We provide application of derivatives practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on application of derivatives skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

#### List of application of derivatives Questions

Question No | Questions | Class |
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1 | The normal to the curve ( x^{2}=4 y ) passing through (1,2) is A. ( x+y=3 ) в. ( x-y=3 ) c. ( x+y=1 ) D. ( x-y=1 ) | 12 |

2 | Water is running into an underground right circular conical reservoir, which is 10 ( m ) deep and radius of its base is 5 m. If the rate of change in the volume of water in the reservoir is ( frac{3}{2} pi m^{3} / m i n quad, ) then the rate ( ( ) in ( mathrm{m} / mathrm{min} ) ) at which water rises in it, when the water level is ( 4 boldsymbol{m}, ) is: ( A cdot frac{3}{8} ) в. ( frac{1}{8} ) c. ( frac{1}{4} ) D. ( frac{3}{2} ) | 12 |

3 | 3. The minimum value of the expression sin a + sin ß + sin y, where a, b, y are real numbers satisfying a + B+ y = it is a. positive b. zero c. negative d. -3 (IIT-JEE 1995) | 12 |

4 | The approximate value of ( (33)^{1 / 5} ) is A . 2.0125 B. 2.1 c. 2.0 D. none of these | 12 |

5 | Find the greatest value of ( boldsymbol{f}(boldsymbol{x})= ) ( x^{2} log x ) on ( [1, e] ) ( mathbf{A} cdot mathbf{0} ) B ( cdot e^{2} ) c. ( e / 2 ) D. | 12 |

6 | Illustration 2.39 If A = 4 sin 0 + cos²e, then which of the following is not true? a. Maximum value of A is 5. b. Minimum value of A is – 4. c. Maximum value of A occurs when sin 0= 1/2. d. Minimum value of A occurs when sin 0= 1. | 12 |

7 | Find the absolute maximum and minimum values of function given by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-mathbf{4} boldsymbol{x}+mathbf{8} ) in the interval ( [mathbf{1}, mathbf{5}] ) | 12 |

8 | The sides of two squares are ( x ) and ( y ) respectively, such that ( y=x+x^{2} ). The rate of change of area of second square with respect to area of first square is A. ( x^{2}+3 x-1 ) B . ( 2 x^{2}-3 x+1 ) c. ( 2 x^{2}+3 x+1 ) D. ( 1+2 x ) | 12 |

9 | The maximum value of ( 3 cos theta+4 sin theta ) is A . -5 B. 5 c. 25 D. None of these | 12 |

10 | Maximum value of ( frac{log x}{x} ) is ( A cdot frac{2}{e} ) B. ( c cdot frac{1}{e} ) D. | 12 |

11 | The number of stationary points of ( mathbf{f}(mathbf{x})=sin mathbf{x} operatorname{in}[mathbf{0}, mathbf{2} boldsymbol{pi}] ) are A B. 2 ( c cdot 3 ) D. | 12 |

12 | ( operatorname{Given} f(x)left{begin{array}{ll}|x-2|+lambda & text { if } x leq 2 \ x^{2}+1 & text { if } x>2end{array}right. ) If ( f(x) ) has a local minimum at ( x=2 ) then which of the following is most appropriate A. ( lambda leq 5 ) в. ( lambda geq 5 ) c. ( lambda leq 0 ) D. ( lambda geq 0 ) | 12 |

13 | If a ball is thrown vertically upwards and the height ‘s’ reached in time ‘t’ is given by ( s=22 t-11 t^{2}, ) then the total distance travelled by the ball is A. 44 units B. 33 units c. 11 units D. 22 units | 12 |

14 | I: If ( mathrm{f}^{prime}(mathrm{a})<0 ) then the function ( mathrm{f} ) is decreasing at ( mathbf{x}=mathbf{a} ) Il: If ( f ) is decreasing at ( x=a ) then ( mathbf{f}^{prime}(mathbf{a})<mathbf{0} ) Which of the above statements are true ( ? ) A. only B. only II c. both land II | 12 |

15 | Let the function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}+cos boldsymbol{x} ) then ( boldsymbol{f} ) A. Has maximum at ( x=0 ) B. Has minimum at ( x=pi ) c. Is an increasing function D. Is a decreasing | 12 |

16 | A cube of ice melts without changing its shape at the uniform rate of ( 4 mathrm{cm}^{3} / ) min. The rate of change of the surface area of the cube, in ( c m^{2} / )min when the volume of the cube is ( 125 mathrm{cm}^{3} ) is A. -4 в. ( -frac{16}{5} ) c. ( -frac{16}{6} ) D. ( -frac{8}{15} ) | 12 |

17 | Calculate the maximum value of ( boldsymbol{f}(boldsymbol{x})=boldsymbol{3}-(boldsymbol{x}+boldsymbol{2})^{2} ) A . -3 B. – c. 1 D. 3 E. 5 | 12 |

18 | If a particle moves according to the law, ( s=6 t^{2}-frac{t^{3}}{2}, ) then the time at which it is momentarily at rest A. ( t=0 ) only B. ( t=8 ) only c. ( t=0,8 ) D. none of these | 12 |

19 | The equation of tangent to the curve ( boldsymbol{y}=3 x^{2}-x+1 ) at the point (1,3) is A. ( y=5 x+2 ) в. ( y=5 x-2 ) c. ( _{y=frac{1}{5} x+2} ) D. ( _{y=frac{1}{5} x-2} ) | 12 |

20 | The maximum value of ( sin ^{2} x cos ^{3} x ) is A ( cdot frac{6 sqrt{3}}{25 sqrt{5}} ) В. ( frac{9 sqrt{3}}{25 sqrt{5}} ) c. ( frac{9 sqrt{2}}{6 sqrt{5} / 5} ) D. ( frac{sqrt{2}}{sqrt{5}} ) | 12 |

21 | If ( f(x)=x^{x}, ) then ( f(x) ) is increasing in interval: A . ( [0, e] ) B. ( left(frac{1}{e}, inftyright) ) c. [0,1] D. None of these | 12 |

22 | The coordinates of the point(s) on the graph of the function ( f(x)= ) ( frac{x^{3}}{3}-frac{5 x^{2}}{2}+7 x-4 ) where the tangent drawn cut off intercepts from the coordinate axes which are equal in magnitude but opposite in sign is This question has multiple correct options ( mathbf{A} cdotleft(2, frac{8}{3}right) ) B. ( left(3, frac{7}{2}right) ) ( mathbf{c} cdotleft(1, frac{5}{6}right) ) D. none | 12 |

23 | Find the point of inflexion of ( boldsymbol{f}(boldsymbol{x})= ) ( 3 x^{4}-4 x^{3} ) and hence draw the graph of ( boldsymbol{f}(boldsymbol{x}) ) | 12 |

24 | Maximum value of ( r ) where ( frac{c^{2}}{r^{2}}= ) ( frac{mathbf{a}^{2}}{sin ^{2} theta}+frac{b^{2}}{cos ^{2} theta} ) where ( c, a, b ) are constants is A. ( frac{c}{a+b} ) B. ( frac{c}{a-b} ) c. ( frac{a^{2}+b^{2}}{c^{2}} ) D. ( frac{c^{2}}{a^{2}+b^{2}} ) | 12 |

25 | The altitude of a cone is ( 20 mathrm{cm} ) and its semi-vertical angle is ( 30^{0} . ) If the semivertical angle is increasing at the rate of ( 2^{0} ) per second, then the radius of the base is increasing at the rate of A. ( 30 mathrm{cm} / mathrm{sec} ) в. ( frac{160}{3} mathrm{cm} / mathrm{sec} ) c. ( 10 mathrm{cm} / mathrm{sec} ) D. ( 160 mathrm{cm} / mathrm{sec} ) | 12 |

26 | Initially, equation of ellipse was ( 3 x^{2}+ ) ( 4 y^{2}=12 . ) Keeping major axis constant ellipse is bulge to form circle (with major axis as diameter). Its eccentricity changes at a rate ( 0.1 / ) sec. Time taken to form this circle is A . 2 sec B. 3 sec c. 5 sec D. 7 sec | 12 |

27 | ( mathbf{f}(mathbf{x})=|mathbf{x}| ) is minimum at ( mathbf{x}= ) ( mathbf{A} cdot mathbf{1} ) B. ( c cdot-1 ) D. 2 | 12 |

28 | The values of ‘a’ for which ( y=x^{2}+ ) ( a x+25 ) touches ( x ) -axis are ( mathbf{A} cdot pm 10 ) B. ±2 ( c .pm 1 ) D. | 12 |

29 | Find the maxima of function ( 8-7 x^{2} ) | 12 |

30 | The function ( f(x)=x^{2} ) is decreasing in ( A cdot(-infty, infty) ) в. ( (-infty, 0) ) ( c cdot(0, infty) ) D. ( (-2, infty) ) | 12 |

31 | The function ( f(x)=frac{sin x}{x} ) is decreasing in the interval ( ^{mathbf{A}} cdotleft(-frac{pi}{2}, 0right) ) B. ( left(0, frac{pi}{2}right) ) ( ^{mathbf{c}} cdotleft(-frac{pi}{4}, 0right) ) D. None of these | 12 |

32 | If ( x ) and ( y ) are sides of two squares such that ( y=x-x^{2} ). Find the rate of change of area of second square (side ( y ) ) with respect to area of the first square (side ( x) ) when ( x=1 mathrm{cm} ) | 12 |

33 | The slope of tangent to the curve ( x= ) ( boldsymbol{t}^{2}+mathbf{3} boldsymbol{t}-mathbf{8}, boldsymbol{y}=mathbf{2} boldsymbol{t}^{2}-boldsymbol{2} boldsymbol{t}-mathbf{5} ) at the point (2,-1) is : A ( cdot frac{22}{7} ) B. ( frac{6}{7} ) c. -6 D. None of these | 12 |

34 | Suppose ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is differentiable function satisfying ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+ ) ( boldsymbol{f}(boldsymbol{y})+boldsymbol{x} boldsymbol{y}(boldsymbol{x}+boldsymbol{y}) ) for every ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) if ( f^{prime}(0)=0, ) then which of the following hold(s) good? This question has multiple correct options ( A . f ) is an odd function B. ( f ) is a bijective mapping c. ( f ) has a minima but no maxima D. ( f ) has an inflection point | 12 |

35 | A particle starts moving from rest from a fixed point in a fixed direction. The distance s from the fixed point at a time ( t ) is given by ( s=t^{2}+a t-b+17, ) where ( a ) and ( b ) are real numbers. If the particle comes to rest after ( 5 s ) at a distance of ( s=25 ) units from the fixed point, then value of ( a ) and ( b ) are respectively. A .10,-33 B. -10,-33 c. -8,33 D. -10,33 | 12 |

36 | ( boldsymbol{f}:(mathbf{0}, infty) rightarrowleft(-frac{boldsymbol{pi}}{2}, frac{boldsymbol{pi}}{2}right) ) be defined as ( f(x)=arctan (x) ) The above function can be classified as A. injective but not surjective B. surjective but not injective c. neither injective nor surjective D. both injective as well as surjective | 12 |

37 | Find the equation of the tangent and the normal to the following curve at the indicated point. ( c^{2}left(x^{2}+y^{2}right)=x^{2} y^{2} ) at ( left(frac{x}{cos theta}, frac{c}{sin theta}right) ) | 12 |

38 | A particle moves along a straight line according to the law ( s=16-2 t+3 t^{3} ) where ( s ) metres is the distance of the particle from a fixed point at the end of ( t ) second. The acceleration of the particle at the end of 2 s is A ( cdot 3.6 m / s^{2} ) B. ( 36 m / s^{2} ) c. ( 36 k m / s^{2} ) D. ( 360 m / s^{2} ) | 12 |

39 | A function ( f ) such that ( f^{prime}(2)=f^{prime prime}(2)= ) 0 and ( f ) has a local maximum of -17 at ( x ) ( =2 ) is : A ( cdot(x-2)^{4} ) B . ( 3-(x-2)^{4} ) c. ( -17-(x-2)^{4} ) D. None of these | 12 |

40 | The radius of a sphere is given by ( r=2 t ) The rate of change of surface area at ( t= ) 1 is equal to ( A cdot 8 pi ) B. ( 32 pi ) c. ( 16 pi ) D. ( 4 pi ) | 12 |

41 | When the temperature of medium is ( 20^{circ} mathrm{C} ) a certain substance cools from ( 100^{circ} mathrm{C} ) to ( 60^{circ} mathrm{C} ) in 10 minutes. Its temperature after 40 minutes from the beginning is A ( cdot 15^{circ} mathrm{C} ) B . ( 20^{circ} mathrm{C} ) ( mathbf{c} cdot 25^{circ} C ) D. ( 30^{circ} mathrm{C} ) | 12 |

42 | The radius of a circle is increasing uniformly at the rate of ( 5 mathrm{cm} / mathrm{sec} ). Find the rate at which the area of the circle is increasing when the radius is ( 6 mathrm{cm} ) | 12 |

43 | The difference between the greatest and the least values of the function ( f(x)= ) ( int_{0}^{x}left(a t^{2}+1+cos tright) d t, a>0, x in[2,3] ) is: A ( cdot frac{19}{3} a+1+sin 3-sin 2 ) в. ( frac{18}{3} a+1+2 sin 3 ) c. ( frac{18}{3} a-1+2 sin 3 ) D. none of these | 12 |

44 | Let ( f(x)=7 e^{sin ^{2} x}-e^{cos ^{2} x}+2, ) then the value of ( sqrt{7 f_{m i n}+f_{m a x},} ) is | 12 |

45 | The tangent to the curve ( x= ) ( a sqrt{cos 2 theta} cos theta, y=a sqrt{cos 2 theta} sin theta ) at the point corresponding to ( boldsymbol{theta}=boldsymbol{pi} / boldsymbol{6} ) is A. parallel to the ( x ) -axis B. parallel to the ( y ) -axis c. parallel to line ( y=x ) D. none of these | 12 |

