We provide inverse trigonometric functions practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on inverse trigonometric functions skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.
List of inverse trigonometric functions Questions
| Question No | Questions | Class |
|---|---|---|
| 1 | + 43. The number of integer x satisfying sin-‘ x – 2 cos-‘(1 – 13 – x D = is b. 2 h d a. c. 1 3 d. 4 |
12 |
| 2 | 29. sin-‘(sin 5) > x2 – 4x holds if a. x=2 – 19 – 21 b. x=2 + 19 – 2 1 c. x>2 + 19 – 21 d. xe (2 – 49 – 2 7, 2+ 9 – 2 ) |
12 |
| 3 | Range of ( boldsymbol{f}(boldsymbol{x})=sin ^{-1} boldsymbol{x}+tan ^{-1} boldsymbol{x}+ ) ( cos ^{-1} x ) is ( mathbf{A} cdot[0, pi] ) В. ( left[frac{pi}{4}, frac{3 pi}{4}right] ) с. ( [-pi, 2 pi] ) D. None of these |
12 |
| 4 | 88. The solution set of the equation sin- 1- x2 – 1 – x² + cos’x = cot-1 V –sin-‘x is a. [-1, 1]- {0} c. [-1,0) U {1} b. (0, 1] U {-1} d. [-1,1] |
12 |
| 5 | The value of ( tan ^{-1}left(frac{x}{y}right)- ) ( tan ^{-1}left(frac{x-y}{x+y}right), x, y>0 ) is ( A cdot frac{pi}{4} ) B. ( -frac{pi}{4} ) ( c cdot frac{pi}{2} ) D. ( -frac{pi}{2} ) |
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| 6 | If ( boldsymbol{alpha}=mathbf{3} sin ^{-1} frac{mathbf{6}}{mathbf{1 1}} ) and ( boldsymbol{beta}=mathbf{3} cos ^{-1} frac{mathbf{4}}{mathbf{9}} ) where the inverse trigonometric functions take only the principal values then the correct option(s) is(are) This question has multiple correct options ( mathbf{A} cdot cos beta>0 ) B. ( sin beta0 ) D. ( cos alpha<0 ) |
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| 7 | The principle value of ( cos ^{-1}left(frac{-1}{2}right) ) is A ( cdot frac{-pi}{3} ) в. ( frac{2 pi}{3} ) c. ( frac{4 pi}{3} ) D. |
12 |
| 8 | Solve ( : boldsymbol{y}=sin ^{-1}(sec boldsymbol{x}) ) | 12 |
| 9 | Prove that: ( sin ^{-1}left(frac{3}{5}right)+cos ^{-1}left(frac{12}{13}right)= ) ( sin ^{-1}left(frac{56}{65}right) ) |
12 |
| 10 | The equation ( sin ^{-1} x-cos ^{-1} x= ) ( cos ^{-1}left(frac{sqrt{3}}{2}right) ) has A. No solution B. Unique solution c. Infinite solution D. None of these |
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| 11 | If the non-zero numbers ( x, y, z ) are ( A P ) and ( tan ^{-1} x, tan ^{-1} y, tan ^{-1} z ) are also in ( A P, ) then A. ( x y=y z ) B ( cdot z^{2}=x y ) c. ( x=y=z ) D . ( x^{2}=y z ) |
12 |
| 12 | Find the projection of the vector ( hat{mathbf{i}}-widehat{boldsymbol{j}} ) on the vector ( hat{mathbf{i}}+widehat{boldsymbol{j}} ) | 12 |
| 13 | Solve: ( operatorname{cosec}^{-1}(cos x) ) is real ( , ) if A. ( x in[-1,1] ) в. ( x in R ) c. ( x ) is an odd multiple of ( frac{pi}{2} ) D. x is a multiple of ( pi ) |
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| 14 | Solve ( cot ^{-1} cot left(frac{5 pi}{4}right) ) |
12 |
| 15 | The value of ( cos left(2 cos ^{-1} 0.8right) ) is A . 0.48 B. 0.96 ( c .0 .6 ) D. 0.28 |
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| 16 | 50. The least and the greatest values of (sin x)² + (cos x)3 -13 13 a. I a 22 8²8 c. 32 8 d. none of these |
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| 17 | Show that ( tan ^{-1}left(frac{1}{2}right)+tan ^{-1}left(frac{1}{3}right)=frac{pi}{4} ) |
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| 18 | Find the number of values of ( x ) of the form ( 6 n, ) where ( n ) is an integer, in the domain of the function ( boldsymbol{f}(boldsymbol{x})= ) ( boldsymbol{x} ln |boldsymbol{x}-1|+frac{sqrt{mathbf{6 4}-boldsymbol{x}^{2}}}{sin boldsymbol{x}} ) |
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| 19 | The number of real solutions of ( tan ^{-1}(sqrt{x(x+1)}+ ) ( sin ^{-1} sqrt{left(x^{2}+x+1right)}=frac{pi}{2} ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. infinite |
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| 20 | Prove that ( sin ^{-1}left(frac{8}{17}right)+sin ^{-1}left(frac{3}{5}right)= ) ( cos ^{-1}left(frac{36}{85}right) ) |
12 |
| 21 | Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x ) [ begin{array}{c} tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} \ x y1 end{array} ] (a) ( tan ^{-1} frac{1}{4}+2 tan ^{-1} frac{1}{5}+tan ^{-1} frac{1}{6}+ ) [ tan ^{-1} frac{1}{x}=frac{pi}{4} ] (b) ( tan ^{-1}(x-1)+tan ^{-1} x+ ) [ tan ^{-1}(x+1)=tan ^{-1} 3 x ] |
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| 22 | If ( 6 operatorname{Sin}^{-1}left(x^{2}-6 x+12right)=2 pi, ) then the value of ( x, ) is |
12 |
| 23 | Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x ) [ begin{array}{c} tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} \ x y1 \ operatorname{atan}^{-1}left(sqrt{frac{a-b}{a+b}} tan frac{theta}{2}right)= \ cos ^{-1} frac{a cos theta+b}{a+b cos theta} end{array} ] |
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| 24 | If ( f(x)=2 tan ^{-1} x+ ) ( sin ^{-1}left(frac{2 x}{1+x^{2}}right), x>1, ) then ( f(5) ) is equal to: A . ( pi ) в. ( mathbf{c} cdot 4 tan ^{1}(5) ) D. ( tan ^{-1}left(frac{64}{155}right) ) |
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| 25 | Find the value of ( x ) which satisfy equation ( cos left(2 sin ^{-1} xright)=frac{1}{3} ) A ( cdot x=frac{1}{sqrt{5}} ) and ( x=frac{-1}{sqrt{5}} ) B . ( x=frac{1}{sqrt{3}} ) and ( x=frac{-1}{sqrt{3}} ) c. ( x=frac{1}{sqrt{2}} ) and ( x=frac{-1}{sqrt{2}} ) D. None of these |
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| 26 | If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2}, ) the value of ( quad x^{100}+y^{100}+z^{100}- ) ( frac{9}{x^{101}+y^{101}+z^{101}} ) is ( mathbf{A} cdot mathbf{0} ) B. 1 c. 2 D. 3 |
12 |
| 27 | 86. Which of the following is the solution set of the equation 2cos-‘x= cot” 23″-1 a. (0,1) c. (-1,0) b. (-1,1) – {0} d. [-1,1] |
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| 28 | 13 59. The value of a such that sinsinho, sin’ cara Ta are 59. The value of a such that sin the angles of a triangle is VIO – 1 a. T |
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| 29 | Find the value of ( x ) If ( , sin ^{-1} x+sin ^{-1} 2 x=frac{pi}{3} ) |
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| 30 | ( fleft(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1 ) then find the value of ( x ) |
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| 31 | Find the value of ( boldsymbol{x} ) If ( , tan ^{-1}(x-1)+tan ^{-1} x+tan ^{-1}(x+ ) 1) ( =tan ^{-1} 3 x ) |
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| 32 | Illustration 5.67 Solve sin-‘ x + sin- (1 – x) = cos-‘x. |
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| 33 | For the principal value: ( cos ^{-1}left(frac{1}{2}right)-2 sin ^{-1}left(-frac{1}{2}right) ) |
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| 34 | Find the principle value of ( tan ^{-1}(-sqrt{3}) ) ( mathbf{A} cdot pi / 3 ) в. ( -pi / 3 ) c. ( pi / 6 ) D. ( -pi / 6 ) |
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| 35 | Prove ( tan ^{-1} frac{3}{4}+tan ^{-1} frac{3}{5}-tan ^{-1} frac{8}{19}= ) ( frac{pi}{4} ) |
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| 36 | ( f tan ^{-1} frac{x-1}{x-2}+tan ^{-1} frac{x+1}{x+2}=frac{pi}{4}, ) then find the value of ( x ) |
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| 37 | If ( -1<x<0, ) then ( cos ^{-1} x ) is equal to This question has multiple correct options A ( cdot sec ^{-1} frac{1}{x} ) B . ( pi-sin ^{-1} sqrt{1-x^{2}} ) ( pi+tan ^{-1} frac{sqrt{1-x^{2}}}{x} ) D. ( cot ^{-1} frac{x}{sqrt{1-x^{2}}} ) |
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| 38 | ( sec ^{-1} 2 ) | 12 |
| 39 | If ( sin ^{-1} x=frac{pi}{5} ) for ( operatorname{somex} in[-1,1] ) then find the value of ( cos ^{-1} x ) |
12 |
| 40 | The value of ( cos left[frac{1}{2} cos ^{-1} cos left(-frac{14 pi}{5}right)right] ) is This question has multiple correct options ( ^{A} cdot cos left(-frac{7 pi}{5}right) ) B cdot ( sin left(frac{pi}{10}right) ) c. ( cos left(frac{2 pi}{5}right) ) D. ( -cos left(frac{3 pi}{5}right) ) |
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| 41 | The domain of ( boldsymbol{f}(boldsymbol{x})=frac{sin ^{-1} boldsymbol{x}}{boldsymbol{x}} ) is ( mathbf{A} cdot[-1,1] ) B. {0} ( c cdot[-1,0) ) D. None of these |
12 |
| 42 | If ( left[sin ^{-1} cos ^{-1} sin ^{-1} tan ^{-1} thetaright]=1, ) where [.] denotes the greatest integer function, the ( theta ) lies in the interval A. [tan sin cos ( 1, text { sin tan } cos sin 1] ) B. [sin tan cos ( 1, text { tan } sin cos sin 1] ) c. ( [tan sin cos 1, tan sin cos sin 1] ) D. None of these |
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| 43 | 24. The value of sin-(cos(cos(cosx) + sin-‘(sin x))), where Xe is equal to Bla b. – 1 B dond. -** |
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| 44 | The value of ( k ) if the equation ( k x+ ) ( sin ^{-1}left(x^{2}-8 x+17right)+cos ^{-1}left(x^{2}-right. ) ( 8 x+17)=frac{9 pi}{2} ) has atleast one solution is ( mathbf{A} cdot 2 pi ) в. ( pi ) ( c cdot 1 ) D. |
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| 45 | Show that: ( cos ^{-1} frac{4}{5}+cos ^{-1} frac{12}{13}= ) ( cos ^{-1} frac{33}{65} ) |
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| 46 | Illustration 5.22 Find the number of solutions of 2tan-‘tan x) = 6 – X. |
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| 47 | SE 4. Find the sum cosec. V10 + cosec- 50 + cosec – 7170 + … + cosec Vln? +1) (x2 + 2n +2). |
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| 48 | 13. Which of the following pairs of function/functions has same graph? a. y= tan (cos-‘x); y=V1- b. y = tan (cot- x); y = c. y = sin ( tan “x); y= + d. y = cos(tan-“x); y = sin(cot-‘x) |
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| 49 | Solve ( : sin ^{-1}left(frac{2 pi}{4}right) ) | 12 |
| 50 | prove that [ begin{array}{l} 2 tan ^{-1}left[tan frac{alpha}{2} tan left(frac{pi}{4}-frac{beta}{2}right)right]= \ tan ^{-1} frac{sin alpha cos beta}{cos alpha+sin beta} end{array} ] |
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| 51 | ; g(x) = sin ‘ x + cos x are 46. f(x) = tan-x+tan identical functions if a. XER c. x + [-1, 1] b. x > 0 d. x 6 (0,1] |
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| 52 | Find the principal value of: ( sin ^{-1}left(frac{sqrt{mathbf{3}}+1}{2 sqrt{2}}right) ) |
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| 53 | ( sin ^{-1}(1-x)-2 sin ^{-1} x=frac{pi}{2}, ) then ( x ) is equal to: |
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| 54 | If ( boldsymbol{y}=boldsymbol{s} boldsymbol{e} boldsymbol{c}^{-1}left[frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right]+boldsymbol{s i n}^{-1}left[frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}right] ) ( z=operatorname{cosec}^{-1}left[frac{2 x+3}{3 x+2}right]+ ) ( cos ^{-1}left[frac{3 x+2}{2 x+3}right] ) then This question has multiple correct options A. ( y=pi / 2 ) 2 B. ( z=pi / 2 ) ( mathbf{c} cdot y+z=pi ) D. ( y+z=pi / 2 ) |
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| 55 | 41. If sin ‘ x | + |cos + x1 = , then x e a. R c. [0, 1] b. [-1, 1] d. 0 |
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| 56 | The function ( boldsymbol{f}:left[-frac{mathbf{1}}{mathbf{2}}, frac{mathbf{1}}{mathbf{2}}right] rightarrowleft[-frac{boldsymbol{pi}}{mathbf{2}}, frac{boldsymbol{pi}}{mathbf{2}}right] ) defined by ( sin ^{-1}left(3 x-4 x^{3}right) ) is A. both one-one onto B. onto but not one-one c. one-one but not onto D. niether one-one nor onto |
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| 57 | Let ( cos ^{-1}left(4 x^{3}-3 xright)=a+b cos ^{-1} x ) ( x inleft(frac{1}{2}, 1right], ) then the value of ( lim _{y rightarrow a} b cos (y) ) is A. ( -frac{1}{3} ) в. -3 ( c cdot frac{1}{3} ) D. 3 |
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| 58 | Prove: ( sin ^{-1}left(frac{1}{x}right)=operatorname{cosec}^{-1} x, forall x geq 1 ) or ( x leq ) -1 ( cos ^{-1}left(frac{1}{x}right)=sec ^{-1} x, forall x geq 1 ) or ( x leq ) -1 ( tan ^{-1}left(frac{1}{x}right)=cot ^{-1} x, quad forall x>0 ) |
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| 59 | If ( sec ^{-1} frac{1}{sqrt{1-x^{2}}}+cot ^{-1}left(frac{sqrt{1-x^{2}}}{x}right)= ) ( sin ^{-1}(k) ) then ( k= ) B. ( 2 x sqrt{1-x^{2}} ) c. ( sqrt{1-x^{2}} ) D. ( 2 x ) |
12 |
| 60 | 3. Find the range of f(x) = cot- (2x – x?). | 12 |
| 61 | 14. The value of lim cos (tan-‘(sin(tan-? x))) is equal to ( xo a. -1 d. |
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| 62 | 14. If 0<a, <a2 <… <an, then prove that 17- |
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| 63 | Prove that ( 2 tan ^{-1} frac{1}{2}-tan ^{-1} frac{1}{7}=frac{pi}{4} ) Prove that ( 3 sin ^{-1} x=sin ^{-1}(3 x- ) ( left.4 x^{3}right), x inleft[frac{-1}{2}, frac{1}{2}right] ) |
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| 64 | тл 87. The number of solution of equation sin ‘x+n sin ‘(1 – x) = ****, where n > 0, m = 0, is 2 a. 3 b. 1 c. 2 d. None of these |
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| 65 | Find the value of ( x, ) if ( tan ^{-1}left(frac{2 x}{1-x^{2}}right)+cot ^{-1}left(frac{1-x^{2}}{2 x}right)= ) ( frac{2 pi}{3}, x>0 ) |
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| 66 | ( cos left(tan ^{-1} frac{3}{4}right) ) | 12 |
| 67 | If ( tan ^{-1} 2 x+tan ^{-1} 3 x=frac{pi}{4}, ) Then ( x ) is equal to A. -1 в. ( frac{1}{6} ) c. ( _{-1, frac{1}{6}} ) D. |
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| 68 | Assertion If ( a^{2}+b^{2}=c^{2}, c neq, a b neq 0 ) then the non zero solution of the equation ( sin ^{-1} frac{boldsymbol{a} boldsymbol{x}}{boldsymbol{c}}+sin ^{-1} frac{boldsymbol{b} boldsymbol{x}}{boldsymbol{c}}=boldsymbol{operatorname { s i n }}^{-1} boldsymbol{x} ) is ( pm ) Reason ( sin ^{-1} x+sin ^{-1} y=sin ^{-1}(x+y) ) 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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| 69 | Find the principle value of : ( tan ^{-1}left(frac{1}{sqrt{3}}right) ) |
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| 70 | Find the principal value of: ( cot ^{-1}left(-frac{1}{sqrt{3}}right) ) |
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| 71 | The value of ( 3 tan ^{-1} frac{1}{2}+2 tan ^{-1} frac{1}{5}+ ) ( sin ^{-1} frac{142}{65 sqrt{5}} ) is A ( cdot frac{pi}{4} ) B. ( frac{pi}{2} ) c. ( pi ) D. none of these |
12 |
| 72 | Prove that ( frac{1}{2} cos ^{-1}left(frac{1-x}{1+x}right)=tan ^{-1} sqrt{x} ) | 12 |
| 73 | Range of ( sin ^{-1} x-cos ^{-1} x ) is ( ^{mathbf{A}} cdotleft[frac{-3 pi}{2}, frac{pi}{2}right] ) В ( cdotleft[frac{-5 pi}{3}, frac{pi}{3}right. ) ( ^{mathbf{c}} cdotleft[frac{-3 pi}{2}, piright] ) D. ( [0, pi] ) |
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| 74 | If the range for ( boldsymbol{y}= ) ( left(cot ^{-1} xright)left(cot ^{-1}(-x)right) ) is ( mathbf{0}<boldsymbol{y} leq frac{boldsymbol{pi}^{boldsymbol{a}}}{boldsymbol{b}} ) Find the value of ( a+b ) A . 2 в. 4 c. 5 D. 6 |
12 |
| 75 | Find the value of ( tan ^{2}left(frac{1}{2} sin ^{-1} frac{2}{3}right) ) | 12 |
| 76 | (JEE Adv. 2013) | 12 |
| 77 | Find the principal value of the following ( cot left(tan ^{-1} x+cot ^{-1} xright) ) |
12 |
| 78 | ( sum_{m=1}^{n} tan ^{-1}left(frac{2 m}{m^{4}+m^{2}+2}right) ) is equal to A ( cdot tan ^{-1}left(n^{2}+n+1right)-frac{pi}{4} ) B cdot ( tan ^{-1}left(n^{2}+n+1right)+frac{pi}{4} ) c. ( tan ^{-1}left(n^{2}+n-1right)-frac{pi}{4} ) D ( cdot tan ^{-1}left(n^{2}-n-1right)-frac{pi}{4} ) |
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| 79 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{cos ^{-1} sin (boldsymbol{x}+boldsymbol{pi} / mathbf{3})} ) Then This question has multiple correct options ( mathbf{A} cdot fleft(frac{5 pi}{9}right)=e^{5 pi / 18} ) B ( cdot fleft(frac{8 pi}{9}right)=e^{13 pi / 18} ) ( ^{mathbf{C}} cdot fleft(-frac{7 pi}{4}right)=e^{pi / 12} ) D ( quad fleft(-frac{7 pi}{4}right)=e^{11 pi / 12} ) |
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| 80 | The inequality ( sin ^{-1} x>cos ^{-1} x ) vholds for A. all values of ( x ) B ( cdot x inleft(0, frac{1}{sqrt{2}}right) ) ( ^{c} cdot_{x inleft(frac{1}{sqrt{2}}, 1right)} ) D. no value of ( x ) |
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| 81 | Зл а. Тоr — Зл |
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| 82 | Find the range of f(x) = sin- x + tan-‘x+ Illustration 5.45 cos-‘x. |
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| 83 | Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x ) [ begin{array}{l} tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} \ x y1 end{array} ] (a) Find whether ( x=2 ) satisfies the equation [ begin{array}{c} tan ^{-1} frac{x+1}{x-1}+tan ^{-1} frac{x-1}{x}= \ tan ^{-1}(-7) end{array} ] If not, then how should the equation be re-written? (b) ( tan ^{-1} frac{4}{3}+tan ^{-1} frac{5}{6}+tan ^{-1} frac{39}{2}- ) ( pi=dots ) (c) If ( x_{1}, x_{2}, x_{3}, x_{4} ) are roots of equation [ x^{4}-x^{3} sin 2 beta+x^{2} cos 2 beta-x cos beta- ] ( sin beta=0, ) then prove that [ sum_{i=1}^{4} tan ^{-1} x_{1}=frac{pi}{2}-beta ] |
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| 84 | If ( sin left(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1, ) then ( x ) is equal to A . B. 0 ( c cdot frac{4}{5} ) D. |
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| 85 | ( cos ^{-1}left[cos left(left(-frac{17}{15}right) piright)right] ) is equal to A ( cdot frac{17 pi}{15} ) в. ( frac{13 pi}{15} ) ( c cdot frac{3 pi}{15} ) D. ( -frac{17 pi}{15} ) |
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| 86 | If ( sin ^{-1} frac{3}{x}+sin ^{-1} frac{4}{x}=frac{pi}{2}, ) then ( x ) is equal to ( A cdot 3 ) B. 5 ( c cdot 7 ) D. 11 |
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| 87 | f ( 2 tan ^{-1} x=cos ^{-1}(3 / sqrt{13}) ) then the value of ( 60 x^{4}-540 x^{2}+360 x+9261 ) is equal to |
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| 88 | ( int frac{x}{left(x^{2}+4right) sqrt{x^{2}+1}} d x= ) ( frac{1}{sqrt{k}} tan ^{-1} sqrt{frac{x^{2}+1}{3}}+c . ) what is ( k ? ) |
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| 89 | The value of ( sin ^{-1}left(sin 5 frac{pi}{3}right)= ) A ( cdot-frac{pi}{3} ) B . ( mathbf{c} cdot frac{4 pi}{3} ) D. ( frac{3 pi}{3 pi} ) |
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| 90 | Write the value of ( tan ^{-1}left(frac{1}{x}right) ) for ( x<0 ) in terms of ( cot ^{-1}(x) ) | 12 |
| 91 | If ( x ) takes negative permissible value, then ( sin ^{-1} x= ) B. ( -cos ^{-1} sqrt{1-x^{2}} ) ( mathrm{c} cdot cos ^{-1} sqrt{x^{2}-1} ) D・ ( pi-cos ^{-1} sqrt{1-x^{2}} ) |
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| 92 | If two angles of a triangle are ( tan ^{-1}(2) ) and ( tan ^{-1}(3), ) then the third angle is ( ^{A} cdot frac{pi}{4} ) в. ( c cdot frac{pi}{3} ) D. |
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| 93 | 2x 2x Hlustration 5.15 if sin ‘ = tan 2, then find the If sin , then find the tan-1_ Illustration 5.75 value of x. 1+12 I-r2, then fin |
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| 94 | Illustration 5.18 Evaluate the following: i. sin-‘(sin 10) ii. sin-‘(sin 5) iii. cos(cos 10) iv. tan-‘(tan(-6)) |
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| 95 | 10 44. If tan-x + 2 cot-‘x = ***, then x is equal to 13 – b. 3 d. 2 c. √3 |
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| 96 | Solve : ( cos ^{-1}left(log _{2} xright)=0 ) |
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| 97 | The value of ( sin left(2 sin ^{-1} mathbf{0 . 8}right) ) A ( cdot frac{1}{25} ) в. ( frac{25}{24} ) c. ( frac{24}{25} ) D. none |
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| 98 | 60. The number of solutions of the equation tan-‘(1 + x) + tan-‘(1 – x) = a. 2 b. 3 c. 1 an d. 0 |
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| 99 | 73. If sin 3 sin 28 = tan-‘x, then x = 5 + 4 cos 20 a. tan 30 c. (1/3) tano b. 3 tano d. 3 cote |
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| 100 | If ( cot ^{-1}left(frac{1}{x+1}right)+cot ^{-1}left(frac{1}{x-1}right)= ) ( tan ^{-1} 3 x-tan ^{-1} x ) then ( boldsymbol{x}= ) A. ( pm 1 / 2 ) B. ( -1, pm 1 / 3 ) c. 2,±1 D. ( -1 . pm 1 / 2 ) |
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| 101 | Illustration 5.2 Solve sin-‘x>-1. |
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| 102 | ( sin ^{-1} x+sin ^{-1} frac{1}{x}+cos ^{-1} x+ ) ( cos ^{-1} frac{1}{x}= ) A . ( pi ) в. c. ( frac{3 pi}{2} ) D. None of these |
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| 103 | ( sin left[2 cos ^{-1} cot left(2 tan ^{-1} xright)right]=0 ) if This question has multiple correct options ( mathbf{A} cdot x=-1-sqrt{2} ) B. ( x=1+sqrt{2} ) ( mathbf{c} cdot x=1-sqrt{2} ) D. ( x=sqrt{2}-1 ) |
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| 104 | The solution set of the equation ( tan ^{-1} x-cot ^{-1} x=cos ^{-1}(2-x) ) is A ( .(0,1) ) в. (-1,1) ( c cdot[1,3) ) D. (1,3) |
