldsymbol{6}}+boldsymbol{x}^{0}right)right. ) A ( cdot cos left(3 x^{0}right) ) B. ( frac{pi}{60} sin left(3 x^{0}right) ) c. ( frac{pi}{60} cos left(3 x^{0}right) ) D. ( -frac{pi}{60} sin left(3 x^{0}right) ) |
12 |
| 1477 | Let ( y=left(1+x^{2}right) tan ^{-1}(x-x) ) and ( f(x)=frac{1}{2 x} frac{d y}{d x}, ) then ( f(x)+cot ^{-1} x ) is equal to ( mathbf{A} cdot mathbf{0} ) в. ( frac{pi}{2} ) c. ( -frac{pi}{2} ) D. |
12 |
| 1478 | If the function ( f(x)=x^{3}+e^{x} ) and ( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}^{-1}(boldsymbol{x}), ) then the value of ( boldsymbol{g}^{prime}(mathbf{1}) ) is |
12 |
| 1479 | Solve: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} mathbf{7} boldsymbol{x} ) | 12 |
| 1480 | 14. Let f(x)=x sin 7tx, x>0. Then for all natural numbers n, ) vanishes at (JEE Adv. 2013) (a) A unique point in the interval | n9n+ + 1 (b) A unique point in the interval 2 (c) A unique point in the interval (n, n+1) (d) Two points in the interval (n, n+1) |
12 |
| 1481 | Differentiate the following w.r.t. ( x ) ( e^{sin ^{-1} x} ) | 12 |
| 1482 | If ( boldsymbol{y}+sin boldsymbol{y}=cos boldsymbol{x}, ) find ( frac{boldsymbol{d} boldsymbol{y} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |
| 1483 | Consider the functions defined implicitly by the equation ( y^{3}-3 y+ ) ( mathbf{x}=mathbf{0} ) on various intervals in the real line. If ( x in(-infty,-2) cup(2, infty), ) the equation implicitly defines a unique real valued differentiable function ( mathbf{y}= ) ( f(x) ). If ( x in(-2,2) ), the equation implicitly defines a unique real valued differentiable function ( mathbf{y}=mathbf{g}(mathbf{x}) ) satisfying ( mathbf{g}(mathbf{0})=mathbf{0} ) f ( mathrm{f}(-10 sqrt{2})=2 sqrt{2}, ) then ( mathrm{f}^{prime prime}(-10 sqrt{2})= ) A ( cdot frac{4 sqrt{2}}{7^{3} 3^{2}} ) в. ( -frac{4 sqrt{2}}{7^{3} 3^{2}} ) c. ( frac{4 sqrt{2}}{7^{3} 3} ) D. ( -frac{4 sqrt{2}}{7^{3} 3} ) |
12 |
| 1484 | If Rolle’s theorem holds for the function ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{mathbf{3}}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{c} boldsymbol{x}, boldsymbol{x} in[-mathbf{1}, mathbf{1}], ) at the point ( x=frac{1}{2}, ) then ( 2 b+c ) equals: A . -1 B. c. -3 D. |
12 |
| 1485 | Suppose ( f ) is differentiable on ( R ) and ( boldsymbol{a} leq boldsymbol{f}^{prime}(boldsymbol{x}) leq boldsymbol{b} ) where ( boldsymbol{x} in boldsymbol{R} ) where ( boldsymbol{a}, boldsymbol{b}>mathbf{0} . ) If ( boldsymbol{f}(mathbf{0})=mathbf{0}, ) then A ( cdot f(x) leq min (a x, b x) ) B . ( f(x) geq min (a x, b x) ) c. ( a leq f(x) leq b ) D. ( a x leq f(x) leq b x ) |
12 |
| 1486 | By Rolles theorem for ( f(x)=(x- ) ( a)^{m}(x-b)^{n} ) on ( [a, b] ; m, n ) being positive integer. Find the value of ( c ) which lies between ( a ) & b. A ( cdot c=frac{m b+n a}{m+n} ) в. ( c=frac{m b-n a}{m+n} ) c. ( _{c}=frac{n b+m a}{m+n} ) D. ( c=frac{n b-m a}{m+n} ) |
12 |
| 1487 | If ( f(x)=x+log x ) find ( f^{prime}(x) ) | 12 |
| 1488 | ( f(x)=x^{2}left(1-cos left(frac{2}{x}right)right) ) for ( x neq 0 ) and ( f(0)=k . ) If ( f(x) ) is continuos at ( x=0 ) then find ( k ) |
12 |
| 1489 | Find derivative of ( f(x) ) ( f(x)=x sin x ) |
12 |
| 1490 | If ( f ) is a continuous function on the real line. Given that ( x^{2}+(f(x)-2) x- ) ( sqrt{3} cdot f(x)+2 sqrt{3}-3=0 . ) Then the value of ( f(sqrt{3}) ) A. can not be determined B. ( 2(1-sqrt{3} ) c. zero D. ( frac{2(sqrt{3}-2)}{sqrt{3}} ) |
12 |
| 1491 | If ( x^{2}+y^{2}=a^{2} ) and ( k=1 / a, ) then ( k ) is equal to? A ( cdot frac{y prime}{sqrt{1+y^{prime}}} ) B. ( frac{|y prime prime|}{sqrt{left(1+y^{prime 2}right)^{3}}} ) c. ( frac{2 y prime}{sqrt{1+y prime}} ) D. ( frac{y prime}{2 sqrt{left(1+y^{prime 2}right)^{3}}} ) |
12 |
| 1492 | Discuss the applicability of Rolle’s theorem to ( f(x)=log left[frac{x^{2}+a b}{(a+b) x}right], ) in the interval ( [a, b] ) A. Yes Rolle’s theorem is applicable and the stationary point is ( x=sqrt{a b} ) B. No Rolle’s theorem is not applicable due to the discontinuity in the given interval C. Yes Rolle’s theorem is applicable and the stationary point is ( x=a b ) D. none of these |
12 |
| 1493 | Which of the following functions is differentiable at x=0? (a) cos(xl) + bx (b) cos(xl) – bx (20015) c) sin (xl) + 1x (d) sin(xD) – x |
12 |
| 1494 | If ( x^{2}+y^{2}=R^{2} ) and ( K=frac{1}{R} ) then ( K= ) A. ( frac{y_{1}}{x sqrt{1+y_{1}^{2}}} ) в. ( frac{left|y_{2}right|}{sqrt{left(1+y_{1}^{2}right)^{3}}} ) c. ( frac{2left|y_{2}right|}{sqrt{1+y_{1}^{2}}} ) D. ( frac{3left|y_{2}right|}{sqrt{left(1+y_{1}^{3}right)^{3}}} ) |
12 |
| 1495 | VJ (0) 13. The function f(x)= he function f(x)=x2-x21(where [y] is the greatest integer ss than or equal to y), is discontinuous at (1999 – 2 Marks) (a) all integers (6) all integers except 0 and 1 © all integers except 0 (d) all integers except 1 |
12 |
| 1496 | The function ( y=sin ^{-1}(cos x) ) is not differentiable at This question has multiple correct options ( mathbf{A} cdot x=pi ) В. ( x=-2 pi ) c. ( x=2 pi ) D. None of these |
12 |
| 1497 | ( operatorname{Let} boldsymbol{f}(boldsymbol{x})= ) ( frac{boldsymbol{x}(mathbf{1}+boldsymbol{a} cos boldsymbol{x})-boldsymbol{b} sin boldsymbol{x}}{boldsymbol{x}^{3}}, boldsymbol{x} neq mathbf{0} ) and ( f(0)=1 . ) The value of ( a ) and ( b ) so that ( f ) is a continuous function are- A ( cdot frac{5}{2}, frac{3}{2} ) В. ( frac{5}{2},-frac{3}{2} ) c. ( -frac{5}{2},-frac{3}{2} ) D. None of these |
12 |
| 1498 | 24. If velocity of particle is given by v = 24, then it acceleration (dv/dt) at any time t will be given by… |
12 |
| 1499 | State whether the given statement is True or False.