46 | A curve ( C ) has the property that if the tangent drawn at any point ‘P’ on C meets the coordinate axes at ( A ) and ( B ) and ( P ) is midpoint of ( A B . ) If the curve passes through the point (1,1) then the equation of the curve is? A ( . x y=2 ) в. ( x y=3 ) c. ( x y=1 ) D. ( x y=4 ) | 12 |

47 | The function ( f(x)=x^{9}+3 x^{7}+64 ) is increasing on A. ( R ) в. ( (-infty, 0) ) ( c cdot(0, infty) ) D. ( R_{0} ) | 12 |

48 | 3. If the function f(x) = 2x} – 9ax² +12a+x+1, where a>0, attains its maximum and minimum at p and a respectively such that p2 = q , then a equals [2003] (a) – (b) 3 (c) 1 (d) 2 . | 12 |

49 | Find the equation of the normal to the curve ( x^{2}=4 y ) which passes through the point (1,2) | 12 |

50 | A quadratc function in x has the value 10 when ( x=1 ) and has minimum value 1 when ( x=-2 ) the function is ( A cdot 2 x^{2}+3 x+5 ) B. ( 3 x^{2}+2 x+5 ) c. ( x^{2}+3 x+6 ) D. ( x^{2}+4 x+5 ) | 12 |

51 | A balloon, which always remains spherical, has a variable diameter ( frac{3}{2}(2 x+1) . ) Find the rate of change of its volume with respect to ( x ) | 12 |

52 | The surface area of a sphere when its volume is increasing at the same rate as its radius is ( mathbf{A} cdot mathbf{1} ) B. ( frac{1}{2 sqrt{pi}} ) c. ( 4 pi ) D. ( frac{4 pi}{3} ) | 12 |

53 | 10. A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm /min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases is [2005] (a) – 36 cm/min. (b) o 18 TC cm/min. c) – cm/min. 547 cm/min. (d) cm/min 6T cm/min | 12 |

54 | Find an angle ( theta, 0<theta<frac{pi}{2}, ) which increases twice as fast as its sine. | 12 |

55 | The surface area of a solid sphere is increased by ( 21 % ) without changing its shape.Find the percentage increase in its : (i)radius (ii)volume | 12 |

56 | The graph of the curve ( x^{2}=3 x-y-2 ) is A ( cdot ) between the lines ( x=1 ) and ( x=frac{3}{2} ) B. between the lines ( x=1 ) and ( x=2 ) C. strictly below the line ( 4 y=1 ) D. none of these | 12 |

57 | The angle which the tangent to a curve at any point ( (x, y) ) on it makes with axis of ( x ) is ( tan ^{-1}left(x^{2}-2 xright) ) for all values of ( x ) and it passes through the point (2,0) Determine the point on it whose ordinate is maximum. A ( .(2,8 / 3) ) в. ( (0,4 / 3) ) c. ( (1,2 / 3) ) D. ( (-1,4 / 3) ) | 12 |

58 | 6. Find the shortest distance of the point (0,c) from the parabola y=x2 where 0<c< 5. (1982 – 2 Marks | 12 |

59 | ( f(x)=left{begin{array}{l}k-2 x, text { if } x leq-1 \ 2 x+3, text { if } x>-1end{array}right}, ) if ( f ) has a local minimum at ( x=-1, ) then ( k= ) | 12 |

60 | Find the angle of intersection of the following curve: ( x^{2}+y^{2}=2 x ) and ( y^{2}=x ) | 12 |

61 | Consider a real valued function ( boldsymbol{f} ) ( boldsymbol{R} rightarrow boldsymbol{R} ) satisfying ( boldsymbol{f}left(frac{boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}}{mathbf{5}}right)= ) ( frac{2 f(x)+3 f(y)}{5} forall x, y in R quad ) and ( f(0)= ) ( mathbf{2} ; boldsymbol{f}^{prime}(mathbf{0})=mathbf{1} ) Minimum distance of a point on graph of ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) from origin is less than or equal to A. ( sqrt{2} ) units B. ( frac{1}{2} ) units c. ( frac{1}{sqrt{5}} ) units D. ( frac{1}{sqrt{7}} ) units | 12 |

62 | If ( f(x)=(x-a)^{2 n}(x-b)^{2 m+1} ) where ( m cdot n in N, ) then A. ( x=a ) is a point of minimum B. ( x=a ) is a point of maximum c. ( x=a ) is not a point of maximum or minimum D. No value of k satisfies the requirement | 12 |

63 | Maximum value of ( 1+ ) ( 8 sin ^{2}left(x^{2}right) cos ^{2}left(x^{2}right) ) is A. 3 B. – ( c cdot-8 ) D. | 12 |

64 | Find points on the curve ( frac{x^{2}}{9}+frac{y^{2}}{16}=1 ) at which the tangents are Parallel to ( x ) -axis are ( (a, pm b) ).Find ( a+b ) | 12 |

65 | If ( boldsymbol{x} sqrt{mathbf{1}+boldsymbol{y}}+boldsymbol{y} sqrt{mathbf{1}+boldsymbol{x}}=mathbf{0} ) and ( boldsymbol{x} neq boldsymbol{y} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) A. ( frac{1}{1+x} ) B. ( frac{1}{(1+x)^{2}} ) c. ( frac{-1}{(1+x)^{2}} ) D. ( frac{-1}{1+x} ) | 12 |

66 | Find the point on the curve ( y^{2}=2 x ) which is at a minimum distance from the point (1,4) | 12 |

67 | If ( boldsymbol{f}(mathbf{0})=mathbf{0} ) and ( boldsymbol{f}^{prime prime}(boldsymbol{x})>mathbf{0} ) for all ( boldsymbol{x}>mathbf{0} ) then ( frac{boldsymbol{f}(boldsymbol{x})}{boldsymbol{x}} ) A ( cdot ) decreases on ( (0, infty) ) B . increases on ( (0, infty) ) c. decreases on ( (1, infty) ) D. neither increases nor decreases on ( (0, infty) ) | 12 |

68 | If ( x ) be real then the minimum value of ( 40-12 x+x^{2} ) is? A . 28 B. 4 ( c cdot-4 ) D. | 12 |

69 | Find the value of ( a ) if ( f(x)=2 e^{x}- ) ( a e^{-x}+(2 a+1) x-3 ) is increasing for all values of ( x ) | 12 |

70 | Find the value of ( frac{d y}{d x} ) at ( theta=frac{pi}{4} ) if ( x= ) ( boldsymbol{a} e^{theta}(sin theta-cos theta) ) and ( y=a e^{theta}(sin theta+ ) ( cos theta) ) | 12 |

71 | An equation for the line that passes through (10,-1) and is perpendicular to ( y=frac{x^{2}}{4}-2 ) is A. ( 4 x+y=39 ) в. ( 2 x+y=19 ) c. ( x+y=9 ) D. ( x+2 y=8 ) | 12 |

72 | If ( log _{10}left(x^{3}+y^{3}right) ) ( log _{10}left(x^{2}+y^{2}-x yright) leq 2, ) then the maximum value of ( x y, ) for all ( x geq ) ( mathbf{0}, boldsymbol{y} geq mathbf{0}, ) is A . 2500 в. 3000 c. 1200 D. 3500 | 12 |

73 | Consider the function ( boldsymbol{f}(boldsymbol{x})= ) [ begin{array}{cl} sqrt{boldsymbol{x}} ln boldsymbol{x} text { when } & boldsymbol{x}>mathbf{0} \ mathbf{0} text { for } & boldsymbol{x}=mathbf{0} end{array} ] Let the inflection points of the graph of [ boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) text { be } boldsymbol{x}=boldsymbol{k} text { .Find } boldsymbol{k} ? ] | 12 |

74 | The lines tangent to the curves ( y^{3}- ) ( x^{2} y+5 y-2 x=0 ) and ( x^{4}-x^{3} y^{2}+ ) ( 5 x+2 y=0 ) at the origin intersect at an angle ( theta ) equal to A ( cdot frac{pi}{6} ) B. c. D. | 12 |

75 | If the cube equation ( x^{3}-p x+q ) has three distinct real roots, where ( p>0 ) and ( boldsymbol{q}>mathbf{0} ) Then, which one of the following is correct? A ( cdot ) the cubic has maxima at both ( sqrt{frac{p}{3}} ) and ( -sqrt{frac{p}{3}} ) B. The cubic has minima at ( sqrt{frac{p}{3}} ) and maxima at ( -sqrt{frac{p}{3}} ) c. The cubic has minima at ( -sqrt{frac{p}{3}} ) and maxima at ( sqrt{frac{p}{3}} ) D. The cubic has minima at both ( sqrt{frac{p}{3}} ) and ( -sqrt{frac{p}{3}} ) | 12 |

76 | ( left(frac{2 a x}{a^{2}+x^{2}}right)+frac{1}{3}left(frac{2 a x}{a^{2}+x^{2}}right)^{3}+ ) ( frac{1}{5}left(frac{2 a x}{a^{2}+x^{2}}right)^{5}+dots= ) ( A cdot log (a+x) ) B. ( log (a-x) ) ( ^{mathbf{C}} cdot log left(frac{a+x}{a-x}right) ) ( D ) | 12 |

77 | 3x(x +1) 38. Prove that for sin x + 2x > . Explain the identity if any used in the proof. (2004 – 4 Marks) | 12 |

78 | Find the intervals in which the function ( f ) given by ( f(x)=x^{3}+frac{1}{x^{3}}, x neq 0 ) is (i) increasing (ii) decreasing. | 12 |

79 | fthe rate of change in ( y=frac{x^{3}}{3}-2 x^{2}+ ) ( 5 x+7 ) is two times the rate of change in ( boldsymbol{x}, ) then ( boldsymbol{x}= ) A .2,3 в. 1,3 c. 1,2 D. 2,5 | 12 |

80 | 32. Iff:R → R is a twice differentiable function such that f”(x) > 0 for all x e R, and f f f(1)=1, then (JEE Adv. 2017) (a) f'(1)50 (b) 0<f'l)s * <f'(1) 1 | 12 |

81 | Find minimum value ( 4 x+frac{9}{x} quad x>0 ) | 12 |

82 | ( text { If }boldsymbol{f}(boldsymbol{x})=boldsymbol{a} sec boldsymbol{x}-boldsymbol{b} tan boldsymbol{x}, boldsymbol{a}rangle boldsymbol{b}rangle boldsymbol{0}, ) then the minimum value of ( f(x) ) is A ( cdot sqrt{a^{2}+b^{2}} ) B. ( 2 sqrt{a^{2}-b^{2}} ) c. ( sqrt{a^{2}-b^{2}} ) D. ( a-b ) | 12 |

83 | If ( 1^{0}=alpha ) radians then the approximate value of ( cos 60^{0} 1^{prime} ) is A ( cdot frac{1}{2}+frac{alpha sqrt{3}}{120} ) в. ( frac{1}{2}-frac{alpha}{120} ) ( ^{mathrm{C}} cdot frac{1}{2}-frac{alpha sqrt{3}}{120} ) D. none of these | 12 |

84 | A particle moves along a straight line and its velocity at a distance ‘x’ from the origin is ( k sqrt{a^{2}-x^{2}} . ) Then acceleration of the particle is ( A cdot k ) B. ( -k^{2} ) c. ( k x ) ( mathbf{D} cdot-k^{2} x ) | 12 |

85 | 11. Let f:R → R be given by fire be given by f (x) = (x – 1) (x – 2) (x – 5). Define F(x) = S f (t)dt,x>0. 0 Then when of the following options is/are correct? (JEE Adv. 2019) (a) Fhas a local maximum at x=2 (b) Fhas a local minimum at x=1 (c) Fhas two local maxima and one local minimum in (0,00) (d) F(x) O for all x € (0,5) | 12 |

86 | Two trains are running in opposite directions with the same speed. If the length of each train is 120 metres and they cross each other in 12 seconds, then the speed of each train (in ( mathrm{km} / mathrm{hr} ) ) is A . 10 B. 18 ( c . ) 36 D. 72 | 12 |

87 | Sum of the maximum and minimum values of ( 12 cos ^{2} x-6 sin x cos x+ ) ( 2 sin ^{2} x ) is ( mathbf{A} cdot mathbf{0} ) B. 7 c. 14 D. 15 | 12 |

88 | If ( y=frac{a x+b}{(x-1)(x-4)} ) has a turning value at (2,-1) find ( a & b ) A ( . a=0, b=0 ) В. ( a=1, b=0 ) c. ( a=0, b=1 ) D. ( a=1, b=1 ) | 12 |

89 | 11. The maximum and minimum values of x3 – 18×2 +96x in interval (0,9) are (a) 160,0 (b) 60,0 (c) 160, 128 (d) 120, 28 | 12 |

90 | 6. The value of a for which the sum of the squares of he sum of the squares of the roots of the equation x value is (a) I (b) 0 (a-2) x-a-1=0 assume the least [2005] (c) 3 (d) 2 | 12 |

91 | A mans walks along a straight path at a speed of ( 4 f t / ) sec.A scarch light is located on the ground ( 20 f t ) from the path and is kept focused on man. At what rate is the scarch light rotating when the man is ( 15 f t ) from the point on the path closest to the search light. A . 0.128 rad/sec B. 1.6rad/sec c. ( 0.8 mathrm{rad} / mathrm{sec} ) D. ( 0.24 mathrm{rad} / mathrm{sec} ) | 12 |

92 | The temperature has dropped by 15 degree Celsius in the last 30 days. If the rate of temperature drop remains the same, how much more will the temperature drop in the next 10 days? | 12 |

93 | Show that ( boldsymbol{f}(boldsymbol{x})= ) ( frac{x}{1+x tan x}, x epsilonleft(0, frac{pi}{2}right) ) is maximum when ( boldsymbol{x}=cos boldsymbol{x} ) | 12 |

94 | 22. The curve y = ax3 + bx2 + cx + 5, touches the x-axis at P(-2, 0) and cuts the y axis at a point Q, where its gradient is 3. Find a, b, c. (1994 – 5 Marks) | 12 |

95 | Find the point on the curve ( y^{2}=8 x ) for which the abscissa and ordinate change at the same rate. | 12 |