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| 105 | The value of ( x ) where ( x>0 ) ( tan left(sec ^{-1} frac{1}{x}right)=sin left(tan ^{-1} 2right) ) is A ( cdot sqrt{5} ) в. ( frac{sqrt{5}}{3} ) c. D. |
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| 106 | If ( x=n pi-tan ^{-1} 3 ) is a solution of the equation ( 12 tan 2 x+frac{sqrt{10}}{cos x}+1=0 ) then A. ( n ) is any integer B. n is an even integer c. ( n ) is a positive integer D. ( n ) is an odd integer |
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| 107 | Find the value of ( tan ^{-1} sqrt{3}-sec ^{-1}(-2) ) is equal to ( A ) B. ( -frac{pi}{3} ) c. D. ( frac{2 pi}{3} ) |
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| 108 | Prove that ( : tan ^{-1}left[frac{6 x-8 x^{3}}{1-12 x^{2}}right]- ) ( tan ^{-1}left[frac{4 x}{1-4 x^{2}}right]=tan ^{-1} 2 x,|2 x|<frac{1}{sqrt{3}} ) |
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| 109 | Write the principal value of ( cos ^{-1}left(frac{1}{2}right)- ) ( 2 sin ^{-1}left(-frac{1}{2}right) ) | 12 |
| 110 | If ( 4 sin ^{-1} x+cos ^{-1} x=pi, ) then find the value of ( boldsymbol{x} ) |
12 |
| 111 | Match the following | 12 |
| 112 | Two angles of a triangle are ( cot ^{-1} 2 ) and ( cot ^{-1} 3 . ) Then the third angle A. в. ( frac{3 pi}{4} ) c. D. |
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| 113 | The value of ( sin ^{-1}left(frac{3}{5}right)+tan ^{-1}left(frac{1}{7}right) ) ( A cdot 0 ) в. ( c cdot frac{pi}{3} ) D. |
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| 114 | Which of the following quantities is/are positive? This question has multiple correct options A ( cdot cos left(tan ^{-1}(tan 4)right) ) B. ( sin left(cot ^{-1}(cot 4)right) ) c. ( tan left(cos ^{-1}(cos 5)right) ) D. ( cot left(sin ^{-1}(sin 4)right) ) |
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| 115 | 37. If sin la + sin ‘ b + sin c = 1, then the value of a (1-a?) +b/(1-6?) +c/(1-c?) will be a. 2abc b. abc nh d. – abc c. – abc |
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| 116 | If ( cos ^{-1} x+cos ^{-1} y=2 pi ) then ( sin ^{-1} x+sin ^{-1} y= ) ( A cdot pi ) в. ( -pi ) ( c cdot frac{pi}{2} ) D. None of these |
12 |
| 117 | If ( cos ^{-1}left(frac{1}{x}right)=theta ) then ( tan theta= ) A ( frac{1}{sqrt{x^{2}-1}} ) B. ( sqrt{x^{2}+1} ) c. ( sqrt{1-x^{2}} ) D. ( sqrt{x^{2}-1} ) |
12 |
| 118 | Find the value of ( x .left(tan ^{-1} xright)^{2}+ ) ( left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} ) |
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| 119 | Write the value of ( cot ^{-1}(-x) ) for all ( x epsilon R ) in terms of ( cot ^{-1} x ) |
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| 120 | Evaluate ( cos ^{-1} x+ ) ( cos ^{-1}left{frac{x}{2}+frac{1}{2} sqrt{3-3 x^{2}}right}, x epsilonleft[frac{1}{2}, 1right] ) A ( cdot-frac{pi}{6} ) B. ( +frac{pi}{6} ) ( c cdot-frac{pi}{3} ) D. ( +frac{pi}{3} ) |
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| 121 | If ( x ) and ( y ) are positive and ( x y>1 ), then what is ( tan ^{-1} x+tan ^{-1} y ) equal to? A ( cdot tan ^{-1}left(frac{x+y}{1-x y}right) ) в. ( pi+tan ^{-1}left(frac{x+y}{1-x y}right) ) c. ( pi-tan ^{-1}left(frac{x+y}{1-x y}right) ) D. ( tan ^{-1}left(frac{x-y}{1+x y}right) ) |
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| 122 | Solve for ( x ) : ( tan ^{-1} x=frac{1}{2} cot ^{-1} x ) |
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| 123 | If value of ( mathbf{x} ) which satisfy equation ( left(cot ^{-1} xright)^{2}-3left(cot ^{-1} xright)+2>0 ) is ( xcot b ) Find the value of ( a+b ) A . 1 B. 2 ( c .3 ) D. 4 |
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| 124 | If ( sin ^{-1}(1-x)-2 sin ^{-1} x=pi / 2, ) then ( x ) equals- в. ( _{0, frac{1}{2}} ) c. 0 D. None of these |
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| 125 | If ( sec ^{-1} x+sec ^{-1} y+sec ^{-1} z=3 pi ) then ( boldsymbol{x} boldsymbol{y}+boldsymbol{y} boldsymbol{z}+boldsymbol{z} boldsymbol{x}= ) ( mathbf{A} cdot mathbf{0} ) B. – ( c .3 ) ( D ) |
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| 126 | The value of ( cos ^{-1}left(cos frac{5 pi}{4}right) ) is? A ( cdot frac{-3 pi}{4} ) в. ( frac{3 pi}{4} ) c. ( frac{-5 pi}{4} ) D. ( frac{5 pi}{4} ) |
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| 127 | The domain of the function ( f(x)= ) ( sqrt{cos ^{-1}left(frac{1-|x|}{2}right)} ) A ( cdot(-3,3) ) в. [-3,3] C ( cdot(-infty,-3) cup(-3, infty) ) D. ( (-infty,-3) cup(3, infty) ) |
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| 128 | 20. The sum of the solutions of the equation 2 sin-‘ Vx2 +x+1 +cos” Vx2 + x = 31 is a. 0 b. – 1 c. 1 d. 2 |
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| 129 | ( operatorname{Let} cos ^{-1}(x)+cos ^{-1}(2 x)+cos ^{-1}(3 x) ) be ( pi ) If ( x ) satisfies the equation ( a x^{3}+ ) ( b x^{2}+c x-1=0, ) then the value of ( (b-a-c) ) is |
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| 130 | Illustration 5.63 Find the value of 4 tan- tan 1 S 99 |
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| 131 | ( tan ^{-1} x+2 cot ^{-1} x=frac{2 pi}{3}, ) then the value of ( x ) is- A ( cdot sqrt{2} ) B. 3 ( c cdot sqrt{3} ) D. ( frac{sqrt{3}-1}{sqrt{3}+1} ) |
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| 132 | Solve for ( boldsymbol{x} ) ( 2 tan ^{-1}(cos x)=tan ^{-1}(2 operatorname{cosec} x) ) |
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| 133 | The ascending order of ( boldsymbol{A}= ) ( sin ^{-1}left(log _{3} 2right), B=cos ^{-1}left(log _{3}left(frac{1}{2}right)right) ) and ( C=tan ^{-1}left(log _{1 / 3} 2right) ) is A. ( mathrm{C}, mathrm{B}, mathrm{A} ) в. В, А, С c. ( mathrm{c}, mathrm{A}, mathrm{B} ) D. B, C, A |
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| 134 | The value of ( tan left{frac{1}{2} cos ^{-1}left(frac{sqrt{5}}{3}right)right} ) is A ( cdot frac{3+sqrt{5}}{2} ) B. ( 3+sqrt{5} ) c. ( frac{1}{2}(3-sqrt{5}) ) D. None of these |
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| 135 | Find the value of : [ begin{array}{l} cos left[frac{pi}{6}+2 tan ^{-1}(1)right]+ \ sin left[3 sin ^{-1}left(frac{1}{2}right)+2 cos ^{-1}left(frac{1}{2}right)right] end{array} ] |
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| 136 | Which of the following is the solution set of the equation ( 2 cos ^{-1}(x)= ) ( cot ^{-1}left(frac{2 x^{2}-1}{2 x sqrt{1-x^{2}}}right) ? ) ( mathbf{A} cdot(0,1) ) B . ( (-1,1)-{0} ) c. (-1,0) D cdot [-1,1] |
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| 137 | Evaluate: [ begin{array}{l} tan ^{-1}left(-frac{1}{sqrt{3}}right)+tan ^{-1}(-sqrt{3})+ \ tan ^{-1}left(sin left(-frac{pi}{2}right)right) end{array} ] |
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| 138 | The value of ( tan left(2 tan ^{-1} 1 / 5-pi / 4right) ) is? A. ( -7 / 17 ) в. ( +7 / 17 ) c. ( -12 / 17 ) D. ( -+2 / 17 ) |
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| 139 | f ( x y+y z+z x=1 ) then find the value of ( tan ^{-1} x+tan ^{-1} y+tan ^{-1} z ) |
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| 140 | The number of integral values of ( k ) for which the equation ( sin ^{-1} x+ ) ( tan ^{-1} x=2 k+1 ) has a solution is A . 1 B. 2 ( c .3 ) D. 4 |
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| 141 | If ( boldsymbol{x} boldsymbol{epsilon}[-1,0), ) then find the value of ( cos ^{-1}left(2 x^{2}-1right)-2 sin ^{-1} x ) ( mathbf{A} cdot-pi / 2 ) в. ( +pi / 2 ) c. ( -pi ) D. ( +pi ) |
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| 142 | Solve: ( sin left(tan ^{-1} xright),|x|<1 ) is equal to A ( cdot frac{x}{sqrt{1-x^{2}}} ) в. ( frac{1}{sqrt{1-x^{2}}} ) c. ( frac{1}{sqrt{1+x^{2}}} ) D. ( frac{x}{sqrt{1+x^{2}}} ) |
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| 143 | Simplify ( tan ^{-1}left(frac{6 x}{1-8 x^{2}}right) ) ( A cdot tan ^{-1} 2 x+tan ^{-1} 4 x ) B. ( tan ^{-1} 2 x-tan ^{-1} 4 x ) c. ( -tan ^{-1} 2 x-tan ^{-1} 4 x ) D. ( 2 tan ^{-1} 2 x-tan ^{-1} 4 x ) |
12 |
| 144 | Find the principle value of ( cos ^{-1}left[cos left(frac{7 pi}{3}right)right] ) | 12 |
| 145 | If ( cos ^{-1}left(4 x^{3}-3 xright)=2 pi-3 cos ^{-1} x ) then ( x ) lies in interval A ( cdotleft[-1,-frac{1}{2}right] ) в. ( |x|<frac{1}{2} ) ( mathbf{c} cdotleft[frac{1}{2}, 1right. ) D. None of these |
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| 146 | If ( sin ^{-1} x+4 cos ^{-1} x=pi, ) then ( x= ) A. ( 1 / 2 ) в. ( frac{1}{sqrt{2}} ) c. ( frac{sqrt{3}}{2} ) D. |
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| 147 | The principal value of ( sin ^{-1}left(frac{-1}{2}right) ) is A ( cdot frac{-pi}{6} ) В. ( frac{5 pi}{6} ) c. ( frac{7 pi}{6} ) D. none of these |
12 |
| 148 | Prove that ( operatorname{cosec}left(tan ^{-1}left(cos left(cot ^{-1}left(sec left(sin ^{-1} aright)right)right)right)right) ) ( sqrt{mathbf{3}-boldsymbol{a}^{2}}, ) where ( boldsymbol{a} in[mathbf{0}, mathbf{1}] ) |
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| 149 | ( cos ^{-1}(x)=cot ^{-1}left(frac{x}{sqrt{1-x^{2}}}right) ) where is in the common domain of the functions. A. True B. False |
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| 150 | Solve: ( tan ^{-1} 2 x+tan ^{-1} 3 x=frac{pi}{4} ) | 12 |
| 151 | Find the principal value of the following: ( operatorname{cosec}^{-1}(-sqrt{2}) ) |
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| 152 | If ( sin left(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1, ) then find the value of ( x ) A . -1 B. ( c cdot frac{1}{5} ) D. |
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| 153 | The domain of ( sin ^{-1}left[log _{2}left(x^{2} / 2right)right] ) is A . [2,1] в. [1,2] c. [-2,-1]( cup[1,2] ) D. [-2,0] |
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| 154 | Which of the following is/are the value of ( cos left[frac{1}{2} cos ^{-1}left(cos left(-frac{14 pi}{5}right)right)right] ? ) This question has multiple correct options A ( cdot cos left(-frac{7 pi}{5}right) ) B cdot ( sin left(frac{pi}{10}right) ) c. ( cos left(frac{2 pi}{5}right) ) D. ( cos left(-frac{3 pi}{5}right) ) |
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| 155 | The range of ( a r c sin x+a r c cos x+ ) ( arctan x ) is |
12 |
| 156 | The range of ( tan ^{-1} x ) A ( cdot(-pi, pi) ) B. D. ( left(-frac{pi}{2}, frac{pi}{2}right) ) |
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| 157 | ( sum_{r=1}^{n} tan ^{-1}left(frac{2^{r-1}}{1+2^{2 r-1}}right) ) is equal to: ( mathbf{A} cdot tan ^{-1}left(2^{n}right) ) B cdot ( tan ^{-1}left(2^{n}right)-frac{pi}{4} ) c. ( tan ^{-1}left(2^{n+1}right) ) ( mathbf{D} cdot tan ^{-1}left(2^{n+1}right)-frac{pi}{4} ) |
12 |
| 158 | The domain of the function ( sin ^{-1}left(log _{2}left(frac{x}{3}right)right) ) is A ( cdot frac{1}{2}, 3 ) B. ( frac{1}{2}, 4 ) c. ( frac{3}{2}, 6 ) D. ( frac{1}{2}, 2 ) |
12 |
| 159 | 42. If (sin x)2 – (cos- ‘x)2 = an? then find the range of a. dc. 1-1, 1] d. -1, |
12 |
| 160 | 1. The principal value of sin “sin 25) is (1986- 2 Marks) (a) 20 (6) 24 ( 47 (a) none |
12 |
| 161 | Write the following in the simplest form: ( tan ^{-1}left{frac{sqrt{1+x^{2}}-1}{x}right}, x neq 0 ) |
12 |
| 162 | 83. If sin ta+sin-‘b + sin c = , then av1-a? +bv1 – b +cV1-c? is equal to a. a + b + c b. a-b2c2 c. 2abc d. 4abc |
12 |
| 163 | ( cos ^{-1}left{cos left(frac{5 pi}{4}right)right} ) is given by ( ^{text {A }} cdot frac{5 pi}{4} ) в. ( frac{3 pi}{4} ) c. ( frac{-pi}{4} ) D. none of these |
12 |
| 164 | 33. The value of water cours to slevo . a. (a – B) (02 +B2) c. (a+B) (o? +B) b. (a+B) (02-B) d. none of these |
12 |
| 165 | If the domain of the function ( f(x)= ) ( sqrt{3 cos ^{-1}(4 x)-pi} ) is [a,b] then the value of ( (4 a+64 b) ) is |
12 |
| 166 | 12. The value of k (k > 0) such that the length of the longest interval in which the function f(x) = sinsin kx + cos(cos kx) is constant is te/4 is/are a. 8 b. 4 c. 12 d. 16 |
12 |
| 167 | Write the principal value of ( cos ^{-1}left(frac{1}{2}right)-2 sin ^{-1}left(-frac{1}{2}right) ) | 12 |
| 168 | Evaluate ( sin left(frac{pi}{6}+cos ^{-1} frac{1}{4}right) ) | 12 |
| 169 | If ( x>0 ) and ( cos ^{-1}left(frac{12}{x}right)+ ) ( cos ^{-1}left(frac{35}{x}right)=frac{pi}{2}, ) then ( x ) is ( A cdot 7 ) B. 39 c. 37 D. -37 |
12 |
| 170 | If ( 3 tan ^{-1} x+cot ^{-1} x=pi, ) then ( x ) equals: ( mathbf{A} cdot mathbf{0} ) B. c. -1 D. |
12 |
| 171 | The value of ( cos left{cos ^{-1}left(-frac{sqrt{3}}{2}right)+frac{pi}{6}right} ) is- ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot 0 ) D. |
12 |
| 172 | The value of ( sin ^{-1}left(cot left(sin ^{-1} sqrt{frac{2-sqrt{3}}{4}}+cos ^{-1} frac{sqrt{2}}{2}right)right. ) is A . 0 B. ( frac{pi}{2} ) ( c cdot frac{pi}{3} ) D. none of these |
12 |
| 173 | The number of values of ( x ) for which [ begin{array}{l} sin ^{-1}left(x^{2}-frac{x^{4}}{3}+frac{x^{6}}{9} dotsright)+ \ cos ^{-1}left(x^{4}-frac{x^{8}}{3}+frac{x^{12}}{9} dotsright)=frac{pi}{2}, text { where } \ 0 leq|x|<sqrt{3}, text { is } end{array} ] |
12 |
| 174 | Simplify ( tan ^{-1}left[frac{boldsymbol{a} cos boldsymbol{x}-boldsymbol{b} sin boldsymbol{x}}{boldsymbol{b} cos boldsymbol{x}+boldsymbol{a} sin boldsymbol{x}}right], ) if ( frac{boldsymbol{a}}{boldsymbol{b}} tan boldsymbol{x}>-1 ) |
12 |
| 175 | 15. If cos’x + cos’y + cos’z = 1, then a. x² + y2 + x2 + 2xyz = 1 b. 2(sin “x+sin ‘y + sin ‘z) = cos ‘x+cos ly+ cos’z c. xy + yz + zx = x + y + z-1 d. X+ – + y + – + z + – 126 |
12 |
| 176 | ( M C Q: sin left(sin ^{-1} frac{5 pi}{6}right)+ ) ( cos ^{-1}left(cos frac{5 pi}{3}right)+tan ^{-1}left(tan frac{7 pi}{3}right) ) ( A cdot frac{5 pi}{6} ) в. ( frac{pi}{3} ) c. ( frac{7 pi}{6} ) ( D cdot frac{29 pi pi pi}{6} ) |
12 |
| 177 | If ( sin ^{-1}left(frac{x}{5}right)+operatorname{cosec}^{-1}left(frac{5}{4}right)=frac{pi}{2}, ) then the value of ( x ) ( A cdot 3 ) B . 2 c. 1 ( D ) |
12 |
| 178 | ( cos left(tan ^{-1} frac{3}{4}right)=? ) ( A cdot frac{3}{5} ) ( B cdot frac{4}{5} ) ( c cdot frac{4}{9} ) D. none of these |
12 |
| 179 | 38. If a sinx – b cos= x=c, then a sin’x + b cos ‘x is equal a. O Ttab +c(b-a) atb Tab + c(a – b) atba |
12 |
| 180 | ( cot ^{-1}(sqrt{cos alpha})-tan ^{-1}(sqrt{cos alpha})=x ) then ( sin x ) is equal to ( A cdot tan ^{2} frac{alpha}{2} ) B ( cdot cot ^{2} frac{alpha}{2} ) c. ( tan alpha ) D. ( cot frac{alpha}{2} ) |
12 |
| 181 | If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) and ( boldsymbol{f}(mathbf{1})=mathbf{1}, boldsymbol{f}(boldsymbol{p}+boldsymbol{q})= ) ( boldsymbol{f}(boldsymbol{p}) cdot boldsymbol{f}(boldsymbol{q}) quad forall boldsymbol{p}, boldsymbol{q} in boldsymbol{R} operatorname{th} operatorname{en} boldsymbol{x}^{f(1)}+ ) ( boldsymbol{y}^{f(2)}+boldsymbol{z}^{f(boldsymbol{3})}-frac{boldsymbol{x}+boldsymbol{y}+boldsymbol{z}}{boldsymbol{x}^{f(1)}+boldsymbol{y}^{f(2)}+boldsymbol{z}^{f(3)}}= ) ( A cdot O ) в. ( c cdot 2 ) ( D ) |
12 |
| 182 | If ( cot ^{-1} x+tan ^{-1} 3=frac{pi}{2} ) then ( x= ) A ( -frac{1}{3} ) в. ( frac{1}{4} ) ( c cdot 3 ) D. |
12 |
| 183 | ( fleft(sin ^{-1} xright)^{2}+left(cos ^{-1} xright)^{2}=frac{17 pi^{2}}{36}, ) find ( boldsymbol{x} ) |
12 |
| 184 | 5. If cot-1 ,ne N, then the maximum value of n is a. 6 c. 5 b. 7 d. none of these |
12 |
| 185 | If ( M ) denotes the maximum value of ( left(1+sec ^{-1} xright)left(1+cos ^{-1} xright) & m ) denotes the maximum value of ( left(1+operatorname{cosec}^{-1} xright)left(1+sin ^{-1} xright), ) then ( left[frac{M}{m}right] ) is (where [.] denotes greatest integer function) |
12 |
| 186 | Prove that ( 3 sin ^{-1}=sin ^{-1}(3 x-1) ) ( left.4 x^{3}right), x epsilonleft[frac{-1}{2}, frac{1}{2}right] ) |
12 |
| 187 | The number of triplets ( (x, y, z) ) satisfies the equation ( boldsymbol{f}(boldsymbol{x}, boldsymbol{y}, boldsymbol{z})=sin ^{-1} boldsymbol{x}+ ) ( sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) is ( mathbf{A} cdot mathbf{1} ) B. 2 c. 0 D. Infinite |
12 |
| 188 | The set of values of ( x ) for which ( tan ^{-1} frac{x}{sqrt{1-x^{2}}}=sin ^{-1} x ) holds is A. ( R ) в. (-1,1) ( mathbf{c} cdot[0,1] ) D. [-1,0] |
12 |
| 189 | Find the principal value of the following ( cos ^{-1}left(cos frac{7 pi}{6}right) ) |
12 |
| 190 | Find the value of: ( sin left(2 tan ^{-1} frac{1}{4}right)+ ) ( cos left(tan ^{-1} 2 sqrt{2}right) ) |
12 |
| 191 | Assertion STATEMENT 1: Let ( boldsymbol{f}(boldsymbol{x})= ) ( sin ^{-1}left(frac{2 x}{1+x^{2}}right), f^{prime}(2)=-frac{2}{5} ) Reason STATEMENT ( 2: sin ^{-1}left(frac{2 x}{1+x^{2}}right)=pi ) ( 2 tan ^{-1} x forall x>1 ) A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1 C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE |
12 |
| 192 | for 0<x< 12, then x equals (a) 12 (b) I (c) -1/2 (20015) (d) 1 |
12 |
| 193 | Solve the equation ( 3 sin ^{-1}left(frac{2 x}{1+x^{2}}right)- ) ( 4 cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right)+2 tan ^{-1}left(frac{2 x}{1-x^{2}}right)=frac{pi}{3} ) |
12 |
| 194 | Illustration 5.61 II aj, az, az, …,a,, is an A.P. with common Ifaj, az, az, …, an is an A difference d, then prove that d (n-1)d + tan-1 1+ an-1an It aan |
12 |
| 195 | If ( tan ^{-1} frac{sqrt{1+x^{2}}-sqrt{1-x^{2}}}{sqrt{1+x^{2}}+sqrt{1-x^{2}}}=alpha, ) then ( x^{2}= ) ( mathbf{A} cdot cos 2 alpha ) B. ( sin 2 alpha ) ( c cdot tan 2 alpha ) D. ( cot 2 alpha ) |
12 |
| 196 | The value of ( sin ^{-1}left[cot left[sin ^{-1}(sqrt{frac{2-sqrt{3}}{4}})+cos ^{-1}right.right. ) is |
12 |
| 197 | Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x ) [ tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} ] ( x y1 ] Prove (a) ( sin ^{-1} frac{4}{5}+sin ^{-1} frac{5}{13}+sin ^{-1} frac{16}{65}=frac{pi}{2} ) (b) ( sin ^{-1} frac{3}{5}+sin ^{-1} frac{8}{17}=cos ^{-1} frac{36}{85} ) ( (c) sin ^{-1} frac{3}{5}+cos ^{-1} frac{12}{13}=cos ^{-1} frac{33}{65} ) |
12 |
| 198 | Solve ( : cos ^{-1} sqrt{frac{1+cos x}{2}} ) | 12 |
| 199 | The value of p for which system has a solution is A . 1 B. 2 c. 0 D. – |
12 |
| 200 | Find ( x, ) If ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} ) A . -1 B. ( c cdot 0 ) ( D ) |
12 |
| 201 | Solve ( boldsymbol{y}= ) ( tan ^{-1}left(frac{3 x-x^{3}}{1-3 x^{2}}right),-frac{1}{sqrt{3}}<x<frac{1}{sqrt{3}} ) |
12 |
| 202 | Evaluate: ( sin ^{-1}left(sin frac{5 pi}{6}right) ) |
12 |
| 203 | ( sin ^{-1} mathbf{6} boldsymbol{x}+boldsymbol{operatorname { s i n }}^{-1} boldsymbol{6} sqrt{mathbf{3}} boldsymbol{x}=-boldsymbol{pi} / 2 ) if ( mathbf{x} ) is equal to A . – ( -1 / 12 ) B. 1/6 ( c cdot 1 / 12 ) D. -1/6 |
12 |
| 204 | Which of the following is the solution set of the equation ( 2 cos ^{-1} x= ) ( cot ^{-1}left(frac{2 x^{2}-1}{2 x sqrt{1-x^{2}}}right) ) A ( .(0,1) ) B . ( (-1,1)-{0} ) c. (-1,0) D. [-1,1] |
12 |
| 205 | Write the value of ( 2 sin ^{-1} frac{1}{2}+ ) ( cos ^{1}left(-frac{1}{2}right) ) |
12 |
| 206 | Find the principal value of: ( sec ^{-1}(sqrt{2})+2 operatorname{cosec}^{-1}(-sqrt{2}) ) |
12 |
| 207 | ( 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 . 0 B. ( c cdot frac{1}{2} ) ( D ) |
12 |
| 208 | If ( 4 sin ^{-1} x+cos ^{-1} x=pi, ) then ( x ) is equal to: A ( cdot frac{1}{2} ) B. 2 c. 1 D. |
12 |
| 209 | 3. Which of the following is/are the value of COS cos(cos” (cos(-1947)] (211 d. -cos C. COS |
12 |
| 210 | Solve ( sin ^{-1}(cos x) ) |
12 |
| 211 | ( sin ^{-1} frac{3}{5}+sin ^{-1} frac{4}{5} ) is equal to A ( cdot frac{pi}{2} ) в. ( c cdot frac{pi}{4} ) D. |
12 |
| 212 | Evaluate the following: ( cos ^{-1}(cos 12) ) |
12 |
| 213 | Write the principal value of : ( left[cos ^{-1} frac{sqrt{3}}{2}+cos ^{-1}left(-frac{1}{2}right)right] ) |
12 |
| 214 | Illustration 5.7 Find the range of f(x) = 13 tan ‘x- cos’O – cos(-1). |
12 |
| 215 | f ( sum_{i=1}^{2 n} sin ^{-1} x_{i}=n pi, ) then ( sum_{i=1}^{2 n} x_{i} ) is equal to ( mathbf{A} cdot n / 2 ) B. ( 2 n ) c. ( frac{n(n+1)}{2} ) D. none of these |
12 |
| 216 | If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}+dots dots dots dots inftyright)+ ) ( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots dots dots inftyright)=frac{pi}{2} ) and ( 0<x<sqrt{2} ) then ( x= ) A ( cdot frac{1}{2} ) B. ( c cdot-frac{1}{2} ) D. – |
12 |
| 217 | The principle value of ( tan ^{-1}(-sqrt{3}) ) is A ( cdot frac{2 pi}{3} ) в. ( frac{4 pi}{3} ) c. ( frac{-pi}{3} ) D. none of these |
12 |
| 218 | If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) then the value of ( x^{9}+y^{9}+z^{9}-frac{1}{x^{9} y^{9} z^{9}} ) is equal to ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. |
12 |
| 219 | Illustration 5.33 Prove that |
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| 220 | If ( tan ^{-1} 4=4 tan ^{-1} x, ) then ( x^{5}-7 x^{3}+ ) ( 5 x^{2}+2 x ) is equal to |