Derivative of ( y=2 x^{5} ) with respect to ( x ) |
12 |
| 1500 | Letf:R R be a function such that f (x + y) = f(x) + f(y), X, y E R. If f(x) is differentiable at x=0, then (2011) (a) f(x) is differentiable only in a finite interval containing zero (b) f(x) is continuous x eR (c) f'(x) is constant x ER (d) f(x) is differentiable except at finitely many points. |
12 |
| 1501 | Differentiate the following w.r.t. ( x ) ( log left(cos e^{x}right) ) |
12 |
| 1502 | If ( y=f(x) ) is continuous on [0,6] differentiable on ( (0,6), f(0)=-2 ) and ( f(6)=16, ) then at some point between ( boldsymbol{x}=mathbf{0} ) and ( boldsymbol{x}=mathbf{6}, mathbf{f}^{prime}(mathbf{x}) ) must be equal to? A . -18 B. -3 ( c .3 ) D. 14 |
12 |
| 1503 | If ( boldsymbol{y}=boldsymbol{x} sqrt{1-boldsymbol{x}^{2}}+sin ^{-1} boldsymbol{x}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is | 12 |
| 1504 | ( boldsymbol{y}=boldsymbol{A} cos boldsymbol{n} boldsymbol{x}+boldsymbol{B} sin boldsymbol{n} boldsymbol{x} ) Prove that ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{n}^{2} boldsymbol{y}=mathbf{0} ) |
12 |
| 1505 | In the law of mean, the value of ( theta ) satisfies the condition ( mathbf{A} cdot theta>0 ) B . ( theta1 ) ( 0.0<theta<1 ) |
12 |
| 1506 | Let [.] denote the greatest integer function and ( f(x)=left[tan ^{2} xright] . ) Then ( mathbf{A} cdot lim _{x rightarrow 0} f(x) ) does not exis B. ( f(x) ) is continuous at ( x=0 ) c. ( f(x) ) is not differentiable at ( x=0 ) |
12 |
| 1507 | If ( y=left(tan ^{-1} xright)^{2} ) and ( left(x^{2}+1right)^{2} frac{d^{2} y}{d x^{2}}+ ) ( 2 xleft(x^{2}+1right) frac{d y}{d x}=k, ) then the value of ( k ) is A . 3 B. 2 c. 1 D. |
12 |
| 1508 | If ( y=x sqrt{a^{2}+x^{2}}+ ) ( a^{2} log (x+sqrt{a^{2}+x^{2}}) ) then ( frac{d y}{d x}= ) ( 2 sqrt{a^{2}+x^{2}} ) |
12 |
| 1509 | Find the derivative of ( f(x)= ) ( frac{x+cos x}{tan x} w . r . t . x ) | 12 |
| 1510 | ( f frac{a_{0}}{n+1}+frac{a_{1}}{n}+frac{a_{2}}{n-1}+dots+frac{a_{n-1}}{2}+ ) ( a_{n}=0, ) then the equation ( a_{0} x^{n}+ ) ( a_{1} x^{n-1}+cdots+a_{n-1} x+a_{n}=0 ) has, in the interval ( (mathbf{0}, mathbf{1}) ) A. Exactly one root B. Atleast one root c. Atmost one root D. No root |
12 |
| 1511 | If ( boldsymbol{x}=boldsymbol{a} boldsymbol{t}^{2}, quad boldsymbol{y}= ) 2at then ( boldsymbol{d}^{2} boldsymbol{y} / boldsymbol{d} boldsymbol{x}^{2}= ) A ( cdot frac{-1}{t^{2}} ) в. ( frac{1}{t^{2}} ) c. 0 D. ( frac{1}{2 a^{3}} ) |
12 |
| 1512 | Differentiate ( boldsymbol{y}=cos (2 x-5) ) with respect to ( x ) |
12 |
| 1513 | Obtain the differential equation of the family of circles ( x^{2}+y^{2}+2 g x+ ) ( 2 f y+c=0 ; ) where ( g, f ) and ( c ) are arbitrary constants. ( ^{A} cdotleft[1+left(y^{prime}right)^{2}right] y^{prime prime}-3 y^{prime}left(y^{prime prime}right)^{2}=0 ) ( ^{mathrm{B}}left[1+left(y^{prime prime}right)^{3}right] y-2 y^{prime}left(y^{prime prime}right)^{2}=0 ) ( ^{mathbf{C}}left[1+left(y^{prime prime}right)^{2}right] y^{prime prime}-3 y^{prime}left(y^{prime prime}right)^{2}=0 ) D. None of these |
12 |
| 1514 | 8. f(x) and g(x) are two differentiable functions on [0,2] such that f”(x)-g”(x) = 0, f'(1) = 2g'(1) = 4 f(2)=3g(2)=9 then f(x)-g(x) at x =3/2 is [2002] (a) o (6) 2 (c) 10 (d) 5 |
12 |
| 1515 | ( lim _{n rightarrow infty}left(frac{(n+1)(n+2) dots 3 n}{n^{2 n}}right)^{frac{1}{n}} ) equal to :- |
12 |
| 1516 | 12. x/1+ y + y/1+ x = 0, then dy dar (a) 1+ x (b) (1+x)? (c) -(1 + x)-1 (d) -(1 + x)-2 I do |
12 |
| 1517 | Answer the following question in one word or one sentence or as per exact requirement of the question. If ( boldsymbol{x}<mathbf{2}, ) then write the value of ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(sqrt{boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{x}+boldsymbol{4}}) ) |
12 |
| 1518 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left[frac{boldsymbol{x}^{mathbf{1} / 3}+boldsymbol{a}^{mathbf{1} / 3}}{mathbf{1}-boldsymbol{x}^{mathbf{1} / 3} boldsymbol{a}^{mathbf{1} / 3}}right] ) | 12 |
| 1519 | If ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{2} ) and ( boldsymbol{y}_{2}=boldsymbol{A} boldsymbol{y}^{-3} ) then ( mathbf{A}= ) A . -2 B. – ( c cdot 0 ) ( D ) |
12 |
| 1520 | The value of ( k ) for which the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}frac{1-cos 4 x}{8 x^{2}} & , x neq 0 \ k & , x=0end{array}right. ) continuous at ( boldsymbol{x}=mathbf{0}, ) is ( mathbf{A} cdot k=0 ) B. ( k=1 ) c. ( k=-1 ) D. None of the above |
12 |
| 1521 | Let ( f ) be differentiable for all ( x ). If ( f(1)= ) -2 and ( f^{prime}(x) geq 2 ) for ( x in[1,6], ) then ( mathbf{A} cdot mathbf{f}(6) geq 8 ) B. ( f(6)<8 ) ( mathrm{c} cdot mathrm{f}(6)<5 ) D. ( f(6)=5 ) |
12 |
| 1522 | Find derivative of ( sin ^{-1}left(x^{2}right) ) using first principle. A ( cdot frac{2 x}{sqrt{1-x^{2}}} ) в. ( frac{2 x}{sqrt{1-x}} ) c. ( frac{2 x}{sqrt{1-x^{4}}} ) D. ( frac{x}{sqrt{1-x^{4}}} ) |
12 |
| 1523 | If the derivative of the functions ( f(x)= ) ( left{begin{array}{cc}b x^{2}+a x+4 ; & x geq-1 \ a x^{2}+b ; & x<-1end{array}right} ) is everywhere continuous then A ( . a=2, b=3 ) В. ( a=3, b=2 ) C ( . a=-2, b=-3 ) D. ( a=-3, b=-2 ) |
12 |
| 1524 | If ( boldsymbol{f}(boldsymbol{x})=mathbf{1} ) for ( boldsymbol{x}<mathbf{0}=mathbf{1}+sin boldsymbol{x} ) for ( 0 leq x<pi / 2, ) then at ( x=0, ) then show that the derivative ( f^{prime}(x) ) does not exist. |
12 |
| 1525 | ( operatorname{Let} F(x)=f(x) g(x) h(x) ) for all real ( x ) where ( boldsymbol{f}(boldsymbol{x}), boldsymbol{g}(boldsymbol{x}), boldsymbol{h}(boldsymbol{x}) ) are differentiable functions. At some point ( boldsymbol{x}_{0}, ) if ( boldsymbol{F}^{prime}left(boldsymbol{x}_{0}right)=mathbf{2 1} boldsymbol{F}left(boldsymbol{x}_{0}right), boldsymbol{f}^{prime}left(boldsymbol{x}_{0}right)= ) ( 4 fleft(x_{0}right), g^{prime}left(x_{0}right)=-7 gleft(x_{0}right) ) and ( h^{prime}left(x_{0}right)=lambda hleft(x_{0}right), ) then ( lambda= ) A . 12 B. -12 ( c cdot 24 ) D. -24 |
12 |
| 1526 | Compute the derivative of ( 6 x^{100}- ) ( boldsymbol{x}^{boldsymbol{5} boldsymbol{5}}+boldsymbol{x} ) |
12 |
| 1527 | ( boldsymbol{x}=boldsymbol{e}^{boldsymbol{theta}}(sin boldsymbol{theta}+cos boldsymbol{theta}), boldsymbol{y}=boldsymbol{e}^{boldsymbol{theta}}(sin boldsymbol{theta}-boldsymbol{1}) ) ( cos boldsymbol{theta}) ) Fine ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) |
12 |
| 1528 | If ( y=3 cos x, ) then ( frac{d y}{d x} ) at ( x=frac{pi}{2} ) is A. -3 B. 3 ( c .0 ) D. – |
12 |
| 1529 | Find drivative of ( boldsymbol{y}=(2-sin x)left(e^{x}+right. ) ( left.x^{3}+2right) ) with respect to ( x ) |
12 |
| 1530 | The degree and order of differential equation ( left(frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}right)^{2}=left(boldsymbol{y}+frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{frac{1}{2}} ) which of the following? |
12 |
| 1531 | Verify Rolle’s theorem for the following function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-mathbf{5} boldsymbol{x}+mathbf{9}, boldsymbol{x} varepsilon[mathbf{1}, boldsymbol{4}] ) |
12 |