96 | If ( g(x)=7 x^{2} e^{-x^{2}} forall x in R, ) then ( g(x) ) has This question has multiple correct options A. local maximum at ( x=0 ) B. local minima at ( x=0 ) c. local maximum at ( x=1 ) D. two local maxima and one local minima | 12 |

97 | Let ( f ) be the function ( f(x)=cos x- ) ( left(1-frac{x^{2}}{2}right), ) then find the interval in which ( f(x) ) is strictly increasing. ( A cdot(-infty, infty) ) в. ( (-2, infty) ) c. ( [0, infty) ) () D. ( (0, infty) ) | 12 |

98 | The function f(x) = 2x + x +21-1x + 21-2 xhas a local minimum or a local maximum atx= (JEE Adv. 2013) | 12 |

99 | ( frac{1}{2}-frac{1}{2^{2} cdot 2}+frac{1}{2^{3} cdot 3}-frac{1}{2^{4} cdot 4}+ldots infty= ) A ( . log 2 ) B. ( log 3 ) ( mathrm{c} cdot log frac{3}{2} ) D. ( log _{3} frac{2}{3} ) | 12 |

100 | ( operatorname{Let} g^{prime}(x)>0 ) and ( f^{prime}(x)f(f(x-1)) ) в. ( f(g(x-1))>f(g(x+1)) ) ( mathbf{c} . g(f(x+1))>g(f(x-1)) ) D. ( g(g(x+1))>g(g(x-1)) ) | 12 |

101 | Let ( f(x)=x^{3} e^{-3 x}, x>0 . ) Then the maximum value of ( f(x) ) is A ( cdot e^{-3} ) B. ( 3 e^{-3} ) ( mathbf{c} cdot 27 e^{-9} ) D. ( infty ) | 12 |

102 | Let the interval ( I[-1,4] ) and ( f: I rightarrow R ) be a function such that ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{boldsymbol{3}}- ) ( 3 x ). Then the range of the function is A . [2,52] в. [-2,2] ( mathbf{c} cdot[-2,52] ) D. none of these | 12 |

103 | Find the interval of increase and decrease of the following functions. ( f(x)=frac{x}{ln x} ) | 12 |

104 | If the radius of a sphere is measured as ( 7 m ) with an error of ( 0.02 m, ) then find the approximate error in calculating its volume | 12 |

105 | For what value of ( x, 8 x^{2}-7 x+2 ) has the minimum value? | 12 |

106 | To examine the function ( f(x)=2 x^{3}- ) ( 15 x^{2}+36 x+10 ) for maxima and minima, if any. Also find the maximum and minimum value. | 12 |

107 | Assertion The points on the curve ( y^{2}=x+sin x ) at which tangent is parallel to x-axis lies on a straight line. Reason Tangent is parallel to ( x ) -axis then ( frac{d y}{d x}= ) 0 or ( frac{d x}{d y}=infty ) 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 |

108 | The angle made by the tangent line at (1 3) on the curve ( y=4 x-x^{2} ) with ( O X ) is A ( cdot tan ^{-1}(2) ) B. ( tan ^{-1}(1 / 3) ) c. ( tan ^{-1}(3) ) D. ( pi / 4 ) | 12 |

109 | Show that ( f(x)=x^{3}-15 x^{2}+75 x- ) 50 is an increasing function for all ( x in ) ( boldsymbol{R} ) | 12 |

110 | The maximum value of ( frac{17}{3}-left(x-frac{4}{5}right)^{2} ) is A ( cdot 4 / 5 ) B. ( -4 / 5 ) c. ( 17 / 3 ) D. ( -17 / 3 ) | 12 |

111 | Equation of a straight line passing through (1,4) if the sum of its positive intercepts on the coordinate axis is the smallest is A. ( 2 x+y-6=0 ) B . ( x+2 y-9=0 ) c. ( y+2 x+6=0 ) D. none | 12 |

112 | 24. Let (h, k) be a fixed point, where h>0,k>0. A straight line passing through this point cuts the positive direction of the coordinate axes at the points P and Q. Find the minimum area of the triangle OPO, O being the origin.(1995-5 Marks) | 12 |

113 | A man is moving away from a tower 41.6 ( mathrm{m} ) high at a rate of ( 2 mathrm{m} / mathrm{s} ). If the eye level of the man is ( 1.6 mathrm{m} ) above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 ( mathrm{m} ) from the foot of the tower, is ( A cdot-frac{4}{125} mathrm{rad} / mathrm{s} ) B. ( -frac{2}{25} ) rad/s c. ( -frac{1}{625} ) rad/s D. none of these | 12 |

114 | ( f(x)=frac{x}{log x}-frac{log 5}{5} ) is decreasing in ( A cdot(e, infty) ) B. (0,1) ( mathrm{U}(1, mathrm{e}) ) ( c cdot(0,1) ) D. (1, e) | 12 |

115 | If y=a In x + bx2 + x has its extremum values at x=-1 and x = 2, then (1983 – 1 Mark) (a) a=2, b=-1 (b) a=2,b= – (c) a=- 2, b= 1 (d) none of these | 12 |

116 | Find the minimum value of ( e^{(2 x-2 x+1) sin ^{2} x} ) | 12 |

117 | For ( boldsymbol{f}(boldsymbol{x})=sin ^{2} boldsymbol{x}, boldsymbol{x} in(mathbf{0}, boldsymbol{pi}) ) point of inflection is A ( cdot frac{pi}{6} ) в. ( frac{2 pi}{4} ) ( c cdot frac{3 pi}{4} ) D. ( frac{4 pi}{3} ) | 12 |

118 | The volume of a cube is increasing at the rate of ( 8 c m^{3} / s . ) How fast is the surface area increasing when the length of an edge is ( 12 mathrm{cm} ? ) | 12 |

119 | The total cost ( C(x) ) associated with the production of x units of an item is given by ( C(x)=0.05 x^{3}-0.02 x^{2}+30 x+ ) 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. | 12 |

120 | The contentment obtained after drinking ( x- ) units of a new drink at a trial function is given by the function ( C(x)=2 x^{3}-x^{2}+7 x+2 . ) If the marginal contentment is defined as rate of change of ( C ) with respect to the number of units consumed at an instant, then find the marginal contentment when 4 units of drink are consumed | 12 |

121 | Write the set of values of ( a ) for which the function ( f(x)=a x+b ) is decreasing for all ( boldsymbol{x} in boldsymbol{R} ) | 12 |

122 | Illustration 2.39 If A = 4 sin 0 + cos²e, then which of the following is not true? a. Maximum value of A is 5. b. Minimum value of A is – 4. c. Maximum value of A occurs when sin 0= 1/2. d. Minimum value of A occurs when sin 0= 1. | 12 |

123 | Equation of tangent at that point of the curve ( boldsymbol{y}=mathbf{1}-boldsymbol{e}^{frac{x}{2}}, ) where it meets ( mathbf{y} ) -axis ( mathbf{A} cdot x+2 y=0 ) в. ( 2 x+y=0 ) c. ( x-y=2 ) D. None of these | 12 |

124 | A closed conical vessel is filled with water fully and is placed with its vertex down. The water is let out at a constant speed. After 21 minutes, it was found that the height of the water column is half of the original height. How much more time in minutes does it empty the vessel? A . 21 B. 14 ( c cdot 7 ) D. 3 | 12 |

125 | The equation of the tangent to the curves ( x=t cos t ) and ( y=t sin t ) at the origin is? ( mathbf{A} cdot x=0 ) в. ( y=0 ) c. ( x+y=0 ) D. ( x-y=0 ) | 12 |

126 | Find the points of maxima, minima and the intervals of monotonicity of the following function: ( boldsymbol{y}=boldsymbol{x}^{4}+boldsymbol{4} boldsymbol{x}^{3}-boldsymbol{8} boldsymbol{x}^{2}+boldsymbol{3} ) | 12 |

127 | ( operatorname{Let} mathbf{h}(mathbf{x})=mathbf{f}(mathbf{x})-[mathbf{f}(mathbf{x})]^{2}+[mathbf{f}(mathbf{x})]^{3} ) for every real number x then A. h is increasing whenever fis increasing B. h is increasing whenever f is decreasing c. ( h ) is decreasing whenever ( f ) is increasing D. Nothing can be said in general | 12 |

128 | 23. The circle x2 + y2 = 1 cuts the x-axis at P and Q. Another circle with centre at Q and variable radius intersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum area of the triangle OSR. (1994 – 5 Marks) ( 11) | 12 |

129 | 26. J Bulleenlulaule alivu Let f:R → R be a positi → R be a positive increasing function with R (3x) lim 2= 1. Then lim [2010] = *-700 f(x) (a) ſ 6) f(x) 3 (d) 1 | 12 |

130 | Determine the intervals over which the function is decreasing, increasing, and constant A. Increasing ( [3, infty) ; ) Decreasing ( (-infty, 3] ) B. Increasing ( (-infty, 3] ); Decreasing ( [3, infty) ) c. Increasing ( (-infty, 3] ; ) Decreasing ( (-infty, 3 ) D. Increasing [3 ( infty) ; ) Decreasing ( [3, infty) ) | 12 |

131 | If there is an error of ( 0.1 % ) in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere | 12 |

132 | ( frac{a-b}{a}+frac{1}{2}left(frac{a-b}{a}right)^{2}+frac{1}{3}left(frac{a-b}{a}right)^{3}+= ) A ( cdot log _{e}left(frac{a}{b}right) ) ( ^{mathrm{B}} cdot log _{e}left(frac{b}{a}right) ) ( mathbf{c} cdot log _{e}(a+b) ) ( mathbf{D} cdot log _{e}(a-b) ) | 12 |

133 | Find the angle of intersection of the following curve: ( x^{2}=27 y ) and ( y^{2}=8 x ) | 12 |

134 | Find the absolute maximum and the absolute minimum value of the following function in the given intervals. ( f(x)=3 x^{4}-8 x^{3}+12 x^{2}-48 x+25 ) in ( [mathbf{0}, mathbf{3}] ) | 12 |

135 | Show that ( frac{log x}{x} ) has a maximum value at ( boldsymbol{x}=boldsymbol{e} ) | 12 |

136 | Two measurements of a cylinder are varying in such a way that the volume is kept constant. If the rates of change of the radius ( (r) ) and height ( (h) ) are equal in magnitude but opposite in sign, then A ( . r=2 h ) B . ( h=2 r ) c. ( h=r ) D. ( h=4 r ) | 12 |

137 | If ( f(x)=frac{x^{3}}{3}-frac{5 x^{2}}{2}+6 x+7 ) increas in the interval ( T ) and decreases in the interval ( S ), then which one of the following is correct? A ( . T=(-infty, 2) cup(3 infty) ) and ( S=(2,3) ) в. ( T=Phi ) and ( S=(-infty, infty) ) c. ( T=(-infty, infty) ) and ( S=Phi ) D. ( T=(2,3) ) and ( S=(-infty, 2) cup(3, infty) ) | 12 |

138 | The function ( f(x)=frac{1}{x} ; x>0 ) on its domain is A. increasing B. decreasing c. constant D. information insufficient | 12 |

139 | A body travels a distance ( s ) in ( t ) seconds. It starts from rest and ends at rest. In the first part of the journey, it move with constant acceleration ( f ) and in the seconds part with constant retardation ( r . ) The value of ( t ) given by ( ^{mathrm{A}} cdot_{2 s}left(frac{1}{f}+frac{1}{r}right) ) в. ( frac{2 s}{frac{1}{f}+frac{1}{r}} ) c. ( sqrt{2 s(f+r)} ) D. ( sqrt{2 sleft(frac{1}{f+r}right)} ) | 12 |

140 | Approximate value of ( tan ^{-1}(0.999) ) is A . 0.7847 B. 0.748 c. 0.787 D. 0.847 | 12 |

141 | The total ( operatorname{cost} C_{(x)} ) and the total revenue ( R_{(x)} ) associated with production and sale of ( x ) units of an item are given by ( C(x)=0.1 x^{2}+30 x+1000 ) and ( R(x)=0.2 x^{2}+36 x-100 . ) Find the marginal cost and the marginal revenue when ( x=20 ) | 12 |

142 | The absolute maximum and minimum value of ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x}+cos boldsymbol{x}, boldsymbol{x} in[mathbf{0}, boldsymbol{pi}] ) are respectively A ( cdot sqrt{2},-1 ) B. ( sqrt{2}, 1 ) c. ( sqrt{2},-sqrt{2} ) D. ( sqrt{3}, sqrt{2} ) | 12 |

143 | A boat goes ( 60 k m ) upstream and downs stream in ( 6 h r ) and ( 2 h r ) respectively. Determine the product of speed of the stream and that of the boat in still water | 12 |

144 | Consider the following in respect of the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}2+boldsymbol{x}, & boldsymbol{x} geq 0 \ 2-boldsymbol{x}, & boldsymbol{x}<0end{array}right. ) 1. ( lim _{x rightarrow 1} f(x) ) does not exist. 2. ( f(x) ) is differentiable ( a t x=0 ) 3. ( f(x) ) is continuous at ( x=0 ) Which of the above statements is/are correct? A. 1 only B. 3 only c. 2 and 3 only D. 1 and 3 only | 12 |

145 | Show that ( y=log (1+x)-frac{2 x}{2+x}, x> ) ( -1, ) is an increasing function of ( x ) throughout its domain. | 12 |

146 | Table | 12 |

147 | If errors of ( 1 % ) each are made in the base radius and height of a cylinder, then the percentage error in its volume is A . ( 1 % ) B. 2% c. ( 3 % ) D. none of these | 12 |

148 | A man 2 metres tall walks away from a lamp post 5 metres height at the rate of ( 4.8 mathrm{km} / mathrm{hr} . ) The rate of increase of the length of his shadow is A. ( 1.6 mathrm{km} / mathrm{hr} ) B. ( 6.3 mathrm{km} / mathrm{hr} ) c. ( 5 mathrm{km} / mathrm{hr} ) D. 3.2 km/hr | 12 |

149 | Let ( S ) be a square with sides of length ( x ) If we approximate the change in size of the area of ( S ) by ( left.h cdot frac{d A}{d x}right|_{x=x_{0}}, ) when the sides are changed from ( x_{0} ) to ( x_{o}+h ) then the absolute value of the error in our approximation, is ( mathbf{A} cdot h^{2} ) B. ( 2 h x_{0} ) c. ( x_{0}^{2} ) D. ( h ) | 12 |