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| 221 | If ( cos ^{-1} x+cos ^{-1} y=frac{pi}{2} ) then prove that ( cos ^{-1} x=sin ^{-1} y ) |
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| 222 | Solution of the equation ( tan left(cos ^{-1} xright)=sin left(cot ^{-1} frac{1}{2}right) ) A ( cdot x=pm frac{sqrt{7}}{3} ) B. ( x=pm frac{sqrt{5}}{3} ) c. ( x=pm frac{3 sqrt{5}}{2} ) D. None of these |
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| 223 | The value of ( sec left[sin ^{-1}left(sin frac{50 pi}{9}right)+cos ^{-1} cos left(frac{31 tau}{9}right.right. ) is equal to A ( cdot sec frac{10 pi}{9} ) в. ( sec 9 pi ) c. -1 D. |
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| 224 | Illustration 5.64 If (x – 1) (x² + 1) > 0, then find the value tan |
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| 225 | Prove that: ( 2 tan ^{-1}left(frac{1}{5}right)+sec ^{-1}left(frac{5 sqrt{2}}{7}right)+ ) ( 2 tan ^{-1}left(frac{1}{8}right)=frac{pi}{4} ) |
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| 226 | Evaluate the following: ( cos ^{-1}(cos 4) ) |
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| 227 | The value of ( sin ^{-1}left(cos frac{53 pi}{5}right) ) is A ( cdot frac{3 pi}{5} ) в. ( frac{-3 pi}{5} ) c. ( frac{pi}{10} ) D. ( frac{-pi}{10} ) |
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| 228 | ( tan ^{-1}left(frac{x+1}{x-1}right)+tan ^{-1}left(frac{x+1}{x}right)= ) ( tan ^{-1}(2)+pi ) |
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| 229 | Find the principal value of ( cos ^{-1}left(cos frac{7 pi}{6}right) ) | 12 |
| 230 | Write the principal value of ( sin ^{-1}left(-frac{1}{2}right) ) ( ? ) | 12 |
| 231 | Solve ( cos left[tan ^{-1}left[sin left(cot ^{-1} xright)right]right] ) A ( cdot sqrt{frac{x^{2}+2}{x^{2}+3}} ) B. ( sqrt{frac{x^{2}+2}{x^{2}+1}} ) c. ( sqrt{frac{x^{2}+1}{x^{2}+2}} ) D. None of these |
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| 232 | The domain of the function ( sin ^{-1} 2 x ) is: ( mathbf{A} cdot[0,1] ) B . [-1,1] c. [-2,2] D. ( left[frac{-1}{2}, frac{1}{2}right] ) |
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| 233 | The value of ( cos ^{-1}left(-frac{1}{2}right)+ ) ( sin ^{-1}left(-frac{sqrt{mathbf{3}}}{mathbf{2}}right) ) is ( A cdot frac{pi}{3} ) B. c. ( frac{2 pi}{3} ) D. none of these |
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| 234 | The number of integer ( boldsymbol{x} ) satisfying ( sin ^{-1}|x-2|+cos ^{-1}(1-|3-x|)=frac{pi}{2} ) is ( mathbf{A} cdot mathbf{1} ) B . 2 ( c cdot 3 ) ( D ) |
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| 235 | For ( tan ^{-1}left(frac{1-x}{1+x}right), 0 leq x leq 1 ) What is the sum of the smallest and the largest values of function. A ( cdot frac{pi}{4} ) в. ( frac{pi}{2} ) c. ( frac{3 pi}{4} ) D. ( frac{3 pi}{2} ) |
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| 236 | Evaluate ( cos ^{-1}left(cos left(frac{pi}{4}right)right) ) A. B. ( -frac{pi}{4} ) c. ( frac{3 pi}{4} ) D. ( -frac{3 pi}{4} ) |
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| 237 | If ( f:left(-frac{pi}{2}, frac{pi}{2}right) rightarrow(-infty, infty) ) is defined by ( f(x)=tan x, ) then ( f^{-1}(2+sqrt{3})= ) ( A cdot frac{pi}{12} ) в. c. ( frac{5 pi}{12} ) D. |
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| 238 | The value of ( tan left{2 tan ^{-1} frac{1}{5}-frac{pi}{4}right} ) is A . B. ( c cdot frac{7}{17} ) D. none of these |
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| 239 | Prove that ( cot ^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right)= ) ( frac{x}{2} ; x inleft(0, frac{pi}{4}right) ) |
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| 240 | Prove the following: ( cos ^{-1}left(frac{12}{13}right)+sin ^{-1}left(frac{3}{5}right)= ) ( sin ^{-1}left(frac{56}{65}right) ) |
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| 241 | f ( sin left{sin ^{-1} frac{1}{5}+cos ^{-1} xright}=1, ) then ( x ) is equal to ( A cdot 1 ) B. ( c cdot frac{4}{5} ) D. |
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| 242 | The domain of ( sin ^{-1}[x], ) where ( [x] ) is greatest integer function, given by A ( cdot[-1,1] ) B . [-1,2) ( mathbf{c} cdot{-1,0,1} ) D. None of these |
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| 243 | From the mast head of a ship the angle of depression of a boat is ( tan ^{-1}left(frac{5}{12}right) ) If the mast head is 100 metres. The distance of the boat from the ship is ( mathbf{A} cdot 120 m ) B. ( 180 m ) c. ( 240 m ) D. None of these |
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| 244 | The principal value of ( sin ^{-1}left{sin frac{5 pi}{6}right} ) is A ( cdot frac{pi}{6} ) в. ( frac{5 pi}{6} ) c. ( frac{7 pi}{6} ) D. none of these |
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| 245 | Show that: ( sin ^{-1}left(frac{12}{13}right)+cos ^{-1}left(frac{4}{5}right)+ ) ( tan ^{-1}left(frac{63}{16}right)=pi ) |
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| 246 | 17. If tan-‘(x²+3[x] – 4) + cot -‘ (4:+ sin ‘ sin 14) = -, then the value of sin ‘sin 2x is a. 6-21 b. 21-6 c. 1-3 c. 3 – 1 |
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| 247 | i) Solve for ( x: tan ^{-1}(x-1)+ ) ( tan ^{-1} x+tan ^{-1}(x+1)=tan ^{-1} 3 x ) ii) Prove that ( tan ^{-1}left(frac{6 x-8 x^{3}}{1-12 x^{2}}right)- ) ( tan ^{-1}left(frac{4 x}{1-4 x^{2}}right)=tan ^{-1} 2 x ;|2 x|< ) ( frac{1}{sqrt{3}} ) |
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| 248 | ( sin ^{-1}(sin (4))=? ) | 12 |
| 249 | If ( 0 leq x leq 1, ) then ( sin left{tan ^{-1} frac{1-x^{2}}{2 x}+cos ^{-1} frac{1-x^{2}}{1+x^{2}}right} ) equal to ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot 0 ) D. none of these |
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| 250 | The formula ( cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right)= ) ( 2 tan ^{-1} x ) holds only for ( mathbf{A} cdot x in R ) B . | ( x mid leq 1 ) c. ( x in(-1,1) ) D・ ( x in(0, infty) ) |
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| 251 | If ( theta=sin ^{-1} x+cos ^{-1} x-tan ^{-1} x, x geq ) ( 0, ) then the smallest interval in which ( theta ) lies is- ( ^{mathrm{A}} cdot frac{pi}{2} leq theta leq frac{3 pi}{4} ) В. ( quad 0 leq theta leq frac{pi}{4} ) ( ^{mathrm{c}}-frac{pi}{4} leq theta leq 0 ) ( stackrel{pi}{4} leq theta leq frac{pi}{2} ) |
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| 252 | Find ( int tan ^{-1} frac{x}{sqrt{a^{2}-x^{2}}} d x ;|x|langle a ) |
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| 253 | If ( a x+bleft(sec left(tan ^{-1} xright)right)=c ) and ( a y+ ) ( bleft(sec left(tan ^{-1} yright)right)=c, ) then the value of ( frac{boldsymbol{x}+boldsymbol{y}}{mathbf{1}-boldsymbol{x} boldsymbol{y}} ) is, A ( cdot frac{2 a b}{a^{2}-c^{2}} ) в. ( frac{2 a c}{a^{2}-c^{2}} ) c. ( frac{c^{2}-b^{2}}{a^{2}+b^{2}} ) D. none of these |
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| 254 | 2. The trigonometric equation sin – x=2 sina has a solution for [2003] (a) Jelz te (b)}<lakte (©) all real values of a (d) lal< |
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| 255 | Solve: ( sin ^{-1}(cos x) ) |
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| 256 | Write the principal values of the following: ( sin ^{-1}left(-frac{1}{2}right)+cos ^{-1}left(-frac{1}{2}right) ) |
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| 257 | If ( tan ^{-1} x+tan ^{-1} y=frac{2 pi}{3}, ) then ( cot ^{-1} x+cot ^{-1} y ) is equal to ( ^{A} cdot frac{pi}{2} ) в. c. D. ( frac{sqrt{3}}{2} ) E . ( pi ) |
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| 258 | 6x – Illustration 5.77 If cos-1 find the values of x. 1+ 9×2 – 5 + 2 tan-‘3x, then |
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| 259 | If ( tan alpha=frac{m}{m+1} ) and ( tan beta=frac{1}{2 m+1} ) find the possible values of ( (boldsymbol{alpha}+boldsymbol{beta}) ) ( A cdot 30 ) B. 90 ( c cdot 60 ) D. ( 45^{circ} ) |
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| 260 | Find the value of ( sin ^{-1}left(cos frac{33 pi}{5}right) ) | 12 |
| 261 | Write ( tan ^{-1}left[frac{sqrt{1+x^{2}}-1}{x}right], x neq 0 ) the simplest form. |
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| 262 | 1+r2 45. The number of solutions of the equation cos – cos x = + sin ‘x is b. 1 c. 2 d. 3 a. 0 |
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| 263 | Show that ( sin ^{-1} frac{12}{13}+cos ^{-1} frac{4}{5}+ ) ( cot ^{-1} frac{63}{16}=frac{pi}{2} ) |
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| 264 | Assertion If ( boldsymbol{x}^{2}-boldsymbol{p} boldsymbol{x}+boldsymbol{q}=mathbf{0} ) where ( boldsymbol{p} ) is twice the tangent of the arithmetic mean of ( sin ^{-1} x ) and ( cos ^{-1} x ; ) q is the geometric mean of ( tan ^{-1} x ) and ( cot ^{-1} x ) then ( x=1 ) Reason ( tan left(sin ^{-1} x+cos ^{-1} xright)=1 ) 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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| 265 | The following graph represents: ( mathbf{A} cdot cot ^{-1}(x+1) ) ( B cdot sin ^{-1}left(x^{4}+1right) ) ( mathbf{c} cdot tan ^{-1}left(x^{3}+1right) ) ( D cdot cos ^{-1}left(x^{2}+1right) ) |
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| 266 | Find ( y=left(sin ^{-1} xright)^{x^{2}}, ) then ( y^{prime}(0)=? ) | 12 |
| 267 | If the equation ( sin ^{-1}left(x^{2}+x+1right)+ ) ( cos ^{-1}(a x+1)=frac{pi}{2} ) has exactly two distinct solutions then value of ( a ) could not be This question has multiple correct options |
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| 268 | ( cos left[2 sin ^{-1} sqrt{frac{1-x}{2}}right]= ) ( A cdot x ) B. ( frac{1}{x} ) ( c cdot 2 x ) D. 3x |
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| 269 | Find the value of ( sin left(cot ^{-1} xright) ) | 12 |
| 270 | ( sec ^{2}left(tan ^{-1} 2right)+operatorname{cosec}^{2}left(cot ^{-1} 3right) ) is equal to ( mathbf{A} cdot mathbf{5} ) B. 13 c. 15 D. 6 |
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| 271 | Calculate ( (192-214) ) ( sin ^{-1}+2 tan ^{-1}(-sqrt{3}) ) |
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| 272 | If the equation ( sin ^{-1}left(x^{2}+x+1right)+ ) ( cos ^{-1}(lambda x+1)=frac{pi}{2} ) has exactly two solutions for ( lambda epsilon[a, b) ) then the value of ( (a+b) ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 3 |
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| 273 | Find the value of ( cos ^{-1}left(cos frac{2 pi}{3}right)+ ) ( sin ^{-1}left(sin frac{2 pi}{3}right) ) |
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| 274 | Find ( sin ^{-1}left(frac{sqrt{mathbf{3}}+1}{2 sqrt{2}}right)= ) | 12 |
| 275 | Let ( cos ^{-1}left(4 x^{3}-3 xright)=a+b cos ^{-1} x ) ( x epsilonleft[-frac{1}{2},-1right) ) then the value of ( a+b pi ) is A ( .2 pi ) в. ( 3 pi ) ( c ) D. ( -2 pi ) |
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| 276 | Write the value of ( 2 sin ^{-1} frac{1}{2}+ ) ( cos ^{1}left(-frac{1}{2}right) ) |
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| 277 | Solve for ( boldsymbol{x}: cos ^{-1}left(frac{boldsymbol{x}^{2}-mathbf{1}}{boldsymbol{x}^{2}+mathbf{1}}right)+ ) ( frac{mathbf{1}}{mathbf{2}} tan ^{-1} frac{mathbf{2} boldsymbol{x}}{mathbf{1}-boldsymbol{x}^{2}}=frac{mathbf{2} boldsymbol{pi}}{mathbf{3}} ) |
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| 278 | Illustration 5.34 Prove that T sinu {vl 45.7 -5) 5.5 * 0<x<1 sin-1 + sin-' x 1,0<x<1 2 |
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| 279 | Solve the equation ( tan ^{-1}left[frac{1-x}{1+x}right]= ) ( frac{1}{2} tan ^{-1} x,(x>0) ) |
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| 280 | 15. Let f (x) = sin x + cos x + tan x + sin- ‘ x + cos- ‘ x + tan- ‘x. Then find the maximum and minimum values of f(x) |
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| 281 | Find the value of ( x ) which satisfy euqation ( : tan ^{-1} 2 x+tan ^{-1} 3 x=pi / 4 ) A. ( x=-1 / 6 ) В. ( x=+1 / 6 ) c. ( x=-1 ) D. ( x=+1 ) |
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| 282 | If the equation ( sin ^{-1}left(x^{2}+x+1right)+ ) ( cos ^{-1}(lambda x+1)=frac{pi}{2} ) has exactly two solutions, then ( lambda ) can not have the integral value(s) This question has multiple correct options A . -1 B. 0 c. 1 D. 2 |
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| 283 | Solve: ( cos ^{-1}left(frac{x-x^{-1}}{x+x^{-1}}right) ) |
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| 284 | If ( sin ^{-1}left(frac{2 a}{1+a^{2}}right)-cos ^{1}left(frac{1-b^{2}}{1+b^{2}}right)= ) ( tan ^{-1}left(frac{2 x}{1-x^{2}}right), ) then what is the value of ( x ? ) ( A cdot frac{a}{b} ) B. ( a b ) ( c cdot frac{b}{a} ) D. ( frac{a-b}{1+a b} ) |
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| 285 | Find the value of ( x ) for which; ( operatorname{cosec}^{-1}(cos x) ) is real A. ( x=-pi ) B . ( x=pi ) c. ( x=2 pi ) D. All of the above |
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| 286 | ( cos ^{-1}left(frac{pi}{3}+sec ^{-1}(-2)right)= ) ( A cdot-1 ) B. ( c cdot 0 ) D. None of these |
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| 287 | The value of ( sin ^{-1}left[cot left(sin ^{-1} sqrt{left(frac{2-sqrt{3}}{4}right)}right)+cos ^{-1}right. ) is A . 0 в. ( frac{pi}{4} ) ( c cdot frac{pi}{6} ) D. ( frac{pi}{2} ) |
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| 288 | 62. If cot-x + cot ‘y + cot’z = , x, y, z > 0 and xy < 1, then x + y + z is also equal to b. XYZ 1 1 1 a. -+-+- X Y Z c. xy + yz + zx d. none of these |
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| 289 | Prove that ( sin ^{-1}(2 x sqrt{1-x^{2}})= ) ( 2 cos ^{-1} x, frac{1}{sqrt{2}} leq x leq 1 ) |
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| 290 | ( tan ^{-1}(2)+tan ^{-1}(3)= ) | 12 |
| 291 | Find the principal value of: ( cos ^{-1}left(sin frac{4 pi}{3}right) ) |
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| 292 | 64. If tan- x + tan ‘y + tan-‘z = “, then N a. x + y + z – xyz = 0 c. xy + yz + zx + 1 = 0 to b. x+y+z + xyz = 0 18 d. xy + yz + zx – 1 = 0 |
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| 293 | If ( frac{1}{sqrt{2}}<x<1, ) then ( cos ^{-1} x+ ) ( cos ^{-1}left(frac{x+sqrt{1-x^{2}}}{sqrt{2}}right) ) is equal to A ( cdot 2 cos x^{-1} ) B. ( 2 cos ^{-1} x ) ( c cdot frac{pi}{4} ) ( D ) |
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| 294 | 5. Which of the following quantities is/are positive? a. cos(tan-‘(tan 4)) b. sin(cot-‘(cot 4)) c. tan(cos(cos 5)) d. cot(sin-‘(sin 4)) 1) |
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| 295 | Illustration 5.53 If x;€ [0, 1] Vi = 1, 2, 3, …, 28 then find the maximum value of Vsin x ſcos x2 + ſsin x2 cos xz. +/sin- xz ſcos x4 + … +& sin ‘ x 28 /costx. |
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| 296 | If ( sin ^{-1} x=frac{pi}{5}, ) for some ( x in(-1,1) ) then find the value of ( cos ^{-1} x ) |
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| 297 | 7. Let tan “y= tan “x + tan” (1232) hen a value of y is: [JEE M 2015) who 3x – X3 3x + x 1+3×2 1+ 3×2 3x + x 3x – x3 1- 32 m (d) 1 – 3x² |
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| 298 | 2. Find all the solution of 4 cos xsin x – 2 sinx = 3 sin x (1983 – 2 Marks |
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| 299 | If ( frac{1}{2} sin ^{-1}left[frac{3 sin 2 theta}{5+4 cos 2 theta}right]=tan ^{-1} x ) then ( boldsymbol{x}= ) ( mathbf{A} cdot tan 3 theta ) B. ( 3 tan theta ) c. ( (1 / 3) tan theta ) D. ( 3 cot theta ) |
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| 300 | 90. The equation 3 cos-“x – Ax-* = 0 has a. one negative solution b. one positive solution c. no solution d. more than one solution |
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| 301 | 16. If sin-la- + … + cos'(1 + b + b2 + …) = then 2a-3 a. b = 3a-2 b. b= 3a 2a c. a=2-31 c. a= d. d. a=3–26 2-36 a= 3-26 |
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| 302 | 65. If x2 + y2 + z2 = r2, then tan-“| + tan (ar) + tan-” ) is equal to To a. b. c. O d. none of these |
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| 303 | Find the value of ( x ) if ( sin left{sin ^{-1} frac{1}{5}+cos ^{-1} xright}=1 ) |
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| 304 | Find the value of ( sin ^{-1}left[sin left(-frac{17 pi}{8}right)right] ) | 12 |
| 305 | The solution of ( sin ^{-1}left(frac{2 a}{1+a^{2}}right)-cos ^{-1}left(frac{1-b^{2}}{1+b^{2}}right)= ) ( 2 tan ^{-1} x ) A ( frac{a-b}{1-a b} ) B. ( frac{1+a b}{a-b} ) c. ( frac{a b-1}{a+b} ) D. ( frac{a-b}{1+a b} ) |
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| 306 | ( sin ^{-1}left(sin left(frac{2 x^{2}+4}{1+x^{2}}right)right)<pi-3 ) if A . ( -1 leq x leq 0 ) в. ( 0 leq x leq 1 ) c. ( -1<x1 ) |
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| 307 | ( operatorname{Let} f(x)=cos left(tan ^{-1} 2 xright) ) ( sin left{tan ^{-1}left(frac{1}{2 x+1}right)right} ) and ( boldsymbol{a}=cos left(tan ^{-1}left(sin left(cot ^{-1} 2 xright)right)right) ) and ( boldsymbol{b}= ) ( cos left(frac{pi}{2}+cos ^{-1} 2 xright) ) The value of ( x ) for which ( f(x)=0 ) is ( A cdot-frac{1}{4} ) B. ( frac{1}{4} ) c. 0 D. ( frac{1}{2} ) |
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| 308 | Evaluate the following: ( tan ^{-1}(tan 2) ) |
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| 309 | Value of ( tan ^{-1}left{frac{sin 2-1}{cos 2}right} ) is A ( cdot frac{pi}{2}-1 ) B cdot ( 1-frac{pi}{4} ) c. ( 2-frac{pi}{2} ) D. ( frac{pi}{4}-1 ) |
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| 310 | ( frac{1}{2} tan ^{-1} x=cos ^{1}left{frac{1+sqrt{1+x^{2}}}{2 sqrt{1+x^{2}}}right}^{frac{1}{2}} ) | 12 |
| 311 | The number of real solution of the equation ( tan ^{-1} sqrt{x^{2}-3 x+2}+ ) ( cos ^{-1} sqrt{4 x-x^{2}-3}=pi ) is ( A ) B. 2 ( c cdot c ) D. infinite |
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| 312 | Consider the following: 1. ( sin ^{-1} frac{4}{5}+sin ^{-1} frac{3}{5}=frac{pi}{2} ) 2. ( tan ^{-1} sqrt{3}+tan ^{-1} 1=-tan ^{-1}(2+ ) ( sqrt{mathbf{3}}) ) Which of the above is/are correct? A. 1 only B. 2 only c. Both 1 and 2 D. Neither 1 nor 2 |
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| 313 | If ( cos ^{-1}left(frac{1}{x}right)=theta ) then the value of ( tan theta ) is A ( cdot frac{1}{sqrt{x^{2}-1}} ) B. ( sqrt{x^{2}-1} ) c. ( sqrt{1-x^{2}} ) D. ( sqrt{1+x^{2}} ) |
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| 314 | Evaluate : ( tan left(2 tan ^{-1} frac{1}{5}right) ) A ( cdot frac{5}{6} ) в. ( frac{5}{12} ) c. ( frac{7}{12} ) D. none of these |
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| 315 | The number of solutions of the equation ( tan ^{-1}left(frac{1}{2 x+1}right)+tan ^{-1}left(frac{1}{4 x+1}right)= ) ( tan ^{-1}left(frac{2}{x^{2}}right) ) is ( A ) B. ( c cdot 2 ) D. 3 |
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| 316 | Find the value of ( cos ^{-1}left(cos frac{13 pi}{6}right) ) | 12 |
| 317 | Solve: ( tan ^{-1}left(frac{2 x}{1-x^{2}}right) ) |
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| 318 | If ( boldsymbol{x}<mathbf{0}, ) then ( tan ^{-1} boldsymbol{x} ) is equal to This question has multiple correct options A ( cdot-pi+cot ^{-1} frac{1}{x} ) B. ( sin ^{-1} frac{x}{sqrt{1+x^{2}}} ) c. ( -cos ^{-1} frac{1}{sqrt{1+x^{2}}} ) D. ( -operatorname{cosec}^{-1} frac{sqrt{1+x^{2}}}{x} ) |
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| 319 | Evaluate: ( tan ^{-1}(1)+cos ^{-1}left(frac{1}{2}right)+ ) ( sin ^{-1}left(frac{1}{2}right) ) which lies in the interval ( [mathbf{0}, boldsymbol{pi}] ) |
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| 320 | tan-‘ x tan+ 2x tan- 3x 85. Let tan- 3x tan-‘x tan- 2×1 = 0, then the number tan 2x tan- 3x tan- x of values of x satisfying the equation is a. 1 b. 2 c. 3 d. 4 |
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| 321 | Find the value of ( cos ^{-1}left(cos frac{5 pi}{3}right)+ ) ( sin ^{-1}left(sin frac{5 pi}{3}right) ) |
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| 322 | ( tan left[2 tan ^{-1} frac{1}{5}-frac{pi}{4}right]=? ) ( A cdot frac{7}{17} ) в. ( frac{-7}{17} ) ( c cdot frac{7}{12} ) D. ( frac{-7}{12} ) |
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| 323 | Solve the equation for ( x ) ( sin ^{-1} x+sin ^{-1}(1-x)=cos ^{-1} x, x neq 0 ) | 12 |
| 324 | For the principal value: ( tan ^{-1}left{2 sin left(4 cos ^{-1} frac{sqrt{3}}{2}right)right} ) |
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| 325 | If ( sin ^{-1} x+sin ^{-1} y=frac{2 pi}{3}, ) then find the value of ( cos ^{-1} x+cos ^{-1} y ) |
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| 326 | If ( boldsymbol{alpha} boldsymbol{epsilon}left(-frac{boldsymbol{pi}}{2}, boldsymbol{0}right), ) then find the value of ( tan ^{-1}(cot alpha)-cot ^{-1}(tan alpha) ) |