| 1532 | ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=boldsymbol{a}_{5} boldsymbol{x}^{5}+boldsymbol{a}_{4} boldsymbol{x}^{4}+boldsymbol{a}_{3} boldsymbol{x}^{3}+ ) ( a_{2} x^{2}+a_{1} x, ) where ( a_{i}^{prime} s ) are real and ( f(x)=0 ) has a positive root ( alpha_{0} . ) Then This question has multiple correct options A ( cdot f^{prime}(x)=0 ) has a root ( alpha_{1} ) such that ( 0<alpha_{1}<alpha_{0} ) B . ( f^{prime}(x)=0 ) has at least one real root C ( cdot f^{prime prime}(x)=0 ) has at least one real root D. All of the above |
12 |
| 1533 | Illustration 2.23 Find the derivatives of y=(x + 1) (x+3). |
12 |
| 1534 | If ( boldsymbol{y}=log _{e}left(frac{boldsymbol{x}^{2}}{e^{2}}right), ) then ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} ) equals ( A cdot-frac{1}{x} ) B. ( -frac{1}{x^{2}} ) c. ( frac{2}{x^{2}} ) D. ( -frac{2}{x^{2}} ) |
12 |
| 1535 | 35. If the function. g(x)= JkVx+1, 0553 [mx+2, 3 **ss is differentiable, then the value ofk+ mis : JEE M 20151 (2) 10 (6) 4 (0) 2 (2) 16 |
12 |
| 1536 | If ( f(x)=x^{2}-6 x+8 ) and there exists a point ( c ) in the interval [2,4] such that ( boldsymbol{f}^{prime}(boldsymbol{c})=mathbf{0}, ) then what is the value of ( boldsymbol{c} ? ) A . 2.5 B. 2.8 ( c cdot 3 ) D. 3.5 |
12 |
| 1537 | The value of ( f(0) ) so that the function ( f(x)=frac{2 x-sin ^{-1} x}{2 x+tan ^{-1} x} ) is continuous at each point in its domain, is equal to A . 2 в. ( frac{1}{3} ) c. ( frac{2}{3} ) D. ( frac{-1}{3} ) |
12 |
| 1538 | If the function ( g(x) ) is defined by ( boldsymbol{g}(boldsymbol{x})=frac{boldsymbol{x}^{200}}{200}+frac{boldsymbol{x}^{199}}{199}+frac{boldsymbol{x}^{198}}{198}+ldots .+ ) ( frac{x^{2}}{2}+x+5, ) then ( g^{prime}(0)= ) ( A ) в. 200 ( c .100 ) D. 5 |
12 |
| 1539 | The interval on which ( f(x)=sqrt{1-x^{2}} ) is continuous is: A. ( (0, infty) ) (i) В. ( (1, infty) ) c. [-1,1] D. ( (-infty,-1) ) |
12 |
| 1540 | If for all ( x, y ) the function ( f ) is defined by ( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})+boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y})=1 ) and ( boldsymbol{f}(boldsymbol{x})>mathbf{0}, ) then A ( cdot f^{prime}(x) ) does not exist B ( cdot f^{prime}(x)=0 ) for all ( x ) c. ( f^{prime}(0)<f^{prime}(1) ) D. None of these |
12 |
| 1541 | If ( x^{3}+y^{3}=3 a x y, ) find ( frac{d y}{d x} ) | 12 |
| 1542 | et f: R → R be a continuous function defined by 28 ex + 2ex [2010] Statement-1:f some c ER Statement -2:0<f(x) s, for all x ER (a) Statement -1 is true, Statement -2 is true ; Statement-2 is not a correct explanation for Statement -1. (b) Statement -1 is true, Statement -2 is false. (C) Statement-1 is false, Statement -2 is true . (d) Statement – 1 is true, Statement 2 is true; Statement -2 is a correct explanation for Statement -1. |
12 |
| 1543 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1} ) and ( boldsymbol{g}(boldsymbol{x})= ) ( left{begin{array}{l}max {f(t)}, quad 0 leq t leq x quad 0 leq x leq 1 \ 3-x, quad 1<x leq 2end{array}right. ) Then in the interval ( [0,2], g(x) ) is This question has multiple correct options A. Continuous for all ( x ) B. Differentiable for all ( x ) c. Discontinuous at ( x=1 ) D. Not differentiable at ( x=1 ) |
12 |
| 1544 | If the functions ( f(x) ) and ( g(x) ) are continuous on ( [a, b] ) and differentiable on ( (a, b), ) then in the interval ( (a, b), ) the equation ( left|begin{array}{ll}boldsymbol{f}^{prime}(boldsymbol{x}) & boldsymbol{f}(boldsymbol{a}) \ boldsymbol{g}^{prime}(boldsymbol{x}) & boldsymbol{g}(boldsymbol{a})end{array}right|=frac{1}{boldsymbol{a}-boldsymbol{b}}left|begin{array}{ll}boldsymbol{f}(boldsymbol{a}) & boldsymbol{f}(boldsymbol{b}) \ boldsymbol{g}(boldsymbol{a}) & boldsymbol{g}(boldsymbol{b})end{array}right| ) A. has at least one root B. has exactly one root c. has at most one root D. no root |
12 |
| 1545 | Differentiate w.r.t. ( boldsymbol{x} ) ( boldsymbol{y}=e^{3 x-2} sin 3 x ) |
12 |
| 1546 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{x}^{2}+boldsymbol{x}+1 ) and ( boldsymbol{g}(boldsymbol{x})= ) ( left{begin{array}{lll}max (boldsymbol{f}(boldsymbol{t})) & text { for } & mathbf{0} leq boldsymbol{t} leq boldsymbol{x} \ boldsymbol{3}-boldsymbol{x}+boldsymbol{x}^{2} & text { for } & mathbf{1}<boldsymbol{x} leq mathbf{2}end{array}right. ) then A. ( g(x) ) is continuous and derivable at ( x=1 ) B. ( g(x) ) is continuous but not derivable at ( x=1 ) c. ( g(x) ) is neither continuous nor derivable at ( x=1 ) D. ( g(x) ) is derivable but not continuous at ( x=1 ) |
12 |
| 1547 | 28. If f (x) is continuous and differentiable function and f(1/n)=0 n land nel, then (2005) (a) f(x)=0, x € (0, 1] (b) 10=0, 0) = 0 (C) FO)=0= f'(O), X € (0,1] (d) f0 = 0 and f’o need not to be zero |
12 |
| 1548 | Illustration 2.33 If y= cos x”, then find |
12 |
| 1549 | If ( y=frac{x sin ^{-1} x}{sqrt{1-x^{2}}}, ) prove that ( (1- ) ( left.x^{2}right) frac{d y}{d x}=x+frac{y}{x} ) |
12 |
| 1550 | 27. If y= tan x. cosx then will be … |
12 |
| 1551 | Let ( f ) and ( g ) be functions satisfying ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} boldsymbol{g}(boldsymbol{x}), boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+ ) ( boldsymbol{f}(boldsymbol{y}), boldsymbol{g}(mathbf{0})=mathbf{0}, boldsymbol{g}^{prime}(mathbf{0})=mathbf{4}, boldsymbol{g} ) and ( boldsymbol{g}^{prime} ) are continuous at 0 Then A. ( f(x)=0 ) for all ( x ) B. ( f(x)=x ) for all ( x ) c. ( f(x)=x+4 ) for all ( x ) D. ( f(x)=4 x ) for all ( x ) |
12 |
| 1552 | Find the derivative of ( boldsymbol{y}= ) ( frac{1}{4} ln frac{x^{2}-1}{x^{2}+1} ) | 12 |
| 1553 | For the function ( boldsymbol{f}(boldsymbol{x})= ) ( (x-1)(x-2)(x-3) ) in [0,4] value of c’ in Lagrange’s mean value theorem is A ( cdot 2 pm frac{2}{sqrt{3}} ) B. ( _{1-frac{sqrt{21}}{6}} ) c. ( 1+frac{sqrt{21}}{6} ) D. ( 4-2 sqrt{3} ) |
12 |
| 1554 | If ( boldsymbol{x}=boldsymbol{a}left(boldsymbol{1}-cos ^{3} boldsymbol{theta}right), boldsymbol{y}=boldsymbol{a} sin ^{3} boldsymbol{theta}, ) prove that ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}=frac{boldsymbol{3} mathbf{2}}{mathbf{2 7} boldsymbol{a}} ) at ( boldsymbol{theta}=frac{boldsymbol{pi}}{boldsymbol{6}} ) |
12 |
| 1555 | f ( boldsymbol{y}=boldsymbol{a} cos (sin 2 boldsymbol{x})+boldsymbol{b} sin (sin 2 boldsymbol{x}) ) then ( boldsymbol{y}^{prime prime}+(2 tan 2 boldsymbol{x}) boldsymbol{y}^{prime}= ) ( A ) B ( cdot 4left(cos ^{2} 2 xright) y ) c. ( -4left(cos ^{2} 2 xright) y ) ( D cdot-left(cos ^{2} 2 xright) y ) |
12 |
| 1556 | If ( sin x=frac{2 t}{1+t^{2}}, ) tany ( =frac{2 t}{1-t^{2}}, ) then ( d y ) is equal to ( A_{n} ) A . – B. 2 ( c cdot 0 ) D. |
12 |
| 1557 | If ( f(x)=tan x, ) find ( f^{prime}(x) ) and hence find ( boldsymbol{f}^{prime}left(frac{boldsymbol{pi}}{boldsymbol{4}}right) ) | 12 |
| 1558 | Find the derivative of ( sin ^{2} x ) with respect to ( x ) using product rule | 12 |
| 1559 | State whether the following statement is true or false. Enter 1 for true and 0 for false ( f(x) ) is differentiable at a point ( P, ) if there exists a unique tangent at point ( boldsymbol{P} ) |
12 |
| 1560 | ( boldsymbol{y}=sqrt{frac{boldsymbol{1}-boldsymbol{x}}{mathbf{1}+boldsymbol{x}}} ) Prove that ( left(mathbf{1}-boldsymbol{x}^{2}right) frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{y}=mathbf{0} ) |