150 | The least value of ( 5^{sin x-1}+5^{-sin x-1} ) is A . 10 в. ( frac{5}{2} ) ( c cdot frac{2}{5} ) D. | 12 |

151 | A water tank has the shape of a right circular cone with its vertex down. Its altitude is ( 10 mathrm{cm} ) and the radius of the base is ( 15 mathrm{cm} ). Water leaks out of the bottom at a constant rate of 1 ( c u ) cm / sec. Water is poured into the ( operatorname{tank} ) at a constant rate of ( C ) cu.cm / sec. Compute ( C ) so that the water level will be rising at the rate of ( 4 mathrm{cm} / ) sec at the instant when the water is ( 2 mathrm{cm} ) deep. | 12 |

152 | The condition that ( mathbf{f}(mathbf{x})=mathbf{x}^{mathbf{3}}+mathbf{a x}^{mathbf{2}}+ ) ( mathbf{b x}+mathbf{c} ) is an increasing function for al real values of ( mathbf{x} ) is A ( cdot a^{2}<12 b ) B . a ( ^{2}<3 b ) ( c cdot a^{2}<4 b ) D. ( a^{2}<16 b ) | 12 |

153 | Assertion (A): The points on the curve ( y=x^{3}-3 x ) at which the tangent is parallel to ( x ) -axis are (1,-2) and (-1,2) Reason (R): The tangent at ( left(x_{1}, y_{1}right) ) on the curve ( y=f(x) ) is vertical then ( frac{d y}{d x} ) ( operatorname{at}left(x_{1}, y_{1}right) ) is not defined A. Both A and R are true and R is the correct explanation for A B. Both A and R are true but R is not the correct explanation for ( A ) C. A is true but R is false D. A is false but R is true | 12 |

154 | Two trains of equal lengths take 10 seconds and 15 seconds respectively to cross a telegraph post. If the length of each train is 120 metres, in how much time (in seconds) will they cross each other if travelling in opposite direction? A . 10 B. 12 c. 15 D. 20 | 12 |

155 | The maximum value of ( 4 sin ^{2} x- ) ( 12 sin x+7 ) is. A . 25 B. 4 c. Does not exist D. None of these | 12 |

156 | The equation of the normal to the curve ( x=a cos ^{3} theta, y=a sin ^{3} theta ) at the point ( boldsymbol{theta}=frac{boldsymbol{pi}}{boldsymbol{4}} ) is ( mathbf{A} cdot x=0 ) В. ( y=0 ) c. ( x=y ) D. ( x+y=a ) | 12 |

157 | The percentage error in the surface area of a cube with edge ( x mathrm{cm}, ) when the edge is increased by ( 11 % ) is A . 11 B . 22 c. 10 D. 44 | 12 |

158 | 4. Given A = {x: 55 x 55 and 3 f(x) = cos x-x(1+x); find f(A). (1980) | 12 |

159 | Consider the function ( left{begin{array}{c}x sin frac{pi}{x} text { for } x>0 \ text { 0 for } x=0end{array} ) then the number right. of points in (0,1) where the derivative ( f^{prime}(x) ) vanishes, is A . 0 B. 1 ( c cdot 2 ) D. infinite | 12 |

160 | If the velocity of a body moving in a straight line is proportional to the square root of the distance traversed, then it moves with A. variable force B. constant force c. zero force D. zero acceleration | 12 |

161 | The maximum value of function ( boldsymbol{f}(boldsymbol{x})= ) ( 3 x^{3}-18 x^{2}+27 x-40 ) on the ( operatorname{set} S= ) ( left{boldsymbol{x} in boldsymbol{R}: boldsymbol{x}^{2}+mathbf{3 0} leq mathbf{1 1} boldsymbol{x}right} ) is: A . 122 B . -222 c. -122 D. 222 | 12 |

162 | Match the maxima of the functions ( f(x) ) on L.H.S | 12 |

163 | If ( f ) and ( g ) are two increasing function such that ( f o g ) is defined, then fog will be A. increasing function B. decreasing function c. neither increasing nor decreasing D. None of these | 12 |

164 | The pressure ( P ) and volume ( V ) of a gas are connected by the relation ( boldsymbol{P} boldsymbol{V}^{1 / 4}= ) constant. The percentage increase in the pressure corresponding to a deminition of ( frac{1}{2} % ) in the volume is A ( cdot frac{1}{2} % ) B. ( frac{1}{4} % ) c. ( frac{1}{8} % ) D. none of these | 12 |

165 | Attempt the following: The price P for demand D is given as ( P=183+120 D-3 D^{2} . ) Find D for which the price is increasing. | 12 |

166 | 37. The eccentricity of an ellipse whose centre is at the origin If one of its directices is x=-4, then the equation of the normal to it at [JEE M 2017|| (a) (c) x+2y=4 4x – 2y=1 (b) 2y-x=2 (d) 4x +2y=7 | 12 |

167 | 41. If q denotes the acute angle between the curves, y=10- x2 and y=2 + x2 at a point of their intersection, then tan 0 is equal to: JEE M 2019-9 Jan (M) (b) 15 tla na | 12 |

168 | Find the rate of change of the area of a circle with respect to its radius ( r, ) when ( boldsymbol{r}=mathbf{5} mathrm{cm} ) | 12 |

169 | Find the intervals in which the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{x}+boldsymbol{6} ) is strictly decreasing. | 12 |

170 | Example 2.5 Water pours out at the rate of Q from a tap, into a cylindrical vessel of radius r. Find the rate at which the height of water level rises when the height is h. | 12 |

171 | If ( a ) and ( b ) are the non-zero distinct roots of ( x^{2}+a x+b=0, ) then the least value of ( x^{2}+a x+b ) is A ( cdot frac{3}{2} ) B. 9 c. ( -frac{9}{4} ) D. | 12 |

172 | The maximum value of ( boldsymbol{f}(boldsymbol{x})= ) ( frac{log x}{x}(x neq 0, x neq 1) ) is ( A ) B. ( c cdot e^{2} ) D. ( frac{1}{e^{2}} ) | 12 |

173 | The function ( f(x)=int_{-1}^{x} tleft(e^{t}-1right)(t- ) 1) ( (t-2)^{3}(t-3)^{5} d t ) has a local ( operatorname{maximum} operatorname{at} x= ) ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 3 | 12 |

174 | Two candles of the same height are lighted at the same time. The first is consumed in 4 hours and the second in 3 hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second? A ( cdot frac{3}{4} ) hr. B. ( frac{11}{2} ) hr. c. ( 2 h r ) D. ( frac{12}{5} ) h. E ( cdot frac{21}{2} ) hr. | 12 |

175 | Solve: ( frac{d y}{d x}+2 y tan x=sin x, ) given that ( y=0, ) when ( x=frac{pi}{3} . ) Show that maximum value of ( y ) is ( frac{1}{8} ) | 12 |

176 | Using differentials the approximate value of ( sqrt{401} ) is A . 20.100 B. 20.025 c. 20.030 D. 20.125 | 12 |

177 | A point on the parabola ( y^{2}=18 x ) at which the ordinate increases at twice the rate of the abscissa is A ( cdot(2,4) ) В ( cdot(2,-4) ) ( ^{mathrm{c}} cdotleft(-frac{9}{8}, frac{9}{2}right) ) D. ( left(frac{9}{8}, frac{9}{2}right) ) | 12 |

178 | Two towns ( A ) and ( B ) are ( 60 mathrm{km} ) apart. ( A ) school is to be built to serve 150 students in town ( A ) and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school be built is A. town B B. ( 45 mathrm{km} ) from town c. town A D. ( 45 mathrm{km} ) from town ( mathrm{B} ) | 12 |

179 | The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20,000 in 1999 and 25000 in the year ( 2004, ) what will be the population of the village in ( 2009 ? ) | 12 |

180 | A particle is moving in a straight line such that its distance at any time t is given by ( mathbf{S}=frac{boldsymbol{t}^{4}}{mathbf{4}}-mathbf{2} boldsymbol{t}^{mathbf{3}}+mathbf{4} boldsymbol{t}^{2}+mathbf{7} ) then its acceleration is minimum at ( t= ) ( A ) B. 2 ( c cdot 1 / 2 ) D. 3/2 | 12 |

181 | 7. If from a wire of length 36 metre a rectangle of greatest area is made, then its two adjacent sides in metre are (a) 6, 12 (b) 9,9 (C) 10,8 (d) 13,5 | 12 |

182 | Find the approximate value of ( boldsymbol{f}(mathbf{2 . 0 1}) ) where ( boldsymbol{f}(boldsymbol{x})=mathbf{4} boldsymbol{x}^{2}+mathbf{5} boldsymbol{x}+mathbf{2} ) | 12 |

183 | Assertion (A): Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function such that ( boldsymbol{f}(boldsymbol{X})=boldsymbol{X}^{3}+boldsymbol{X}^{2}+ ) ( 3 X+sin X, ) then ( f ) is one to one Reason ( (R): f(x) ) is neither increasing nor decreasing function. A. Both (A) and (R) are true and (R) is the correct explanation of (A) B. Both (A) and (R) are true and (R) is not the correct explanation of (A). c. (A) is true but (R) is false. D. (A) is false but (R) is true | 12 |

184 | Set up an equation of a tangent to the graph of the following function. ( boldsymbol{y}=boldsymbol{3}^{x}+boldsymbol{3}^{-2 x} ) at the points with abscissa ( boldsymbol{x}=mathbf{1} ) | 12 |

185 | The greatest value of the function ( mathbf{f}(mathbf{x})=sin left{boldsymbol{x}[boldsymbol{x}]+mathbf{e}^{[x]}+frac{boldsymbol{pi}}{2}-1right} ) for all ( boldsymbol{x} in[mathbf{O}, infty) ) is ( A cdot-1 ) B. ( c ) D. | 12 |

186 | 15. The minimum value of (x2+299) is (a) 75 (b) 50 (c) 25 (d) 55 | 12 |

187 | The slope of the tangent to the curve ( boldsymbol{x}=boldsymbol{t}^{2}+boldsymbol{3} boldsymbol{t}-mathbf{8}, boldsymbol{y}=boldsymbol{2} boldsymbol{t}^{2}-boldsymbol{2} boldsymbol{t}-mathbf{5} ) at the point (2,-1) is A ( cdot frac{22}{7} ) B. ( frac{6}{7} ) ( c cdot frac{7}{6} ) D. ( -frac{6}{7} ) | 12 |

188 | A man 1.5 m tall walks away from a lamp post ( 4.5 ~ m ) high at a rate fo ( 4 k m / h r . ) How fast is the farther end of shadow moving on the pavement? A. ( 4 mathrm{km} / mathrm{hr} ) B. ( 2 mathrm{km} / mathrm{hr} ) ( mathrm{c} .6 mathrm{km} / mathrm{hr} ) D. ( 5 mathrm{km} / mathrm{hr} ) | 12 |

189 | If there is an error of ( a % ) in measuring the edge of a cube, then the percentage error in its surface area is ( mathbf{A} cdot 2 a ) B. ( frac{a}{2} ) ( c .3 a ) D. None of the above | 12 |

190 | Write the differential equation representing the family of curves ( y= ) ( m x, ) where ( m ) is an arbitrary constant. | 12 |

191 | The point on the curve ( x y^{2}=1 ) which is nearest to the origin is A. (1,1) ) 1 в. (1,-1) c. ( left(4, frac{1}{2}right) ) D. ( left(frac{1}{2^{frac{1}{3}}}, 2^{frac{1}{6}}right) ) | 12 |

192 | Find the equation of the tangent to the curve ( boldsymbol{y}=sqrt{mathbf{3} boldsymbol{x}-mathbf{2}} ) which is parallel to the line ( 4 x-2 y+5=0 ) | 12 |

193 | The radius of a circular disc is given as ( 24 c m ) with a maximum error in measurement of ( 0.02 mathrm{cm} . ) Estimate the maximum error in the calculated area of the disc and compute the relative error by using differentials. | 12 |

194 | Answer the following question in one word or one sentence or as per exact requirement of the question. Write the maximum value of ( f(x)= ) ( boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x}<mathbf{0} ) | 12 |

195 | If the real-valued function ( f(x)=x^{3}+ ) ( 3left(a^{2}-1right) x+1 ) be invertible then the set of possible real values of a is A ( cdot(-infty,-1) cup(1+infty) ) в. (-1,1) c. [-1,1] D ( cdot(-infty,-1] cup[1,+infty) ) | 12 |

196 | Find the maximum and minimum values, if any of the following function given by: ( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{boldsymbol{3}}+mathbf{1} ) | 12 |

197 | The velocity of a body varies with time as ( V=3 t^{2}+2 . ) Find the instantaneous acceleration ( t=3 ) sec A ( cdot 31 m / s^{2} ) B . ( 18 mathrm{m} / mathrm{s}^{2} ) c. ( 3 m / s^{2} ) D . ( -2 / 3 mathrm{m} / mathrm{s}^{2} ) | 12 |

198 | 9. 1+x Find the coordinates of the point on the curve y=- 2 where the tangent to the curve has the greatest slope. (1984 – 4 Marks) | 12 |

199 | If ( boldsymbol{y}=boldsymbol{4}^{l o g_{2} sec x}-boldsymbol{9}^{l o g_{3} t a n x}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= ) ( mathbf{A} cdot mathbf{0} ) B. c. 2 D. 3 | 12 |

200 | The function ( x^{x} ) decreases in the interval- A ( cdot(0, e) ) в. (0,1) ( ^{mathbf{c}} cdotleft(0, frac{1}{e}right) ) D. None of the above. | 12 |

201 | All possible values of ( x ) for which the function ( f(x)=x ln x-x+1 ) is positive is A. ( (1, infty) ) (n) B. ( left(frac{1}{e}, inftyright) ) c. ( [e, infty) ) D. ( (0, infty)-{1} ) | 12 |