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| 327 | ♡i 69. The sum of series sec-‘ V2 + sec 1 V10 + sec ! V50 (n? + 1)(n? – 2n+2) is V (n? – n+1) +…..+ sec-1 a. tan-1 c. tan-‘(n+1) b. tan ‘n d. tan-‘(n-1) |
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| 328 | Find the domain of the following function: ( boldsymbol{f}(boldsymbol{x})=cos ^{-1} sqrt{log [boldsymbol{x}] frac{|boldsymbol{x}|}{boldsymbol{x}}}, ) where, ( [cdot] ) denotes the greatest integer function. |
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| 329 | Solve the following: ( tan left(frac{1}{2} sin ^{-1} frac{3}{4}right) ) |
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| 330 | 20. 2 tan(tan-‘(x) + tan-‘(x)), where x e R-{-1,1), is 1. is equal to 2x a. b. tan(2 tan-‘x) od 8 c. tan (cot-‘(-x) – cot-‘(x))o d. tan(2 cot-‘ x) . 8 mo d |
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| 331 | 57. If y=tan –+tan-+b, (0<b<1) and 0<ys, then the maximum value of b is a. 1/2 b. 1/3 d. 2/3 c. 1/4 |
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| 332 | Find the value of ( cos ^{-1}left(cos frac{5 pi}{3}right) ) | 12 |
| 333 | 3. The number of real solutions of tan x(x+1)+sin – Vx2 + x +1 = n/2 is (1999 – 2 Marks) (a) zero (b) one (c) two (d) infinite |
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| 334 | The set of values of ‘ ( x^{prime} ) for which the formula ( 2 sin ^{-1} x=sin ^{-1}(2 x sqrt{1-x^{2}}) ) is true, is A. (-1,0) B. [0,1] c. ( left[-frac{sqrt{3}}{2}, frac{sqrt{3}}{2}right] ) D. ( left[-frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right] ) |
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| 335 | ( cos ^{-1}left(frac{3+5 cos x}{5+3 cos x}right)= ) A ( cdot tan ^{-1}left(frac{1}{2} tan frac{x}{2}right) ) B ( cdot 2 tan ^{-1}left(-frac{1}{2} tan frac{x}{2}right) ) c. ( frac{1}{2} tan ^{-1}left(2 tan frac{x}{2}right) ) D. ( 2 tan ^{-1}left(frac{1}{2} tan frac{x}{2}right) ) |
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| 336 | Show that ( 2 tan ^{-1} frac{3}{5}=tan ^{-1} frac{15}{8} ) | 12 |
| 337 | Assertion STATEMENT 1: Domain of ( tan ^{-1} x ) and ( cot ^{-1} x ) is ( R ) Reason STATEMENT 2: ( boldsymbol{f}(boldsymbol{x})=tan boldsymbol{x} ) and ( boldsymbol{g}(boldsymbol{x})= ) ( cot x ) are unbounded function A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT1 B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT1 C. STATEMENT1 is TRUE and STATEMENT 2 is FALSE D. STATEMENT1 is FALSE and STATEMENT 2 is TRUE |
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| 338 | Assertion Consider ( boldsymbol{f}(boldsymbol{x})=sin ^{-1}left(sec left(tan ^{-1} boldsymbol{x}right)+right. ) ( cos ^{-1}left(operatorname{cosec}left(cot ^{-1} xright)right. ) Statement-1: Domain of ( f(x) ) is a singleton. Reason Statement-2: Range of the function ( boldsymbol{f}(boldsymbol{x}) ) is a singleton. 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- lis true, Statement-2 is false. D. Statement-1 is false, Statement-2 is true |
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| 339 | The solution set of the equation ( sin ^{-1} sqrt{1-x^{2}}+cos ^{-1} x= ) ( cot ^{-1} frac{sqrt{1-x^{2}}}{x}-sin ^{-1} x ) is? ( mathbf{A} cdot[-1,1]-{0} ) в. (0,1]( cup{-1} ) c. [-1,0)( cup{1} ) D. [-1,1] |
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| 340 | If ( cot ^{-1}left(frac{1}{x+1}right)+cot ^{-1}left(frac{1}{x-1}right)= ) ( tan ^{-1} 3 x-tan ^{-1} x ) then ( boldsymbol{x}= ) A. ( pm 1 / 2 ) B. ( -1, pm 1 / 3 ) c. 2,±1 D. ( -1 . pm 1 / 2 ) |
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| 341 | Assertion ( (A) ) If ( 0<x<frac{pi}{2} ) then ( sin ^{-1}(cos x)+cos ^{-1}(sin x)=pi-2 x ) Reason ( (mathrm{R}) cos ^{-1} x=frac{pi}{2}-sin ^{-1} x forall x in ) ( [mathbf{0}, mathbf{1}] ) A. Both ( A ) and ( R ) are true and ( R ) is the correct explanation of ( A ) B. Both A and R are true but R is not correct explanation of ( A ) C. ( A ) is true but Ris false D. A is false but ( R ) is true |
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| 342 | If ( f(x)=sin ^{-1}left{frac{sqrt{3}}{2} x-frac{1}{2} sqrt{1-x^{2}}right} ) ( -frac{1}{2} leq x leq 1, ) then ( f(x) ) is equal to : ( ^{mathbf{A}} cdot sin ^{-1}left(frac{1}{2}right)-sin ^{-1}(x) ) B ( cdot sin ^{-1} x-frac{pi}{6} ) ( c cdot sin ^{-1} x+frac{pi}{6} ) D. none of these |
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| 343 | Assertion ( mathrm{f} sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2}, ) then ( frac{3 sum_{r=1}^{2008}left(x^{r}+y^{r}right)}{2 sum_{r=1}^{2008}left(x^{r} y^{r}right)}=3 ) Reason ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) is possible only if ( boldsymbol{x}=boldsymbol{y}=boldsymbol{z}=mathbf{1} ) 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 false but Reason is true |
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| 344 | Evaluate the following: ( cos ^{-1}(cos 5) ) |
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| 345 | Find the principal values of ( sin ^{-1}left(-frac{1}{sqrt{2}}right) ) | 12 |
| 346 | Illustration 5.29 Simplify sin cot’ tan cos’x, x > 0. |
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| 347 | f ( 0 leq x leq 1, ) then ( tan left{frac{1}{2} sin ^{-1} frac{2 x}{1+x^{2}}+frac{1}{2} cos ^{-1} frac{2 x}{1+x^{2}}right} ) ( A ) B. c. ( frac{2 x}{1+x^{2}} ) ( D ) |
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| 348 | The sum of the solution of the equation ( 2 sin ^{-1} sqrt{x^{2}+x+1}+ ) ( cos ^{-1} sqrt{x^{2}+x}=frac{3 pi}{2} ) is ( mathbf{A} cdot mathbf{0} ) B. – 1 ( c cdot 1 ) ( D ) |
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| 349 | Find the principal value of: ( sin ^{-1}left(-frac{sqrt{3}}{2}right)-2 sec ^{-1}left(2 tan frac{pi}{6}right) ) |
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| 350 | ( cos ^{-1}(44 / 125) ) is equal to A ( .2 alpha ) в. ( 3 alpha ) c. ( pi-3 alpha ) D. ( pi-2 alpha ) |
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| 351 | 1. Solve 2 cos + s = sinº (23 W1–?) 1. Solve = sin |
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| 352 | Find the principal value: ( tan ^{-1}left(-frac{1}{sqrt{3}}right) ) |
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| 353 | Illustration 5.9 If sin-‘(x2 – 4x + 5) + cos='(02- 2y + 2) = then find the value of x and y. |
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| 354 | Evaluate the following: ( sin ^{-1}(sin 2) ) |
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| 355 | 35. The value of 2 tan-‘ (cosec tan ‘x – tan cot ‘x) is equal to a. cot’ x b. cot!! c. tan-‘x d. none of these |
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| 356 | The number of solutions of the equation ( tan ^{-1}(x-1)+tan ^{-1}(x)+tan ^{-1}(x+ ) 1) ( =tan ^{-1}(3 x) ) is : ( mathbf{A} cdot mathbf{1} ) B . 2 ( c .3 ) D. 4 |
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| 357 | llustration 5.20 Solve cos'(cos x) > sin (sin x),x € [0,21]. |
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| 358 | Let ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x}+cos boldsymbol{x}+tan boldsymbol{x}+ ) ( arcsin x+arccos x+arctan x . ) If ( mathrm{M} ) and ( mathrm{m} ) are maximum and minimum values of ( f(x) ) then their arithmetic mean is equal to A ( cdot frac{pi}{2}+cos 1 ) B . ( frac{pi}{2}+sin 1 ) C ( cdot frac{pi}{4}+tan 1+cos 1 ) D. ( frac{pi}{4}+tan 1+sin 1 ) |
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| 359 | ff ( quad A=tan ^{-1}left(frac{x sqrt{3}}{2 k-x}right) ) and ( B= ) ( tan ^{-1}left(frac{2 x-k}{k sqrt{3}}right) . ) Then, ( A-B ) is equal to A ( cdot frac{pi}{2} ) в. c. D. None of these |
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| 360 | The value ( csc left(cos ^{-1}(-12 / 13)right) ) is? ( mathbf{A} cdot+12 / 5 ) в. ( -12 / 5 ) ( mathbf{c} cdot+13 / 5 ) D. ( -13 / 5 ) |
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| 361 | Illustration 5.54 Prove that cos– + cos! 13 -cos-1 33 5 |
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| 362 | Consider the function ( boldsymbol{y}= ) ( log _{a}(x+sqrt{x^{2}+1}), a>0, a neq 1 . ) The inverse of the function A. does not exist B cdot is ( x=log _{a}(y+sqrt{y^{2}+1}) ) ( mathbf{c} cdot ) is ( x=sin (y ln a) ) D ( quad ) is ( x=cosh left(-y ln frac{1}{a}right) ) |
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| 363 | The value of ( cos ^{-1}left(cos frac{5 pi}{3}right)+ ) ( sin ^{-1}left(sin frac{5 pi}{3}right) ) is A ( cdot frac{pi}{2} ) В. ( frac{5 pi}{2} ) ( c cdot frac{10 pi}{2} ) D. |
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| 364 | Let ( boldsymbol{f}:[mathbf{0}, boldsymbol{4} boldsymbol{pi}] rightarrow[mathbf{0}, boldsymbol{pi}] ) be defined by ( f(x)=cos ^{-1}(cos x) . ) The number of points ( boldsymbol{x} in[mathbf{0}, mathbf{4} boldsymbol{pi}] ) satisfying the equation ( f(x)=frac{10-x}{10} ) is |
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| 365 | 80. If the equation x3 + bx2 + cx + 1 = 0, (b<c), has only one real root a, then the value of 2 tan-' (cosec a) + tan-' (2 sina sec-a) is a. -T b. – – – 52 odd. a com |
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| 366 | 12. If sin-‘x = 0 + B and sin-y= 0-B, then 1 + xy is equal to a. sin? e + sin? B b . sin? 8+ cos2 B3 c. cos? 0+ cos? O d. cos² + sin? ß |
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| 367 | 1 + x cos- x Prove that cos- Illustration 5.32 -1<x< 1. |
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| 368 | Solve:( tan ^{-1}left(frac{6 x}{1-8 x^{2}}right) ) | 12 |
| 369 | Write the function in the simplest form: ( tan ^{-1} frac{1}{sqrt{x^{2}-1}},|x|>1 ) |
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| 370 | The value of ( 2 tan ^{-1}(-2) ) is equal to This question has multiple correct options ( ^{mathbf{A}} cdot sin ^{-1}left(-frac{4}{5}right) ) B. ( -sin ^{-1}left(frac{4}{5}right) ) ( mathbf{C} cdot sin ^{-1}left(frac{4}{5}right)-pi ) D ( -cos ^{-1}left(frac{4}{5}right)-frac{pi}{2} ) |
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| 371 | Solve the equation for ( x: sin ^{-1} frac{5}{x}+ ) ( sin ^{-1} frac{12}{x}=frac{pi}{2}, x neq 0 ) | 12 |
| 372 | Find the principal value of: ( sec ^{-1}(2) ) |
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| 373 | The range of the function ( f(x)= ) ( sin ^{-1}left(x^{2}-2 x+2right) ) ( A cdot phi ) B. ( left[-frac{pi}{2}, frac{pi}{2}right] ) c. ( frac{pi}{2} ) D. none of these |
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| 374 | ( fleft(frac{1}{2 i^{2}}right)=t, ) then ( tan t ) equals | 12 |
| 375 | The set for which ( 2 cos ^{-1} x= ) ( cos ^{-1}left(2 x^{2}-1right) ) is valid is ( mathbf{A} cdot x in[0,1] ) B ( cdot x in(0,1) ) ( mathbf{c} cdot x in[0,1) ) D. ( x in(0,1] ) |
12 |
| 376 | The value of ( sin left(tan ^{-1} x+cot ^{-1} xright) ) is | 12 |
| 377 | The value of ( sin ^{-1}left(sin 2010^{0}right)+ ) ( cos ^{-1}left(cos 2010^{0}right)+tan ^{-1}left(tan 2010^{0}right) ) is ( A cdot frac{pi}{6} ) B . ( mathbf{c} cdot frac{2 pi}{3} ) D. ( frac{5 pi}{6} ) |
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| 378 | Illustration 5.49 If sec !x= cosecly, then find the value of cos – – + cos- |
12 |
| 379 | Value of ( x ) for which ( cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right)= ) ( 2 tan ^{-1} x ) satisfied is ( x epsilon[a, infty) ) Find the value of ( a ) A ( . a=-infty ) B . ( a=-1 ) ( mathbf{c} cdot a=0 ) D. ( a=1 ) |
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| 380 | The value of ( lim _{|x| rightarrow infty} cos left(tan ^{-1}left(sin left(tan ^{-1} xright)right)right) ) is equal to A . -1 B. ( sqrt{2} ) c. ( -frac{1}{sqrt{2}} ) D. ( frac{1}{sqrt{2}} ) |
12 |
| 381 | Illustration 5.11 If cos’2+ cos’u + cos-‘ y=31, then find the value of λμ + μγ+ γλ. |
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| 382 | If ( sin ^{-1} frac{x}{5}+operatorname{cosec}^{-1} frac{5}{4}=frac{pi}{2}, ) then ( x ) is equal to: ( A ) B. 4 ( c cdot 3 ) D. 5 |
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| 383 | Illustration 5.69 Ifx e then show that cos” (3 (1 + cos2x) + Vísinºx – 48 cos?r) sin x) = x – cos-‘(7 cos x) |
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| 384 | Write the following in simplest form: ( tan ^{-1}left(frac{sqrt{left(1+x^{2}right)}-1}{x}right) ) |
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| 385 | Find the principal value of: ( sin ^{-1}left(cos frac{3 pi}{4}right) ) |
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| 386 | The principal value of ( tan ^{-1}left(cot frac{43 pi}{4}right) ) is A ( cdot-frac{3 pi}{4} ) B. ( frac{3 pi}{4} ) ( c cdot-frac{pi}{4} ) ( D cdot frac{pi}{4} ) |
12 |
| 387 | Find the real solution of the equation ( tan ^{-1} sqrt{x(x+1)}+sin ^{-1} sqrt{x^{2}+x+1}= ) | 12 |
| 388 | Itsin-1 26 anx, then x is 76. If sin-11 – 2a) (1+a?) equal to [a, b e (0, 1)] (1+6² o ab b. ltab 1 + ab 1+ ab b atb 1- ab 1- ab |
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| 389 | 36. If tan-1 V1 + x2 – 1 – = 4°, then a. x= tan 2° c. x =tan(1/4) b. x = tan 4° d. x = tan 8° |
12 |
| 390 | If cos(2 sin-‘x) = -, then find the values Illustration 5.23 of x. |
12 |
| 391 | Solve ( : tan ^{-1}left(frac{x-1}{x-2}right)+ ) ( tan ^{-1}left(frac{x+1}{x+2}right)=frac{pi}{4} ) |
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| 392 | For all values of ( x, ) the values of ( 3- ) ( cos x+cos left(x+frac{pi}{3}right) ) lie in the interval A ( .[-2,3] ) B. [-2,1] c. [2,4] D. [1,5] |
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| 393 | 9. For the equation cos-x + cos2x + =0, the number of real solution is Toita. 1 dl olb. 2 10 9gan de c. O d. ) ) 200 |
12 |
| 394 | 21. Complete solution set of tan”(sin’x) > 1 is – (-a) 6) (Ta’ tal-c03 c. (-1,1) – {0} d. None of these |
12 |
| 395 | The range of values of p for which the equation ( sin cos ^{-1}left(cos left(tan ^{-1} xright)right)=p ) has a solution is ( A cdotleft(-frac{1}{sqrt{2}}, frac{2}{sqrt{2}}right) ) в. [0,1) c. ( left(frac{1}{sqrt{21}}right) ) D. (-1,1) |
12 |
| 396 | If ( tan ^{-1}(x+1)+tan ^{-1}(x-1)= ) ( tan ^{-1}left(frac{8}{31}right), ) then ( x ) is equal ( A cdot frac{1}{2} ) B. ( -frac{1}{2} ) ( c cdot frac{1}{4} ) D. |
12 |
| 397 | If ( cot ^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right)=frac{x}{m}, x in ) ( left(0, frac{pi}{4}right) ).Find ( m ) |
12 |
| 398 | Solve: ( cos ^{-1}left(sin frac{4 pi}{3}right) ) A. ( -frac{5 pi}{6} ) в. ( c cdot frac{7 pi}{6} ) D. ( frac{5 pi}{6} ) |
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| 399 | The number of integral values of k for which the equation ( sin ^{-1} x+ ) ( tan ^{-1} x=2 k+1 ) has a solutions is: A . 1 B. 2 ( c .3 ) D. 4 |
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| 400 | Number of real value of ( x ) satisfying the equation, arctan ( sqrt{x(x+1)}+ ) ( arcsin sqrt{x(x+1)+1}=frac{pi}{2} ) is A . B. ( c cdot 2 ) D. more than 2 |
12 |
| 401 | f ( tan ^{-1}left(frac{x-1}{x-2}right)+tan ^{-1}left(frac{x+1}{x+2}right)= ) 4 then ( x ) is |
12 |
| 402 | The number of real solutions of the equation ( tan ^{-1} sqrt{x(x+1)}+ ) ( sin ^{-1} sqrt{x^{2}+x+1}=frac{pi}{2} ) is A. One B. Four c. Two D. Infinitely many |
12 |
| 403 | ( sin ^{-1} sin 15+cos ^{-1} cos 20+ ) ( tan ^{-1} tan 25=? ) A . 1.04719754 в. 11.04719754 c. 111.04719754 D. 1111.04719754 |
12 |
| 404 | Simplify ( tan ^{-1} sqrt{2}-cot ^{-1}(1 / sqrt{2}) ) | 12 |
| 405 | If ( tan left(2 tan ^{-1}left(frac{1}{5}right)-frac{pi}{4}right)=-frac{lambda}{17}, ) then ( lambda ) is equal to |
12 |
| 406 | Illustration 5.65 Find the value of cot- – + sin 13 |
12 |
| 407 | The principal value of ( cos ^{-1}left(-sin frac{7 pi}{6}right. ) ( A cdot frac{5 pi}{3 pi} ) B. ( frac{7 pi}{6} ) ( c cdot frac{pi}{3} ) D. none of these |
12 |
| 408 | The value of sin ( sin left{tan ^{-1}left(tan frac{7 pi}{6}right)+cos ^{-1}left(cos frac{7 pi}{3}right)right} ) is A . в. ( c cdot-1 ) D. None of these |
12 |
| 409 | The number of real solutions of the equation [ begin{array}{l} sin ^{-1}left(sum_{i=1}^{infty} x^{i+1}-x sum_{i=1}^{infty}left(frac{x}{2}right)^{i}right)=frac{pi}{2}- \ cos ^{-1}left(sum_{i=1}^{infty}left(-frac{x}{2}right)^{i}-sum_{i=1}^{infty}(-x)^{i}right) text { lying in } end{array} ] the interval ( left(-frac{1}{2}, frac{1}{2}right) ) is |
12 |
| 410 | Illustration 5.62 Find the value of tan (l+rtph r=0 |
12 |
| 411 | Find the principal value: ( tan ^{-1}left(frac{1}{sqrt{3}}right) ) |
12 |
| 412 | Illustration 5.1 Find the principal value of the following: (i) cosec-‘(2) (ii) tan-‘ (-13) (ii) cos( – – – |
12 |
| 413 | Find tan- in terms of sin, Illustration 5.28 where x e (0, a). ſa² – x |
12 |
| 414 | Illustration 5.47 Solve sin- – + sin Nia |
12 |
| 415 | Illustration 5.44 Solve |
12 |
| 416 | Illustration 5.39 Find the range of y = (cot- x)(cot ‘(-x)). |
12 |
| 417 | illustration 5.76 ut sin (za) + 2 tun (13) Illustration 5.76 If sin- + 2 tan- N18 x² + 4 independent of x, find the values of x. |
12 |
| 418 | For the principal value: ( sin ^{-1}left(-frac{1}{2}right)+2 cos ^{-1}left(-frac{sqrt{3}}{2}right) ) |
12 |
| 419 | The value of ( cos left[frac{1}{2} cos ^{-1}left[cos left[-frac{14 pi}{5}right]right]right] ) is/are – This question has multiple correct options ( ^{A} cdot cos left[-frac{7 pi}{5}right] ) B. ( sin left[frac{pi}{10}right. ) ( ^{mathbf{c}} cdot cos left[frac{2 pi}{5}right] ) D. ( -cos left[frac{3 pi}{5}right] ) |
12 |
| 420 | Prove that ( tan ^{-1} frac{63}{16}=sin ^{-1} frac{5}{13}+ ) ( cos ^{-1} frac{3}{5} ) |
12 |
| 421 | Write the following into simple test form; (1) ( sin left{2 tan ^{-1} sqrt{frac{1-n}{1+n}}right} ) |
12 |
| 422 | If ( cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=pi ), then prove that ( x^{2}+y^{2}+z^{2}+2 x y z=1 ) |
12 |
| 423 | Illustration 5.4 Solve for x if (cot- x)2 – 3 (cot-‘x) +2>0. |
12 |
| 424 | Show that: [ tan ^{-1}left(frac{sqrt{1+x}-sqrt{1-x}}{sqrt{1+x}+sqrt{1-x}}right)=frac{pi}{4} ] ( frac{1}{2} cos ^{-1} x,-frac{1}{sqrt{2}} leq x leq 1 ) |
12 |
| 425 | ( tan left(2 cos ^{-1} frac{3}{5}right)= ) ( A cdot frac{8}{3} ) B . ( frac{24}{25} ) ( c cdot frac{7}{25} ) ( D cdot frac{-24}{7} ) |
12 |
| 426 | 31. The value of tan(sin-‘(cos(sin-+ x))) tan(cos-‘(sin (cosx))), where x € (0, 1), is equal to a. 0 Obt n b. 1 c. -1 d. none of these |
12 |
| 427 | Illustration 5.56 Find the value of tan-1 tan 2A + I tan (cot A) + tan-‘(cot? A), for 0 <A<*. |
12 |
| 428 | The value of ( sec ^{-1}left(sec frac{8 pi}{5}right) ) is A ( cdot frac{2 pi}{5} ) в. ( frac{3 pi}{5} ) ( c cdot frac{8 pi}{5} ) D. none of these |
12 |
| 429 | The value of ( tan ^{-1}left(frac{1}{2} tan 2 Aright)+ ) ( tan ^{-1}(cot A)+tan ^{-1}left(cot ^{3} Aright) ) for ( 0< ) ( boldsymbol{A}<frac{boldsymbol{pi}}{boldsymbol{4}} ) is? ( mathbf{A} cdot 4 tan ^{-1} 1 ) B. ( 2 tan ^{-1} 2 ) ( c cdot 0 ) D. None |
12 |
| 430 | Illustration 5.27 Solve sin-‘(1 – x) – 2 sin-‘x = | 12 |
| 431 | 8. If cos” ()+cos (23) = (**), then OS x is equal to: JEEM 2019-9 Jan (M) (6) 1145 12 (0) VIAG 12 (a) Vi45 11 |
12 |
| 432 | The set of values of parameter ( a ) so that the equation ( left(sin ^{-1} xright)^{3}+left(cos ^{-1} xright)^{3}= ) ( a pi^{3} ) has a solution. ( mathbf{A} cdotleft[frac{-1}{32}, frac{7}{8}right] ) в. ( left[frac{1}{32},, frac{9}{8}right] ) ( ^{mathbf{c}} cdotleft[0, frac{7}{8}right] ) D. ( left[frac{1}{32}, frac{7}{8}right] ) |
12 |
| 433 | Find the value of ( tan ^{-1}left[2 cos left(2 sin ^{-1} frac{1}{2}right)right] ) |
12 |
| 434 | ( tan left(cot ^{-1} xright)=cot left(tan ^{-1} xright) ) | 12 |
| 435 | 78. Ifx = 2 tang 4 xy=sin” – where xe (0, 0), then xı + x2 is equal to a. 0 c. b. 21 d. none of these |
12 |
| 436 | ( tan ^{-1}left(frac{5-x}{6 x^{2}-5 x-3}right) ) | 12 |
| 437 | 2. 2 tan- ‘(- 2) is equal to a. – b. – 1+ cos-1 5 c. – – 2 + tan-1 d. – T + cot- Cot-1 |
12 |
| 438 | Evaluate: ( sum_{r=1}^{infty} tan ^{-1}left(frac{2}{1+(2 r+1)(2 r-1)}right) ) |
12 |
| 439 | Evaluate ( tan left[frac{1}{2} cos ^{-1}left(frac{3}{sqrt{11}}right)right] ) | 12 |
| 440 | If ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8}, ) then ( x ) A . -1 B. c. 1 D. 2 |
12 |
| 441 | The value of ( lim _{|x| rightarrow infty} cos left(tan ^{-1}left(sin left(tan ^{-1} xright)right)right) ) is equal to A . -1 B. ( sqrt{2} ) ( c cdot-frac{1}{sqrt{2}} ) D. ( frac{1}{sqrt{2}} ) |
12 |
| 442 | ( tan ^{-1}left[frac{cos x}{1+sin x}right] ) is equal to A ( cdot frac{pi}{4}-frac{x}{2}, ) for ( x epsilonleft(-frac{pi}{2}, frac{3 pi}{2}right) ) B. ( frac{pi}{4}-frac{x}{2}, ) for ( x inleft(-frac{pi}{2}, frac{pi}{2}right) ) ( ^{mathbf{C}} cdot frac{pi}{4}-frac{x}{2}, ) for ( x inleft(frac{3 pi}{2}, frac{5 pi}{2}right) ) D ( cdot frac{pi}{4}-frac{x}{2}, ) for ( x inleft(-frac{3 pi}{2},-frac{3 pi}{2}right) ) |
12 |