12 |
| 1561 | If ( f(x)=frac{x^{2}-9}{x^{2}-2 x-3}, x neq 3 ) is continuous at ( x=3, ) then which one of the following is correct? A. ( f(3)=0 ) B. ( f(3)=1.5 ) c. ( f(3)=3 ) D. ( f(3)=-1.5 ) |
12 |
| 1562 | Differentiate w.r.t ( boldsymbol{x} ) ( e^{operatorname{cosec}^{2} x} ) |
12 |
| 1563 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{x}+boldsymbol{p}(boldsymbol{0} geq boldsymbol{x} geq mathbf{2}) ) where ( p ) is a constant. The value ( c ) of mean value theorem is : A ( cdot frac{sqrt{3}}{2} ) B. ( frac{sqrt{6}}{2} ) c. ( frac{sqrt{3}}{3} ) D. ( frac{sqrt{2}}{3} ) E ( cdot frac{2 sqrt{3}}{3} ) |
12 |
| 1564 | Find the derivative of the following functions from first principle ( (x-1)(x-2) ) | 12 |
| 1565 | If ( boldsymbol{f}(boldsymbol{x})= ) [ left{begin{array}{cc} frac{x^{2}}{2}, & text { if } 0 leq x leq 1 \ 2 x^{2}-3 x+frac{3}{2}, & text { if } 1<x leq 2 end{array},text { Show }right. ] that ( f ) is continuous at ( x=1 ) |
12 |
| 1566 | Solve : ( int frac{x^{2}+1}{(x+1)^{2}} d x ) | 12 |
| 1567 | Differentiate the given function w.r.t. ( x ) ( boldsymbol{y}=sqrt{e^{sqrt{x}}}, boldsymbol{x}>0 ) | 12 |
| 1568 | ( mathbf{f}_{boldsymbol{f}(boldsymbol{x})}=left{begin{array}{ll}frac{boldsymbol{x}^{2}-boldsymbol{9}}{boldsymbol{x}-mathbf{3}}+boldsymbol{alpha} & , text { for } boldsymbol{x}>mathbf{3} \ mathbf{5} & , text { for } boldsymbol{x}=mathbf{3} \ mathbf{2} boldsymbol{x}^{2}+mathbf{3} boldsymbol{x}+boldsymbol{beta} & , text { for } boldsymbol{x}<mathbf{3}end{array}right. ) is continuous at ( x=3, ) find ( alpha ) and ( beta ) |
12 |
| 1569 | If ( boldsymbol{y}=sec left(tan ^{-1} xright), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{1} ) is equal to. A ( cdot frac{1}{sqrt{2}} ) B. ( frac{1}{2} ) ( c cdot 1 ) D. ( sqrt{2} ) |
12 |
| 1570 | If ( y=sin left(m sin ^{-1} xright) ) then ( left(1-x^{2}right) y^{prime prime}-x y^{prime} ) is equal to A ( cdot m^{2} y ) в. ( m y ) c. ( -m^{2} y ) D. None of these |
12 |
| 1571 | If ( y ) is expressed in terms of a variable ( x ) as ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), ) then ( boldsymbol{y} ) is called A. Explicit function B. Implicit function c. Linear function D. Identity function |
12 |
| 1572 | Solve : ( int frac{1+x cos x}{xleft(1-x^{2} e^{2 sin x}right)} d x= ) ( k ell n sqrt{frac{x^{2} e^{2 sin x}}{1-x^{2} e^{2 sin x}}}+C ) then ( k ) is equal to |
12 |
| 1573 | If the derivatives of ( tan ^{-1}(a+b x) ) takes the value 1 at ( x=0, ) prove that ( 1+ ) ( a^{2}=b ) |
12 |
| 1574 | If ( f(x)= ) ( left{begin{array}{c}frac{1-sin x}{(pi-2 x)^{2}} cdot frac{log sin x}{log left(1+pi^{2}-4 pi x+x^{2}right)} \ kend{array}right. ) is continuous at ( x=frac{pi}{2}, ) then ( k ) is equal to. A ( cdot-frac{1}{16} ) B. ( -frac{1}{32} ) ( c cdot-frac{1}{64} ) D. ( -frac{1}{28} ) |
12 |
| 1575 | Find all points of discontinuity of ( boldsymbol{f} ) where ( f ) is defined by ( f(x)= ) ( left{begin{array}{ll}2 x+3, & x leq 2 \ 2 x-3, & x>2end{array}right. ) |
12 |
| 1576 | ( f(x)=x^{4}-3 x^{2}+4 ) in the interval [-4 4]. Is Rolle’s theorem applicable? |
12 |
| 1577 | Solve: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(sin ^{-1}left{frac{sqrt{mathbf{1}+boldsymbol{x}}+sqrt{mathbf{1}-boldsymbol{x}}}{mathbf{2}}right}right) ) A ( cdot frac{-1}{2 sqrt{1-x^{2}}} ) в. ( frac{1}{2 sqrt{1-x^{2}}} ) c. ( frac{1}{sqrt{1-x^{2}}} ) D. ( frac{-1}{sqrt{1-x^{2}}} ) |
12 |
| 1578 | If the function ( mathbf{f}(boldsymbol{x})= ) ( frac{log (1+boldsymbol{a} boldsymbol{x})-log (1-boldsymbol{b} boldsymbol{x})}{boldsymbol{x}} ) for ( boldsymbol{x} neq mathbf{0} ) is continuous at ( x=0 ) then ( f(0)= ) A ( . a-b ) B. ( a+b ) ( mathbf{c} cdot log a+log b ) D. ( log a-log b ) |
12 |
| 1579 | The set of all points of continuity of ( f o ) fo( f, ) where ( f(x)=operatorname{sgn} x ) is A ( . R sim{0} ) в. ( R sim{1,0,1} ) c. ( R sim{-1,1} ) D. none of these |
12 |
| 1580 | Consider ( mathbf{f}(mathbf{x})=left{begin{array}{l}frac{x^{2}}{|x|}, mathbf{x} neq 0 \ mathbf{0}, mathbf{x}=0end{array}right. ) Then find the continuity of the function ( f(x) ) ( mathbf{A} cdot f(mathrm{x}) ) is discontinuous every where B. ( f(x) ) is continuous only at ( x=0 ) C ( . f(x) ) is discontinuous everywhere except at ( x=0 ) D. ( f(x) ) is continuous everywhere |
12 |
| 1581 | Illustration 2.21 If y= 3x + 2x, then find dyldx. | 12 |
| 1582 | Find ( frac{d y}{d x} ) while: ( boldsymbol{x}^{boldsymbol{y}}+boldsymbol{y}^{boldsymbol{x}}=boldsymbol{a}^{boldsymbol{b}} ) |
12 |
| 1583 | ( operatorname{If}left(1+3 x+3 x^{2}right)^{20}=a_{0}+a_{1} x+ ) ( boldsymbol{a}_{2} boldsymbol{x}^{2}+boldsymbol{a}_{3} boldsymbol{x}^{3}+boldsymbol{a}_{4} boldsymbol{x}^{4}+boldsymbol{a}_{5} boldsymbol{x}^{5}+ldots ldots+ ) ( a_{40} x^{40}, ) then find the value of ( 2 a_{2}- ) ( 6 a_{3}+12 a_{4}-20 a_{5} dots dots+1560 a_{40} ) A . 3450 B. 3350 ( c .3540 ) D. 2150 |
12 |
| 1584 | Differentiate with respect to ( x: e^{(5 x+2)} ) ( A cdot 5 e^{5 x+2} ) B. ( 10 e^{5 x+2} ) ( mathbf{c} cdot 25 e^{5 x+2} ) D. ( e^{5 x+2} ) |
12 |
| 1585 | Differentiate the following ( x^{2}(3 x-2)^{4} cos x ) | 12 |
| 1586 | ( frac{d}{d x}left{cos ^{-1} x+sin ^{-1} sqrt{1-x^{2}}right}= ) A . B. c. ( frac{2}{sqrt{1-x^{2}}} ) D. ( frac{-2}{sqrt{1-x^{2}}} ) |
12 |
| 1587 | Find the derivative of ( left(5 x^{3}+3 x-1right)(x-1) ) |
12 |
| 1588 | If ( f(x)=cos ^{2} x+cos ^{2}left(frac{pi}{3}+xright) ) ( cos x cos left(frac{pi}{3}+xright) ) then ( 4 fleft(frac{pi}{8}right) ) is equal to |
12 |
| 1589 | ( boldsymbol{f}(boldsymbol{x})=(sin boldsymbol{x}+cos boldsymbol{x}) ) Find ( boldsymbol{f}^{prime}(boldsymbol{x}) ) | 12 |
| 1590 | Let then ( boldsymbol{f}(boldsymbol{x})= ) ( (x-4)(x-5)(x-6)(x-7) ) then A ( cdot f^{prime}(x)=0 ) has four real roots ( mathbf{B} cdot ) three roots of ( f^{prime}(x)=0 ) lie in (4,5)( cup(5,6) cup(6,7) ) C ( cdot ) the euation ( f^{prime}(x)= ) has only two roots ( >d ). three roots of ( f^{prime}(x)=0(3,4) cup(4,5) cup(5,6) ) D. three roots of ( f^{prime}(x)=0(3,4) cup(4,5) cup(5,6) ) |
12 |
| 1591 | Applying mean value theorem on ( boldsymbol{f}(boldsymbol{x})=log boldsymbol{x} ; boldsymbol{x} in[mathbf{1}, boldsymbol{e}] ) the value of ( boldsymbol{c}= ) ( mathbf{A} cdot log (e-1) ) в. ( e-1 ) ( mathbf{c} cdot 1-e ) ( D ) |
12 |
| 1592 | If ( boldsymbol{y}=cot ^{-1}left[frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right] ) where ( 0<x<frac{pi}{2}, ) then ( frac{d y}{d x} ) is equal to A. ( -frac{1}{2} ) B. 2 ( c cdot sin x+cos x ) ( mathbf{D} cdot sin x-cos x ) |
12 |
| 1593 | Let ( boldsymbol{f}: mathbb{R} rightarrow(0,1) ) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0,1)( ? ) This question has multiple correct options ( mathbf{A} cdot f(x)+int_{0}^{frac{pi}{2}} f(t) sin t d t ) B . ( e^{x}-int_{0}^{x} f(t) sin t d t ) C ( cdot x-int_{0}^{frac{pi}{2}-x} f(t) cos t d t ) D. ( x^{9}-f(x) ) |
12 |
| 1594 | At what point on the curve ( y=x(x-4) ) on [0,4] is the tangent parallel to ( X ) -axis. | 12 |