202 | 8. A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm /min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is (a) 5cm/min – cm/min 547 (b) cm/min 367 cm/min cm/min 187 | 12 |

203 | The equation to the normal to the hyperbola ( frac{x^{2}}{16}-frac{y^{2}}{9}=1 ) at (-4,0) is A ( .2 x-3 y=1 ) B. ( x=0 ) ( mathbf{c} cdot x=1 ) D. ( y=0 ) | 12 |

204 | A cylindrical water tank of diameter 1.4 ( m ) and height 2.1 ( m ) is being fed by a pipe of diameter ( 3.5 mathrm{cm} ) through which water flows at the rate of ( 2 m / s ) Calculate the time, it takes to fill the ( operatorname{tank} ) | 12 |

205 | The perimeter of a sector is ( P . ) The area of the sector is maximum when its radius is ( A cdot 1 / sqrt{P} ) B. ( P / 2 ) c. ( P / 4 ) D. ( sqrt{P} ) | 12 |

206 | 3. The area of a rectangle will be maximum for the given perimeter, when rectangle is a (a) Parallelogram (b) Trapezium (c) Square (d) None of these | 12 |

207 | Given ( f^{2}(x)+g^{2}(x)+h^{2}(x) leq 9 ) and ( boldsymbol{U}(boldsymbol{x})=boldsymbol{3} boldsymbol{f}(boldsymbol{x})+mathbf{4} boldsymbol{g}(boldsymbol{x})+mathbf{1 0 h}(boldsymbol{x}), ) where ( f(x) cdot g(x) ) and ( h(x) ) are continuous ( forall x in ) R. If maximum value of ( U(x) ) is ( sqrt{N} ) Then find ( N ) | 12 |

208 | ( frac{1}{2.3}+frac{1}{4.5}+frac{1}{6.7}+ldots infty= ) A ( cdot 1+log _{e} 2 ) B . ( 1-log _{mathrm{e}} 2 ) c. ( 2+log _{mathrm{e}} 2 ) D. ( 2-log _{mathrm{e}} 2 ) | 12 |

209 | Find the set of values of ( a ) for which ( f(x)=x+cos x+a x+b ) is increasing on ( boldsymbol{R} ) | 12 |

210 | 8. On the interval [0, 1] the function x25 (1 – x)75 takes its maximum value at the point (1995) (a) 0 (6) – | 12 |

211 | Assume that a shperical randrop evaporates at a rate proportional to its a surface area. if it’s radius is originally 3 ( mathrm{mm}, ) and ( 1 mathrm{minute} ) later has been reduced to ( 2 mathrm{mm} ). Find an expression for the radius of the raindrop at any time. | 12 |

212 | 16. Ifx=-1 and x=2 are extreme points of f(x) = a log|x+3x² + x then JEEM 2014 (a) a=2,8 =- (e) =-6,8 = 1 (b) a = 2, B= (d) a = -6,8 = | 12 |

213 | Find the interval of increase and decrease of the following functions. ( f(x)=x^{2} e^{-x} ) | 12 |

214 | The slope of the tangent to the curve ( boldsymbol{x}=mathbf{3} boldsymbol{t}^{2}+mathbf{1}, boldsymbol{y}=boldsymbol{t}^{3}-mathbf{1} ) at ( boldsymbol{x}=mathbf{1} ) is A ( cdot frac{1}{2} ) B. ( c cdot-2 ) ( D cdot infty ) | 12 |

215 | Show that ( f(x)=tan ^{-1}(sin x+cos x) ) is a decreasing function on the interval ( left(frac{pi}{4}, frac{pi}{2}right) ) | 12 |

216 | Show that the function given by ( f(x)= ) ( 3 x+17 ) is strictly increasing on ( R ) | 12 |

217 | The point at which the tangent to the curve ( y=x^{3}+5 ) is perpendicular to the line ( x+3 y=2 ) are A. (6,1),(-1,4) В. (6,1)(4,-1) c. (1,6),(1,4) D. (1,6),(-1,4) | 12 |

218 | Write the angle between the curves ( y= ) ( e^{-x} ) and ( y=e^{x} ) at their point of intersection. | 12 |

219 | xed*, x 50 27. Let f(x)= x + ax2 – x?, x>0 (1996 – 3 Marks) Where a is a positive constant. Find the interval in which f'(x) is increasing. | 12 |

220 | Identify a local maxima for: ( boldsymbol{y}=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}+boldsymbol{2} ) ( mathbf{A} cdot x=2 ) B. ( x=1 ) c. ( x=-2 ) D. ( x=-1 ) | 12 |

221 | The minimum of ( f(x)=frac{1+x+x^{2}}{1-x+x^{2}} ) occurs at ( x= ) ( A cdot-1 ) B. ( c cdot 2 ) ( D ldots-2 ) | 12 |

222 | If ( boldsymbol{alpha}=cos 10^{circ}-sin 10^{circ}, beta=cos 45^{circ}- ) ( sin 45^{circ}, gamma=cos 70^{circ}-sin 70^{circ} ) then the descending order of ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) is ( mathbf{A} cdot alpha, beta, gamma ) в. ( gamma, beta, alpha ) c. ( alpha, gamma, beta ) ( mathbf{D} cdot beta, alpha, gamma ) | 12 |

223 | ABCD is a rectangle in which ( A B=10 ) ( mathrm{cms}, mathrm{BC}=8 mathrm{cms} . ) A point ( mathrm{P} ) is taken on AB such that ( P A=x ). Then the minimum value of ( P C^{2}+P D^{2} ) is obtained when ( x ) ( = ) A . 10 B. 5 ( c cdot 8 ) D. 4 | 12 |

224 | Water is flowing out at the rate of ( 6 m^{3} / ) min from a reservoir shaped like a hemispherical bowl of radius ( R=13 ) ( mathrm{m} ) The volume of water in the hemispherical bowl is given by ( v= ) ( frac{pi}{3} cdot y^{2}(3 R-y) ) when the water is ( y ) meter deep Find at what rate is the water level changing when the water is ( 8 mathrm{m} ) deep. ( ^{mathbf{A}} cdot-frac{1}{12 pi} m / m i n ) B. ( -frac{1}{18 pi} m / ) min c. ( -frac{1}{24 pi} m / ) min D. ( -frac{1}{30 pi} m / ) min | 12 |

225 | Function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{mathbf{1}+|boldsymbol{x}|} ) is A. increasing function B. decreasing function C. neither increasing nor decreasing D. not differentiable at ( x=0 ) | 12 |

226 | If the function ( boldsymbol{f}(boldsymbol{x})= ) ( left(a^{2}-3 a+2right) cos frac{x}{2}+(a-1) x ) possesses critical points, then ( a ) belongs to the interval ( mathbf{A} cdot(-infty, 0) cup(4, infty) ) B . ( (-infty, 0] cup[4, infty) ) ( mathbf{c} cdot(-infty, 0] cup{1} cup[4, infty) ) D. None of these | 12 |

227 | Show that ( f(x)=frac{1}{x} ) is decreasing function on ( (mathbf{0}, infty) ) | 12 |

228 | If error in measuring the edge of a cube is ( k % ) then the percentage error in estimating its volume is ( A cdot k ) B. ( 3 k ) ( c cdot frac{k}{3} ) D. none of these | 12 |

229 | The interval in which ( y=x^{2} e^{-x} ) is increasing is: ( A cdot(-infty, infty) ) в. (-2,0) c. ( (2, infty) ) D. (0,2) | 12 |

230 | The volume of a ball increases at ( 4 pi c . c / )sec. The rate of increases of radius when the volume is ( 288 pi c . c . s ) is A ( cdot frac{1}{6} mathrm{cm} / mathrm{sec} ) B ( cdot frac{1}{36} mathrm{cm} / mathrm{sec} ) c. ( frac{1}{9} mathrm{cm} / mathrm{sec} ) D. ( frac{1}{49} mathrm{cm} / mathrm{sec} ) | 12 |

231 | If ( f(x)=a-(x-3)^{89}, ) then greatest value of ( boldsymbol{f}(boldsymbol{x}) ) is ( mathbf{A} cdot mathbf{3} ) B. ( a ) c. no maximum value D. none of these | 12 |

232 | Answer the following question in one word or one sentence or as per exact requirement of the question. Write the point where ( f(x)=x log _{e} x ) | 12 |

233 | Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( sqrt{49.5} ) | 12 |

234 | Find the equation of the quadratic function ( boldsymbol{f} ) whose graph increases over the interval ( (-infty,-2) ) and decreases over the interval ( (-2,+infty), f(0)=23 ) and ( boldsymbol{f}(mathbf{1})=mathbf{8} ) A ( cdot f(x)=3(x+2)^{2}+35 ) B. ( f(x)=-3(x+2)^{2}-35 ) C. ( f(x)=-3(x-2)^{2}+35 ) D. ( f(x)=-3(x+2)^{2}+35 ) | 12 |

235 | The minimum value of ( f(x)=mid 2 x+ ) ( mathbf{5} mid+mathbf{6} ) is: A . 2 B. 3 c. 5 D. 6 E . 8 | 12 |

236 | The minimum & maximum value of ( f(x)=sin (cos x)+cos (sin x) forall-frac{pi}{2} leq ) ( x leq frac{pi}{2} ) are respective A. ( cos 1 ) and ( 1+sin 1 ) 1 B. ( sin 1 ) and ( 1+cos 1 ) c. ( cos 1 ) and ( cos left(frac{1}{sqrt{2}}right)+sin left(frac{1}{sqrt{2}}right) ) D. None of these | 12 |

237 | 5. If the edge of a cube increases at the rate of 60 cm per second, at what rate the volume is increasing when the edge is 90 cm (a) 486000 cu cm per sec (b) 1458000 cu cm per sec (c) 43740000 cu cm per sec (d) None of these | 12 |

238 | Show that ( f(x)=tan ^{-1} x-x ) is a decreasing function on ( boldsymbol{R} ) | 12 |

239 | Find an angle ( theta, 0<theta<frac{pi}{2}, ) which increases twice as fast as its sine. A ( .60^{circ} ) B. ( 120^{circ} ) ( c .90^{circ} ) D. ( 45^{circ} ) | 12 |

240 | The length ( x ) of rectangle is decreasing at the rate of ( 5 mathrm{cm} / mathrm{minute} ) and the width ( y ) is increasing at the rate of 4 ( mathrm{cm} / mathrm{minute} . ) When ( boldsymbol{x}=mathbf{8} mathrm{cm} ) and ( boldsymbol{y}=mathbf{6} ) ( mathrm{cm}, ) find the rates of change of the area of the rectangle. | 12 |

241 | The curve ( y=a x^{3}+b x^{2}+c x+5 ) touches the ( x ) -axis at ( P(-2,0) ) and cuts the ( y ) -axis at a point ( Q, ) where its gradient is ( 3 . ) Find ( a, b, c ) A ( cdot a=-frac{1}{5}, b=1, c=3 ) B. ( a=-frac{1}{4}, b=-1, c=4 ) c. ( _{a=-frac{1}{4}, b=0, c=3} ) D. ( a=-frac{1}{3}, b=1, c=-3 ) | 12 |

242 | The least vlue of the function ( f(x)= ) ( boldsymbol{a x}+frac{boldsymbol{b}}{boldsymbol{x}}(boldsymbol{a}>mathbf{0}, boldsymbol{b}>mathbf{0}, boldsymbol{x}>mathbf{0}) ) is | 12 |

243 | What is the nature of the graph: ( y=frac{4}{x} ) A. Rectangular hyperbola in first and third quadrant B. Rectangular hyperbola in first and second quadrant C. Hyperbola but not rectangular hyperbola D. Rectangular hyperbola in second and fourth quadrant | 12 |

244 | The distance moved by a particle in time ( t ) seconds is given by ( s=t^{3}- ) ( 6 t^{2}-15 t+12 . ) The velocity of the particle when acceleration becomes zero is A . 15 в. -27 ( c cdot 6 / 5 ) D. none of these | 12 |

245 | If the error committed in measuring the radius of the circle is ( 0.05 % ), then the corresponding error in calculating the area is: ( mathbf{A} cdot 0.05 % ) B. ( 0.025 % ) c. ( 0.25 % ) D. ( 0.1 % ) | 12 |

246 | ( f(x)=tan ^{-1} x-x ) is ( _{-1} ) ( boldsymbol{R} ) A. increasing to ( R ) B. decreasing to ( R ) C . increasing on ( R^{+} ) D. increasing on ( (-infty, 0) ) | 12 |

247 | Assertion The largest term in the sequence ( a_{n}= ) ( frac{n^{2}}{n^{3}+200}, n in N ) is the ( 7^{t h} ) term Reason The function ( f(x)=frac{x^{2}}{x^{3}+200} ) attains local maxima at ( x=7 ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |

248 | The Point ( (s) ) on the cure ( y^{3}+3 x^{2}= ) ( 12 y ) where the tangent is vertical (parallel to y-axis), is/are. ( mathbf{A} cdotleft[pm frac{4}{sqrt{3}},-2right] ) B ( cdotleft(pm frac{sqrt{11}}{3}, 1right) ) D ( cdotleft(pm frac{4}{sqrt{3}}, 2right) ) | 12 |

249 | Using the differentials, the approximate value of ( (627)^{1 / 4} ) is A . 5.002 B. 5.003 c. 5.005 D. 5.004 | 12 |

250 | Find the greatest and the least values of the following functions. ( f(x)=sin x+cos 2 x ) on the interval ( [mathbf{0}, boldsymbol{pi}] ) | 12 |

251 | Find ( a ) for which ( f(x)=a(x+sin x)+ ) ( a ) is increasing on ( boldsymbol{R} ) | 12 |

252 | Find the angle of intersection of the following curve: ( x^{2}+4 y^{2}=8 ) and ( x^{2}-2 y^{2}=2 ) | 12 |

253 | The maximum value of ( f(x)=2 x^{3}- ) ( 21 x^{2}+36 x+20 ) in ( 0 leq x leq 2 ) is A . 37 B. 44 ( c .32 ) D. 30 | 12 |

254 | Find the maximum or minimum value of the quadratic expression ( 2 x-7- ) ( 5 x^{2} ) | 12 |