| 443 | Find the principle value of: ( cos ^{-1}left(-frac{1}{2}right) ) | 12 |
| 444 | 10. If p > q> 0 and pr <- l< qr, then find the value of tan-1 P-9 + tan-1 9-" + tan-1 " – p. 1 + pq 1 +gr 1 +rp |
12 |
| 445 | 10 10 Illustration 5.52 Find the value of tan ºr r=1s=1 |
12 |
| 446 | The value of ( sin ^{-1}left(cos frac{33 pi}{5}right) ) is A ( cdot frac{3 pi}{5} ) в. ( frac{7 pi}{5} ) c. ( frac{pi}{10} ) D. ( -frac{pi}{10} ) |
12 |
| 447 | If ( sin ^{-1} x+sin ^{-1} y=frac{pi}{2} ) and ( sin 2 x= ) ( cos 2 y, ) then This question has multiple correct options A ( cdot x=frac{pi}{8}+sqrt{frac{1}{2}-frac{pi^{2}}{64}} ) B. ( y=sqrt{frac{1}{2}-frac{pi^{2}}{64}}-frac{pi}{12} ) c. ( _{x}=frac{pi}{12}+sqrt{frac{1}{2}-frac{pi^{2}}{64}} ) D. ( y=sqrt{frac{1}{2}-frac{pi^{2}}{64}}-frac{pi}{8} ) |
12 |
| 448 | 6. Ifx <0, then tan x is equal to lon a. – 1 + cot-1 I a. b. – T + cot x sin- v1 + x –1 d. – cosec |
12 |
| 449 | If ( frac{(x+1)^{2}}{x^{3}+x}=frac{A}{x}+frac{B x+C}{x^{2}+1}, ) then ( sin ^{-1} A+tan ^{-1} B+sec ^{-1} C= ) A ( cdot frac{pi}{2} ) B. ( c cdot c ) D. ( frac{5 pi}{6} ) |
12 |
| 450 | Prove that: ( tan ^{-1} frac{1}{4}+tan ^{-1} frac{2}{9}=frac{1}{2} cos ^{-1} frac{3}{5} ) |
12 |
| 451 | Given that ( 0 leq x leq frac{1}{2} ) the value of ( tan left[sin ^{-1}left(frac{x}{sqrt{2}}+sqrt{frac{1-x^{2}}{2}}right)-sin ^{-1} xright. ) is ( A ) B. ( c cdot 1 sqrt{3} ) D. ( sqrt{3} ) |
12 |
| 452 | 5. Which of the following quantities is/are positive? a. cos(tan-‘(tan 4)) b. sin(cot-‘(cot 4)) c. tan(cos(cos 5) d. cot(sin-‘(sin 4)) |
12 |
| 453 | Solve : ( cot ^{-1}left(frac{sqrt{1-sin x}+sqrt{1+sin x}}{sqrt{1-sin x}-sqrt{1+sin x}}right) ) A ( cdot_{pi}-frac{x}{2} ) в. ( _{pi+frac{x}{2}} ) c. ( frac{x}{2} ) D . ( 2 pi ) |
12 |
| 454 | Find the value of ( x ) which satisfy equation ( : 2 tan ^{-1}(cos x)= ) ( tan ^{-1}(2 csc x) ) A ( cdot x=n pi+frac{pi}{4} ) B cdot ( x=n pi-frac{pi}{4} ) c. ( x=n pi+frac{pi}{2} ) D. ( x=n pi-frac{pi}{2} ) |
12 |
| 455 | If ( theta=sin ^{-1} x+cos ^{-1} x-tan ^{-1} x, 1 leq ) ( x<infty, ) the smallest interval in which ( theta ) lies is |
12 |
| 456 | If ( cos ^{-1} x-sin ^{-1} x=0, ) then ( x ) is equal to- ( ^{mathrm{A}} pm frac{1}{sqrt{2}} ) B. 1 ( ^{mathrm{c}} pm frac{1}{sqrt{3}} ) D. ( frac{1}{sqrt{2}} ) |
12 |
| 457 | ( mathbf{s} frac{sin ^{-1}}{cos ^{-1}}=tan ^{-1} ) a valid relation? | 12 |
| 458 | 1 COS The value of tam cor” ()un (1) – n (1) Bosch (a) none (1983-1 Mark) |
12 |
| 459 | Illustration 5.6 Find the values of (i) sin’ (2) (ii) cos-1 Vx– x+1 (iii) tan- (iv) sec 1+x4 (m) tant (iv) see “(x+) X |
12 |
| 460 | If ( cos ^{-1}left(frac{x}{2}right)+cos ^{-1}left(frac{y}{3}right)=theta, ) if ( 9 x^{2}- ) ( 12 x y cos theta+4 y^{2}=m sin ^{2} theta . ) Find ( m . ) |
12 |
| 461 | Write the simplest form of ( tan ^{-1}left(frac{cos x-sin x}{cos x+sin x}right), 0<x<frac{pi}{2} ) | 12 |
| 462 | Find the range of ( boldsymbol{f}(boldsymbol{x})= ) ( cos ^{-1} sqrt{log _{[x]} frac{|x|}{x}}, ) where [.]denotes the greatest integer. A ( cdotleft{frac{pi}{2}right} ) в. ( left{frac{pi}{4}right} ) ( c cdotleft{frac{pi}{6}right} ) D. ( left{frac{pi}{8}right} ) |
12 |
| 463 | There is flag-staff at the top of 10 metres high tower. If the flag-staff makes an angle ( tan ^{-1}(1 / 8) ) at a point 24 metres away from the tower, then the height of the flag staff in metres is A ( .26 / 7 ) B. 27/8 c. ( 27 / 6 ) D. 26 /3 |
12 |
| 464 | Match the column | 12 |
| 465 | If ( tan ^{-1} y=4 tan ^{-1} x, ) then ( frac{1}{y} ) is zero for A ( . x=1 pm sqrt{2} ) B. ( x=sqrt{2} pm sqrt{3} ) c. ( x=3 pm 2 sqrt{2} ) D. all value of ( x ) |
12 |
| 466 | Find the principal value of: ( tan ^{-1}(-1)+cos ^{-1}left(-frac{1}{sqrt{2}}right) ) |
12 |
| 467 | Solve ( sin ^{-1}left{frac{sin x+cos x}{sqrt{2}}right},-frac{3 pi}{4}< ) ( x<frac{pi}{4} ) |
12 |
| 468 | Find the principal value of the following: ( sec ^{-1}(-2) ) |
12 |
| 469 | 5. The value of cot( cosec t an |
12 |
| 470 | A value of ( x ) satisfying the equation ( sin left[cot ^{-1}(1+x)right]=cos left[tan ^{-1} xright], ) is. A. ( -frac{1}{2} ) B. ( frac{1}{2} ) c. -1 ( D ) |
12 |
| 471 | 56. = tan’ (2 tan’) – tan- – tan 0 then tan = a. 2 c. 2/3 b. -1 d. 2 |
12 |
| 472 | Show that ( 2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) is constant for ( x geq 1 . ) Also find that constant. |
12 |
| 473 | If ( cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=3 pi ) then ( x y+y x+z x ) is equal to A . 1 B. ( c .-3 ) D. 3 |
12 |
| 474 | Number of solutions of the equation ( sin left(frac{1}{3} cos ^{-1} xright)=1 ) are A. only one B. no solution c. only three D. at least two |
12 |
| 475 | If ( left(sin ^{-1} xright)^{2}+left(sin ^{-1} yright)^{2}+left(sin ^{-1} zright)^{2}= ) ( frac{3 pi^{2}}{4}, ) then find the minimum value of ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z} ) |
12 |
| 476 | Illustration 5.14 Find x satisfying (tan-‘x] + [cot–x] = 2, where [:] represents the greatest integer function. |
12 |
| 477 | There exists a positive real number ( x ) satisfying ( cos left(tan ^{-1} xright)=x ). The value of ( cos ^{-1}left(frac{x^{2}}{2}right) ) is A ( cdot frac{pi}{10} ) в. c. ( frac{2 pi}{5} ) D. ( frac{4 pi}{5} ) |
12 |
| 478 | The solution of the equation ( 2 cos ^{-1} x=sin ^{-1}(2 x sqrt{1-x^{2}}) ) A ( cdot[-1,0] ) B ( cdot[0,1] ) c. [-1,1] D. ( left[frac{1}{sqrt{2}}, 1right] ) |
12 |
| 479 | Write the given trigonometric expression in its simplest form. ( tan ^{-1}left(frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}right), a>0 ; frac{-a}{sqrt{3}} leq ) ( boldsymbol{x} leq frac{boldsymbol{a}}{sqrt{mathbf{3}}} ) |
12 |
| 480 | Prove: ( 2 sin ^{-1} frac{3}{5}=tan ^{-1} frac{24}{7} ) | 12 |
| 481 | If sin sin- Illustration 5.43 value of x. ??? + cos-‘x = 1, then find the |
12 |
| 482 | Prove that ( : sin ^{-1}left(frac{5}{13}right)+ ) ( cos ^{-1}left(frac{4}{5}right)=frac{1}{2} sin ^{-1}left(frac{3696}{4225}right) ) |
12 |
| 483 | If ( tan ^{-1}(1+x)+tan ^{-1}(1-x)=frac{pi}{2} ) then ( x=? ) A . B. – c. D. |
12 |
| 484 | A tower stands at the top of a hill whose height is three times the height of the tower. The tower is found to subtend an angle of ( mid tan ^{-1}(1 / 7) ) at a point ( 2 k m ) away on the horizontal throught the foot of the hill. Then the height of the tower is ( ^{mathbf{A}} cdot frac{1}{2} k m ) or ( frac{1}{3} k m ) B. ( frac{1}{3} k m ) or ( frac{2}{3} k m ) c. ( frac{2}{3} k m ) or ( frac{1}{2} k m ) D. ( frac{3}{4} k m ) or ( frac{1}{2} k m ) |
12 |
| 485 | 4:(**”)un jo sguvą 51 | 12 |
| 486 | The value of ( left{s i n^{-1}left[x^{2}+frac{1}{2}right]+cos ^{-1}left[x^{2}-frac{1}{2}right]right} ) where {} and ( [.] ) denote fractional part function and greatest integer function respectively ( mathbf{A} cdot pi-3 ) B. ( 4-pi ) ( c cdot 2-frac{pi}{2} ) D. o |
12 |
| 487 | Find the number of terms of the AP ( 121,117,113, dots,-3 ? ) A . 32 B. 30 c. 28 D . 26 |
12 |
| 488 | Illustration 5.72 If f(x) = sin-‘ x then prove that lim f(3x – 4x?) = 1 – 3 lim sin-‘ x. |
12 |
| 489 | If ( a<frac{1}{32}, ) then the number of solutions of ( left(sin ^{-1} xright)^{3}+left(cos ^{-1} xright)^{3}=a pi^{3}, ) is ( A cdot 0 ) B. ( c cdot 2 ) D. infinite |
12 |
| 490 | Find the inverse of the following functions: ( boldsymbol{f}(boldsymbol{x})=sin ^{-1}left(frac{boldsymbol{x}}{mathbf{3}}right), boldsymbol{x} in[-mathbf{3}, mathbf{3}] ) ( operatorname{then} boldsymbol{f}^{-1}(boldsymbol{pi} / mathbf{2}) ) |
12 |
| 491 | Consider the function ( boldsymbol{f}(boldsymbol{x})= ) ( sin left(sin ^{-1} 2 xright)+sec left(sec ^{-1} 3 xright)+ ) ( tan left(tan ^{-1} 4 xright) ) and ( g(x)=9 x, ) then which of the following is correct? This question has multiple correct options A ( cdot ) fog ( (x) ) and ( g o f(x) ) are equal function B . ( f(x) ) is an odd function C. Number of integers in range of ( f(x) ) are 4 D. Number of integers in domain of ( f(x) ) are 2 |
12 |
| 492 | If ( sin ^{-1}(1-x)-2 sin ^{-1} x=frac{pi}{2}, ) then ( x ) is equal to A ( cdot 0, frac{1}{2} ) в. ( _{1, frac{1}{2}} ) ( c cdot frac{1}{2} ) D. |
12 |
| 493 | If ( x=cos ^{2}left(tan ^{-1}left(sin left(cot ^{-1} 3right)right)right), ) then ( 1331 x^{3}-3630 x^{2}+3300 x+7369= ) ( m ) then find the sum of the second and third digits of ( boldsymbol{m} ) |
12 |
| 494 | Solve the equation: ( tan ^{-1} x+ ) ( 2 cot ^{-1} x=frac{2 pi}{3} ) | 12 |
| 495 | What is ( tan ^{-1}left(frac{1}{2}right)+tan ^{-1}left(frac{1}{3}right) ) equa to? A ( cdot frac{pi}{2} ) в. ( frac{pi}{3} ) c. ( frac{pi}{4} ) D. |
12 |
| 496 | (2008) 6. If0<x<1, then VI+x? [{x cos (cot" x) + sin (cot 1 x)}2 – 1]1/2 = (2) vita (b) x (c) x V1 + x² (d) V1 + x² |
12 |
| 497 | If we consider only the principle values of the inverse trigonometric functions then the value of tan cos SUD (1994) |
12 |
| 498 | Find the principal value of: ( operatorname{cosec}^{-1}left(2 tan frac{11 pi}{6}right) ) |
12 |
| 499 | Illustration 5.57 Simplify 3 sin 2a | 5 + 3 cos 2a tan a – + tan-1 1, where – I < a < 1 2 |
12 |
| 500 | Which one of the following quantities is negative? A ( cdot cos left(tan ^{-1}(tan 4)right) ) B. ( sin left(cot ^{-1}(cot 4)right) ) c. ( tan left(cos ^{-1}(cos 5)right) ) D・cot ( left(sin ^{-1}(sin 4)right) ) |
12 |
| 501 | Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x ) [ tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} ] ( x y1 ) (a) If ( tan ^{-1} frac{sqrt{1+x^{2}}-sqrt{1-x^{2}}}{sqrt{left(1+x^{2}right)}+sqrt{left(1+x^{2}right)}}= ) ( alpha . ) then prove that ( x^{2}=sin 2 alpha ) (b) If ( frac{operatorname{mtan}(alpha-theta)}{cos ^{2} theta}=frac{n tan theta}{cos ^{2}(alpha-theta)} ) then prove that ( boldsymbol{theta}= ) ( frac{1}{2}left[alpha-tan ^{-1}left(frac{n-m}{n+m} tan alpharight)right] ) (c) ( cos ^{-1} frac{cos alpha+cos beta}{1+cos alpha cos beta}= ) ( 2 tan ^{-1}left(tan frac{alpha}{2} tan frac{beta}{2}right) ) |
12 |
| 502 | Evaluate ( sin ^{-1}(cos x) ) | 12 |
| 503 | If value of ( x ) which satisfy equation ( sin ^{-1} frac{2 x}{1+x^{2}}=tan ^{-1} frac{2 x}{1-x^{2}}, ) is ( -a< ) ( boldsymbol{x}<boldsymbol{b} ) Find the value of ( a+b ) A . -1 B. c. 1 D. |
12 |
| 504 | If the sum of maximum and minimum values of ( boldsymbol{E}=left(sin ^{-1} xright)^{2}+ ) ( 2 pi cos ^{-1} x+pi^{2} ) is ( frac{a pi^{2}}{b}, ) where ( a ) and ( b ) are coprime, then the value of ( (a-b) ) is |
12 |
| 505 | ( $ $ operatorname{lint}|,| operatorname{left}left[x sin ^{wedge}{-1} x mid, text { lleft } mid{1right. ) ( left|operatorname{sqrt}left{left(1|,-|, mathbf{x}^{wedge} 2right}|,| r i g h t midright} wedge{-1}right| ) |right ( ] wedge mid, $ $ d x ) |
12 |
| 506 | f ( tan ^{-1}left(frac{x-1}{x-2}right)+tan ^{-1}left(frac{x+1}{x+2}right)= ) find the value of ( x ) ( overline{mathbf{4}} ) |
12 |
| 507 | Prove that ( frac{pi}{2}-sin ^{-1} x= ) ( operatorname{cosec}^{-1} frac{1}{sqrt{1-x^{2}}} ) |
12 |
| 508 | ff ( y=left(tan ^{-1} xright)^{2} ) then show ( left(x^{2}+1right)^{2} y_{2}+2 xleft(x^{2}+1right) y_{1}=2 ) |
12 |
| 509 | Illustration 5.68 Solve |
12 |
| 510 | If the value of ( tan ^{-1}left(tan frac{3 pi}{4}right) ) is ( -frac{pi}{k} ) then ( k ) is |
12 |
| 511 | Simplify: ( sin ^{-1}(2 x sqrt{1-x^{2}})= ) ( dots . .left(|x|<frac{1}{sqrt{2}}right) ) |
12 |
| 512 | 47. The value of a for which ax? + sin (x2 – 2x + 2) + cos (r? – 2x + 2) = 0 has a real solution is |
12 |
| 513 | 84. If f (x) = sin- is equal to SxS 1, then f(x) c. sin x +* c. sin-2x + d. none of these |
12 |
| 514 | If the number ( 93215 x 2 ) is completely divisible by ( 11, ) then ( x ) is equal to ( A cdot 2 ) B. 3 ( c cdot 1 ) D. 4 |
12 |
| 515 | Solve: ( cot ^{-1} x+cot ^{-1} 2=frac{pi}{2} ) |
12 |
| 516 | If ( quad x inleft(frac{3 pi}{2}, 2 piright), ) then the value of the expression ( sin ^{-1}left[cos left{cos ^{-1}(cos x)+sin ^{-1}(sin x)right}right. ) ( ^{A} cdot-frac{pi}{2} ) в. c. 0 D. |
12 |
| 517 | ( f(x)=cos ^{-1}left(frac{sqrt{2 x^{2}+1}}{x^{2}+1}right), ) then range of ( boldsymbol{f}(boldsymbol{x}) ) is ( mathbf{A} cdot[0, pi] ) B cdot ( left(0, frac{pi}{4}right. ) ( c cdotleft(0, frac{pi}{3}right. ) D ( cdotleft[0, frac{pi}{2}right) ) |
12 |
| 518 | ( tan left(3 tan ^{-1} 3right)+cos left(3 cos ^{-1}(1 / 3)right)+1 ) is equal to ( A cdot 1 ) B. 9/13 c. ( 4 / 27 ) D. 295/351 |
12 |
| 519 | Prove that: (i) ( 2 tan ^{-1} sqrt{frac{b}{a}}=cos ^{-1}left(frac{a-b}{a+b}right) ) (ii) Find the principal value of ( cos ^{-1}left(-frac{1}{sqrt{2}}right) ) |
12 |
| 520 | The number of solutions of the equation ( sin ^{-1}left(frac{1+x^{2}}{2 x}right)=frac{pi}{2} sec (x- ) 1) is |
12 |
| 521 | Number of value ( x ) satisfying the equation ( sin ^{-1}left(frac{5}{x}right)+sin ^{-1}left(frac{12}{x}right)= ) ( frac{pi}{2} ) is A . B. ( c cdot 2 ) D. more than 2 |
12 |
| 522 | ( operatorname{Let} cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=pi ) then prove that ( x^{2}+y^{2}+z^{2}+2 x y z= ) 1 |
12 |
| 523 | The number of solution of the equation ( tan ^{-1}(x-1)+tan ^{-1}(x)+tan ^{-1}(x+ ) 1) ( =tan ^{-1}(3 x) ) is ( mathbf{A} cdot mathbf{1} ) B . 2 ( c .3 ) D. 4 |
12 |
| 524 | Itxt 2. Find the domain for f(x) = sin 2x |
12 |
| 525 | If ( sin ^{-1}(6 x)+sin ^{-1}(6 sqrt{3} x)=-frac{pi}{2} ) then the value of ( x ) is A ( cdot frac{1}{12} ) B. ( -frac{1}{12} ) c. ( -frac{1}{4 sqrt{3}} ) D. ( frac{1}{4 sqrt{3}} ) |
12 |
| 526 | Illustration 5.66 Solve sin-‘x + sin 2x = | 12 |
| 527 | Illustration 5.15 Find the number of solutions of the equation cos(cos=’x) = cosec(cosec-‘x). |
12 |
| 528 | Find the number of solutions of the equation ( cos left(cos ^{-1} xright)= ) ( operatorname{cosec}left(operatorname{cosec}^{-1} xright) ) A. B. ( c cdot 2 ) D. |
12 |
| 529 | The value of ( tan left(sin ^{-1}left(cos left(sin ^{-1} xright)right)right) tan left(cos ^{-1}(sin right. ) where ( boldsymbol{x} boldsymbol{epsilon}(boldsymbol{0}, boldsymbol{1}), ) is equal to ( mathbf{A} cdot mathbf{0} ) B. ( c cdot-1 ) D. none of these |
12 |
| 530 | If ( sin ^{-1} x+sin ^{-1} y=frac{2 pi}{3}, ) then ( cos ^{-1} x+cos ^{-1} y ) is equal to A ( cdot frac{2 pi}{3} ) в. c. D. |
12 |
| 531 | Find the principal value of ( sin ^{-1}left(-frac{1}{2}right) ) | 12 |
| 532 | Indicate the relation which can hold in their respective domain for infinite values of ( x ) This question has multiple correct options ( mathbf{A} cdot tan left|tan ^{-1} xright|=|x| ) B cdot ( cot left|cot ^{-1} xright|=|x| ) C ( cdot tan ^{-1}|tan x|=|x| ) D ( cdot sin left|sin ^{-1} xright|=|x| ) |
12 |
| 533 | Evaluate ( cos left[cos ^{-1}left(-frac{sqrt{3}}{2}right)+frac{pi}{6}right] ) | 12 |
| 534 | Find the principal value of ( tan ^{-1}(-sqrt{3}) ) | 12 |
| 535 | ( sin ^{-1}left(cos left(sin ^{-1} xright)right)+ ) ( cos ^{-1}left(sin left(cos ^{-1} xright)right) ) is equal to A. в. ( frac{pi}{2} ) c. ( frac{3 pi}{4} ) D. |
12 |
| 536 | n Illustration 5.71 Find the value -1/1+ /(k – 1)k(k+1)(k + 2) lim k(k+1) COS no k=2 |
12 |
| 537 | Number of solution of equation ( left|sin ^{-1} x-cos ^{-1} xright|=x+2 ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 3 |
12 |
| 538 | The value of ( sin h^{-1}(3) ) ( A cdot log (1+sqrt{2}) ) B ( cdot log (2+sqrt{3}) ) ( mathbf{c} cdot log (3+sqrt{5}) ) D ( cdot log (3+sqrt{10}) ) |
12 |
| 539 | If ( cot ^{-1}left(frac{1}{a}right)+cot ^{-1}left(frac{1}{b}right)+ ) ( cot ^{-1}left(frac{1}{c}right)=frac{pi}{2}, ) then ( mathbf{A} cdot a+b+c=a b c ) B. ( a b+b c+c a=1 ) c. ( a b+b c+c a=a b c ) D. none of these |
12 |
| 540 | Exhaustive set of values of parameter ( boldsymbol{a} ) so that ( sin ^{-1} x-tan ^{-1} x=a quad ) has a solution is A. ( left[-frac{pi}{6}, frac{pi}{6}right] ) B. ( left[-frac{pi}{4}, frac{pi}{4}right] ) ( mathbf{c} cdotleft[-frac{pi}{2}, frac{pi}{2}right] ) D. none of these |
12 |
| 541 | Prove that: [ tan ^{-1}left[frac{sqrt{1+x}-sqrt{1-x}}{sqrt{1+x}+sqrt{1-x}}right]=frac{pi}{4}- ] ( frac{1}{2} cos ^{-1} x,-frac{1}{sqrt{2}} leq x leq 1 ) |
12 |
| 542 | . If a = tan-1/ 4x – 4.83 ) ut (1-672 2 , = 2 sin-1 lain-1 2x (+72) and tan – = k, then a. a+=for x € 1, b. a= ß for x € (-k, k) c. a+B=- = for x e 1,) d. a+B=0 for x € (-k, k) |
12 |
| 543 | Show that ( tan ^{-1} 1 / 2+tan ^{-1} 2 / 11+ ) ( tan ^{-1} 4 / 3=pi / 2 ) |
12 |
| 544 | Write the value of ( cos ^{-1}left(cos frac{5 pi}{4}right) ) | 12 |
| 545 | Find the set of values of parameter a so that the equation (sin-2x)3 + (cos’x)’ = at has a solution. |
12 |
| 546 | If ( tan ^{-1} x=frac{pi}{10} ) for some ( x in R, ) then find the value of ( cot ^{-1} x ) |
12 |
| 547 | 12. Solve the equation tan ” * +1+tan **= tan” (-7). x -1 |
12 |
| 548 | T 1 -+-COS 4 2 X + tan cos’x, x = 0, is equal 4 2 b. 2x d. none of these |
12 |
| 549 | If ( cos ^{-1} lambda+cos ^{-1} mu+cos ^{-1} gamma=3 pi ) then find the value of ( lambda mu+mu gamma+gamma lambda ) A . 1 B . 2 ( c .3 ) D. 4 |
12 |
| 550 | 83. If sin ‘x+ sin ‘y + sin- z = T, then x4 + y4 + 24 + 4x?y?z? = K (x+ y2 + y22 + 2?x?), where K is equal to a. 1 b. 2 c. 4 d. none of these |
12 |
| 551 | If value of ( x ) which satisfy ( sin ^{-1} x leq ) ( cos ^{-1} x ) is ( x epsilonleft[a, frac{b}{sqrt{c}}right] ) Find the value of ( a+b+c ) A. B. ( c cdot 2 ) D. |
12 |
| 552 | Find the value of ( x ) if ( sin (arcsin x)= ) ( frac{sqrt{2}}{4} ) A ( cdot frac{sqrt{2}}{4} ) B. ( frac{sqrt{7}}{7} ) c. ( frac{sqrt{2}}{2} ) D. ( frac{2 sqrt{2}}{3} ) |
12 |
| 553 | The value of the expression ( 2 sec ^{-1} 2+ ) ( sin ^{-1} frac{1}{2} ) is A ( cdot frac{pi}{6} ) в. ( frac{5 pi}{6} ) c. ( frac{7 pi}{6} ) D. |
12 |
| 554 | The value of ( sin ^{-1}(sin 10) ) is A . 10 в. ( 10-3 pi ) ( c .3 pi-10 ) D. none of these |
12 |
| 555 | Find the minimum value the function ( f(x)=frac{pi^{2}}{16 cot ^{-1}(-x)}-cot ^{-1} x ) ( ^{A} cdot-frac{pi}{4} ) B. ( -frac{pi}{2} ) c. D. |
12 |
| 556 | Illustration 5.36 If x € (-1, 0), then find the value of cos-(2×2 – 1)-2 sin-‘x. |
12 |
| 557 | Assertion Statement 1 Range of ( boldsymbol{f}(boldsymbol{x})=tan ^{-1} boldsymbol{x}+ ) ( sin ^{-1} x+cos ^{-1} x ) is ( (0, pi) ) Reason Statement ( 2 f(x)=tan ^{-1} x+ ) ( sin ^{-1} x+cos ^{-1} x=frac{pi}{2}+tan ^{-1} x ) for ( boldsymbol{x} boldsymbol{epsilon}(-mathbf{1}, mathbf{1}] ) A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1 B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE |
12 |
| 558 | ( tan ^{-1}(-2) ) is equal to A ( cdot-cos ^{-1}left(frac{-3}{5}right) ) в. ( _{pi}+cos ^{-1} frac{3}{5} ) c. ( -frac{pi}{2}+tan ^{-1} xleft(-frac{3}{5}right) ) D. none of these |
12 |
| 559 | Solve the following equations. ( sin ^{-1} frac{5}{x}+sin ^{-1} frac{x}{2}=frac{pi}{2} ) | 12 |
| 560 | 2x 8. The maximum value of f(x) = tan osa. 18° c. 22.5° sus b. 36° ismert.. d. 15° 2 in se |
12 |
| 561 | 27. If f(x)= x11 + x – x? + x + 1 and f(sin-‘(sin 8)) = a, a is a constant, then f (tan-(tan 8)) is equal to .. a. a b. a- 2 c. a + 2 d. 2-a . |
12 |
| 562 | f ( sin ^{-1}left(frac{x}{5}right)+operatorname{cosec}^{-1}left(frac{5}{4}right)=frac{pi}{2} ) then ( a ) value of ( x ) is: ( A ) B. 3 ( c cdot 4 ) D. 5 |
12 |
| 563 | f ( sin ^{-1}left(tan frac{17 pi}{4}right)-sin ^{-1}(sqrt{frac{3}{x}}) ) ( left(frac{pi}{6}right)=0, ) then ( x ) is a root of the equation A ( cdot x^{2}-x-6=0 ) B . ( x^{2}+x-6=0 ) c. ( x^{2}-x-12=0 ) D. ( x^{2}+x-12=0 ) |
12 |
| 564 | The solution set of the equation ( sin ^{-1} x=2 tan ^{-1} x ) ( mathbf{A} cdot{1,2} ) B ( cdot{-1,2} ) ( mathbf{c} cdot{-1,1,0} ) D. ( left{1, frac{1}{2}, 0right} ) |
12 |
| 565 | The value of ( cos left(tan ^{-1} tan 4right) ) is A ( frac{1}{sqrt{17}} ) B. ( -frac{1}{sqrt{17}} ) ( c cdot cos 4 ) D. – cos 4 |
12 |
| 566 | 39. The solution of the inequality log12 sin x > log1/2 cos x a. XEO, b. x 1 1 1 XE c. xe0T2) d. None of these |
12 |
| 567 | 61. The sum of roots of the equation 1 tan-1 – + tan 1+2x – + ton- 1 = tan-1 – 1+ 4x – a. 2 c. 4 b. 3 d. none of these |
12 |
| 568 | ( fleft(frac{1}{3}right)+cos ^{-1} x=frac{pi}{2}, ) then find ( x ) | 12 |
| 569 | If ( 2 sinh ^{-1}left(frac{a}{sqrt{1-a^{2}}}right)=log left(frac{1+X}{1-X}right) ) then ( X= ) ( A ) B. c. ( sqrt{1-a^{2}} ) D. ( frac{1}{sqrt{1-a^{2}}} ) |
12 |
| 570 | If ( boldsymbol{alpha} leq 2 sin ^{-1} boldsymbol{x}+cos ^{-1} boldsymbol{x} leq boldsymbol{beta}, ) then A ( cdot alpha=frac{pi}{2}, beta=frac{pi}{2} ) B. ( quad alpha=frac{pi}{2}, beta=frac{3 pi}{2} ) c. ( alpha=0, beta=pi ) |
12 |
| 571 | 89. The number of real solutions of the equation V1 + cos2x = V2 sin? (sin x), -ASxSt, is a. 0 c. 2 b. 1 d. infinite |
12 |
| 572 | If ( sin ^{-1} x=frac{pi}{5}, ) for some ( x in(-1,1) ) then find the value of ( cos ^{-1} x ) A. ( -frac{3 pi}{10} ) в. ( frac{3 pi}{10} ) c. ( -frac{7 pi}{10} ) D. ( frac{7 pi}{10} ) |
12 |
| 573 | 10. The number of real solutions of the equation tan! Vx2 – 3x + 2 + cos 14x – x² – 3 = is a. one b. two c. zero d. infinite |