| 1595 | f ( f: R rightarrow R ) is defined by ( f(x)= ) ( left{begin{array}{ccc}frac{boldsymbol{x}+mathbf{2}}{boldsymbol{x}^{2}+mathbf{3} boldsymbol{x}+mathbf{2}} & boldsymbol{i f} & boldsymbol{x} in boldsymbol{R}-{-mathbf{1},-mathbf{2}} \ -mathbf{1} & boldsymbol{i f} & boldsymbol{x}=-mathbf{2} \ mathbf{0} & boldsymbol{i f} & boldsymbol{x}=-mathbf{1}end{array}right. ) then ( f ) is continuous on the set. ( A ) в. ( R-{-2} ) c. ( R-{-1} ) D. ( R-{-1,-2} ) |
12 |
| 1596 | For the function ( boldsymbol{f}(boldsymbol{x})=left|boldsymbol{x}^{2}-mathbf{5} boldsymbol{x}+boldsymbol{6}right| ) the right hand derivative ( f^{prime}(2+) ) is equal to. |
12 |
| 1597 | If it is possible to make ( f(x) ) continuous ( operatorname{at} x=2 ) then ( f(2) ) is equal to A. 0 B. 2 ( c cdot 3 ) ( D ) |
12 |
| 1598 | Function ( f(x)=left{begin{array}{ll}5 x-4 & text { for } 0<x leq 1 \ 4 x^{2}-3 x & text { for } 1<x<2 \ 3 x+4 & text { for } x geq 2end{array}right. ) A. continuous at ( x=1 ) and ( x=2 ) B. continuous at ( x=1 ) but not derivable at ( x=z ) c. continuous at ( x=2 ) but not derivable at ( x=1 ) D. none of these |
12 |
| 1599 | Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem: ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-mathbf{5} boldsymbol{x}-boldsymbol{3} ) on ( [mathbf{1}, boldsymbol{3}] ) |
12 |
| 1600 | If ( f(x)=sin x, ) find ( frac{d y}{d x} ) A ( cdot cos x ) B. ( -cos x ) c. ( cot x ) D. ( -c o t^{2} x ) |
12 |
| 1601 | If ( boldsymbol{y}=frac{1}{4}(boldsymbol{x} pm boldsymbol{A})^{2} ) Hence prove: ( boldsymbol{y}_{1}^{2}=boldsymbol{y} ) |
12 |
| 1602 | Let ( f(x) ) be a function satisfying ( f(x+ ) ( boldsymbol{y})=mathbf{f}(mathbf{x}) mathbf{f}(mathbf{y}) ) for all ( boldsymbol{x}, boldsymbol{y} in mathbf{R} ) and ( mathbf{f}(mathbf{x})=mathbf{1}+mathbf{x} mathbf{g}(mathbf{x}), ) where ( lim _{x rightarrow 0} mathbf{g}(mathbf{x})=mathbf{1} ) then ( f^{prime}(x) ) is equal to A. ( x g(x) ) в. ( mathrm{g}^{prime}(mathrm{x}) ) c. ( f(x) ) D. |
12 |
| 1603 | ff ( boldsymbol{y}=tan ^{-1} frac{1}{1+x+x^{2}}+ ) ( tan ^{-1} frac{1}{x^{2}+3 x+3}+ ) ( tan ^{-1} frac{1}{x^{2}+5 x+7}+ldots+ ) upto ( n ) terms then ( y^{prime}(0) ) is equal to A ( cdot-frac{1}{1+n^{2}} ) B. ( -frac{n^{2}}{1+n^{2}} ) c. ( frac{n}{1+n^{2}} ) D. none of these |
12 |
| 1604 | If ( y=frac{x^{4}-x^{2}+1}{x^{2}+sqrt{3} x+1} ) and ( frac{d y}{d x}=a x+b ) then the value of ( a+b ) is equal to A ( cdot cot frac{5 pi}{8} ) B. ( cot frac{5 pi}{12} ) c. ( tan frac{5 pi}{12} ) D. ( tan frac{5 pi}{8} ) |
12 |
| 1605 | Find the second derivative of ( sin 3 x cos ) ( 5 x ) |
12 |
| 1606 | Differentiate the following function w.r.t ( x ) ( sqrt{x+frac{1}{x}} ) |
12 |
| 1607 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{boldsymbol{x}}{log boldsymbol{x}^{2}+mathbf{1}}right) ) | 12 |
| 1608 | Differentiate: ( frac{e^{x}}{sin x} ) | 12 |
| 1609 | Differentiate : ( y^{x}=x^{y} ) |
12 |
| 1610 | is continuous but not derivable at ( x=0 ) A. ( m in[0,3] ) B . ( m in(0,2) ) c. ( m in(0,1] ) D. ( m=0,1 ) |
12 |
| 1611 | ( boldsymbol{i} boldsymbol{f} boldsymbol{x} sqrt{boldsymbol{1}+boldsymbol{y}}+boldsymbol{y} sqrt{boldsymbol{1}+boldsymbol{x}}= ) ( 0, ) then ( frac{d y}{d x} ) is equal to A ( cdot frac{1}{(1+x)^{2}} ) B. ( -frac{1}{(1+x)^{2}} ) c. ( frac{1}{left(1+x^{2}right)} ) D. ( frac{1}{(1+x)} ) |
12 |
| 1612 | Differentiate with respect to ‘t’ ( e^{-w t} ) | 12 |
| 1613 | Verify LMVT for the function ( f(x)=x+ ) ( frac{1}{x}, x in[1,3] ) |
12 |
| 1614 | Differentiate w.r.t. ( x ) in ( tan ^{-1}left(frac{5 x}{1-6 x^{2}}right) ) | 12 |
| 1615 | 19. Letf: R R be a function defined by f(x)=max {x,x}. The set of all points where f() is NOT differentiable is (20015) (a) {-1,1} (b) -1,0; (c) {0,1; (d){-1,0,1) 20 VL1 |
12 |
| 1616 | Find ( frac{d y}{d x}, ) if ( y=sqrt{cos (3 x+1)} ) | 12 |
| 1617 | ( operatorname{Let} f(x)=(x+1) 2^{-left(frac{1}{[x]}+frac{1}{x}right)} ) and ( boldsymbol{f}(mathbf{0})=mathbf{0} ) A. ( f ) is continuous at ( x=0 ) B. ( lim _{x rightarrow 0^{+}} f(x) ) exists C. ( lim _{x rightarrow 0^{+}} f(x) ) does not exist D. ( lim _{x rightarrow 0} f(x) neq lim _{x rightarrow 0^{-}} f(x) ) |
12 |
| 1618 | If ( 2^{x}+2^{y}=2^{x+y}, ) then find ( frac{d y}{d x} ) | 12 |
| 1619 | If ( y=sec ^{-1}left(frac{1}{2 x^{2}-1}right) ) then ( frac{d y}{d x}=? ) A ( cdot frac{-2}{left(1+x^{2}right)} ) B. ( frac{-2}{left(1-x^{2}right)} ) c. ( frac{-2}{sqrt{1-x^{2}}} ) D. none of these |
12 |
| 1620 | 7. Iff(x) is a twice differentiable function and given that (1) = 1;/2) = 4,/3)=9, then (2005) (a) F”(x) = 2 for xe (1,3) (b) f(x)=f()=5 for some x = (2,3) c) S”(x)=3 for xe (2,3) (d) {“(x)=2 for some x = (1,3) |
12 |
| 1621 | If the function ( f ) defined on ( left(-frac{1}{3}, frac{1}{3}right) ) by ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{1}{x} log _{e}left(frac{1+3 x}{1-2 x}right), & text { when } x neq 0 \ k, & text { when } x=0end{array}right. ) is continuous, then k is equal to |
12 |
| 1622 | [ begin{array}{rlr} text { If } boldsymbol{f}(boldsymbol{x}) & =frac{sin 4 boldsymbol{x}}{mathbf{5} boldsymbol{x}}+boldsymbol{a} & text { for } boldsymbol{x}>mathbf{0} \ & =boldsymbol{x}+mathbf{4}-boldsymbol{b} & text { for } boldsymbol{x}<mathbf{0} \ & =mathbf{1} & text { for } boldsymbol{x}=mathbf{0} end{array} ] is continuous at ( x=0 . ) Find ( a & b ) |
12 |
| 1623 | ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}a tan ^{-1}left(frac{1}{x-4}right) text { if } 0 leq x<4 \ b tan ^{-1}left(frac{2}{x-4}right) text { if } 4<x<6 \ sin ^{-1}(7-x)+a frac{pi}{4} quad text { if } sin ^{-1}(7-2)end{array}right. ) and ( f(4)=pi / 2 ) is continuous on (0,8) then A . (1,1) B. (1,-1) c. (-1,1) D. (-1,-1) |
12 |
| 1624 | ( int_{0}^{1} frac{e^{x}}{1+e^{2 x}} d x ) ( mathbf{A} cdot tan ^{-1} e-frac{pi}{4} ) B ( cdot tan ^{-1} e+frac{pi}{4} ) ( mathrm{C} cdot tan e-frac{pi}{4} ) D. None of these |
12 |
| 1625 | Find the value of the constant ( k ) so that the given function is continuous at the indicated point: ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}boldsymbol{k} boldsymbol{x}+mathbf{1}, text { if } boldsymbol{x} leq mathbf{5} \ boldsymbol{3} boldsymbol{x}-mathbf{5}, text { if } boldsymbol{x}>mathbf{5}end{array} text { at } boldsymbol{x}=mathbf{5}right. ) |
12 |
| 1626 | If ( x^{y}+y^{x}=a^{b} ) then show that ( frac{d y}{d x}= ) ( -left[frac{boldsymbol{y} boldsymbol{x}^{boldsymbol{y}-mathbf{1}}+boldsymbol{y}^{boldsymbol{x}} log boldsymbol{y}}{boldsymbol{x}^{boldsymbol{y}} log boldsymbol{x}+boldsymbol{x} boldsymbol{y}^{boldsymbol{x}-1}}right] ) | 12 |
| 1627 | Examine the Rolles theorem is applicable to the followng function. Find the number of points the following function is not continous? ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] ) for ( boldsymbol{x} boldsymbol{epsilon}[boldsymbol{2}, boldsymbol{2}] ) |
12 |
| 1628 | Assertion Statement -1: If ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{c}boldsymbol{x} cos boldsymbol{x} cdot sin left(frac{1}{boldsymbol{x} cos boldsymbol{x}}right), quad text { whenever de } \ mathbf{0}end{array}right. ) then ( f(x) ) is continuous Reason Statement – ( 2: lim _{x rightarrow infty} frac{sin x}{x}=0 ) A. Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement – B. Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement- c. Statement- -1 is True, Statement-2 is False D. Statement- -1 is False, Statement-2 is True |