255 | Approximate change in the volume V of a cube of side x metres caused by increasing the side by ( 3 % ) is? | 12 |

256 | If the tangent at any point on the curve ( x^{4}+y^{4}=c^{4} ) cuts off intercepts ( a ) and ( b ) on the coordinate axes, the value of ( boldsymbol{a}^{-mathbf{4} / mathbf{3}}+boldsymbol{b}^{-mathbf{4} / mathbf{3}} ) is A ( cdot c^{-4 / 3} ) B ( cdot c^{-1 / 2} ) c. ( c^{1 / 2} ) D. ( c^{4 / 3} ) | 12 |

257 | For the curve ( y=3 sin theta cos theta, x= ) ( e^{theta} sin theta, 0 leq theta leq pi ; ) the tangent is parallel to ( x ) -axis when ( theta ) is ( mathbf{A} cdot mathbf{0} ) в. ( frac{pi}{2} ) c. D. | 12 |

258 | Find the value of a for which the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{3}-boldsymbol{3}(boldsymbol{a}+boldsymbol{2}) boldsymbol{x}^{2}+ ) ( 9(a+2) x-1 ) is decreasing for all ( x in ) ( boldsymbol{R} ) | 12 |

259 | 26. If(x) is differentiable and strictly increasing function, then the value of lima f(x²)-f(x) (2004) (2) 1 (b) 0 (0) 1 (d) 2. F(x)-f(0) is | 12 |

260 | A cylindrical gas container is closed at the top and open at the bottom; if the iron plate of the top is ( 5 / 4 ) times as thick as the plate forming the cylindrical sides, find the radio of the radius to the height of the cylinder using minimum material for the same capacity. A . 5 ( bar{A} ) в. ( frac{4}{5} ) ( c cdot frac{5}{2} ) D. ( frac{2}{5} ) | 12 |

261 | Show that the function ( f(x)=2 x+ ) ( cot ^{-1} x-log {x+sqrt{left(1+x^{2}right)}} ) increasing ( forall mathrm{R} ) | 12 |

262 | An inverted cone has a depth of ( 40 mathrm{cm} ) and a base of radius ( 5 mathrm{cm} ). Water is poured into it at a rate of 1.5 cubic centimetres per minute. Find the rate at which the level of water in the cone is rising when the depth is 4 cm. | 12 |

263 | The radius of a circular plate is increasing at the ratio of 0.20 cm/sec.At what rate is the area increasing when the radius of the plate is ( 25 mathrm{cm} ) | 12 |

264 | A particle moves along the y-axis so that its position at time ( 0 leq t leq 20 ) is ( operatorname{given} operatorname{by} y(t)=5 t-frac{t^{2}}{3} cdot ) At what time does the particle change direction? A. 5 seconds B. 7.5 seconds c. 10 seconds D. 15 seconds E. 18 seconds | 12 |

265 | ( P(x, y) ) is a point on a straight line which makes intercepts a and b on the ( x, y ) axes respectively, then the minimum value of ( x^{2}+y^{2}= ) A ( cdot frac{a^{2}+b^{2}}{a^{2} b^{2}} ) B. ( frac{a^{2}-b^{2}}{a^{2}+b^{2}} ) c. ( frac{a^{2}-b^{2}}{2left(a^{2}+b^{2}right)} ) D. ( frac{mathrm{a}^{2} b^{2}}{left(mathrm{a}^{2}+b^{2}right)} ) | 12 |

266 | Find the max.value of the total surface of a right circular cylinder which can be inscribed in a sphere of radius a. A ( cdot pi a^{2}(1+1 / sqrt{5}) ) B ( cdot pi / 2 a^{2}(1+sqrt{5}) ) c. ( pi a^{2}(1+sqrt{5}) ) D cdot ( pi a^{2}(2+sqrt{5}) ) | 12 |

267 | Find the approximate change in the surface area of a cube of side ( x ) metres caused by decreasing the side by 1 percent | 12 |

268 | If ( f(x)=x^{3}-6 x^{2}+9 x+3 ) then ( f(2) ) ( A cdot 3 ) B. 4 ( c cdot 5 ) D. None of these | 12 |

269 | Match the points on the curve ( 2 y^{2}= ) ( x+1 ) with the slope of normals at those points and choose the correct answer. A ( . i-b, i i-d, i i i-c, i v-a ) B . ( i-b, i i-a, i i i-d, i v-c ) c. ( i-b, i i-c, i i i-d, i v-a ) D. ( i-b, i i-d, i i i-a, i v-c ) | 12 |

270 | Find the tangents and normal to the curve ( boldsymbol{y}(boldsymbol{x}-mathbf{2})(boldsymbol{x}-mathbf{3})-boldsymbol{x}+mathbf{7}=mathbf{0}, mathbf{a t} ) point (7,0) are A. ( x-20 y-7=0,20 x+y-140=0 ) в. ( x+20 y-7=0,20 x-y-140=0 ) D. ( 7 x+20 y-1=0,20 x-7 y-100=0 ) | 12 |

271 | It is given that at ( x=1 ), the function ( x^{4}-62 x^{2}+a x+9 ) attains its maximum value, on the interval [0,2]. Find the value of a. | 12 |

272 | The length ( x ) of a rectangle is decreasing at the rate of ( 5 mathrm{cm} / mathrm{minute} ) and the width y is increasing at the rate of ( 4 mathrm{cm} / mathrm{minute} . ) When ( x=8 mathrm{cm} ) and ( y=6 mathrm{cm} ) find the rates of change of perimeter, and (b) the area of the rectangle. | 12 |

273 | Find the value of ( 64.75-75.97+36.82 ) A . 25.60 B. 23.98 c. 29.76 D. 21.96 | 12 |

274 | Let ( f(x) ) be a derivable function, ( f^{prime}(x)>f(x) ) and ( f(0)=0 . ) Then? A. ( f(x)>0 ) for all ( x>0 ) B. ( f(x)0 ) c. No sign of ( f(x) ) can be ascertained D. ( f(x) ) is a constant function | 12 |

275 | For ( boldsymbol{x} in boldsymbol{R}-{boldsymbol{n} boldsymbol{pi}}, ) the graph is: ( boldsymbol{y}= ) ( x+sin x ) A. Monotonically Increasing B. Monotonically Decreasing c. Either ( A ) or ( B ) D. None of these | 12 |

276 | The maximum value of the function ( f(x)=sin left(x+frac{pi}{6}right)+cos left(x+frac{pi}{6}right) ) in the interval ( left(0, frac{pi}{2}right) ) occurs at A ( cdot frac{pi}{12} ) в. ( c cdot frac{pi}{4} ) D. ( frac{pi}{3} ) | 12 |

277 | f ( boldsymbol{y}=boldsymbol{a} log |boldsymbol{x}|+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{x} ) has its extremum values at ( x=-1 ) and ( x=2 ) then the value of ( -2 a b ) is | 12 |

278 | Answer the following question in one word or one sentence or as per exact requirement of the question. Write the minimum value of ( f(x)= ) ( boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x}>mathbf{0} ) | 12 |

279 | The shortest distance between line y-x=1 and curvex=y? [2011] is | 12 |

280 | The circumference of a circle is measured as ( 56 mathrm{cm} ) with an error 0.02 ( mathrm{cm} . ) The percentage error in its area is A ( cdot frac{1}{7} ) в. ( frac{1}{28} ) c. ( frac{1}{14} ) D. ( frac{1}{56} ) | 12 |

281 | The equation of the tangent to the curve ( y=x+frac{4}{x^{2}}, ) that is parallel to the ( x- ) axis, is ( mathbf{A} cdot mathbf{y}=1 ) В. ( mathrm{y}=2 ) ( mathbf{c} cdot mathbf{y}=3 ) D. ( mathrm{y}=0 ) | 12 |

282 | At an extreme point of a function ( boldsymbol{f}(boldsymbol{x}) ) the tangent to the curve is A. Parallel to the ( x ) -axis B. Perpendicular to the ( x ) -axis c. Inclined at an angle ( 45^{circ} ) to the ( x ) -axis D. Inclined at an angle ( 60^{circ} ) to the ( x ) -axis | 12 |

283 | Find the equation of normal to the curve ( x^{2}=4 y ) passing through the point (1,2) A ( . x+y=3 ) B. ( x-y=3 ) c. ( 2 x-y=4 ) D. ( 2 x-3 y=1 ) | 12 |

284 | The image of the interval [-1,3] under the mapping ( f(x)=4 x^{3}-12 x ) is A. [-2,0] B. [-8, 72] c. [-8,0] D. [-8,-2] | 12 |

285 | Illustration 2.34 Find the minimum value of 2 cos 0 + sino + V2 tan ® in (0) | 12 |

286 | The first and second order derivatives of a function ( f(x) ) exist at all points in ( (a, b) ) with ( mathbf{f}^{prime}(mathbf{c})=mathbf{0}, ) where ( mathbf{a}<mathbf{c}<mathbf{b} . ) Further more, if ( mathrm{f}^{prime}(mathrm{x})mathbf{0} ) for all points on the immediate right of ( c ), then at ( mathbf{x}=mathbf{c}, mathbf{f}(mathbf{x}) ) has a A. local maximum B. local minimum c. point of inflexion D. none of these | 12 |

287 | 25. A curve y=f(x) passes through the point P(1,1). The normal to the curve at Pis a(y – 1) + (x – 1) = 0. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the y-axis, the curve and the normal to the curve at P. (1996 – 5 Marks) | 12 |

288 | If the function ( f(x)=frac{K sin x+2 cos x}{sin x+cos x} ) is strictly increasing for all values of ( x ) then ( mathbf{A} cdot K1 ) c. ( K2 ) | 12 |

289 | If the function ( f(x)=frac{a x+b}{(x-1)(x-4)} ) has a local maxima at ( (2,-1), ) then A. ( b=1, a=0 ) В. ( a=1, b=0 ) c. ( b=-1, a=0 ) D. ( a=-1, b=0 ) | 12 |

290 | Minimum and maximum ( z=5 x+2 y ) subject to the following constraints: ( x-2 y leq 2 ) ( 3 x+2 y leq 12 ) ( -3 x+2 y leq 3 ) ( boldsymbol{x} geq mathbf{0}, boldsymbol{y} geq mathbf{0} ) | 12 |

291 | The tangent to the curve ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}- ) ( sin theta) ; y=a(1+cos theta) ) at the points ( boldsymbol{theta}=(mathbf{2 k + 1}) boldsymbol{pi}, boldsymbol{k} in boldsymbol{Z} ) are parallel to ( mathbf{A} cdot y=x ) в. ( y=-x ) ( mathbf{c} cdot y=0 ) D. ( x=0 ) | 12 |

292 | The point on the curve ( y=x^{2} ) which is nearest to (3,0) is A . (1,-1) B. (-1,1) c. (-1,-1) D. (1,1) | 12 |

293 | Find the condition tht the curves ( 2 x= ) ( y^{2} ) and ( 2 x y=k ) intersect orthogonally. | 12 |

294 | The curve ( y=a x^{3}+b x^{2}+c x+d ) has a point on inflexion at ( x=1 ) then ( mathbf{A} cdot a+b=0 ) B . ( a+3 b=0 ) c. ( 3 a+b=0 ) D. ( 3 a+b=1 ) | 12 |

295 | The changes in a function y and the independent variable ( x ) are related as ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{2} . ) Find ( y ) as a function of ( boldsymbol{x} ) | 12 |

296 | Consider the following statements: ( f(x)=ln x ) is an increasing function on ( (0, infty) ) 2. ( f(x)=e^{x}-x(ln x) ) is an increasing function on ( (1, infty) ) Which of the above statements is lare correct? A. 1 only B. 2 only c. Both 1 and 2 D. Neither 1 nor 2 | 12 |

297 | ff ( boldsymbol{y}=boldsymbol{a} boldsymbol{l} boldsymbol{n}|boldsymbol{x}|+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{x} ) has its extremum values at ( x=-1 ) and ( x=2 ) then A ( . a=2, b=-1 ) в. ( a=2, b=-frac{1}{2} ) c. ( a=-2, b=frac{1}{2} ) D. none of these | 12 |

298 | The maximum value of the function ( f(x)=x^{3}+2 x^{2}-4 x+6 ) exists at A. ( x=-2 ) B. ( x=1 ) c. ( x=2 ) D. ( x=-1 ) | 12 |

299 | The interval(s) of decrease of of the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2} log 2 mathbf{7}- ) ( 6 x log 27+ ) ( left(3 x^{2}-18 x+24right) log left(x^{2}-6 x+8right) ) is This question has multiple correct options A ( cdot(3-sqrt{1+1 / 3 e}, 2) ) B. ( (4,3+sqrt{1+1 / 3 e}) ) c. ( (3,4+sqrt{1+1 / 3 e}) ) D. none of these | 12 |

300 | Find the rate of change of the area of a circle with respect to its radius ( r ) when (i) ( r=3 mathrm{cm} ) (ii) ( r=4 mathrm{cm} ) | 12 |

301 | The minimum value of the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} log boldsymbol{x} ) is A. ( -frac{1}{e} ) B. ( -e ) ( c cdot frac{1}{e} ) D. | 12 |

302 | A ladder ( 13 mathrm{m} ) long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate ( 1.5 mathrm{m} / mathrm{sec} . ) How fast is the angle ( theta ) between the ladder and the ground is changing when the foot of the ladder is ( 12 mathrm{m} ) away from the wall. | 12 |

303 | The points on the curve ( 12 y=x^{3} ) whose ordinate and abscissa change at the same rate, are A ( cdot(-2,-2 / 3),(2,2 / 3) ) в. ( (-2,2 / 3),(2 / 3,2) ) c. ( (-2,-2 / 3) ) only D. ( (2 / 3,2) ) only | 12 |

304 | 37. If a and b are positive quantities such that a > b, the minimum value of a seco- b tan is a. 2ab b. Ja² – 6² c. a-b d. a² +6² | 12 |

305 | 14. If the normal to the curve y=f(x) at the point (3,4) makes an 3 T angle with the positive x-axis, then f'(3) = (20008) (6) 3 (c) 1 / 2 (d) 1 A 4 (a) –1 | 12 |