12 |
| 574 | Find the principal value of: ( sec ^{-1}left(2 sin frac{3 pi}{4}right) ) |
12 |
| 575 | 58. If x, y, z are natural numbers such that cot ‘x + cotly = cot-‘z then the number of ordered triplets (x, y, z) that satisfy the equation is a. O b. 1 c. 2 d. Infinite solutions |
12 |
| 576 | Evaluate the following: i. ( sin left(cot ^{-1} xright) ) ii. ( sin left(frac{pi}{2}-sin ^{-1}left(-frac{sqrt{3}}{2}right)right) ) ( ^{mathbf{A}} cdot(mathrm{i}) frac{1}{sqrt{x^{2}+1}},(mathrm{ii}) frac{1}{2} ) B. ( quad(i) frac{1}{sqrt{x^{2}-1}},left(text { ii) } frac{-sqrt{3}}{2}right. ) ( c cdot(i) frac{-1}{sqrt{x^{2}+1}},left(text { ii) } frac{1}{2}right. ) D. ( left(text { i) } frac{-1}{sqrt{x^{2}-1}} text { , (ii) } frac{+sqrt{3}}{2}right. ) |
12 |
| 577 | Assertion If ( alpha, beta ) are the roots of the equation ( 18left(tan ^{-1} xright)^{2}-9 pi tan ^{-1} x+pi^{2}=0 ) ( operatorname{then} boldsymbol{alpha}+boldsymbol{beta}=frac{boldsymbol{4}}{sqrt{mathbf{3}}} ) Reason ( sec ^{2}left(cos ^{-1}left(frac{1}{4}right)right)+ ) ( operatorname{cosec}^{2}left(sin ^{-1}left(frac{1}{5}right)right)=41 ) 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 |
| 578 | The value of ( sin ^{-1}left[cos left[sin ^{-1}left(frac{1}{2} tan frac{pi}{3}right)right]right] ) is ( mathbf{A} cdot 2 tan ^{-1}(2-sqrt{3}) ) B ( cdot 2 tan ^{-1}(sqrt{2}-1) ) ( mathbf{c} cdot 2 tan ^{-1}(sqrt{2}+1) ) D. |
12 |
| 579 | 68. The value of tan-14 … equals + tan-1 19 + tan 14 20 + tan-1 4+ 39 a. tan-‘1 + tan- tan-11 b. tan1 + cott3 c. cot ‘1 + cot-1 + cott d. cot ! 1+tan 3 |
12 |
| 580 | ( cos left[2 cos ^{-1} frac{1}{5}+sin ^{-1} frac{1}{5}right]= ) | 12 |
| 581 | in? – 10n +21.6) 10. If cot-1 E N, then n can be a. 3 c. 4 b. 2 d. 8 |
12 |
| 582 | ( cos ^{-1}left(cos left(frac{5 pi}{4}right)right) ) is given by ( mathbf{A} cdot 5 pi / 4 ) B . ( 3 pi / 4 ) ( mathbf{c} .-pi / 4 ) D. None of these |
12 |
| 583 | If ( sum_{i=1}^{2 n} sin ^{-1} x_{i}=n pi, ) then ( sum_{i=1}^{2 n} x_{i} ) is equal to A ( . n ) B. ( 2 n ) c. ( frac{n(n+1)}{2} ) D. None of these |
12 |
| 584 | The principal value of ( tan ^{-1}left(cot frac{3 pi}{4}right) ) is : A. ( -frac{3 pi}{4} ) в. ( frac{3 pi}{4} ) ( c cdot-frac{pi}{4} ) D. |
12 |
| 585 | Prove that ( : cos ^{-1}left(frac{3}{5}right)+cos ^{-1}left(frac{4}{5}right)= ) ( frac{pi}{2} ) |
12 |
| 586 | ( cot ^{-1} frac{x y+1}{x-y}+cot ^{1} frac{y z+1}{y-z}+ ) ( cot ^{-1} frac{x z+1}{z-x} ) ( A ) B. – ( c cdot 0 ) D. none of these |
12 |
| 587 | If ( sin ^{-1}(1-x)-2 sin ^{-1} x=pi / 2, ) then ( x ) is equals? ( mathbf{A} cdot{0,-1 / 2} ) B cdot ( {1 / 2,0} ) ( c cdot{0} ) D. (-1,0) |
12 |
| 588 | If ( alpha, beta ) are the roots of the equation ( left(tan ^{-1}(x / 5)right)^{2}+(sqrt{3}- ) 1) ( tan ^{-1}(x / 5)-sqrt{3}=0,|alpha|>|beta| ) then This question has multiple correct options A ( cdot alpha+beta=-5 pi / 12 ) в. ( |alpha-beta|=35 pi / 12 ) ( mathbf{c} cdot alpha beta=-25 pi^{2} / 12 ) D. ( 3 alpha+4 beta=0 ) |
12 |
| 589 | Find the value of : ( sec ^{2}left(tan ^{-1} 2right)+csc ^{2}left(cot ^{-1} 3right) ) A . 11 B. 15 c. 17 D. 21 |
12 |
| 590 | ( int frac{x^{2}}{sqrt{1-x^{6}}} d x=frac{1}{k} sin ^{-1}left(x^{k}right)+C . ) what | 12 |
| 591 | ( tan left(cot ^{-1} xright) ) is equal to: A ( cdot frac{pi}{2}-x ) B. ( cot left(tan ^{-1} xright) ) ( c cdot tan x ) D. |
12 |
| 592 | ( sin ^{-1}(117 / 125) ) is equal to This question has multiple correct options A ( .2 alpha ) в. ( 3 alpha ) c. ( pi / 2-2 alpha ) D. ( pi-3 alpha ) |
12 |
| 593 | Illustration 5.42 If sin-‘x = 7d/5, for some x € (-1, 1), then find the value of cos-‘x. |
12 |
| 594 | The equation ( 2 cos ^{-1} x+sin ^{-1} x=frac{11 pi}{6} ) has A. No solution B. One solution c. Two solutions D. Three solutions |
12 |
| 595 | Illustration 5.30 Prove that cosec(tan’ (cos(cot’ (sec(sin-‘ a))))) = 13 – a?, where a € [0, 1]. |
12 |
| 596 | Solve the equation:cos ( ^{-1}left(frac{x^{2}-1}{x^{2}+1}right)+ ) ( sin ^{-1}left(frac{2 x}{x^{2}+1}right)+tan ^{-1}left(frac{2 x}{x^{2}-1}right)= ) 3 A ( cdot frac{1}{sqrt{3}} ) B. ( cot 20^{circ} ) ( c cdot frac{-1}{sqrt{3}} ) D. ( tan 20 ) |
12 |
| 597 | Prove: ( 3 cos ^{-1} x=cos ^{-1}left(4 x^{3}-3 xright), x inleft[frac{1}{2}, 1right] ) |
12 |
| 598 | The value of ( x ) satisfying ( tan left(sec ^{-1} xright)=sin left(cos ^{-1} frac{1}{sqrt{5}}right) ) is ( ^{A}+frac{3}{sqrt{5}} ) ( ^{mathrm{B}} pm frac{5}{sqrt{3}} ) c. ( =frac{sqrt{2}}{3} ) D. ( pm frac{3}{5} ) |
12 |
| 599 | If ( boldsymbol{A}=tan ^{-1}left(frac{boldsymbol{x} sqrt{mathbf{3}}}{mathbf{2 K}-boldsymbol{x}}right) ) and ( boldsymbol{B}= ) ( tan ^{-1}left(frac{2 x-K}{K sqrt{3}}right), ) then the value of ( A-B ) is ( A ) B. 45 ( c .60 ) D. 30 |
12 |
| 600 | Find the principal value of the following ( tan ^{-1}left(tan frac{3 pi}{4}right) ) |
12 |
| 601 | ( operatorname{Let} tan ^{-1} y=tan ^{-1} x+ ) ( tan ^{-1}left(frac{2 x}{1-x^{2}}right) ) where ( |x|<frac{1}{sqrt{3}} . ) Then a value of y is A ( cdot frac{3 x-x^{3}}{1-3 x^{2}} ) В. ( frac{3 x+x^{3}}{1-3 x^{2}} ) c. ( frac{3 x+x^{3}}{1+3 x^{2}} ) D. ( frac{3 x-x^{3}}{1+3 x^{2}} ) |
12 |
| 602 | 19. The range of values of p for which the sin cos (cos(tan-‘x) = p has a solution is a (te ta 1) b. (0,1) d. (-1,1) d. (-1,1) |
12 |
| 603 | Find the principal value of: ( operatorname{cosec}^{-1}left(2 cos frac{2 pi}{3}right) ) |
12 |
| 604 | If ( boldsymbol{x}=mathbf{2} cos ^{-1}left[frac{1}{2}right]+sin ^{-1}left[frac{mathbf{1}}{sqrt{mathbf{2}}}right]+ ) ( tan ^{-1}(sqrt{3}) ) and ( y= ) ( cos left[frac{1}{2} sin ^{-1}left[sin frac{x}{2}right]right] ) then which of the following statements holds good? A ( cdot y=cos frac{3 pi}{16} ) В. ( y=cos frac{5 pi}{16} ) c. ( x=4 cos ^{-1} y ) D. none of these |
12 |
| 605 | The principal value of ( sin ^{-1}left{cos left(sin ^{-1} frac{sqrt{3}}{2}right)right} ) A. ( frac{pi}{6} ) в. ( frac{pi}{3} ) ( c cdot-frac{pi}{3} ) D. none of these |
12 |
| 606 | Find the value of ( sin ^{-1} x+sin ^{-1} frac{1}{x}+ ) ( cos ^{-1} x+cos ^{-1} frac{1}{x} ) |
12 |
| 607 | Assertion ( sin ^{-1}left[x-frac{x^{2}}{2}+frac{x^{3}}{4} dotsright]=pi / 2 ) ( cos ^{-1}left[x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4} ldots .right] ) for ( 0<|x|< ) ( sqrt{2} ) has a unique solution. Reason ( tan ^{-1} sqrt{x(x+1)}+ ) ( sin ^{-1} sqrt{x^{2}+x+1}=pi / 2 ) has no solution for ( -sqrt{2}<x<0 ) 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 |
| 608 | (2x²+4) Illustration 5.19 Solve sin-| sin < 1 – 3 (1+x2 |
12 |
| 609 | If ( tan ^{-1} frac{a+x}{a}+tan ^{-1} frac{a-x}{a}=frac{pi}{6} ) then ( x^{2}=? ) A ( cdot 2 sqrt{3} a ) an B. ( sqrt{3} a ) ( c cdot 2 sqrt{3} a^{2} ) ( ^{2} ) D. None of these |
12 |
| 610 | If ( sin ^{-1} x+sin ^{-1} y=frac{2 pi}{3}, ) then ( cos ^{-1} x+cos ^{-1} y= ) A ( cdot frac{pi}{6} ) в. ( c cdot frac{pi}{3} ) D. |
12 |
| 611 | ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} Rightarrow x= ) ( A cdot-1 ) B. ( c cdot c ) D. ( pi sqrt{frac{5}{8}} ) |
12 |
| 612 | Evaluate : ( int x^{2} tan ^{-1} frac{x}{2} d x ) |
12 |
| 613 | Illustration 5.25 Find the value of sin -cos |
12 |
| 614 | – а – х а 55. If tan-4° + tan-1″ а a. 2/за с. 2/за” b. За d. none of these |
12 |
| 615 | Find the value of ( x ) if ( tan ^{-1} x+2 cot ^{-1} x=frac{2 pi}{3} ) |
12 |
| 616 | represents the graph of the function ( f(x)=lim _{n rightarrow infty} frac{2}{pi} tan ^{-1}(n x) ? ) ( A ) B. ( c ) ( D ) |
12 |
| 617 | The domain of function ( f(x)=sin ^{-1} 5 x ) is A ( cdotleft(-frac{1}{5}, frac{1}{5}right) ) B. ( left[-frac{1}{5}, frac{1}{5}right] ) c. ( R ) D. ( left(0, frac{1}{5}right) ) |
12 |
| 618 | ( cos ^{-1}left(cos left(frac{-17 pi}{5}right)right) ) is equal to A. ( -frac{17 pi}{5} ) в. ( frac{3 pi}{5} ) ( c cdot frac{2 pi}{5} ) D. none of these |
12 |
| 619 | Express the following in the simplest form ( tan ^{-1}left(frac{cos x}{1+sin x}right), frac{-pi}{2}<x<frac{pi}{2} ) |
12 |
| 620 | 1. COS Find the value of : cos(2cos-tx + sin- x) at x = , where Oscos-?xst and -1/2 <sin x S /2. (1981 – 2 Marks) |
12 |
| 621 | Find the value of ( cos left(sec ^{-1} x+right. ) ( left.operatorname{cosec}^{-1} xright),|x| geq 1 ) |
12 |
| 622 | If ( A=tan ^{-1} frac{1}{7} ) and ( B=tan ^{-1} frac{1}{3} ) then This question has multiple correct options A ( cdot cos 2 A=frac{24}{25} ) B. ( cos 2 B=frac{4}{5} ) c. ( cos 2 A=sin 4 B ) D. ( tan 2 B=frac{3}{4} ) |
12 |
| 623 | The greatest and least value of ( left(sin ^{-1} xright)^{2}+left(cos ^{-1} xright)^{2} ) are respectively A ( cdot frac{pi^{2}}{4} a n d 0 ) B . ( frac{pi}{2} a n d-frac{pi}{2} ) C. ( frac{5 pi^{2}}{4} ) and ( frac{pi^{2}}{8} ) D. ( frac{pi^{2}}{4} ) and ( frac{-pi}{4} ) |
12 |
| 624 | If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=pi, ) prove that ( x sqrt{1-x^{2}}+y sqrt{1-y^{2}}+ ) ( z sqrt{1-z^{2}}=2 x y z ) |
12 |
| 625 | ( sec ^{2}left(tan ^{-1} 2right)+operatorname{cosec}^{2}left(cot ^{-1} 3right) ) | 12 |
| 626 | Illustration 5.60 If x > y > z> 0, then find the value of cot-1 *y + 1 – + cot-1 YZ +1 – + cor-1 ZX + 1 x – y Y – Z z – x |
12 |
| 627 | 2. 2 tan- ‘(- 2) is equal to a. – COS b. – Te + cos2 |
12 |
| 628 | Tllustration 2.8 Find the value of cos+ (-1/2). |
12 |
| 629 | 53. The exhaustive set of values of a for which a-cot- 3x = 2tan-‘3x + cos ‘x v3 + sin ‘x 73 may have solution, is TTT (1 370 a. — c. 172 [20 40 [ 31 71 I 2 3 1 6 6 |
12 |
| 630 | Differentiate ( tan ^{-1}left(frac{a cos x-b sin x}{b cos x+a sin x}right) ) | 12 |
| 631 | Find the principal value of: ( sin ^{-1}left(frac{sqrt{mathbf{3}}-1}{2 sqrt{2}}right) ) |
12 |
| 632 | Solve the equation ( sin ^{-1}(3 x)=-frac{1}{3} pi ) giving the solution in an exact form. |
12 |
| 633 | Solve ( int frac{sin ^{-1} sqrt{x}-cos ^{-1} sqrt{x}}{sin ^{-1} sqrt{x}+cos ^{-1} sqrt{x}} d x ) | 12 |
| 634 | Show that ( sin ^{-1}(2 x sqrt{1-x^{2}})=2 sin ^{-1} x ) | 12 |
| 635 | Evaluate: ( tan ^{-1} sqrt{3}-cot ^{-1}(-sqrt{3}) ) ( mathbf{A} cdot mathbf{0} ) B. ( 2 sqrt{3} ) ( c cdot-frac{pi}{2} ) D. |
12 |
| 636 | If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) and ( boldsymbol{f}(mathbf{2})=mathbf{2}, boldsymbol{f}(boldsymbol{a}+boldsymbol{b})= ) ( boldsymbol{f}(boldsymbol{a}) boldsymbol{f}(boldsymbol{b}), forall boldsymbol{a}, boldsymbol{b} boldsymbol{epsilon} boldsymbol{R}, ) then ( boldsymbol{x}^{f(mathbf{2})}, boldsymbol{y}^{f(4)}, boldsymbol{z}^{f(boldsymbol{6})} ) are in This question has multiple correct options A. A.P в. G. c. н. D. None |
12 |
| 637 | Prove that: ( tan ^{-1}left(frac{6 x-8 x^{3}}{1-12 x^{2}}right) ) ( tan ^{-1}left(frac{4 x}{1-4 x^{2}}right)=tan ^{-1} 2 x ;|2 x|< ) ( frac{1}{sqrt{3}} ) |
12 |
| 638 | ( sin left[2 cos ^{-1} cot left(2 tan ^{-1} frac{1}{2}right)right] ) is equal to A ( cdot frac{3 sqrt{7}}{8} ) B. ( frac{5 sqrt{7}}{8} ) ( c cdot frac{5 sqrt{7}}{2} ) D. ( frac{3 sqrt{7}}{2} ) |
12 |
| 639 | Assertion: The value of ( frac{tan ^{-1} frac{4}{3}}{tan ^{-1} frac{1}{2}} ) is equal to 2 Reason: ( forall boldsymbol{x} in[mathbf{0}, mathbf{1}], tan ^{-1}left(frac{mathbf{2} boldsymbol{x}}{mathbf{1 – x}^{2}}right)= ) ( 2 tan ^{-1} x ) A. Both A and R are true and R is the correct explanation of B. Both A and R are true but R is not the correct explanation of A c. ( R ) is true but ( A ) is false D. A is true but R is false |
12 |
| 640 | ( sec h^{-1}left(frac{1}{5}right)= ) ( mathbf{A} cdot log (sqrt{24}+5) ) ( mathbf{B} cdot log 5+sqrt{27} ) ( mathbf{C} cdot log 26+sqrt{5} ) D. ( log 27+sqrt{5} ) |
12 |
| 641 | ( cos ^{-1}left(cos left(2 cot ^{-1}(sqrt{2}-1)right)right) ) is equal to A ( cdot sqrt{2}-1 ) в. ( c cdot frac{3 pi}{4} ) D. none of these |
12 |
| 642 | Write the principal value of ( tan ^{-1}(1)+ ) ( cos ^{-1}left(-frac{1}{2}right) ) | 12 |
| 643 | Solve: ( sin ^{-1}left(frac{2 x}{1+x^{2}}right) ) |
12 |
| 644 | 52. The product of all values of x satisfying the equation sin-cos(2x+ + 10|x|+ 4 = cotcor- (2-18ſx))+ 0$ x2 + 5|x +3 ) coco (9|x|JJ*2 is a. 9 b. -9 c. -3 |
12 |
| 645 | ( cos ^{-1}(sqrt{frac{boldsymbol{a}-boldsymbol{x}}{boldsymbol{a}-boldsymbol{b}}})=sin ^{-1}(sqrt{frac{boldsymbol{x}-boldsymbol{b}}{boldsymbol{a}-boldsymbol{b}}}) ) possible if A ( . a>x>b ) or ( a<xb ) and ( x ) takes any value D. ( a<b ) and ( x ) takes any value |
12 |
| 646 | Illustration 5.41 Prove that 2 tan-‘(cosec tan ‘x- tan cotx) = tan- x (x 0). |
12 |
| 647 | The value of ( x ) for which ( sin left(cot ^{-1}(1+x)right)=cos left(tan ^{-1} xright) ) is ( A cdot frac{1}{2} ) B. ( c ) D. ( -frac{1}{2} ) |
12 |
| 648 | Prove that: ( cos ^{-1}(x)+cos ^{-1}left{frac{x}{2}+frac{sqrt{3-3 x^{2}}}{2}right}=frac{pi}{3} ) |
12 |
| 649 | If ( 2 sin ^{-1} x=sin ^{-1}(2 x sqrt{1-x^{2}}), ) then ( boldsymbol{x} in ) A ( .[-1,1] ) B. ( left[-frac{1}{sqrt{2}}, 1right] ) ( c cdotleft[-frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right] ) D. none of these |
12 |
| 650 | f ( sin ^{-1}left(frac{2 a}{1+a^{2}}right)+sin ^{-1}left(frac{2 b}{1+a^{2}}right)= ) ( 2 tan ^{-1} x, ) then ( x ) is equal to A ( frac{a-b}{1+a b} ) в. ( frac{b}{1+a b} ) c. ( frac{b}{1-a b} ) D. ( frac{a+b}{1-a b} ) |
12 |
| 651 | ( operatorname{Let} f(x)=cos left(tan ^{-1} 2 xright)- ) ( sin left{tan ^{-1}left(frac{1}{2 x+1}right)right} ) and ( a= ) ( cos left(tan ^{-1}left(sin left(cot ^{-1} 2 xright)right)right) ) and ( b= ) cos ( left(frac{pi}{2}+cos ^{-1} 2 xright) cdot ) If ( f(x)=0, ) then ( b= ) A. ( frac{1}{sqrt{2}} ) B. ( -frac{sqrt{3}}{2} ) c. ( frac{sqrt{3}}{2} ) D. ( -frac{1}{sqrt{2}} ) |
12 |
| 652 | Topic-wise SULULUI Ras ve 3. For any positive integer n, define f.,:(0,00) For any positi f, (*) – Ez- tan” (+68+)(x+;-)) for all x 0,00). Here, the inverse trigonometric function tan” xassumes values in Then, which of the following statement(s) is (are) TRUE? (JEE Adv. 2018) (a) (b) :-1 tan? (,0)=55 1 (1+f;(0)) sec (5,0)=10 © For any fixed positive integer n, lim tan (, (x)= *+00 (d) For any fixed positive integer n, lim sec (f(x))=1 Com |
12 |
| 653 | Find the value of ( sin ^{-1} x+sin ^{-1} frac{1}{x}+ ) ( cos ^{-1} x+cos ^{-1} frac{1}{x} ) A. ( -pi ) в. ( +pi ) ( c .-2 pi ) D. ( +2 pi ) |
12 |
| 654 | If ( a sin ^{-1} x-b cos ^{-1} x=c, ) then the value of ( a sin ^{-1} x+b c o s^{-1} x ) (whenever exists) is equal to A . 0 B. ( frac{pi a b+c(b-a)}{a+b} ) ( c cdot frac{pi}{2} ) D. ( frac{pi a b+c(a-b)}{a+b} ) |
12 |
| 655 | 7. sec?(tan-? 2) + cosec?(cot-3) is equal to a. 5 b. 13 190 c. 1575] 100 d. 6 ostalo |
12 |
| 656 | Find general solution of the following equations: ( sin theta=frac{1}{2} ? ) |
12 |
| 657 | If ( left[cot ^{-1} xright]+left[cos ^{-1} xright]=0, ) where [ denotes the greatest integer function, then the complete set of values of ( x ) is A ( .(cos 1,1] ) B. ( (cos 1,-cos 1) ) c. ( (cot 1,1] ) D. none of these |
12 |
| 658 | Find the value of ( tan left[frac{1}{2} cos ^{-1} frac{sqrt{5}}{3}right] ) | 12 |
| 659 | Illustration 5.73 Solve sin ‘ x – cos’ x = sin-‘(3x – 2). COS |
12 |
| 660 | The principal value of ( cos ^{-1}left[frac{1}{sqrt{2}}left(cos left(frac{9 pi}{10}right)-sin left(frac{9 pi}{10}right)right)right] ) is ( mathbf{A} cdot frac{3 pi}{20} ) B. ( frac{7 pi}{200} ) ( mathbf{C} cdot frac{7 pi}{10} ) D. none of these |
12 |
| 661 | The value of ( : tan ^{-1} frac{1}{5}+tan ^{-1} frac{1}{7}+ ) ( tan ^{-1} frac{1}{3}+tan ^{-1} frac{1}{8}-frac{pi}{4} ) |
12 |
| 662 | Write the simplest form of : ( tan ^{-1} frac{sqrt{1+x^{2}}-1}{x}, x neq 0 ) |
12 |
| 663 | 32. There exists a positive real number x satisfying cos(tan-‘x) = x. Then the value of cos! Tu labb. Tebe c. 21 |
12 |
| 664 | If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots inftyright)+ ) ( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots inftyright)=frac{pi}{2} ) and ( 0<x<sqrt{2}, ) then ( x= ) A ( cdot frac{1}{2} ) в. ( c cdot-frac{1}{2} ) D. ( frac{sqrt{3}}{2} ) |
12 |
| 665 | Find ( n ) if ( sin ^{-1} frac{4}{5}+sin ^{-1} frac{5}{13}+ ) ( sin ^{-1}left(frac{16}{65}right)=frac{n pi}{2} ) |
12 |
| 666 | Illustration 5.40 Find the value of sin-‘(sin 5) + cos(cos 10) + tan-‘{tan(-6)} + cot-‘{cot(-10)}. |
12 |
| 667 | TT 377 30. If sin” : [-1, 1] and cos!: (-1, 1] → [0, 1] he two bijective functions, respectively inverses of bijective functions sin: → [-1, 1] and cos : [0, 1] + [-1, 1], then sin!x + cos ix is b. TT 31 o sito d. not a constant |
12 |
| 668 | If ( a_{1}, a_{2}, a_{3} dots dots dots a_{n} ) are in A.P.with common difference ( d, ) then ( tan left[tan ^{-1}left(frac{d}{1+a_{1} a_{2}}right)+tan ^{-1}left(frac{d}{1+a_{2} a_{3}}right)+right. ) ( dots dots dots dots dots+ ) ( left.tan ^{-1}left(frac{d}{1+a_{n-1} a_{n}}right)right]=? ) A ( cdot frac{(n-1) d}{a_{1}+a_{n}} ) B. ( frac{(n-1) d}{1+a_{1} a_{n}} ) c. ( frac{n d}{1+a_{1} a_{n}} ) D. ( frac{a_{n}-a_{1}}{a_{n}+a_{1}} ) |
12 |
| 669 | 18. The number of integral values of k for which the equation sin-‘x + tan-?x=2k + 1 has a solution is a. 110 0 b . 2 acting c. 3 d. 4 |
12 |
| 670 | The principal solution of the equation ( cot x=-sqrt{3} ) is A ( cdot frac{pi}{6} ) в. ( frac{pi}{3} ) c. ( frac{5 pi}{6} ) D. ( -frac{5 pi}{6} ) |
12 |
| 671 | If ( cos ^{-1} x-cos ^{-1} frac{y}{2}=alpha ) where -1 ( 1 leq x leq 1,-2 leq y leq 2, x leq frac{y}{2} ) then for all ( 4 x^{2}-4 x y cos alpha+y^{2} ) is equal to A ( cdot 4 sin ^{2} alpha-2 x^{2} y^{2} ) B. ( 4 cos ^{2} alpha+2 x^{2} y^{2} ) ( mathbf{c} cdot 4 sin ^{2} alpha ) ( D cdot 2 sin ^{2} alpha ) |
12 |
| 672 | Assertion The equation ( 2left(sin ^{-1} xright)^{2}- ) ( 5left(sin ^{-1} x+2right)=0 ) Reason ( sin ^{-1}(sin x)=x ) if ( x epsilon[-1.57,1.57] ) 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 false but Reason is true |
12 |
| 673 | State True or False ( sin ^{-1} 2+cos ^{-1} 2=frac{pi}{2} ) A. True B. False |
12 |
| 674 | If ( tan ^{-1} x+tan ^{-1} y=frac{pi}{4}, x y<1, ) then write the value of ( boldsymbol{x}+boldsymbol{y}+boldsymbol{x} boldsymbol{y} ) |
12 |
| 675 | Prove that ( tan ^{-1}left(frac{sin x}{1+cos x}right),-pi<x<pi ) | 12 |
| 676 | If ( sin ^{-1}left(frac{2 p}{1+p^{2}}right)-cos ^{-1}left(frac{1-q^{2}}{1+q^{2}}right)= ) ( tan ^{-1}left(frac{2 x}{1+x^{2}}right), ) then the value of ( x ) is equal to A ( cdot frac{p+q}{1+p q} ) в. ( frac{p-q}{1-p q} ) c. ( frac{p-q}{p q-1} ) D. ( frac{p-q}{p q+1} ) |
12 |
| 677 | Equations ( 2 sin ^{-1} x+3 sin ^{-1} y=frac{5 pi}{2} ) and ( y=k x-5 ) hold simultaneously when k equals ( A cdot 2 ) B. 4 ( c cdot 6 ) D. no such k exists |
12 |
| 678 | Find the principal value of ( cos ^{-1}left(-frac{1}{2}right) ) | 12 |
| 679 | Find the principal value of the following: ( tan ^{-1}(-1) ) |
12 |
| 680 | illustration 5:26. Find the value of sin(3cot” (22) Illustration 5.24 Find the value of sin CO2 |
12 |
| 681 | Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x ) [ begin{aligned} tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} \ x y1 end{aligned} ] Solve (a) ( cos left(2 sin ^{-1} xright)=1 / 9 ) (b) ( cos ^{-1}(3 / 5)-sin ^{-1}(4 / 5)=cos ^{-1} x ) (c) If ( sin left(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1, ) then prove that ( x ) is equal to ( 1 / 5 ) |
12 |
| 682 | Solve:tan ( ^{-1} mathbf{2} boldsymbol{x}+tan ^{-1} mathbf{3} boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{4}} ) | 12 |
| 683 | If ( alpha ) is a real number for which ( f(x)= ) ( log _{e} cos ^{-1} x ) is defined, then a possible value of ( [boldsymbol{alpha}] ) (where [] denotes the greatest function) is This question has multiple correct options A . 0 B. ( c cdot-1 ) D. – – |
12 |
| 684 | Assertion The value of the determinant ( begin{array}{|ccc|}tan ^{-1} x & cot ^{-1} x & pi / 2 \ sin ^{-1}(4 / 5) & sin ^{-1}(3 / 5) & sin ^{-1} 1 \ cos ^{-1}(3 / 5) & cos ^{-1}(4 / 5) & 1end{array} ) equal to zero for all values of ( x ) Reason ( 2 cos ^{-1} x=cos ^{-1}left(2 x^{2}-1right) ) if ( -1 leq ) ( x leq 1 ) 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 |
| 685 | Illustration 5.31 If x < 0, then prove that cos-"x = 1 – sin- 1 – x?. |
12 |
| 686 | Prove that ( sin ^{-1} frac{3}{5}-sin ^{-1} frac{8}{17}= ) ( cos ^{-1}left(frac{84}{85}right) ) |
12 |
| 687 | Express ( tan ^{-1} x+tan ^{-1} frac{2 x}{1-x^{2}} ) in the terms of ( tan ^{-1} frac{3 x-x^{3}}{1-3 x^{2}} ) |
12 |
| 688 | n Illustration 5.35 Prove that cos- | = 2 tan- x (1+z2n 0<x< . |
12 |
| 689 | If two angle of a triangle are ( sin ^{-1}left(frac{1}{sqrt{5}}right) ) and ( sin ^{-1}left(frac{1}{sqrt{10}}right), ) then third angle is ( A cdot frac{pi}{4} ) B. ( c cdot frac{3 pi}{4} ) D. ( frac{2 pi}{3} ) |
12 |
| 690 | 6. If tan ‘y=4 tan ‘x (14|<tan), find y as an algebraic function of x, and, hence, prove that tan 7/8 is a root of the equation x4 – 6×2 + 1 = 0. |
12 |
| 691 | If ( (x-1)left(x^{2}+1right)>0, ) then find the value of ( sin left(frac{1}{2} tan ^{-1} frac{2 x}{1-x^{2}}-tan ^{-1} xright) ) A . ( -1 / 2 ) B. – ( c cdot 1 / 2 ) D. |
12 |