12 |
| 1629 | Prove that ( f(x)=sin x+sqrt{3} cos x ) has maximum value at ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{6}} ) |
12 |
| 1630 | Verify Rolle’s theorem the function ( f(x)=x^{3}-4 x ) on ( [-2,2] . ) If you think it is applicable in the given interval then find the stationary point? A. Yes Rolle’s theorem is applicable and stationary point is ( x=pm frac{2}{sqrt{3}} ) B. No Rolle’s theorem is not applicable c. yes Rolle’s theorem is applicable and ( x=2 ) or -2 D. none of these |
12 |
| 1631 | ( frac{boldsymbol{d}^{20}(2 cos boldsymbol{x} cos mathbf{3} boldsymbol{x})}{boldsymbol{d} boldsymbol{x}^{20}}= ) A ( cdot 2^{20}left(cos 2 x-2^{20} cos 4 xright) ) B . ( -2^{20}left(cos 2 x+2^{20} cos 4 xright) ) C ( cdot 2^{20}left(sin 2 x+2^{20} sin 4 xright) ) D. ( 2^{20}left(sin 2 x-2^{20} sin 4 xright) ) |
12 |
| 1632 | The values of ( p ) and ( q ) for which the function ( mathbf{f}(mathbf{x})= ) ( left{begin{array}{cl}frac{sin (mathbf{p}+1) mathbf{x}+sin mathbf{x}}{mathbf{x}} & , mathbf{x}0end{array}right. ) is continuous for all ( mathbf{x} ) in ( mathbf{R} ), are A ( cdot p=frac{1}{2}, q=-frac{3}{2} ) B. ( _{mathrm{p}}=frac{5}{2}, mathrm{q}=frac{1}{2} ) ( ^{mathbf{C}} cdot_{mathrm{p}}=-frac{3}{2}, mathrm{q}=frac{1}{2} ) D. ( _{mathrm{p}}=frac{1}{2}, mathrm{q}=frac{3}{2} ) |
12 |
| 1633 | The function given by ( y=| x|-1| ) is differentiable for all real numbers except the points. B. ±1 ( c cdot 1 ) D. – |
12 |
| 1634 | if ( y=e^{x} cos x, ) prove that ( frac{d y}{d x}= ) ( sqrt{2} e^{x} cos left(x+frac{pi}{4}right) ) |
12 |
| 1635 | Find ( frac{d y}{d x}, i f x^{y}=e^{x-y} ) | 12 |
| 1636 | If ( g ) is the inverse function of ( f ) and ( f^{prime}(x)=frac{1}{1+x^{n}}, ) then ( g^{prime}(x) ) is equal to A ( cdot 1+[g(x)]^{n} ) в. ( 1-g(x) ) c. ( 1+g(x) ) D ( cdot-g(x)^{n} ) |
12 |
| 1637 | If ( boldsymbol{y}= ) ( sqrt{sin x+sqrt{sin x+sqrt{sin x+cdots cdot t o infty}}} ) then ( frac{d y}{d x} ) is A ( cdot frac{cos x}{1+2 y} ) B. ( -frac{sin x}{1-2 y} ) c. ( frac{cos x}{1-2 y} ) D. ( frac{cos x}{2 y-1} ) |
12 |
| 1638 | Differentiate the following w.r.t.x: ( sin ^{-1} x+cos ^{-1} x ) |
12 |
| 1639 | If ( boldsymbol{y}=boldsymbol{x}^{4} boldsymbol{e}^{2 x} ) then ( boldsymbol{y}_{10}(boldsymbol{0}) ) is equal to ( A cdot 2^{10} ) B . ( 315 times 2^{10} ) c. ( 195 times 2^{10} ) D. ( 315 times 2^{8} ) |
12 |
| 1640 | Find the ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) by implicit differentiation ( boldsymbol{x}^{2}-mathbf{8} boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}=boldsymbol{8} ) |
12 |
| 1641 | Evaluate: ( 1.2+2.3+3.4+ldots+ ) ( n(n+1)=frac{n}{3}(n+1)(n+2) ) |
12 |
| 1642 | ( boldsymbol{f}(boldsymbol{x})=mathbf{1} /left(mathbf{1}-boldsymbol{e}^{-mathbf{1} / boldsymbol{x}}right), boldsymbol{x} neq mathbf{0} ) If ( mathbf{f} ) is continuous at ( x=0 ) then, Find ( f(0) ) |
12 |
| 1643 | u 2 27. The function given by y=||x-1| is differentiable for all real numbers except the points (2005) (a) {0, 1,-1} (b) +1 (c) 1 (d) -1 |
12 |
| 1644 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of ( boldsymbol{x}^{3}+boldsymbol{x}^{2} boldsymbol{y}+boldsymbol{x} boldsymbol{y}^{2}+boldsymbol{y}^{3}=boldsymbol{8} mathbf{1} ) | 12 |
| 1645 | Examine the following functions for continuity. (i) ( f(x)=x-5 ) (ii) ( f(x)=frac{1}{x-5} ) ( boldsymbol{x} neq mathbf{5} ) (iii) ( f(x)=frac{x^{2}-25}{x+5}, x neq-5 ) ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{5}| ) |
12 |
| 1646 | If ( sqrt{1-x^{2}}+sqrt{1-y^{2}}=a ) find ( frac{d y}{d x} ) | 12 |
| 1647 | 19. If square of x varies as cube of y and x = 3 when y = 4, the value of y at ill be… |
12 |
| 1648 | Let f (x + y)=f(x) + f(y) for all x an is continuous at x = 0, then show that allx. ) + f() for all x and y. If the function f(x) -0, then show that f(x) is continuous at (1981 – 2 Marks) |
12 |
| 1649 | ( f(x)=left{begin{array}{cl}frac{left(1-sin ^{3} xright)}{3 cos ^{2} x}, & xfrac{pi}{2}end{array}right. ) continuous at ( x=frac{pi}{2}, ) then the value of ( left(frac{b}{a}right)^{5 / 3} ) is ( mathbf{A} ) B. ( c cdot 32 ) D. 54 |
12 |
| 1650 | If ( 2 a+3 b+6 c=0, ) then at least one root of the equation ( boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c}=mathbf{0} ) lies in the interval ( mathbf{A} cdot(0,1) ) B ( cdot(1,2) ) ( mathbf{c} cdot(2,3) ) D ( cdot(-1,0) ) |
12 |
| 1651 | The graph of any Quadratic polynomial is such that the chord joining the points ( x=a ) and ( x=b ) is parallel to the tangent line at ( boldsymbol{x}=? ) A. A.M. of ( a ) and ( b ) B. G.M. of ( a ) and ( b ) C . H.M. of ( a ) and ( b ) D. AGP |
12 |
| 1652 | What is the nature of the graph: ( y= ) ( -4 x^{2}+6 ) A. parabola not passing through origin B. Hyperbola not passing through origin c. Ellipse not passing through origin D. it is not a conic |
12 |
| 1653 | Differentiate ( log sqrt{frac{1+cos ^{2} x}{left(1-e^{2 x}right)}} ) w.r.t. ( x ) | 12 |
| 1654 | The value of ( c ) in Lagranges mean value theorem for ( boldsymbol{f}(boldsymbol{x})=boldsymbol{l} boldsymbol{x}^{2}+boldsymbol{m} boldsymbol{x}+ ) ( boldsymbol{n},(boldsymbol{l} neq mathbf{0}) ) on ( [boldsymbol{a}, boldsymbol{b}] ) is A ( cdot frac{a}{2} ) B. ( frac{b}{2} ) c. ( frac{(a-b)}{2} ) D. ( frac{(a+b)}{2} ) |
12 |
| 1655 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} int_{boldsymbol{f}(boldsymbol{x})}^{boldsymbol{g}(boldsymbol{x})} boldsymbol{h}(boldsymbol{t}) boldsymbol{d} boldsymbol{t}= ) A ( cdot g^{prime}(x) h(g(x)) ) B. ( h(g(x))-h(f(x)) ) c. ( h(g(x)) . g^{prime}(x)-h(f(x)) . f^{prime}(x) ) D. none of these |
12 |
| 1656 | Differentiate the following w.r.t. ( x: ) ( e^{x}+e^{x^{2}}+ldots+e^{x^{5}} ) | 12 |
| 1657 | Differentiate: ( log left(cos e^{x}right) ) | 12 |
| 1658 | If ( boldsymbol{y}=log _{e}left(boldsymbol{x}+log _{e}(boldsymbol{x}+ldots .)right), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( left(x=e^{2}-2, y=sqrt{2}right) ) is A ( cdot frac{1}{e^{sqrt{2}}-1} ) B. ( frac{log 2}{2 sqrt{2}left(e^{2}-1right)} ) ( ^{mathbf{c}} cdot frac{sqrt{2} log frac{e}{2}}{left(e^{2}-1right)} ) D. None of these |
12 |
| 1659 | If, ( boldsymbol{f}(boldsymbol{x})= ) ( left[begin{array}{cc}boldsymbol{x} tan ^{-1} boldsymbol{x}+sec ^{-1} frac{1}{x} & , boldsymbol{x} boldsymbol{epsilon}(-1,1)-mathbf{0} \ frac{boldsymbol{pi}}{2} & boldsymbol{i} boldsymbol{f} boldsymbol{x}=mathbf{0}end{array}right] ) then ( boldsymbol{f}^{prime}(mathbf{0}) ) is A. equal to – B. equal to 0 c. equal to 1 D. non existent |
12 |
| 1660 | Differentiate ( (x)^{tan x}+(tan x)^{x} ) w.r.t ( x ) | 12 |
| 1661 | Suppose ( f ) is differentiable at ( x=1 ) and ( lim _{h rightarrow 0} frac{1}{h} f(1+h)=5, ) then ( mathbf{A} cdot f^{prime}(1)=4 ) B ( cdot f^{prime}(1)=3 ) ( mathbf{c} cdot f^{prime}(1)=6 ) D. None of these |
12 |
| 1662 | If ( y=log (sec x+tan x), ) then ( frac{d y}{d x}= ) ( mathbf{A} cdot sec x ) в. ( frac{1}{sec x+tan x} ) C. ( log left(cos x+sec ^{2} xright) ) D. none of these |
12 |
| 1663 | If ( frac{3}{2}+y^{3}=3 a x y, ) then find ( frac{d y}{d x} ) | 12 |