306 | Let ( y=frac{6 x^{3}-45 x^{2}+108 x+2}{2 x^{3}-15 x^{2}+36 x+1} ) ( boldsymbol{x} in(mathbf{0}, mathbf{1 0}) ) Maximum value of ( y ) A . 3 B . 2 c. ( frac{86}{29} ) D. ( frac{82}{29} ) | 12 |

307 | Find the greatest and the least values of the following functions. ( y=x^{3}+3 x^{2}+3 x+2 ) on the interval [-2, 2]. | 12 |

308 | The value of ‘a’ for which the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}-boldsymbol{x}^{3}+cos boldsymbol{a} & boldsymbol{0}<boldsymbol{x}<1 \ boldsymbol{x}^{2} & boldsymbol{x} geq mathbf{1}end{array}right. ) has a local minimum at ( x=1 ), is This question has multiple correct options A . -1 B. 1 ( c cdot 0 ) D. ( -frac{1}{2} ) | 12 |

309 | The point on the curve ( y=sqrt{x-1} ) where the tangent is perpendicular to the line ( 2 x+y-5=0 ) is A. (2,-1) в. (10,3) c. (2,1) () D. (5,-2) | 12 |

310 | Find the minimum value of ( boldsymbol{f}(boldsymbol{x})= ) ( frac{e^{x}}{[x+1]}, x geq 0 ) | 12 |

311 | 29. Let : R R given by x<0; x + 5×4 +10×3 +10x² +3x +1, 1 x²-x+1, 2x – 4x +7x- f(x) 0x<1; 15x<3; x23 (x-2) loge(x-2)-x+- Then which of the following options is/are correct? (JEE Adv. 2019) (a) f'has a local maximum atx=1 (b) fis increasing on (-0,0) (c) f' is NOT differentiable at x=1 (d) f is onto | 12 |

312 | The minimum value of ( 2 x^{3}-9 x^{2}+ ) ( 12 x+4 ) is A . 4 B. 5 ( c .6 ) D. E. 8 | 12 |

313 | 12. Find the point on the curve 4×2 + a²y2 = 4a?, 4<a? <8 that is farthest from the point (0, -2). (1987 – 4 Marks) | 12 |

314 | A police cruiser, approaching a rightangled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is ( 0.6 mathrm{km} ) north of the intersection and the car is ( 0.8 mathrm{km} ) to the east, the police determine with radar that the distance of the car is increasing at ( 20 mathrm{km} / mathrm{h} ). Suppose that the cruiser is moving at ( 60 mathrm{km} / mathrm{h} ) at the instant of measurement. The speed of the car is (in ( mathrm{km} / mathrm{h}) ) A . 70 B. 80 c. 75 D. 60 | 12 |

315 | The maximum value of function ( x^{3}- ) ( 12 x^{2}+36 x+17 ) in the interval [1,10] is A . 17 в. 17 c. 77 D. None of these | 12 |

316 | Let ( x ) be a number which exceeds its square by the greatest possible quantity, then ( x= ) A. ( 1 / 2 ) B . ( 1 / 4 ) c. ( 3 / 4 ) D. ( 1 / 3 ) | 12 |

317 | The maximum of ( mathbf{f}(mathbf{x})=frac{log mathbf{x}}{mathbf{x}^{2}}(mathbf{x}>mathbf{0}) ) occurs at ( mathbf{x}= ) ( A ) B. ( sqrt{e} ) ( c cdot frac{1}{e} ) D. ( frac{1}{sqrt{mathrm{e}}} ) | 12 |

318 | The value of ( a ) in order that ( f(x)= ) ( sqrt{3} sin x-cos x-2 a x+b ) decreases for all real values of ( x, ) is given by ( mathbf{A} cdot a<1 ) в. ( a geq 1 ) ( c cdot a geq sqrt{2} ) D. ( a<sqrt{2} ) | 12 |

319 | If a particle moves according to the law, ( s=6 t^{2}-frac{t^{3}}{2}, ) then the time at which it is momentarily at rest A. ( t=0 ) only B. ( t=8 ) only c. ( t=0,8 ) D. none of these | 12 |

320 | ( mathbf{A} mathbf{x}=mathbf{0}, mathbf{f}(mathbf{x})=cos mathbf{x}-1+frac{mathbf{x}^{2}}{mathbf{2}}-frac{mathbf{x}^{mathbf{3}}}{mathbf{6}} ) A. Has a minimum B. Has a maximum c. Does not have an extremum D. Is not defined | 12 |

321 | Find the least value of secºx + cosecx + Illustration 2.36 secx coseco x. | 12 |

322 | 33. The maximum value of the expression Usin’x + 2a’ – 12a – 1 – cos? x], where a and x are real numbers, is a. 13 b. √2 c. 1 bed. V5 | 12 |

323 | If ( f(x)=max left{sin x, cos ^{-1} xright}, ) then A. ( f ) is differentiable everywhere B. ( f ) is continuous everywhere but not differentiable C tis discountinuous at ( x=frac{n pi}{2}, n in N ) D. ( f ) is non-differentiable at ( x=frac{n pi}{2}, n in N ) | 12 |

324 | The local minimum value of the function ( f^{prime} ) given by ( f(x)=3+|x|, x epsilon R ) is. ( A cdot 3 ) B. ( c cdot-1 ) D. | 12 |

325 | A particle is moving along the curve ( boldsymbol{x}=boldsymbol{a} boldsymbol{t}^{2}+boldsymbol{b} boldsymbol{t}+boldsymbol{c} ). If ( boldsymbol{a} boldsymbol{c}=boldsymbol{b}^{2}, ) then the particle would be moving with uniform A. rotation B. velocity c. acceleration D. retardation | 12 |

326 | The rate at which ice-ball melts is proportional to the amount of ice in it. If half of it melts in 20 minutes, the amount of ice left after 40 minutes compared to it original amount is ( ^{A} cdotleft(frac{1}{8}right) t h ) B. ( left(frac{1}{16}right) ) th ( ^{c} cdotleft(frac{1}{4}right) t h ) D. ( left(frac{1}{32}right) t h ) | 12 |

327 | Use differential to approximate ( sqrt{mathbf{1 0 1}} ) A .10 .00 B. 10.05 ( c cdot 11.00 ) D. Non of these | 12 |

328 | The maximum and minimum values of the function ( y=x^{3}-3 x^{2}+6 ) are A .2,0 в. 6,0 ( c cdot 6,2 ) D. 4,2 | 12 |

329 | ( boldsymbol{x}=boldsymbol{3} cos boldsymbol{Theta}, boldsymbol{y}=boldsymbol{3} sin boldsymbol{Theta}, boldsymbol{t h e n} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? ) ( A cdot-cot theta ) B. ( -sin theta ) ( c cdot sin Theta ) D. ( tan theta ) | 12 |

330 | If ( f(x)=x e^{x(1-x)}, ) then ( f(x) ) is A ( cdot ) increasing on ( left[-frac{1}{2}, 1right] ) B. decreasing on R C. increasing on R D. decreasing on ( left[-frac{1}{2}, 1right] ) | 12 |

331 | A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is ( 10 mathrm{cm} ) | 12 |

332 | = + 9. If a € (0, 1), and f(a) = (a? – a + 1) + osina 91 Va? – a +1 27 cosec? a =, then the least value of f(a)/2 is ſa² –at1 | 12 |

333 | The radius of a circle is increasing at the rate of ( 0.7 mathrm{cm} / mathrm{s} ). What is the rate of increases of its circumference? | 12 |

334 | The maximum value of the function ( 2 x^{3}-15 x^{2}+36 x+4 ) is attained at ( mathbf{A} cdot mathbf{0} ) B. 3 ( c cdot 4 ) D. 2 E. 5 | 12 |

335 | f ( log _{e} 4=1.3868, ) then ( log _{e} 4.01= ) A . 1.3968 B . 1.3898 c. 1.3893 D. none of these | 12 |

336 | Mark the correct alternative of the following. Let ( f(x)=2 x^{3}-3 x^{2}-12 x+5 ) on ( [-2,4] . ) The relative maximum occurs at ( boldsymbol{x}=? ) A . -2 B. – – ( c cdot 2 ) D. 4 | 12 |

337 | Prove that the function given by ( f(x)= ) ( x^{3}-3 x^{2}+3 x-100 ) is inreasing in ( R ) | 12 |

338 | Find the local maxima and local minima of the function ( f(x)=sin x- ) ( cos x, 0<x<2 pi . ) Also find the local maximum and local minimum values. | 12 |

339 | The maximum value of ( left(frac{1}{x}right)^{x} ) is ( e^{1 / e} ) A. True B. False | 12 |

340 | The percentage error in the ( 11^{t h} ) root of the number 28 is approximately times the percentage error in 28 A ( cdot frac{1}{28} ) в. ( frac{1}{11} ) c. 11 D. 28 | 12 |

341 | Consider the cubic ( f(x)=8 x^{3}+ ) ( 4 a x^{2}+2 b x+a, ) where ( a, b, in R ) For ( boldsymbol{a}=mathbf{1}, ) if ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) is strictly increasing ( forall x in R, ) then maximum range of values of ( b ) is ( ^{mathbf{A}} cdotleft(-infty, frac{1}{3}right) ) B. ( left(frac{1}{3}, inftyright) ) ( ^{c} cdotleft[frac{1}{3}, inftyright) ) ( D cdot(-infty, infty) ) | 12 |

342 | Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( (15)^{frac{1}{4}} ) | 12 |

343 | If the radius of a sphere is measured as ( 7 mathrm{m} ) with an error of ( 0.02 mathrm{m}, ) find the approximate error in calculating its volume | 12 |

344 | If ( f(x)=3 x^{2}+15 x+5, ) then the approximate value of ( boldsymbol{f}(mathbf{3} . mathbf{0 2}) ) is. A . 47.66 B. 57.66 c. 67.66 D. 77.66 | 12 |

345 | Find the intervals in which the function ( f(x)=frac{x^{4}}{4}-x^{3}-5 x^{2}+24 x+12 ) is (a) strictly increasing, (b) strictly decreasing. | 12 |

346 | A particle moves along the curve ( x^{2}= ) 2y. At what point, ordinate increases at the same rate as abscissa increases? | 12 |

347 | Which of the following options is the only CORRECT combination? A. ( (I I)(i i i)(S) ) В. ( (I)(i i)(R) ) c. ( (I I I)(i v)(P) ) D. ( (I V)(i)(S) ) | 12 |

348 | Find the values of ( x ) if ( f(x)=frac{x}{x^{2}+1} ) is i) an increasing function ii) a decreasing function | 12 |

349 | An aeroplane at an altitude of 960 meters flying horizontally at ( mathbf{7} 20 k m / h r . ) passes directly over an observer. The rate at which it is approaching the observers when it is 1600 meters directly away from him is A. ( 576 mathrm{km} / mathrm{hr} ) B. ( 676 mathrm{km} / mathrm{hr} ) c. ( 720 mathrm{km} / mathrm{hr} ) D. ( 570 mathrm{km} / mathrm{hr} ) | 12 |

350 | ( operatorname{Let} f(x)=frac{sin 4 pi[x]}{1+[x]^{2}}, ) where ( [x] ) is the greatest integer less than or equal to ( x ) then A. ( f(x) ) is not differentiable at some points B. ( f(x) ) exists but is different from zero c. ( L H D(a t x=0)=0 . R H D(a t x=1)=0 ) D. ( f(x)=0 ) but ( f ) is not a constant function | 12 |

351 | Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( (401)^{frac{1}{2}} ) | 12 |

352 | If ( T=2 pi sqrt{frac{l}{g}}, ) then relative errors in ( T ) and I are in the ratio A . ( 1 / 2 ) B. c. ( 1 / 2 pi ) D. none of these | 12 |

353 | If ( f ) is a real-valued differentiable function such that ( f(x) f^{prime}(x)<0 ) for all real ( x, ) then A. ( f(x) ) must be a increasing function B. ( f(x) ) must be a decreasing function c. ( |f(x)| ) must be a increasing function D. ( |f(x)| ) must be a decreasing function | 12 |

354 | A point is moving along the curve ( y^{3}= ) ( 27 x ). The interval in which the abscissa changes at slower rate than ordinate, is A ( cdot(-3,3) ) B ( cdot(-infty, infty) ) c. (-1,1) D. ( (-infty,-3) cup(3, infty) ) | 12 |

355 | The two curves ( y=x^{2}-1 ) and ( y= ) ( 8 x-x^{2}-9 ) at the point (2,3) have common A. tangent as ( 4 x-y-5=0 ) B. tangent as ( x+4 y-14=0 ) c. normal as ( 4 x+y=11 ) D. normal as ( x-4 y=10 ) | 12 |

356 | The value of ‘a’ for which the function ( f(x)=(a+2) x^{3}-3 a x^{2}+9 a x-1 ) decreases for all real values of ( x ) is B ( cdot(-infty,-3) ) ( c cdot(-infty,-2) ) D. ( (-infty,-3] cup[0, infty) ) | 12 |

357 | The angle at which the curve ( y=k e^{k x} ) intersects the ( y ) – axis is This question has multiple correct options A ( cdot tan ^{-1} k^{2} ) B ( cdot cot ^{-1}left(k^{2}right) ) ( mathrm{c} cdot_{sin }^{-1}left(frac{1}{sqrt{1+k^{4}}}right) ) D・sec- ( 1(sqrt{1+k^{4}}) ) | 12 |

358 | A man ( 1.5 ~ m ) tall walks away from a lamp post ( 4.5 ~ m ) high at a rate of ( 4 k m / h r( ) i) How fast is his shadow lengthening?(ii)How fast is the farther end of shadow moving on the pavement?? | 12 |

359 | Water flows at the rate of 10 m/minute through a cylindrical pipe ( 5 m m ) in diameter. How long would it take to fill a conical vessel whose diameter at the base is ( 40 mathrm{cm} ) and depth ( 24 mathrm{cm} ) ? | 12 |

360 | Let ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}|, ) then ( mathbf{A} cdot f^{prime}(0)=0 ) B. ( f(x) ) has a maximum at ( x=0 ) ( mathbf{c} cdot f(x) ) has a minimum at ( x=0 ) D. ( f(x) ) has no maximum and not minimum | 12 |