| 692 | 16. If [cot-‘ x] + [cos-1 x] = 0, where [.] denotes the greatest integer function, then the complete set of values of x is a. (cos 1, 1] b. (cos 1, cos 1) c. (cot 1, 1] d. none of these |
12 |
| 693 | The range of the function, ( boldsymbol{f}(boldsymbol{x})= ) ( left(1+sec ^{-1} xright)left(1+cos ^{-1} xright) ) is ( A cdot(-infty, infty) ) В ( cdot(-infty, 0] cup[4, infty) ) ( ^{c} cdotleft{0,(1+pi)^{2}right} ) D. ( left[1,(1+pi)^{2}right] ) |
12 |
| 694 | If ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8}, ) then ( boldsymbol{x}= ) A . -1 B. ( c cdot 0 ) D. |
12 |
| 695 | Find the value of sin-‘x+sin! – + cos x Illustration 5.51 + cos-1- |
12 |
| 696 | Find the value of ( sin ^{-1}left(2 cos ^{2} x-1right)+ ) ( cos ^{-1}left(1-2 sin ^{2} xright) ) A ( cdot frac{pi}{2} ) в. c. D. |
12 |
| 697 | 23. The value of the expression sin- sin 221 cos” (cos Spa) + tan” (tan 5x) + sin-” (cos 2)is a. 1772 -2 6. – 2 d. none of these |
12 |
| 698 | The number of solutions of ( sin ^{-1}left(1+b+b^{2}+cdots inftyright)+ ) ( cos ^{-1}left(a-frac{a^{2}}{3}+frac{a^{2}}{9} cdots inftyright)=frac{pi}{2} ) is ( A ) B. 2 ( c .3 ) ( D ) |
12 |
| 699 | ( sin ^{-1}|sin x|=sqrt{sin ^{-1}|sin x|} ) then ( x= ) This question has multiple correct options A ( . n pi-1 ) в. ( n pi ) c. ( n pi+1 ) D. ( n frac{pi}{2}+1 ) |
12 |
| 700 | Evaluate the following: ( sin ^{-1}left(frac{2 pi}{4}right) ) ii. ( cos ^{-1}left(cos frac{7 pi}{6}right) ) iii. ( tan ^{-1}left(tan frac{2 pi}{3}right) ) iv. ( cos left(cos ^{-1}left(frac{sqrt{3}}{2}right)+frac{pi}{6}right) ) A ( cdot ) i. ( -frac{pi}{3} ) ii. ( -frac{5 pi}{6} ) iii. ( frac{pi}{3} ) iv. 1 B . i. ( frac{2 pi}{3} ) ii. ( frac{pi}{6} ) iii. ( -frac{2 pi}{3} ) iv. 1 C ( cdot ) i. ( -frac{2 pi}{3} ) ii. ( -frac{pi}{6} ) iii. ( frac{2 pi}{3} ) iv. -1 D cdot i. ( frac{pi}{3} ) ii. ( frac{5 pi}{6} ) iii. ( -frac{pi}{3} ) iv. -1 |
12 |
| 701 | 34. If sin + x + sin ‘y , then in 1 + x² + y 5 is equal to x² – x² y + y² b. 2 a. 1 d. none of these |
12 |
| 702 | If ( boldsymbol{alpha}, boldsymbol{beta}(boldsymbol{alpha}<boldsymbol{beta}) ) are the roots of the equation ( 6 x^{2}+11 x+3=0 ) then which of the following are real? This question has multiple correct options ( A cdot cos ^{-1} alpha ) B. ( sin ^{-1} beta ) ( mathbf{c} cdot operatorname{cosec}^{-1} alpha ) D. Both ( cot ^{-1} alpha ) and ( cot ^{-1} beta ) |
12 |
| 703 | ff ( y=tan ^{-1}left(frac{x sin alpha}{1-x cos alpha}right) . ) Find ( cot y ) | 12 |
| 704 | If ( a, b, c ) are distinct non-zero real numbers having the same sign, then prove that ( cot ^{-1}left(frac{a b+1}{a-b}right)+cot ^{-1}left(frac{b c+1}{b-c}right)+ ) ( cot ^{-1}left(frac{c a+1}{c-a}right)=pi quad o r ) |
12 |
| 705 | Write the simplest form of ( tan ^{-1}[sqrt{frac{1-cos x}{1+cos x}}] ) | 12 |
| 706 | Find the value of the expression ( sec ^{-1}left(frac{x+1}{x-1}right)+sin ^{-1}left(frac{x-1}{x+1}right) ) | 12 |
| 707 | If ( boldsymbol{A}=frac{1}{1} cot ^{-1}left(frac{1}{1}right)+frac{1}{2} cot ^{-1}left(frac{1}{2}right)+ ) ( frac{1}{3} cot ^{-1}left(frac{1}{3}right) ) and ( B=1 cot ^{-1} 1+ ) ( 2 cot ^{-1} 2+3 cot ^{-1} 3 operatorname{then}|B-A| ) is equal to ( frac{a pi}{b}+frac{c}{d} ) cot ( ^{-1} 3 ) where ( a, b, c, d in N ) and are in their lowest form then ( a+b+c+d ) equal to |
12 |
| 708 | Illustration 5.26 Prove that /1 + sin x + 71-sin x Cor-1 XE 1 + sin x – sin x |
12 |
| 709 | The value of ( sin ^{-1}left(x^{2}-4 x+6right)+ ) ( cos ^{-1}left(x^{2}-4 x+6right) ) for all ( x epsilon R ) is A ( cdot frac{pi}{2} ) в. ( pi ) ( c cdot 0 ) D. none of these |
12 |
| 710 | ( operatorname{Let} sin ^{-1}left(frac{1-x^{2}}{1+x^{2}}right), ) then ( frac{d y}{d x} ) is A ( cdot frac{2}{1+x^{2}} ) в. ( frac{1}{2left(1+x^{2}right)} ) c. ( frac{-2}{1+x^{2}} ) D. ( frac{2}{2-x^{2}} ) |
12 |
| 711 | If ( frac{3 pi}{2} leq x leq frac{5 pi}{2}, ) then ( sin ^{-1}(sin x) ) is equal to- ( A ) B. ( -x ) c. ( 2 pi-x ) D. ( x-2 pi ) |
12 |
| 712 | ( sin left(2 tan ^{-1} sqrt{frac{1-x}{1+x}}right) ) | 12 |
| 713 | 13. Range of f(x) = sin-4x + tan– x + sec-?x is d. none of these |
12 |
| 714 | Assertion Let ( boldsymbol{f}:[mathbf{8} boldsymbol{pi}, mathbf{9} boldsymbol{pi}] rightarrow[-mathbf{1}, mathbf{1}], boldsymbol{f}(boldsymbol{x})=boldsymbol{operatorname { c o s } boldsymbol { x }} ) then Statement-1: ( boldsymbol{f}^{-1}(-1)=mathbf{9} boldsymbol{pi} ) because Reason Statement-2 : ( boldsymbol{f}^{-1}(boldsymbol{x})=mathbf{1 0} boldsymbol{pi}- ) ( cos ^{-1} x forall x infty[-1,1] ) A. Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement- B. Statement-1 is true, Statement-2 is true and Statement-2 is NOT correct explanation for Statement- c. Statement-1 is true, Statement-2 is false D. Statement-1 is false, Statement-2 is true |
12 |
| 715 | If ( cos left(2 sin ^{-1} xright)=frac{1}{9}, ) the value of ( x ) which satify equation is ( pm frac{a}{b} . ) Find the value of ( a+b ) A . 2 B. 3 ( c cdot 4 ) D. |
12 |
| 716 | Assertion The area bounded by the curve ( y= ) ( sin ^{-1} x & ) the line ( x=0 &|y|=frac{pi}{2} ) is ( sqrt{2} ) square units. Reason The domain & principal value branch of ( y=sin ^{-1} x operatorname{are}[-1,1] &left[frac{-pi}{2}, frac{pi}{2}right] ) respectively 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 |
| 717 | ( 2 tan ^{-1}left(frac{1}{3}right)+tan ^{-1}left(frac{1}{7}right) ) is equal to A. ( frac{pi}{6} ) в. ( frac{pi}{4} ) ( c cdot frac{pi}{3} ) ( D cdot frac{pi}{2} ) |
12 |
| 718 | Assertion ( f_{i=1}^{2 n} sin ^{-1} x_{i}=n pi forall n epsilon N ) then ( sum_{i=1}^{2 n} x_{i}= ) ( sum_{i=1}^{2 n} x_{i}^{2}=sum_{i=1}^{2 n} x_{i}^{n}=2 n ) Reason ( -frac{pi}{2} leq sin ^{-1} x leq frac{pi}{2} forall x epsilon[-1,1] ) 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 false but Reason is true |
12 |
| 719 | Find the principal value of ( tan ^{-1} sqrt{3}- ) ( sec ^{-1}(-2) ) |
12 |
| 720 | The value of ( sin ^{-1} x+cos ^{-1} x, forall x in ) [-1,1] is A ( cdot frac{pi}{2} ) в. ( frac{-5 pi}{3} ) c. ( frac{-3 pi}{2} ) D. |
12 |
| 721 | Prove: ( tan ^{-1}left(frac{sqrt{1+x}-sqrt{1-x}}{sqrt{1+x}+sqrt{1-x}}right)= ) ( frac{pi}{4}-frac{1}{2} cos ^{-1} x,-frac{1}{sqrt{2}} leq x leq 1 ) [Hint: ( p u t x=cos 2 theta] ) |
12 |
| 722 | Solve: ( cos ^{-1}(cos x)=pi+x, ) then ( x ) belongs to В. ( (pi, 2 pi) ) D. None of these |
12 |
| 723 | Let ( boldsymbol{f}(boldsymbol{x})=operatorname{cosec}^{-1}left[1+sin ^{2} boldsymbol{x}right], ) where [.] denotes the greatest integer function Then ( f(x) ) equals; ( ^{A} cdotleft{frac{pi}{2}right} ) В ( cdotleft{frac{pi}{2}, operatorname{cosec}^{-1} 2right} ) c. ( left{operatorname{cosec}^{-1} 2right} ) D. none of these |
12 |
| 724 | 6. If x <0, then tan-'x is equal to a. – 1 + cot-1 ] b. sin-1 1+x² a. – cos dit d. – cosec V1 + x² |
12 |
| 725 | If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) then ( frac{sum_{k=1}^{2}left(x^{100 k}+y^{106 k}right)}{sum x^{207} cdot y^{207}} ) is A ( cdot frac{1}{3} ) B. ( frac{4}{3} ) ( c cdot frac{2}{3} ) D. None of these |
12 |
| 726 | If the number ( 93215 x 2 ) is completely divisible by ( 11, ) then ( x ) is equal to ( A cdot 2 ) B. 3 ( c cdot 1 ) D. 4 |
12 |
| 727 | Find the value of ( x ) which satisfy equation ( : sin ^{-1} x+sin ^{-1} 2 x=frac{pi}{3} ) A ( cdot x=frac{1}{2} sqrt{frac{3}{7}} ) B・ ( x=frac{1}{3} sqrt{frac{4}{7}} ) c. ( x=frac{1}{3} sqrt{frac{3}{7}} ) D. ( x=frac{1}{2} sqrt{frac{4}{7}} ) |
12 |
| 728 | If ( sin ^{-1} x+sin ^{-1} y+sin ^{-1} z=frac{3 pi}{2} ) and ( boldsymbol{f}(1)=2, f(x+y)=f(x) f(y) ) for all ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} . ) Then ( boldsymbol{x}^{f(1)}+boldsymbol{y}^{f(2)}+boldsymbol{z}^{f(3)}- ) ( frac{boldsymbol{x}+boldsymbol{y}+boldsymbol{z}}{boldsymbol{x}^{f(1)}+boldsymbol{y}^{f(2)}+boldsymbol{z}^{f(3)}} ) is equal to ( A cdot 0 ) B. 1 ( c cdot 2 ) D. 3 |
12 |
| 729 | 22. The value of sin ‘(sin 12) + cos'(cos 12) is equal to a. zero O b. 24 – 21 c. 41-24 d. none of these |
12 |
| 730 | Find the possible value of ( cos x, ) if ( cot x ) ( +operatorname{cosec} x=5 ) |
12 |
| 731 | The sum of all the solution(s) of the equation ( sin ^{-1} 2 x=cos ^{-1} x ) is |
12 |
| 732 | 2x 18. If 2 tan-‘x+ sin-1- et sin is independent of x, then 1+x o n a. x >1 c. 0<x< 1 b. x<-1 d. – 1<x<0 20 |
12 |
| 733 | Find the principal value of ( tan left(cos ^{-1} frac{1}{2}right) ) | 12 |
| 734 | The trigonometric equation ( sin ^{-1} x= ) ( 2 sin ^{-1} 2 a ) has a real solution, if A ( cdot|a|>frac{1}{sqrt{2}} ) в. ( frac{1}{2 sqrt{2}}<|a|frac{1}{2 sqrt{2}} ) D ( cdot|a| leq frac{1}{2 sqrt{2}} ) |
12 |
| 735 | f ( sin ^{-1}left(frac{x}{13}right)+operatorname{cosec}^{-1}left(frac{13}{12}right)=frac{pi}{2} ) then the value if ( x ) is A . 5 B. 4 c. 12 D. 11 |
12 |
| 736 | Evaluate the following: ( tan ^{-1}(tan 12) ) |
12 |
| 737 | 1. cos’ (cos(2 cot'( 12 – 1))) is equal to a. √2-1 d. none of these |
12 |
| 738 | ( sin ^{-1}left(cos left(sin ^{-1} xright)right)+ ) ( cos ^{-1}left(sin left(cos ^{-1} xright)right) ) is equal to ( A cdot frac{pi}{2} ) B. ( c cdot frac{3 pi}{4} ) D. |
12 |
| 739 | Theorem: For any ( boldsymbol{x} in boldsymbol{R} quad sinh ^{-1} boldsymbol{x}= ) ( log _{e}(x+sqrt{x^{2}+1}) ) |
12 |
| 740 | Find the principal value of: ( tan ^{-1}left(2 cos frac{2 pi}{3}right) ) |
12 |
| 741 | пп Illustration 5.12 If sin-‘x, + sin-x2 + … + sin ‘x, S- ne N, n = 2m + 1, m > 1, then find the value of x1 + x3 + x +…(m+1) terms xż + x + x +…m terms |
12 |
| 742 | If ( 4 cos ^{-1} x+sin ^{-1} x=pi, ) then the value of ( x ) is. A ( cdot frac{1}{2} ) в. ( frac{1}{sqrt{2}} ) ( c cdot frac{sqrt{3}}{2} ) D. ( frac{2}{sqrt{3}} ) |
12 |
| 743 | 1. If a, Ba<B) are the roots of the equation 6×2 + 11x + 3 = 0, then which of the following are real? a. cosa b. sin'B c. cosec-'a d. Both cota and cot B |
12 |
| 744 | Let ( x_{1}, x_{2}, x_{3}, x_{4} ) be four non zero numbers satisfying the equation ( tan ^{-1} frac{a}{x}+tan ^{-1} frac{b}{x}+tan ^{-1} frac{c}{x}+ ) ( tan ^{-1} frac{d}{x}=frac{pi}{2} ) This question has multiple correct options ( ^{mathbf{A}} cdot sum_{i=1}^{4} x_{i}=a+b+c=d ) ( ^{mathrm{B}} cdot sum_{i=1}^{4} frac{1}{x_{1}}=0 ) ( mathbf{c} cdot Pi_{i=1}^{4} x_{i}=a b c d ) D. ( left(x_{1}+x_{2}+x_{3}right)left(x_{2}+x_{3}+x_{4}right)left(x_{3}+x_{4}+x_{1}right)left(x_{4}+x_{1}+right. ) ( left.x_{2}right)=a b c d ) |
12 |
| 745 | The number of solutions of the equation ( 1+x^{2}+2 x sin left(cos ^{-1} yright)=0 ) is ( A ) B. 2 ( c cdot 3 ) ( D cdot 4 ) |
12 |
| 746 | What is ( sin ^{-1} sin frac{3 pi}{5} ) equal to ( ? ) ( mathbf{A} cdot frac{3 pi}{5} ) B. ( frac{2 pi}{5} ) ( c cdot frac{pi}{5} ) D. None of the above |
12 |
| 747 | Evaluate: ( cos left[2 tan ^{-1}left[frac{1}{7}right]right] ) ( A cdot sin left(4 cot ^{-1} 3right) ) B. ( sin left(3 cot ^{-1} 4right) ) ( c cdot cos left(3 cot ^{-1} 4right) ) D. ( cos left(4 cot ^{-1} 4right) ) |
12 |
| 748 | Solve for ( x ; cos ^{-1} sqrt{x}>sin ^{-1} sqrt{x} ) | 12 |
| 749 | Solve ( : 2 tan ^{-1}(-3)= ) This question has multiple correct options ( mathbf{A} cdot-cos ^{-1}(-4 / 5) ) ( mathbf{B} cdot-pi+cos ^{-1}(4 / 5) ) C ( cdot-frac{pi}{2}+tan ^{-1}(-4 / 3) ) D. ( cot ^{-1}(4 / 3) ) |
12 |
| 750 | The value of ( cos ^{-1}left(cos frac{7 pi}{6}right)= ) ( A cdot frac{7 pi}{6} ) в. ( frac{5 pi}{6} ) ( c cdot frac{pi}{3} ) D. |
12 |
| 751 | Solve ( : int frac{tan ^{-1} x}{1+x^{2}} d x ) | 12 |
| 752 | Write in simplest form ( sin ^{-1}left[frac{sqrt{1+x}+sqrt{1-x}}{2}right] ) | 12 |
| 753 | Evaluate the following: ( sin ^{-1}(sin 5) ) |
12 |
| 754 | Which one of the following statement is meaningless? ( ^{mathbf{A}} cdot cos ^{-1}left(ln left(frac{2 e+4}{3}right)right) ) B. ( operatorname{cosec}^{-1}left(frac{pi}{3}right) ) c. ( cot ^{-1}left(frac{pi}{2}right) ) D ( cdot sec ^{-1}(pi) ) |
12 |
| 755 | where x < 1, then x is equal to ماده b. – c ده d. |
12 |
| 756 | 63. If cos ‘x + cos’y + cos’z = , then a. x2 + y2 + z2 + xyz = 0 b. x² + y2 + z2 + 2xyz = 0) c. x2 + y2 + 2 + xyz = 1 d. x2 + y2 + 2 + 2xyz = 1 |
12 |
| 757 | ( cot ^{-1}(2+sqrt{3})= ) ( A cdot frac{pi}{12} ) B. ( frac{pi}{15} ) ( c cdot frac{pi}{5} ) ( D cdot frac{3 pi}{10 pi} ) |
12 |
| 758 | The domain of ( mathbf{f}(mathbf{x})= ) ( cot ^{-1}left(frac{mathbf{x}}{sqrt{mathbf{x}^{2}-left[mathbf{x}^{2}right]}}right) ) is ( ([.] ) denotes the greatest integer function) A. ( (0, infty) ) ) в. ( mathrm{R}-{0 ) c. ( R-{x: x in Z} ) D. ( (-infty, 0) ) |
12 |
| 759 | Solve: ( 3 tan ^{-1} x+cot ^{-1} x=pi ) | 12 |
| 760 | The value of ( sin ^{-1}(sin 2) ) is? A ( .2+n pi ) B. ( 2-pi ) ( c cdot-2+pi ) D. ( 2-2 n pi ) |
12 |
| 761 | 49. If cos? Vp + cos’ V1- p + cos’ V1-9 = 37, then the value of q is a. 1 -la |
12 |
| 762 | ( boldsymbol{alpha}=sin ^{-1}left(cos left(sin ^{-1} xright)right) ) and ( beta= ) ( cos ^{-1}left(sin left(cos ^{-1} xright)right) ) then: ( A cdot tan alpha=cot beta ) B. ( tan alpha=-cot beta ) ( mathbf{c} cdot tan alpha=tan beta ) D. ( tan alpha=-tan beta ) |
12 |
| 763 | Solve ( y=tan ^{-1}left(frac{cos x}{1-sin x}right) ) | 12 |
| 764 | Find the value of ( sin left[frac{1}{2} cot ^{-1}left(frac{-3}{4}right)right] ) | 12 |
| 765 | Solve ( tan x<2 ) | 12 |
| 766 | Find the principal value of the following ( tan ^{-1}left(tan frac{7 pi}{6}right) ) |
12 |
| 767 | Find the principal value of: ( cot ^{-1}(sqrt{3}) ) |
12 |
| 768 | If ( boldsymbol{f}:left(-frac{boldsymbol{pi}}{2}, frac{boldsymbol{pi}}{2}right) rightarrow(-infty, infty) ) is defined by ( f(x)=tan x, ) then ( f^{-1}(sqrt{3})= ) |
12 |
| 769 | How do you simplify ( sin x+cot x cdot cos x ) |
12 |
| 770 | Integrate the function ( tan ^{-1}(sqrt{frac{1-sin x}{1+sin x}}) ) w.r.t d ( x ) | 12 |
| 771 | If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots inftyright)+ ) ( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots inftyright)=frac{pi}{2} ) for ( mathbf{0}<|boldsymbol{x}|<sqrt{mathbf{2}}, ) then ( boldsymbol{x} ) equal ( A cdot frac{1}{2} ) B. ( c cdot frac{-1}{2} ) D. -1 |
12 |
| 772 | If ( tan ^{-1}(2 x)+tan ^{-1}(3 x)=pi / 4 ) then ( x=? ) |
12 |
| 773 | 92. If 227/sin-‘x – 2(a+ 2) 24/sin- x + 8a < 0 for at least one real x, then a. Isa<2 b. a<2 20 c. a € R-{2} d. ae (o 1 u 12,0) d. a e |
12 |
| 774 | If ( tan ^{-1}(cot theta)=2 theta ) then ( theta= ) A ( cdot frac{pi}{3} ) в. ( frac{pi}{4} ) c. D. None of the above |
12 |
| 775 | The number of solutions of the equation ( mathbf{2}left(boldsymbol{operatorname { S i n }}^{-1} boldsymbol{x}right)^{2}-mathbf{5} operatorname{Sin}^{-1} boldsymbol{x}+mathbf{2}=mathbf{0} ) is A . в. ( c cdot 2 ) ( D ) |
12 |
| 776 | Find the domain of the following function: ( boldsymbol{f}(boldsymbol{x})=operatorname{cosec}^{-1}left[mathbf{1}+sin ^{2} boldsymbol{x}right], ) where denotes the greatest integer function |
12 |
| 777 | ( tan left[2 tan ^{-1}left(frac{sqrt{1+x^{2}}-1}{x}right)right]= ) ( A cdot x ) B. ( 2 x ) c. ( x / 2 ) D. ( 3 x ) |
12 |
| 778 | Calculate. ( arctan 1+arccos left(-frac{1}{2}right)+ ) ( arcsin left(-frac{1}{2}right)=? ) |
12 |
| 779 | to our Σ sin- is equal to P= Irort1 c. tan”(Tn) d. tan” (Jn+1) |
12 |
| 780 | If value of ( x ) which satisfy equation ( cos ^{-1} x<2 ) is ( x epsilon(a, b] ) Find the value of ( a+b ) A. ( -1-cos 2 ) B. ( 1-cos 2 ) c. ( -1+cos 2 ) D. ( 1+cos 2 ) |
12 |
| 781 | Solve: ( sin ^{-1} x+sin ^{-1} sqrt{1-x^{2}} ) |
12 |
| 782 | If range of the function ( boldsymbol{f}(boldsymbol{x})= ) ( sin ^{-1} x+2 tan ^{-1} x+x^{2}+4 x+1 ) is ( [p, q], ) then the value of ( (p+q) ) is |
12 |
| 783 | Solve the equation ( 2 tan ^{-1}(cos x)=tan ^{-1}(2 csc x) ) | 12 |
| 784 | Find the principal value of the following ( cos ^{-1} frac{1}{2}+2 sin ^{-1} frac{1}{2} ) |
12 |
| 785 | The value of ( tan left[cos ^{-1} frac{4}{5}+tan ^{-1} frac{2}{3}right] ) is A ( cdot frac{6}{17} ) в. ( frac{7}{16} ) c. ( frac{17}{6} ) D. none of these |
12 |
| 786 | Solve: ( left(tan ^{-1} xright)^{2}+left(cot ^{-1} xright)^{2}=frac{5 pi^{2}}{8} ) | 12 |
| 787 | ( f cot left(cos ^{-1} frac{3}{5}+sin ^{-1} xright)=0, ) find the value of ( boldsymbol{x} ) |
12 |
| 788 | ( tan ^{-1}left(tan frac{2 pi}{3}right)= ) A ( cdot frac{pi}{3} ) B ( cdot frac{2 pi}{3} ) c. ( -frac{pi}{3} ) D. ( -frac{2 pi}{3} ) |
12 |
| 789 | toppr Q Type your question. ( left.right|^{x in L}left|: sin left(log _{e} mid overline{x-1}right)right|^{operatorname{ls} a} ) (Here, the inverse trigonometric function ( sin ^{-1} x ) assumes values in ( left.left[-frac{pi}{2}, frac{pi}{2}right]right) ) Let ( boldsymbol{f}: boldsymbol{E}_{1} rightarrow mathbb{R} ) be the function define by ( f(x)=log _{e}left(frac{x}{x-1}right) ) and ( g: E_{2} rightarrow mathbb{R} ) be the function defined by ( g(x)= ) ( sin ^{-1}left(log _{e}left(frac{x}{x-1}right)right) ) LIST-1 1. ( left(-infty, frac{1}{1-e}right] ) The range of ( f ) is ( quadleft[frac{e}{e-1}, inftyright) ) Q. The range of ( g ) contins R. The domain of ( f ) contains [ begin{array}{ll} text { S. The domain of } g text { is } & 4 .(-infty, 0) cup(0, infty) \ & 5 .left(-infty, frac{e}{e-1}right] \ & text { 6. }(-infty, 0) cupleft(frac{1}{2}, frac{e}{e-1}right] end{array} ] The correct option is A. ( P rightarrow 4 ; Q rightarrow 2 ; R rightarrow 1 ; S rightarrow 1 ) в. ( P rightarrow 3 ; Q rightarrow 3 ; R rightarrow 6 ; S rightarrow 5 ) c. ( P rightarrow 4 ; Q rightarrow 2 ; R rightarrow 1 ; S rightarrow 6 ) D. ( P rightarrow 4 ; Q rightarrow 3 ; R rightarrow 6 ; S rightarrow 5 ) |
12 |
| 790 | ( operatorname{Let} 2 y=left(cot ^{-1}left(frac{sqrt{3} cos x+sin x}{cos x-sqrt{3} sin x}right)right)^{2} ) then ( frac{boldsymbol{a} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is equal to A ( cdot x-frac{pi}{6} ) B. ( x+frac{pi}{6} ) c. ( 2 x-frac{pi}{6} ) D. ( 2 x-frac{pi}{3} ) |
12 |
| 791 | Solve the equation ( tan ^{-1}left(frac{1-x}{1+x}right)= ) ( frac{1}{2} tan ^{-1} x, x>0 ) |
12 |
| 792 | The value of ( cos ^{-1}left(cos left(frac{4 pi}{3}right)right) ) is A ( cdot 2 pi / 3 ) в. ( -2 pi / 3 ) c. ( 4 pi / 3 ) D. ( -4 pi / 3 ) |
12 |
| 793 | Prove that ( tan ^{-1} frac{1}{2}+tan ^{-1} frac{2}{11}= ) ( tan ^{-1} frac{3}{4} ) |
12 |
| 794 | find the value of the following: ( (i) sin ^{-1}left(frac{-1}{2}right) ) ( (i i) cos ^{-1}left(frac{sqrt{3}}{2}right) ) ( (i i i) operatorname{cosec}^{-1}(2) ) ( (i v) tan ^{-1}(-sqrt{3}) ) ( (v) cos ^{-1}left(frac{-1}{2}right) ) ( (v i) tan ^{-1}(-1) ) |
12 |
| 795 | ( cos left(cot ^{-1}left(operatorname{cosec}left(cos ^{-1} aright)right)right)=dots ) (where ( 0<a<1) ) A ( cdot frac{1}{sqrt{2-a^{2}}} ) B. ( sqrt{3-a^{2}} ) c. ( sqrt{2-a^{2}} ) D. ( frac{1}{sqrt{2+a^{2}}} ) |
12 |
| 796 | Show that ( cot ^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right)= ) ( frac{x}{2} ) for ( x inleft(0, frac{pi}{2}right) ) |
12 |
| 797 | Evaluate ( cos ^{-1}(cos 3) ) | 12 |
| 798 | Let ( boldsymbol{f}(boldsymbol{x})=sin ^{-1} boldsymbol{x}+cos ^{-1} boldsymbol{x}, ) then ( frac{boldsymbol{pi}}{2} ) is equal to This question has multiple correct options A ( cdot fleft(-frac{1}{2}right) ) B . ( fleft(k^{2}-2 k+3right), k in R ) c. ( fleft(frac{1}{1+k^{2}}right), k in R ) D. ( f(-2) ) |
12 |
| 799 | Illustration 5.70 Which of the following angles is greater ? COS |
12 |
| 800 | Find the principal value of ( cot ^{-1}(sqrt{3}) ) | 12 |
| 801 | 11. If sin (x – 1) + cos(x – 3) + tan- Tt, then the value of k is = cos-‘k + nie. a. 1 b. – Ja 1 d. none of these 02.10 |
12 |
| 802 | If ( 0<x<1, ) then ( tan ^{-1}left(frac{sqrt{1-x^{2}}}{1+x}right) ) is equal to ( ^{mathbf{A} cdot} frac{1}{2} cos ^{-1} x ) B. ( cos ^{-1} frac{sqrt{1+x}}{2} ) c. ( sin ^{-1} sqrt{frac{1-x}{2}} ) D. ( frac{1}{2} sqrt{frac{1+x}{1-x}} ) |
12 |
| 803 | the number of real solutions of the equation ( tan ^{-1} sqrt{x^{2}-3 x+2}+ ) ( cos ^{-1} sqrt{4 x-x^{2}-3}=pi ) is A. one B. two c. zero D. infinite |
12 |
| 804 | f ( tan ^{-1} frac{x-3}{x-4}+tan ^{-1} frac{x+3}{x+4}=frac{3}{4}, ) then find the value of ( x ) |
12 |
| 805 | Which of the following is/are the value of | 12 |
| 806 | The solutions set of inequality ( cos ^{-1} x<sin ^{-1} x ) is A ( cdot[-1,1] ) B. ( left[frac{1}{sqrt{2}}, 1right] ) c. [0,1] D. ( left(frac{1}{sqrt{2}}, 1right] ) |
12 |
| 807 | Find the value of [ tan left{frac{1}{2} sin ^{-1}left(frac{2 x}{1+x^{2}}right)+frac{1}{2} cos ^{-1}left(frac{1-}{1+}right.right. ] if ( boldsymbol{x}>boldsymbol{y}>1 ) |
12 |
| 808 | ( frac{x}{5} ) | 12 |
| 809 | Find the principal value of ( operatorname{cosec}^{-1}left(frac{2}{sqrt{3}}right) ) | 12 |
| 810 | (sin x) Illustration 5.21 Find the area bounded by y=sin and x-axis for x = [0, 1006]. |
12 |
| 811 | Find the value of ( cos left(sec ^{-1} x+right. ) ( left.csc ^{-1} xright),|x| geq 1 ) |
12 |