| 1664 | Differentiate with respect to ( x ) : ( boldsymbol{y}=cos boldsymbol{x}+sin 2 boldsymbol{x} ) |
12 |
| 1665 | If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{2}}, ) then what is ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=boldsymbol{pi} ) equal to? ( mathbf{A} cdot(1+pi) e^{pi^{2}} ) В . ( 2 pi e^{pi^{2}} ) ( mathbf{c} cdot 2 e^{pi^{2}}^{2} ) D cdot ( e^{pi^{2}} ) |
12 |
| 1666 | ( operatorname{Let} f(x)=left{begin{array}{ll}frac{1-cos 2 x}{2 x^{2}} & : x neq 0 \ k & : x=0end{array}right. ) Then the value of ( k ) for which, ( f(x) ) will be continuous at ( x=0 ) is A . 0 B. ( c cdot 2 ) D. none of these |
12 |
| 1667 | Differentiate with respect to ( x ) : ( e^{x} log sin 2 x ) |
12 |
| 1668 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: ) ( boldsymbol{x}+boldsymbol{y}^{2}=log boldsymbol{y}+boldsymbol{x}^{2} ) |
12 |
| 1669 | Let ( [x] ) be the greatest integer function ( f(x)=frac{sin frac{1}{4} pi[x]}{[x]} ) is- This question has multiple correct options A. not continuous at any point B. continuous at ( frac{3}{2} ) c. discontinuous at 2 D. differentiable at ( frac{4}{3} ) |
12 |
| 1670 | Find the derivatives of the following functions at the indicated points. ( boldsymbol{y}=ln (2-sqrt{2 x+1}), y^{prime}(0)=? ) |
12 |
| 1671 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ; ) if ( boldsymbol{y}=tan ^{-1}left(frac{sin boldsymbol{x}}{mathbf{1}+cos boldsymbol{x}}right) ) | 12 |
| 1672 | Find ( mathbf{k} ) if ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}frac{2^{x+2}-16}{4^{x}-16}, quad text { if } quad x neq 2 \ k, quad text { if } quad x=2end{array}right. ) continuous at ( quad x=2 ) |
12 |
| 1673 | ( boldsymbol{f}(boldsymbol{x})=sec boldsymbol{x}-cos boldsymbol{x}, boldsymbol{x} boldsymbol{epsilon}(mathbf{0}, boldsymbol{pi} / mathbf{2}) ) find ( mathrm{f}^{prime}(mathbf{x}) ) |
12 |
| 1674 | If ( x=a t^{2}, y=2 a t, ) then ( frac{d^{2} y}{d x^{2}}= ) A. ( -frac{1}{t^{2}} ) в. ( frac{1}{2 a t^{3}} ) c. ( -frac{1}{t^{3}} ) D. ( -frac{1}{2 a t^{3}} ) |
12 |
| 1675 | [ f(x)=left{begin{array}{ll} frac{sin a x}{sin b x}, & x neq 0 \ frac{a}{b}, & x=0 end{array}right. ] Test the continuity of function at ( x=0 ) |
12 |
| 1676 | Discuss the applicability of Rolle’s theorem for the following function on the indicated interval: ( boldsymbol{f}(boldsymbol{x})=mathbf{3}[boldsymbol{x}] ) for ( -mathbf{1} leq boldsymbol{x} leq mathbf{1}, ) where ( [boldsymbol{x}] ) denotes the greatest integer not exceeding ( boldsymbol{x} ) |
12 |
| 1677 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) when ( boldsymbol{y}=(boldsymbol{1}+boldsymbol{x})left(boldsymbol{1}+boldsymbol{x}^{2}right)(boldsymbol{1}+ ) ( left.boldsymbol{x}^{4}right)left(boldsymbol{1}+boldsymbol{x}^{6}right) ) |
12 |
| 1678 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{e}^{sqrt{boldsymbol{x}}} ) | 12 |
| 1679 | ( operatorname{Let} f(x)=frac{sqrt{operatorname{sgn}left(alpha x^{2}+alpha x+1right)}}{cot ^{-1}left(x^{2}-alpharight)} ) ( f(x) ) is continuous for all ( x in R, ) then number of integer in the range of ( alpha ), is [Note : sgn k denotes signum function of k. ( mathbf{A} cdot mathbf{0} ) B. 4 c. 5 D. 6 |
12 |
| 1680 | For a differentiable function ( phi(x) ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{e}^{sin phi(boldsymbol{x})} ) | 12 |
| 1681 | Match the columns | 12 |
| 1682 | ( y=e^{x}+e^{-x} ) prove that ( frac{d y}{d x}=sqrt{y^{2}-4} ) | 12 |
| 1683 | Find ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} quad ) If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}-sin boldsymbol{theta}), boldsymbol{y}= ) ( boldsymbol{a}(1+cos boldsymbol{theta}) ) |
12 |
| 1684 | If ( sqrt{frac{boldsymbol{v}}{boldsymbol{mu}}}+sqrt{frac{boldsymbol{mu}}{boldsymbol{v}}}=boldsymbol{6}, ) then ( frac{boldsymbol{d} boldsymbol{v}}{boldsymbol{d} boldsymbol{mu}}= ) A. ( frac{17 mu-v}{mu-17 v} ) в. ( frac{mu-17 v}{17 mu-v} ) c. ( frac{17 mu+v}{mu-17 v} ) D. ( frac{mu+17 v}{17 mu-v} ) |
12 |
| 1685 | Find the second order derivatives of ( e^{x} sin 5 x ) |
12 |
| 1686 | If ( y=frac{(a-x) sqrt{a-x}-(b-x) sqrt{x-b}}{sqrt{a-x}+sqrt{x-b}} ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) wherever defined A ( frac{2 x-(a+b)}{2 sqrt{(a-x)(x-b)}} ) в. ( frac{x}{2 sqrt{(a-x)(x-b)}} ) c. ( frac{2 x-(a+b)}{4 sqrt{(a-x)(x-b)}} ) D. ( frac{2 x+(a+b)}{2 sqrt{(a-x)(x-b)}} ) |
12 |
| 1687 | Let ( boldsymbol{f}:(-mathbf{1}, mathbf{1}) rightarrow boldsymbol{R} ) be a differentiable function satisfying [ begin{array}{c} left(f^{prime}(x)right)^{4}=16(f(x))^{2} text { for all } x in \ (-1,1) \ f(0)=0 end{array} ] The number of such functions is ( A cdot 2 ) B. 3 ( c cdot 4 ) D. more than 4 |
12 |
| 1688 | If ( boldsymbol{y}=(boldsymbol{x}+sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}})^{n} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( A cdot y ) в. ( n y ) c. ( frac{n y}{sqrt{x^{2}+a^{2}}} ) D. ( frac{y}{sqrt{x^{2}+a^{2}}} ) |
12 |
| 1689 | 41. Let S = {TER:f(x) = x-Tem differentiable at t. Then the set Sis equal to : JED (a) {0} (b) {} (c) {0,7} (d) • (an empty set) ER:f(x) = x-T ex – 1)sin x is not ven the set S is equal to : JEE M 2018|| |
12 |
| 1690 | ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} sin boldsymbol{x} ) in the interval ( [mathbf{0}, boldsymbol{pi}] ) Is Rolle’s theorem applicable? |
12 |
| 1691 | Differentiate with respect to ( x ) : ( left(sin ^{-1} x^{4}right)^{4} ) |
12 |
| 1692 | Differentiate ( frac{tan ^{-1} x}{1+tan ^{-1} x} ) w.r.t. ( tan ^{-1} x . ) A. ( frac{1}{left(1+tan ^{-1} xright)^{2}} ) s. ( frac{1}{left(1-tan ^{-1} xright)^{4}} ) c. ( frac{1}{left(1+tan ^{-1} xright)^{4}} ) D. ( frac{1}{left(1-tan ^{-1} xright)^{2}} ) |
12 |
| 1693 | Let a function be defined as ( boldsymbol{f}(boldsymbol{x})= ) ( frac{boldsymbol{x}-|boldsymbol{x}|}{boldsymbol{x}} . ) Then ( boldsymbol{f}(boldsymbol{x}) ) is A. continuous nowhere B. continuous everywhere c. continuous for all ( x ) except ( x=1 ) D. continuous for all ( x ) except ( x=0 ) |
12 |
| 1694 | If ( f(x) ) is continuous function such that ( int_{0}^{x} f(t) d t rightarrow infty ) as ( x rightarrow infty, ) show that every line ( y=m x ) intersect the curve ( boldsymbol{y}^{2}+int_{0}^{x} boldsymbol{f}(boldsymbol{t}) boldsymbol{d} boldsymbol{t}=boldsymbol{a} ) where ( boldsymbol{a} in boldsymbol{R}^{+} ) |
12 |
| 1695 | If ( boldsymbol{y}=boldsymbol{e}^{-boldsymbol{x}} cos boldsymbol{x}, ) show that ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}= ) ( 2 e^{-x} sin x ) |
12 |
| 1696 | 1. If x+ y = 2y, then y as a function of x is (1984-3 Marks) (a) defined for all real x (b) continuous at x = 0 (c) differentiable for all x dy 1 (d) such that = for x<0 dx 3 |
12 |
| 1697 | The derivative of ( ln (x+sin x) ) with respect to ( (x+cos x) ) is A ( cdot frac{1+cos x}{(x+sin x)(1-sin x)} ) B. ( frac{1-cos x}{(x+sin x)(1+sin x)} ) c. ( frac{1-cos x}{(x-sin x)(1+cos x)} ) D. ( frac{1+cos x}{(x-sin x)(1-cos x)} ) |
12 |
| 1698 | Examine the following curve for continuity and differentiability: ( boldsymbol{y}=boldsymbol{x}^{2} ) for ( boldsymbol{x} leq mathbf{0} ; boldsymbol{y}=mathbf{1} ) for ( mathbf{0} boldsymbol{x} leq mathbf{1} ) and ( boldsymbol{y}=mathbf{1} / boldsymbol{x} ) for ( x>1 . ) Also draw the graph of the function. |