| 812 | The largest interval lying in ( left(frac{-pi}{2}, frac{pi}{2}right) ) for which the function ( left[f(x)=4^{-x^{2}}+cos ^{-1}left(frac{x}{2}-1right)+log (c oright. ) is defined, is- A . ( [0, pi] ) в. ( left(frac{-pi}{2}, frac{pi}{2}right) ) c. ( left[-frac{pi}{4}, frac{pi}{2}right) ) D. ( left[0, frac{pi}{2}right) ) |
12 |
| 813 | Find the value of ( sin ^{-1}left(sin frac{3 pi}{5}right) ) | 12 |
| 814 | ( tan left(2 tan ^{-1}left(frac{sqrt{5}-1}{2}right)right)= ) ( A ) B. 3 ( c cdot 2 ) ( D ) |
12 |
| 815 | For ( boldsymbol{x} in(mathbf{0}, boldsymbol{pi} / mathbf{2}) ) ( sin ^{-1}(cos x)=? ) A. ( pi-x ) B ( cdot frac{pi}{2}-x ) c. ( pi-frac{x}{2} ) D . ( pi-2 x ) |
12 |
| 816 | The smallest and the largest values of ( tan ^{-1}left(frac{1-x}{1+x}right), 0 leq x leq 1 ) are A ( .0, pi ) в. ( 0, frac{pi}{4} ) ( c cdot-frac{pi}{4}, frac{pi}{4} ) D. ( frac{pi}{4}, frac{pi}{2} ) |
12 |
| 817 | The trigonometric equation ( sin ^{-1} x= ) ( 2 sin ^{-1} 2 a ) has a real solution if A ( cdot|a|>frac{1}{sqrt{2}} ) в. ( frac{1}{2 sqrt{2}}<|a|frac{1}{2 sqrt{2}} ) D ( cdot|a| leq frac{1}{2 sqrt{2}} ) |
12 |
| 818 | If ( sin ^{-1}left(frac{1}{3}right)+sin ^{-1}left(frac{2}{3}right)=sin ^{-1} x, ) then ( x ) is equal to ( mathbf{A} cdot mathbf{0} ) B. ( frac{sqrt{5}+4 sqrt{2}}{9} ) c. ( frac{5 sqrt{2}-4 sqrt{5}}{9} ) D. |
12 |
| 819 | ff ( y=sin left(cos ^{-1} xright) ) and ( x=99, ) then ( 1 / y^{2} ) is equal to |
12 |
| 820 | Which of the following is/are a rational number? 1 b. cos — -sin |
12 |
| 821 | Calculating the principal value, find the value of ( sin left[2 sin ^{-1}left(frac{4}{5}right)right] ) | 12 |
| 822 | cos x 26. tan nx – for x E |
12 |
| 823 | What is the value of ( cos left{cos ^{-1} frac{4}{5}+cos ^{-1} frac{12}{13}right} ? ) A . ( 63 / 65 ) B. 33/65 c. 22/65 D. ( 11 / 65 ) |
12 |
| 824 | ( f sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots dots dots inftyright)+ ) ( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots dots dots inftyright)=frac{pi}{2} ) and ( 0<x<sqrt{2} ) then ( x= ) ( A cdot frac{1}{2} ) B. ( c cdot-frac{1}{2} ) D. – – |
12 |
| 825 | Find the principal value the following expression: ( sin ^{-1}left(-frac{sqrt{3}}{2}right) ) |
12 |
| 826 | Solve for ( x: ) ( left(tan ^{-1} xright)^{2}+left(cos ^{-1} xright)^{2}=frac{5 pi^{2}}{8} ) |
12 |
| 827 | ( frac{cos ^{-1}(41 / 49)}{sin ^{-1}(2 / 7)}= ) ( A cdot 4 ) B. 3 ( c cdot 2 ) ( D ) |
12 |
| 828 | Evaluate the following: ( sin ^{-1}(sin 10) ) |
12 |
| 829 | Illustration 5.46 (cosec-‘x)? Find the minimum value of (sec- x)2 + |
12 |
| 830 | 75. If 3 tan- – tan’ – = tan , then x is equal to a. 1 c. 3 Set Sb. 2 ons d. 2 |
12 |
| 831 | Solve: ( cot left(cos e c^{-1} frac{5}{3}+tan ^{-1} frac{2}{3}right) ) A ( cdot frac{6}{17} ) в. ( frac{3}{17} ) c. ( frac{4}{17} ) D. ( frac{5}{17} ) |
12 |
| 832 | The value of ( cos ^{-1}left(cos frac{7 pi}{6}right) ) is equal to A ( cdot frac{7 pi}{6} ) в. ( frac{5 pi}{6} ) ( c cdot frac{pi}{3} ) D. |
12 |
| 833 | The value of ( sin left(2 sin ^{-1} 0.8right) ) is equal to. ( mathbf{A} cdot sin ^{-1} 1.2 ) B. ( sin ^{-1}(0.96) ) c. ( sin ^{-1}(0.48) ) D. sin ( 1.6^{circ} ) |
12 |
| 834 | If ( cot ^{-1}left[(cos alpha)^{1 / 2}right]+ ) ( left[tan ^{-1}(cos alpha)^{1 / 2}right]=x, ) then ( sin x ) equals A . B ( cdot cot ^{2}left(frac{alpha}{2}right) ) ( mathbf{c} cdot tan alpha ) D. ( cot left(frac{alpha}{2}right) ) |
12 |
| 835 | 4 If sin + cosec ) , then the values of x is (a) 4 (c) 1 [20071 (b) 5 (d) 3 |
12 |
| 836 | Prove: ( 2 tan ^{-1}left(sqrt{frac{a-b}{a+b}} tan frac{theta}{2}right)=cos ^{-1} ) ( left(frac{a cos theta+b}{a+b cos theta}right) ) |
12 |
| 837 | ( cos ^{-1}left{frac{1}{sqrt{2}}left(cos frac{9 pi}{10}-sin frac{9 pi}{10}right)right}= ) A ( cdot frac{23 pi}{20 pi} ) B. ( frac{7 pi}{10} ) ( c cdot frac{7 pi}{20} ) D. ( frac{17 pi}{20 pi} ) |
12 |
| 838 | Solve: ( sin ^{-1} frac{5}{x}+sin ^{-1} frac{12}{x}=frac{pi}{2} ) | 12 |
| 839 | Differentiate ( cos ^{-1}left(4 x^{2}-3 xright) ; x epsilonleft(frac{1}{2}, 1right) ) | 12 |
| 840 | Find the value of ( tan ^{-1}left(frac{1}{2} tan 2 Aright)+ ) ( tan ^{-1}(cot A)+tan ^{-1}left(cot ^{3} Aright), ) for ( 0< ) ( boldsymbol{A}<frac{boldsymbol{pi}}{boldsymbol{4}} ) ( mathbf{A} cdot-pi / 2 ) в. ( +pi / 2 ) c. ( -pi ) D. ( +pi ) |
12 |
| 841 | If ( cos ^{-1} x= ) ( left{begin{array}{r}a pi-b cos ^{-1}left(2 x^{2}-1right), i f-1 leq x< \ c cos ^{-1}left(2 x^{2}-1right), text { if } 0 leq x leq 1end{array}right. ) Find the value of ( a+b+c ) A . 1 B. 2 ( c .3 ) D. 4 |
12 |
| 842 | 3. If cosx -cos-1y = a, then 4×2 – 4xy cos a + y is equal to [2005] (a) 2 sin 2a (6) 4 (c) 4 sin? a (d) – 4 sin² a |
12 |
| 843 | The value of ( sin left(2 sin ^{-1}(0.8)right) ) is equal to ( A cdot sin 1.2^{circ} ) B. ( sin 1.6^{circ} ) c. 0.48 D. 0.96 |
12 |
| 844 | If ( boldsymbol{f}(boldsymbol{x})= ) ( sin {[boldsymbol{x}+mathbf{5}]+{boldsymbol{x}-{boldsymbol{x}-{boldsymbol{x}}}}} ) for ( boldsymbol{x} inleft(mathbf{0}, frac{boldsymbol{pi}}{mathbf{4}}right) ) is invertible, where ( {.} ) and [.] represent fractional part and greatest integer functions respectively, then ( boldsymbol{f}^{-1}(boldsymbol{x}) ) is This question has multiple correct options ( mathbf{A} cdot sin ^{-1} x ) в. ( frac{pi}{2}-cos ^{-1} x ) ( c cdot sin ^{-1}{x} ) D ( cdot cos ^{-1}{x} ) |
12 |
| 845 | If ( 2 tan h^{-1} x=log y, ) then the value of ( y ) in the terms of ( x ) is A ( .2 x ) в. ( frac{2 x}{1-x^{2}} ) c. ( x^{2} ) D. ( left(frac{1+x}{1-x}right) ) |
12 |
| 846 | The numerical value of tan ( left(2 tan ^{-1} frac{1}{5}-frac{pi}{4}right) ) is | 12 |
| 847 | 40. For 0 < 0 cos? (sin ) is true when 3л |
12 |
| 848 | ( tan ^{-1}(1)+cos ^{-1}left(-frac{1}{2}right)+sin ^{-1}left(-frac{1}{2}right) ) | 12 |
| 849 | Find the value of ( x, ) if: ( tan ^{-1}left(frac{x-2}{x-1}right)+tan ^{-1}left(frac{x+2}{x+1}right)=frac{pi}{4} ) |
12 |
| 850 | If ( tan ^{-1} frac{1-x}{1+x}=frac{1}{2} tan ^{-1} x, ) then ( x= ) ( mathbf{A} cdot mathbf{1} ) B. ( sqrt{3} ) c. ( frac{1}{sqrt{3}} ) D. None of these |
12 |
| 851 | Find the principal value of the following: ( cot ^{-1}(-1) ) |
12 |
| 852 | The value of ( sin left(2 tan ^{-1}(1 / 3)right)+ ) ( cos left(tan ^{-1} 2 sqrt{2}right) ) is ( A cdot 12 / 13 ) B. ( 13 / 14 ) c. ( 14 / 15 ) D. none of these |
12 |
| 853 | Assertion If ( boldsymbol{x}<mathbf{0}, tan ^{-1} boldsymbol{x}+tan ^{-1} frac{mathbf{1}}{boldsymbol{x}}=frac{boldsymbol{pi}}{mathbf{2}} ) Reason ( tan ^{-1} x+cot ^{-1} x=frac{pi}{2} forall x in 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. Assertion is incorrect but Reason is correct |
12 |
| 854 | Find the principal value of: ( sec ^{-1}left(2 tan frac{3 pi}{4}right) ) |
12 |
| 855 | If ( U=cot ^{-1} sqrt{cos 2 theta}-tan ^{-1} sqrt{cos 2 theta} ) then ( sin U ) is equal to ( A cdot sin ^{2} theta ) B. ( cos ^{2} theta ) ( mathbf{c} cdot tan ^{2} theta ) ( mathbf{D} cdot tan ^{2} 2 theta ) |
12 |
| 856 | ( fleft(sin ^{-1} frac{1}{5}+cos ^{-1} xright)=1, ) then find the value of ( x ) |
12 |
| 857 | 72. If cot’ (Vcos a) – tan! (Vcos a ) = x, then sinx is a. tang b. cot a c. tan 10 d. cot 810 |
12 |
| 858 | 28. The trigonometric equation sin-‘x=2 sin ‘a has a solution for a. all real values b. V |
12 |
| 859 | 5. The value of x for which sin (cot -1 (1+x)) = cos (tan- x) is (2004) (a) 112 (b) 1 (C) 0 (d) -1/2 |
12 |
| 860 | If ( tan ^{-1}left(frac{x}{sqrt{a^{p}-x^{q}}}right)= ) ( sin ^{-1}left(frac{x}{a}right), a>0 . ) Find the value of and ( q ) ( mathbf{A} cdot p=1, q=1 ) ( mathbf{B} cdot p=1, q=2 ) ( mathbf{c} cdot p=2, q=1 ) D ( . p=2, q=2 ) |
12 |
| 861 | Find the principal value: ( sin ^{-1}left(tan frac{5 pi}{4}right) ) |
12 |
| 862 | For the principal value: ( sin ^{-1}left(-frac{sqrt{mathbf{3}}}{mathbf{2}}right)+cos ^{-1}left(frac{sqrt{mathbf{3}}}{mathbf{2}}right) ) |
12 |
| 863 | Illustration 5.38 Find the minimum value of the function Ax)= 16 cot-10–cot-‘x. |
12 |
| 864 | Evaluate: ( tan ^{-1}left(frac{sqrt{1+cos x}-sqrt{1-cos x}}{sqrt{1+cos x}+sqrt{1-cos x}}right) ) |
12 |
| 865 | Prove that ( sin ^{-1} frac{3}{5}+sin ^{-1} frac{8}{17}=cos ^{-1} frac{36}{85} ) |
12 |
| 866 | a – b 74. The value 2 ans (Ver mais equilito 74. The value 2 tan-1 tan is equal to Va+h a. cos-1/ a cosO+b b. cos-1 (a+bcos e a cos 0 + b (a + bcos e) c. cos- a cos e (a + bcos ) d. cos (bcoso a cos 0 +b ) |
12 |
| 867 | If ( y=cot ^{-1}(sqrt{cos x})- ) ( tan ^{-1}(sqrt{cos x}) P . T sin y=tan ^{2} x / 2 ) |
12 |
| 868 | Write the value of ( tan ^{-1}left[2 sin left(2 cos ^{-1} frac{sqrt{3}}{2}right)right] ) |
12 |
| 869 | The number of solution of the equation ( 1+x^{2}+2 x sin left(cos ^{-1} yright)=0 ) is : ( A ) B. 2 ( c cdot 3 ) D. 4 |
12 |
| 870 | 8. If (sin-‘x + sin ‘w) (sin-y + sin- z) = re”, then DE 23 WAN, N2, N₃, NEN) a. has a maximum value of 2 b. has a minimum value of 0 c. 16 different D are possible d. has a minimum value of -2 |
12 |
| 871 | 2. The value of sin cot sin-1 12-13 los V 4 +cos-1 V12 +sec-12 4 Rim d. none of these |
12 |
| 872 | If ( I sin ^{-1} x-cos ^{-1} x=frac{pi}{6}, ) then solve for ( boldsymbol{x} ) ? |
12 |
| 873 | Solve: ( tan ^{-1}left(tan frac{7 pi}{6}right) ) |
12 |
| 874 | ( f(x)=tan ^{-1}(sin x+cos x) ) is an increasing function in This question has multiple correct options A ( cdotleft(0, frac{pi}{4}right) ) В ( cdotleft(0, frac{pi}{2}right) ) c. ( left(frac{-pi}{4}, frac{pi}{4}right) ) D. None of these |
12 |
| 875 | If ( cot ^{-1} x+cot ^{-1} y+cot ^{-1} z=frac{pi}{2} ) then ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z} ) equals A ( . x y z ) в. ( x y+y z+z x ) ( mathrm{c} cdot 2 x y z ) D. None of these |
12 |
| 876 | If ( 2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) is independent of ( x ) then A ( cdot x epsilon(1,+infty) ) В . ( x epsilon(-1,1) ) c. ( x epsilon(-infty,-1) ) D. none of these |
12 |
| 877 | ff ( y=2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) then A. ( -pi / 2<y<pi / 2 ) в. ( -3 pi / 2<y<3 pi / 2 ) c. ( -pi<y<pi ) D. ( -pi / 4<y<pi / 4 ) |
12 |
| 878 | The value of ( sin ^{-1} ) ( left{tan left(cos ^{-1} sqrt{frac{2+sqrt{3}}{4}}+cos ^{-1} frac{sqrt{12}}{4}-right.right. ) is ( mathbf{A} cdot mathbf{0} ) в. ( frac{pi}{2} ) ( c cdot-frac{pi}{2} ) D. |
12 |
| 879 | Solve: ( 2 tan ^{-1} frac{3}{4}-tan ^{-1} frac{17}{31} ) |
12 |
| 880 | Simplify ( cot ^{-1} frac{1}{sqrt{x^{2}-1}} ) for ( x<-1 ) ( A cdot cos ^{-1} x ) ( mathbf{B} cdot sec ^{-1} x ) ( mathbf{c} cdot operatorname{cosec}^{-1} x ) D. ( tan ^{-1} x ) |
12 |
| 881 | Simplify: ( sin . cot ^{-1} cot x ) | 12 |
| 882 | The value of ( cos left(tan ^{-1}left(frac{3}{4}right)right) ) is ( A cdot frac{4}{5} ) B. ( c cdot frac{3}{4} ) D. |
12 |
| 883 | Illustration 5.10 If sin- ‘(x2 + 2x + 2) + tan- ‘(x2 – 3x – K) then find the values of k. |
12 |
| 884 | Prove ( 4 tan ^{-1}left(frac{1}{5}right)-tan ^{-1}left(frac{1}{70}right)+ ) ( tan ^{-1}left(frac{1}{99}right)=frac{pi}{4} ) |
12 |
| 885 | Illustration 5.16 Evaluate the following: i. sin(sin at/4) ii. cos'(cos27/3) iii. tan(tan 7/3) |
12 |
| 886 | Prove that ( sin cot ^{-1} tan cos ^{-1} x= ) ( sin operatorname{cosec}^{-1} cot tan ^{-1} x=x ) where ( x in(0,1] ) |
12 |
| 887 | Prove that : ( 3 cos ^{-1} x=cos ^{-1}left(4 x^{3}-3 xright) ) |
12 |
| 888 | Illustration 5.48 If a= sin-‘(cos(sin- x)) and ß= cos-‘(sin(cos+ x)), then find tan a • tan B. |
12 |
| 889 | The number of positive integral solutions of the equation ( tan ^{-1} x+ ) ( cot ^{-1} y=tan ^{-1} 3 ) is : ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 3 |
12 |
| 890 | Find ( x ) if ( tan ^{-1} x+2 cot ^{-1} x=frac{2 pi}{3} ) | 12 |
| 891 | 5. Find the number of positive integral solutions of the 3 equation tan’x + cos = =sin-l √ 1 – 2 TO |
12 |
| 892 | 4. The value of cos -COS is Cool 7 one c. de |
12 |
| 893 | The solution set of the equation ( sin ^{-1} sqrt{1-x}+cos ^{-1} x= ) ( cot ^{-1}left(frac{sqrt{1-x^{2}}}{x}right)-sin ^{-1} x ) A ( cdot[-1,1]-{0} ) B . (0,1]( cup{-1} ) c. [-1,0)( cup{1} ) D. {1} |
12 |
| 894 | Evaluate: ( tan ^{-1}left(frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}right) ) | 12 |
| 895 | xo 71. The value of tan- -cot-1 cose a. 20 c. 0/2 (1-xsin o) x-sin e b. 0 d. independent of e |
12 |
| 896 | If the minimum value of ( left(sec ^{-1} xright)^{2}+ ) ( left(operatorname{cosec}^{-1} xright)^{2} ) is ( frac{pi^{a}}{b} . ) Find the value of ( a+b ) ( mathbf{A} cdot mathbf{6} ) B. 8 c. 10 D. 12 |
12 |
| 897 | The value of ( tan left[frac{1}{2} cos ^{-1}left(frac{2}{3}right)right] ) is A ( cdot frac{1}{sqrt{5}} ) B. ( sqrt{frac{3}{10}} ) c. ( sqrt{frac{5}{2}} ) D. ( 1-sqrt{frac{5}{2}} ) |
12 |
| 898 | If ( sin ^{-1}left(x-frac{x^{2}}{2}+frac{x^{3}}{4}-dots inftyright)+ ) ( cos ^{-1}left(x^{2}-frac{x^{4}}{2}+frac{x^{6}}{4}-dots inftyright)=frac{pi}{2} ) and ( 0<x<sqrt{2} ) then ( x= ) ( A cdot frac{1}{2} ) B. ( c cdot frac{-1}{2} ) D. – |
12 |
| 899 | 48. If sin-1 48. If sin + sin(12) – , then x is equal to + sin-1 X c. 13 |
12 |
| 900 | The value of ( sec left[tan ^{-1} frac{b+a}{b-a}-tan ^{-1} frac{a}{b}right] ) is ( A ) B. ( sqrt{2} ) ( c cdot 4 ) D. |
12 |
| 901 | (4 2. If a= 3sin and B = 3cos , where the inverse trigonometric functions take only the principal values, then the correct option(s) is (are) (JEE Adv. 2015) (a) cosß > 0 b) sinß0 d) cosa < 0 |
12 |
| 902 | Illustration 5.8 Find the value of x for which sec-‘x+sin- |
12 |
| 903 | Evaluate: ( cos ^{-1}left(frac{2 x}{1+x^{2}}right) ) |
12 |
| 904 | If ( tan A=-frac{1}{2} ) and ( tan B=-frac{1}{3} . ) (where ( A, B>0), ) then ( A+B ) can be A ( cdot frac{pi}{4} ) в. ( frac{3 pi}{4} ) c. ( frac{5 pi}{4} ) D. ( frac{7 pi}{4} ) |
12 |
| 905 | If ( boldsymbol{x}>1, ) then the value of ( 2 tan ^{-1} x+sin ^{-1}left(frac{2 x}{1+x^{2}}right) ) is ( ^{A} cdot frac{2 pi}{4} ) в. ( c ) D. ( frac{3 pi}{2} ) |
12 |
| 906 | Show that ( sin ^{-1}(2 x sqrt{1-x^{2}})=2 sin ^{-1} x ) for ( frac{-1}{sqrt{2}} leq x leq frac{1}{sqrt{2}} ) |
12 |
| 907 | 6. Ifx,y,z are in A.P. and tan-1x, tan-ly and tan- z are also in A.P., then [JEE M 2013 (a) x=y=z (b) 2x=3y=62 ©) 6x=3y=22 (d) 6x=4y=3z |
12 |
| 908 | If ( 1<x<sqrt{2}, ) the number of solutions of the equation ( tan ^{-1}(x-1)+tan ^{-1} x+ ) ( tan ^{-1}(x+1)=tan ^{-1} 3 x ) is A. B. ( c cdot 2 ) D. |
12 |
| 909 | Illustration 5.74 IfA=2 tan-‘(2 V2 – 1) and B = 3 sin + sin-1 – , then which is greater ? |
12 |
| 910 | Find the value of ( cos ^{-1}left(frac{1}{2}right)+ ) ( 2 sin ^{-1}left(frac{1}{2}right) ) |
12 |
| 911 | 23. Domain of definition of the function eal valued x, is f(x)= /sin-‘(2x) + ” for real valued x, is (20035) ( ( ) [* |
12 |
| 912 | Illustration 5.55 If two angles of a triangle are tan-‘(2) and tan-‘(3), then find the third angle. |
12 |
| 913 | Simplify: ( sin . cot ^{-1} tan cdot cos ^{-1} x ) | 12 |
| 914 | If ( cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=pi ) then, prove that ( x^{2}+y^{2}+z^{2}+2 x y z=1 ) |
12 |
| 915 | ( 2 tan ^{-1}left(frac{sqrt{a-b}}{a+b} tan frac{x}{2}right)= ) A ( cdot cos ^{-1}left(frac{b+a cos x}{a+b cos x}right) ) B. ( cos ^{-1}left(frac{b+a cos x}{a-b cos x}right) ) c. ( cos ^{-1}left(frac{b-a cos x}{a+b cos x}right) ) D. ( cos ^{-1}left(frac{b-a cos x}{a-b cos x}right) ) |
12 |
| 916 | sin x || 11. Solve the equation VI sin-‘| cos x | + | cos sin- cosx – cos- | sin x |, ” <<<". 2 |
12 |
| 917 | 1. Solve 2 cos-‘x = sin (2x v1 -x?). 1. Solv |
12 |
| 918 | The value of ( cos left(frac{1}{2} cos ^{-1} frac{1}{8}right) ) is A. ( frac{3}{4} ) в. ( -frac{3}{4} ) c. ( frac{1}{16} ) D. |
12 |
| 919 | Find the principal value of ( operatorname{cosec}^{-1}(-sqrt{2}) ) | 12 |
| 920 | ( sin ^{-1}left(frac{sqrt{1+x}+sqrt{1-x}}{2}right) ) | 12 |
| 921 | The principal value of ( sin ^{-1} x ) lies in the interval A ( cdotleft(-frac{pi}{2}, frac{pi}{2}right) ) B. ( left[-frac{pi}{2}, frac{pi}{2}right] ) c. ( left[0, frac{pi}{2}right] ) D. ( [0, pi] ) |
12 |
| 922 | ( tan left(cos ^{-1} xright) ) is equal to A ( cdot frac{x}{1+x^{2}} ) B. ( frac{sqrt{1+x^{2}}}{x} ) c. ( frac{sqrt{1-x^{2}}}{x} ) D. ( sqrt{1-2 x} ) |
12 |
| 923 | ( boldsymbol{y}=boldsymbol{c o t}^{-1} frac{boldsymbol{2} boldsymbol{x}}{1-boldsymbol{x}^{2}}, boldsymbol{x} neq pm mathbf{1} ) | 12 |
| 924 | Prove that ( tan ^{-1} frac{sqrt{1+x^{2}}-1}{x}= ) ( frac{1}{2} tan ^{-1} x, x neq 0 ) |
12 |
| 925 | If ( boldsymbol{alpha}=boldsymbol{2} boldsymbol{s} boldsymbol{i} boldsymbol{n}^{-1}(boldsymbol{2} / boldsymbol{3}) ) and ( boldsymbol{beta}=boldsymbol{2} boldsymbol{t} boldsymbol{a} boldsymbol{n}^{-1} boldsymbol{9} ) then ( 80 operatorname{cosec}^{2} alpha+81 operatorname{cosec}^{2} beta ) is equal to |
12 |
| 926 | Consider ( boldsymbol{x}=mathbf{4} tan ^{-1}left(frac{mathbf{1}}{mathbf{5}}right), boldsymbol{y}= ) ( tan ^{-1}left(frac{1}{70}right) ) and ( z=tan ^{-1}left(frac{1}{99}right) ) What is ( x ) equal to? A ( cdot tan ^{-1}left(frac{60}{119}right) ) в. ( tan ^{-1}left(frac{120}{119}right) ) ( ^{mathbf{c}} cdot tan ^{-1}left(frac{90}{169}right) ) D. ( tan ^{-1}left(frac{170}{169}right) ) |
12 |
| 927 | If ( boldsymbol{alpha} boldsymbol{epsilon}left(mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right), ) then the value of ( tan ^{-1}(cot alpha)-cot ^{-1}(tan alpha)+ ) ( sin ^{-1}(sin alpha)-cos ^{-1}(cos alpha) ) is equal to A ( .2 alpha ) B . ( pi+alpha ) ( c cdot 0 ) D. ( pi-2 alpha ) |
12 |
| 928 | ( cot ^{-1} 9+operatorname{cosec}^{-1} frac{sqrt{41}}{4}=? ) A ( cdot frac{pi}{6} ) в. ( frac{pi}{4} ) ( c cdot frac{pi}{3} ) D. ( frac{3 pi}{4} ) |
12 |
| 929 | ( sin ^{-1}left(3 x-2-x^{2}right)+cos ^{-1}left(x^{2}-4 x+right. ) 3) ( =frac{pi}{4} ) can have a solution for ( x epsilon ) A . [1,2] B ( cdotleft(frac{3+sqrt{5}}{2}, 2+sqrt{2}right) ) ( left(frac{3-sqrt{5}}{2}, 2-sqrt{2}right) ) D ( cdotleft(2-sqrt{2}, frac{3-sqrt{5}}{2}right) cupleft(frac{3-sqrt{5}}{2}, 2+sqrt{2}right) cup{2} ) |
12 |
| 930 | 6. If cosec- (cosec x) and cosec(cosec- x) are equal functions, then the maximum range of value of x is TC .-lul1 b 10,5 1919 L 2 L 2] c. (-, -1] U[1,0) d. [-1, 0) U[0, 1) a. T_T T7 7 |
12 |
| 931 | ( sin cot ^{-1} tan cos ^{-1} x ) is equal to A . ( x ) B. ( sqrt{1-x^{2}} ) c. ( frac{1}{x} ) D. none of these |
12 |
| 932 | cot-(/cos a) – tan-(cosa) = x,then sinx= (a) tana) (6) cot? ) (c) tan a (2) cot (2) 2002] |
12 |
| 933 | A function ( f(x)=sqrt{1-2 x}+x ) is defined from ( D_{1} rightarrow D_{2} ) and is onto. If the set ( D_{1} ) is its complete domain then the set ( D_{2} ) is A ( cdotleft(-infty, frac{1}{2}right] ) в. ( (-infty, 2) ) ( mathbf{c} cdot(-infty, 1) ) D ( cdot(-infty, 1] ) |
12 |
| 934 | Solve ( tan ^{-1}left(frac{1-x}{1+x}right)= ) ( frac{1}{2} tan ^{-1} x,(0<x<1) ) |
12 |
| 935 | Inverse circular functions,Principal values of ( sin ^{-1} x, cos ^{-1} x, tan ^{-1} x ) [ tan ^{-1} x+tan ^{-1} y=tan ^{-1} frac{x+y}{1-x y} ] ( boldsymbol{x} boldsymbol{y}1 ] (a) ( tan ^{-1} frac{1}{2}+tan ^{-1} frac{1}{3}=frac{pi}{4} ) ( (b) tan ^{-1} frac{1}{2}+tan ^{-1} frac{1}{5}+tan ^{-1} frac{1}{8}=frac{pi}{4} ) ( (c) tan ^{-1} frac{3}{4}+tan ^{-1} frac{3}{5}-tan ^{-1} frac{8}{19}=frac{pi}{4} ) |
12 |
| 936 | Show that ( sin ^{-1} frac{4}{5}+sin ^{-1} frac{5}{13}+sin ^{-1} frac{16}{65}=frac{pi}{2} ) |
12 |
| 937 | ( cos ^{-1}left{frac{1}{2} x^{2}+sqrt{1-x^{2}} cdot sqrt{1-frac{x^{2}}{4}}right}= ) ( cos ^{-1} frac{x}{2}-cos ^{-1} x ) holds for A. ( |x| leq 1 ) В. ( x in R ) c. ( 0 leq x leq 1 ) D. ( -1 leq x leq 0 ) |
12 |
| 938 | Simplify:tan ( ^{-1}(1 / 2)+tan ^{-1}(1 / 3) ) | 12 |
| 939 | If ( 2 tan ^{-1} x+sin ^{-1} frac{2 x}{1+x^{2}} ) is independent of ( x, ) then This question has multiple correct options ( mathbf{A} cdot x>1 ) B. ( x<-1 ) c. ( 0<x<1 ) D. ( -1<x<0 ) |
12 |
| 940 | Find the principal value of ( operatorname{cosec}^{-1}(2) ) | 12 |
| 941 | If ( frac{(x+1)^{2}}{x^{3}+x}=frac{A}{x}+frac{B x+C}{x^{2}+1}, ) then ( csc ^{-1}left(frac{1}{A}right)+cot ^{-1}left(frac{1}{B}right)+sec ^{-1} C= ) A ( cdot frac{5 pi}{6} ) B. c. D. ( frac{pi}{2} ) |
12 |
| 942 | ( sin ^{-1} 0 ) is equal to: ( mathbf{A} cdot mathbf{0} ) в. c. ( frac{pi}{2} ) D. ( frac{pi}{3} ) |
12 |
| 943 | If ( f(x)=sin ^{-1} x+sec ^{-1} x ) is defined then which of the following value/values is/are in its range? A ( cdot frac{-pi}{2} ) в. ( frac{pi}{2} ) ( c . pi ) D. ( frac{3 pi}{2} ) |
12 |
| 944 | Find the value of ( cos left[frac{pi}{2}-sin ^{-1}left(frac{1}{3}right)right] ) | 12 |
| 945 | The value of ( sin left(frac{1}{4} sin ^{-1} frac{sqrt{63}}{8}right) ) is A ( cdot frac{1}{2} ) B. ( frac{1}{3} ) ( c cdot frac{1}{2 sqrt{2}} ) D. |
12 |
| 946 | Find the principal value of ( sec ^{-1}left(frac{2}{sqrt{3}}right) ) | 12 |
| 947 | ( operatorname{Let} cos ^{-1}left(4 x^{3}-3 xright)=a+b cos ^{-1} x ) ( x inleft[-frac{1}{2}, frac{1}{2}right], ) then the principal value of ( sin ^{-1}left(sin frac{a}{b}right) ) is ( A cdot-frac{pi}{3} ) в. ( c cdot-frac{pi}{6} ) D. None of these |
12 |
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