12 |
| 1699 | Find the differential coefficient of ( sin x ) by first principle. |
12 |
| 1700 | ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{sin boldsymbol{x}}{boldsymbol{x}}right) ) A. ( frac{x cos x-sin x}{x^{2}} ) B. ( frac{x cos x+sin x}{x^{2}} ) c. ( frac{x cos x+sin x}{x^{3}} ) D. ( frac{x cos x-sin x}{x^{3}} ) |
12 |
| 1701 | Find the derivate of ( e^{sqrt{2 x+1}} ) with respect to ( x ) at ( x=12 ) |
12 |
| 1702 | If ( 2 f(sin x)+f(cos x)=x, ) then ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} boldsymbol{f}(boldsymbol{x}) ) is ( mathbf{A} cdot sin x+cos x ) B. 2 c. ( frac{1}{sqrt{1-x^{2}}} ) D. none of these |
12 |
| 1703 | Find the derivative of following functions using first principle with respect to ( x ) ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sin boldsymbol{x} ) |
12 |
| 1704 | If ( boldsymbol{x}^{boldsymbol{m}} cdot boldsymbol{y}^{boldsymbol{n}}=(boldsymbol{x}+boldsymbol{y})^{boldsymbol{m}+boldsymbol{n}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A. ( frac{y}{x} ) B. ( -frac{y}{x} frac{y}{x} ) c. ( frac{m y}{x} ) D. ( frac{n y}{x} ) |
12 |
| 1705 | A Funtion ( f ) is defined as ( f(x)= ) ( frac{x^{2}-4 x+3}{x^{2}-1} ) for ( x neq 1,=2 ) for ( x=1 ) Is the function continuous at ( x=1 . ? ) A . True B. False |
12 |
| 1706 | ( x sqrt{1+y}+y sqrt{1+x}=0, ) then ( frac{d y}{d x} ) equal to ( mathbf{A} cdot 1+x ) В. ( (1-x)^{-2} ) C. ( -(1+x)^{-1} ) D. ( -(1+x)^{-2} ) |
12 |
| 1707 | If ( left(x^{2}+x y+3 y^{2}right)=1, ) what will be the value of ( (x+6 y)^{3} cdot frac{d^{2} y}{d x^{2}} ? ) |
12 |
| 1708 | If ( y=frac{sqrt{x}(2 x+3)^{2}}{sqrt{x+1}}, ) then ( frac{d y}{d x} ) is equal to A ( cdot yleft[frac{1}{2 x}+frac{4}{2 x+3}-frac{1}{2(x+1)}right] ) в. ( yleft[frac{1}{3 x}+frac{4}{2 x+3}+frac{1}{2(x+1)}right] ) c. ( yleft[frac{1}{3 x}+frac{4}{2 x+3}+frac{1}{x+1}right. ) D. None of these |
12 |
| 1709 | Let ( f(x) ) be defined as follows: [ boldsymbol{f}(boldsymbol{x})=left{begin{array}{l} boldsymbol{x}^{boldsymbol{6}}, boldsymbol{x}^{2}>mathbf{1} \ boldsymbol{x}^{3}, boldsymbol{x}^{2} leq mathbf{1} end{array}right. ] Then ( boldsymbol{f}(boldsymbol{x}) ) is? This question has multiple correct options A. continuous everywhere B. differentiable everywhere c. discontinuous at ( x=-1 ) D. not differentiable at ( x=1 ) |
12 |
| 1710 | ( frac{d}{d x} csc ^{-1}left(frac{1+x^{2}}{2 x}right) ) is equal to A ( cdot frac{-2}{left(1+x^{2}right)}, x neq 0 ) в. ( frac{2}{left(1+x^{2}right)}, x neq 0 ) c. ( frac{2left(1-x^{2}right)}{left(1+x^{2}right)left|1-x^{2}right|}, x neqpm 1,0 ) D. None of the above |
12 |
| 1711 | If ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}(mathbf{1}+|sin boldsymbol{x}|)^{frac{a}{|sin |}}, & -boldsymbol{pi} / boldsymbol{6}<boldsymbol{x}<mathbf{0} \ boldsymbol{b}, & boldsymbol{x}=mathbf{0} \ boldsymbol{e}^{frac{tan 2 x}{tan x}}, & boldsymbol{0}<boldsymbol{x}<boldsymbol{pi} / boldsymbol{6}end{array}right. ) continuous at ( x=0, ) find the values of ( a ) and ( b ) A ( cdot frac{3}{2}, e^{3 / 2} ) B. ( frac{-2}{3}, e^{-3 / 2} ) c. ( frac{2}{3}, e^{2 / 3} ) D. None of these |
12 |
| 1712 | 30. Let f (x) be differentiable on the interval (0,0) such that and lim f(x) – xf(t) – 1 for -=1 for each x > 0. Then – t-x (2007 – 3 marks) 1- X f(x) is 3 x 3 |
12 |
| 1713 | Let ( a, b ) be two distinct roots of a polynomial ( f(x) ). Then there exists at least one root lying between a and b of the polynomial A ( . f(x) ) B. ( f^{prime}(x) ) c. ( f^{prime prime}(x) ) D. ( f^{prime prime prime}(x) ) |
12 |
| 1714 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=sin ^{-1}(boldsymbol{2} boldsymbol{x} sqrt{mathbf{1}-boldsymbol{x}^{2}}) ) ( frac{mathbf{- 1}}{sqrt{mathbf{2}}}<boldsymbol{x}<frac{mathbf{1}}{sqrt{mathbf{2}}} ) |
12 |
| 1715 | If ( y=sin ^{2} x, ) then ( frac{d y}{d x}= ) ( mathbf{A} cdot cos ^{2} x ) B. ( 2 sin x ) ( mathbf{c} cdot sin x cos x ) D. ( sin 2 x ) |
12 |
| 1716 | Let ( boldsymbol{f}(boldsymbol{x})= ) ( lim _{n rightarrow infty} frac{left(x^{2}+2 x+3+sin pi xright)^{n}-1}{left(x^{2}+2 x+3+sin pi xright)^{n}+1} . ) Then A. ( f(x) ) is continuous and differentiable for all ( x in R ) B. ( f(x) ) is continuous but not differentiable for all ( x in R ) ( mathrm{c} cdot f(x) ) is discontinuous at infinite number of points D. ( f(x) ) is discontinuous at finite number of points |
12 |
| 1717 | ( operatorname{Let} f(x)=left{begin{array}{cc}-1, & -2 leq x<0 \ x^{2}-1, & 0<x leq 2end{array} ) and right. ( boldsymbol{g}(boldsymbol{x})=|boldsymbol{f}(boldsymbol{x})|+boldsymbol{f}|boldsymbol{x}| ) then the number of points which ( g(x) ) is non differentiable, is A. at most one point B. 2 c. exactly one point D. infinite |
12 |
| 1718 | ( frac{d}{d x}[log sqrt{frac{1-cos x}{1+cos x}}]= ) A . sec B. ( csc x ) c. ( operatorname{cosec} frac{x}{2} ) D. ( sec frac{x}{2} ) |
12 |
| 1719 | Using Rolle’s theorem, the equation ( boldsymbol{a}_{0} boldsymbol{x}^{boldsymbol{n}}+boldsymbol{a}_{1} boldsymbol{x}^{boldsymbol{n}-1}+ldots+boldsymbol{a}_{boldsymbol{n}}=mathbf{0} ) has atleast one root between 0 and ( 1, ) if A ( cdot frac{a_{0}}{n}+frac{a_{1}}{n-1}+ldots .+a_{n-1}=0 ) в. ( frac{a_{0}}{n-1}+frac{a_{1}}{n-2}+ldots+a_{n-2}=0 ) c. ( n a_{0}+(n-1) a_{1}+ldots .+a_{n-1}=0 ) D. ( frac{a_{0}}{n+1}+frac{a_{1}}{n}+ldots .+a_{n}=0 ) |
12 |
| 1720 | 44. If the function f defined on V2cosx-1 f(x)= cotx-1 ** k, x = 1 is continuous, then k is equal to: JEEM 2019-9 April (M) (2) 2 (6) 1 (1) 1 (2) J |
12 |
| 1721 | Assertion (A):The derivative of ( (log x)^{x} ) w.r.t ( boldsymbol{x} ) is ( (log boldsymbol{x})^{x-1}[mathbf{1}+log boldsymbol{x} log (log boldsymbol{x})] ) Reason ( (mathrm{R}): frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{boldsymbol{f}(boldsymbol{x})^{g(boldsymbol{x})}right}=boldsymbol{f}(boldsymbol{x})^{g(boldsymbol{x})} ) ( left(g(x) frac{f^{prime}(x)}{f(x)}+g^{prime}(x) log (f(x))right) ) A. Both A and R are true R is correct reason of A B. Both A and R are true R is not correct reason of c. A is true but R is false D. A is false but R is true |
12 |
| 1722 | Differentiate with respect to ( x ) : ( e^{-3 x} log (1+x) ) |
12 |
| 1723 | Write the derivative of ( f(x)=|x|^{3} ) at ( boldsymbol{x}=mathbf{0} ) |
12 |
| 1724 | Suppose that on the interval [-2,4] the function ( f ) is differentiable, ( f(2)=1 ) and ( |boldsymbol{f}(boldsymbol{x})| leq mathbf{5} . ) Find the bounding function of ( boldsymbol{f} ) on ( [-mathbf{2}, mathbf{4}], ) using LMVT. A. ( y=-5 x-9 ) and ( y=5 x+11 ) B. ( y=-5 x+9 ) and ( y=5 x+11 ) c. ( y=5 x-9 ) and ( y=5 x-11 ) D. ( y=5 x+9 ) and ( y=5 x-11 ) |
12 |
| 1725 | Illustration 2.18 If y=x”, then find dy/dx. |
12 |
| 1726 | Solve: ( lim _{x rightarrow 2} frac{x^{2}-4}{sqrt{3 x-2}-sqrt{x+2}} ) |
12 |
| 1727 | If ( f(x)=frac{1-cos a x}{1-cos b x} ) for ( x neq 0, ) is continuous at ( boldsymbol{x}=mathbf{0} ) then ( boldsymbol{f}(mathbf{0})= ) A ( cdot frac{a^{2}}{2} ) B. ( frac{a}{b^{2}} ) c. ( frac{a}{b} ) D. ( frac{a^{2}}{b^{2}} ) |
12 |
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