361 | The graph of the equation ( y=x^{2}- ) ( 4 x+5 ) has its lowest point at A ( .(2,1) ) в. (-2,1) c. (-2,-1) D. (2,-1) | 12 |

362 | 29. The intercepts on x-axis made by tangents to the curve, dt, x ER, which are parallel to the line y = 2x, are equal to : (a) #1 (6) #2 (c) 3 [JEE M 2013] (d) 24 | 12 |

363 | ( P(-2,3), Q(3,7) . ) The point ( A ) on the ( x ) axis for which ( mathbf{P A}+mathbf{A Q} ) is least is ( ^{mathrm{A}} cdotleft(frac{-1}{2}, 0right) ) в. ( left(frac{1}{2}, 0right) ) c. ( left(0, frac{-1}{2}right) ) D. ( left(0, frac{1}{2}right) ) | 12 |

364 | ( frac{1}{3}+frac{1}{3}left(frac{1}{3}right)^{3}+frac{1}{5}left(frac{1}{3}right)^{5}+dots= ) A ( cdot frac{1}{2} log _{e} 2 ) B ( cdot 2 log _{e} 2 ) ( mathbf{c} cdot log _{e} 2 ) D. ( log _{e} 3 ) | 12 |

365 | 11. 1 where it is A function is matched below against an interval where I to be increasing. Which of the following pairs is incorrectly matched? [2005] Interval Function (a) – 0,0) x² – 3x² + 3x+3 (b) [2,00) 2×3 – 3×2 – 12x+6 3x² – 2x+1 | 12 |

366 | Answer the following question in one word or one sentence or as per exact requirement of the question. Find the least value of ( f(x)=a x+frac{b}{x} ) where ( a>0, b>0 ) and ( x>0 ) | 12 |

367 | If a point moves along the curve ( y^{2}=x ) At what point on the curve does the ( y ) coordinate change at the same rate as the ( x ) coordinate. | 12 |

368 | Prove that the function ( f(x)=log _{e} x ) is increasing on ( (mathbf{0}, infty) ) | 12 |

369 | Statement-I: The equation ( frac{x^{3}}{4}- ) ( sin pi x+3=2 frac{1}{2} ) has at least one solution in [2,2] Because Statement-II : If ( boldsymbol{f}:[boldsymbol{a}, boldsymbol{b}] rightarrow boldsymbol{R} ) be a function & let ( c ) be a number such that ( f(a)<c<f(b), ) then there is at least one number ( n in(a, b) ) such that ( boldsymbol{f}(boldsymbol{n})=boldsymbol{c} ) A. Statement-l is true, Statement-ll is true ; Statement-I is correct explanation for Statement- is NOT a correct explanation for statement- c. Statement-I is true, Statement-II is false D. Statement-I is false, Statement-II is true | 12 |

370 | The function ( mathbf{f}(mathbf{x})=mathbf{2} log (mathbf{x}-mathbf{3})- ) ( x^{2}+6 x+3 ) increases in the interval A . (3,4) в. ( (-infty, 2) ) ( c cdot(3, infty) ) D. None ot these | 12 |

371 | Function ( f(x)=log _{10} cos x ) is function in ( left(0, frac{pi}{2}right) ) A. Decreasing B. Increasing c. constant D. Increasing and decreasing | 12 |

372 | The side of a square is increased by ( 20 % . ) Find the ( % ) change in its area. A. ( 44 % ) increase B. ( 40 % ) increase c. No change D. None of these | 12 |

373 | The curve ( frac{x^{n}}{a^{n}}+frac{y^{n}}{b^{n}}=2 ) touches the line ( frac{x}{a}+frac{y}{b}=2 ) at the point A ( .(b, a) ) в. ( (a, b) ) c. (1,1) D ( cdotleft(frac{1}{a}, frac{1}{b}right) ) | 12 |

374 | The greatest value of the function ( f(x)=x e^{-x} ) in ( [0, infty), ) is A . B. ( c cdot-c ) D. | 12 |

375 | The minimum and maximum values of ( mathbf{f}(mathbf{x})=sin (cos mathbf{x})+cos (sin mathbf{x}) forall-frac{boldsymbol{pi}}{mathbf{2}} leq ) ( x leq frac{pi}{2} ) are respectively A . ( cos 1 ) and ( 1+sin 1 ) B. ( sin 1 ) and ( 1+cos 1 ) c. ( cos 1 ) and ( cos left(frac{1}{sqrt{2}}right)+sin left(frac{1}{sqrt{2}}right) ) D. ( frac{1}{sqrt{2}} ) | 12 |

376 | Write the set of values of ( a ) for which ( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x}+boldsymbol{a}^{2} boldsymbol{x}+boldsymbol{b} ) is strictly increasing on ( boldsymbol{R} ) | 12 |

377 | The distance, from the origin, of the normal to the curve, ( boldsymbol{x}=mathbf{2} cos boldsymbol{t}+ ) ( mathbf{2} t sin t, y=2 sin t-2 t cos t ) at ( t=frac{pi}{4}, ) is ( A cdot 2 ) B. 4 ( c cdot sqrt{2} ) D. ( 2 sqrt{2} ) | 12 |

378 | The radius of the sphere is measured ( operatorname{as}(10 pm 0.02) c m . ) The error in the measurement of its volume is A ( .25 .1 c c ) B. 25.21cc c. ( 2.51 c c ) D. ( 251.2 c c ) | 12 |

379 | The rate of change of the volume of a cone with respect to the radius of its base is- ( mathbf{A} cdot pi^{2} h ) в. ( frac{4}{3} pi r h ) c. ( frac{4}{3} pi r^{2} h ) D. ( frac{2}{3} pi r h ) | 12 |

380 | The equation of tangent to the curve ( y=e^{-|x|} ) at the point where the curve cuts the line ( x=1 ) is A. ( x+y=e ) В. ( e(x+y)=1 ) c. ( y+e x=1 ) D. none of these | 12 |

381 | 11. Let f(x) = sinºx+2 sin? x, <x<. Find the intervals in which a should lie in order that f(x) has exactly one minimum and exactly one maximum. (1985 – 5 Marks) | 12 |

382 | Let ( x ) be a number which exceeds its square by the greatest quantity. Then ( x ) is equal to A ( cdot frac{1}{2} ) B. ( frac{1}{4} ) ( c cdot frac{3}{4} ) D. none of these | 12 |

383 | Assertion The maximum value of ( (sqrt{-3+4 x-x^{2}}+4)^{2}+(x-5)^{2} ) (where ( 1 leq x leq 3 ) ) is 36 Reason The maximum distance between the point (5,-4) and the point on the circle ( (x-2)^{2}+y^{2}=1 ) is 6 A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |

384 | If ( f(x)=x^{2} e^{2 x}(x>0,) ) then find the local maximum value of ( boldsymbol{f}(boldsymbol{x}) ) ( A cdot frac{2}{e^{2}} ) B. ( frac{-1}{e^{2}} ) ( c cdot e^{2} ) D. ( frac{1}{e^{2}} ) | 12 |

385 | If ( boldsymbol{y}=boldsymbol{x}^{n}, ) then the ratio of relative errors in ( y ) and ( x ) is A . 1: 1 B . 2: 1 ( c cdot 1: n ) D. ( n: 1 ) | 12 |

386 | ff ( y=m log x+n x^{2}+x ) has its extreme values at ( x=2 ) and ( x=1 ) then ( 2 m+10 n ) is equal to A . -1 B. -4 c. -2 D. E. -3 | 12 |

387 | [2011] d²x equals : | 12 |

388 | If there is an error of ( k % ) in measuring the edge of a cube, then the percent error in estimating its volume is ( A cdot k ) B. ( 3 k ) ( c cdot frac{k}{3} ) D. none of these | 12 |

389 | If there is an error of ( 0.01 mathrm{cm} ) in the diameter of a sphere then percentage error in surface area when the radius ( = ) ( 5 c m, ) is A . ( 0.005 % ) B. ( 0.05 % ) ( c cdot 0.1 % ) D. ( 0.2 % ) | 12 |

390 | The minimum value of ( 64 sec theta+ ) ( 27 cos e c theta ) when ( theta operatorname{lies} operatorname{in}left(0, frac{pi}{2}right) ) is A . 125 в. 625 c. 25 D. 1025 | 12 |

391 | On which of the following intervals is the function ( x^{100}+sin x-1 ) decreasing? A ( cdotleft(0, frac{pi}{2}right) ) в. (0,1) c. ( left(frac{pi}{2}, piright) ) D. None of the above. | 12 |

392 | Find the local maxima and local minima, if any, of the following functions. Find the sum of the local maximum and the local minimum values for: ( g(x)=x^{3}-3 x ) | 12 |

393 | The radius of a sphere is changing at the rate of ( 0.1 mathrm{cm} / ) sec. The rate of its surface area when the radius is ( 200 mathrm{cm} ) is A ( cdot 8 pi c m^{2} / ) sec В . ( 12 pi c m^{2} / ) sec c. ( 160 pi c m^{2} / ) sec D. ( 200 pi c m^{2} / ) sec | 12 |

394 | The circumference of a circle is measured as ( 28 c m ) with an error of ( 0.01 mathrm{cm} . ) The percentage error in the area is A ( cdot frac{1}{14} ) в. 0.01 ( c cdot frac{1}{7} ) D. none of these | 12 |

395 | A particle moves in a line with velocity given by ( frac{d s}{d t}=s+1 . ) The time taken by the particle to cover a distance of 9 meter is ( mathbf{A} cdot mathbf{1} ) B. ( log 10 ) ( c cdot 2 log , 10 ) D. 10 | 12 |

396 | The greatest value of ( f(x)=(x+1)^{frac{1}{3}}- ) ( (x-1)^{frac{1}{3}} ) on [0,1] is ( mathbf{A} cdot mathbf{1} ) B. 2 ( c cdot 3 ) D. | 12 |

397 | A point on the parabola ( y^{2}=18 x ) at which the ordinate increase at twice the rate of the abscissa is ( mathbf{A} cdot(9 / 8,9 / 2) ) в. (2,-4) c. ( (-9 / 8,9 / 2) ) D. (2,4) | 12 |

398 | The rate of increase of length of the shadow of a man 2 metres height, due to a lamp at 10 metres height, when he is moving away from it at the rate of ( 2 m / s e c, ) is A ( cdot frac{1}{2} mathrm{m} / mathrm{sec} ) B. ( frac{2}{5} ) m/sec c. ( frac{1}{3} mathrm{m} / mathrm{sec} ) D. ( 5 m / s e c ) | 12 |

399 | The number of stationary points of ( mathbf{f}(mathbf{x})=cos mathbf{x} ) in ( [mathbf{0}, mathbf{2} boldsymbol{pi}] ) are ( mathbf{A} cdot mathbf{1} ) B. 2 ( c .3 ) D. | 12 |

400 | The sum of intercepts of the tangent to the curve ( sqrt{x}+sqrt{y}=sqrt{a} ) upon the coordinates axes is ( mathbf{A} cdot 2 a ) в. c. ( 2 sqrt{2} a ) D. None of these | 12 |

401 | 20. The length of a longest interval in which the function 3 sin x – 4 sinºx is increasing, is (2002) | 12 |

402 | The maximum value of ( sin left(x+frac{pi}{5}right)+cos left(x+frac{pi}{5}right), ) where ( boldsymbol{x} inleft(0, frac{pi}{2}right) ) is attained at ( ^{A} cdot frac{pi}{20} ) в. ( frac{pi}{15} ) c. ( frac{pi}{10} ) D. | 12 |

403 | ( frac{1}{2}-frac{1}{2.2^{2}}+frac{1}{3.2^{3}}+ldots ldots= ) A ( cdot ) [ begin{array}{l}text { B } cdot frac{1}{2} log _{2}left(frac{3}{2}right) \ text { c. } log _{e}left(frac{2}{3}right) \ text { D } cdot frac{1}{2} log _{e}left(frac{2}{3}right)end{array} ] | 12 |

404 | Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a differentiable function for all values of ( x ) and has the peoperty that ( f(x) ) and ( f^{prime}(x) ) have opposite signs for all values of ( x ). Then A. ( f(x) ) is an increasing function B. ( f(x) ) is a decreasing function C ( cdot f^{2}(x) ) is a decreasing function D. ( |f(x)| ) is an increasing function | 12 |

405 | Assertion Equation of tangent to the curve ( boldsymbol{y}= ) ( x^{2}+1 ) at the point where slope of tangent is equal the function value of the curve is ( y=2 x ) Reason ( boldsymbol{f}^{prime}(boldsymbol{x})=boldsymbol{f}(boldsymbol{x}) ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |

406 | A circular disc of radius ( 3 mathrm{cm} ) is being heated. Due to expansion, its radius increases at the rate of ( 0.05 mathrm{cm} / mathrm{s} ). Find the rate at which its area is increasing when radius is ( 3.2 mathrm{cms} ) | 12 |

407 | Prove that the function ( f(x)=log _{e} x ) is increasing on ( (mathbf{0}, infty) ) | 12 |

408 | Water poured into an inverted conical vessel of which the radius of the base is ( 2 mathrm{m} ) and height ( 4 mathrm{m}, ) and the rate of 77 litres/minute. The rate at which the water level is rising at the instant when the depth is ( 70 mathrm{cm} ) is: (use ( pi=22 / 7 ) ) A. ( 10 mathrm{cm} / mathrm{min} ) в. ( 20 mathrm{cm} / mathrm{min} ) c. ( 40 mathrm{cm} / mathrm{min} ) D. None | 12 |

409 | ( cos theta+frac{1}{3} cos ^{3} theta+frac{1}{5} cos ^{5} theta+ldots= ) ( mathbf{A} cdot log (tan theta) ) B. ( log (cot theta) ) ( ^{mathbf{c}} cdot log left(tan frac{theta}{2}right) ) D. ( log left(cot frac{theta}{2}right) ) | 12 |

410 | 16. (20 For all x € (0,1) (a) exx (b) log (1 + x)x | 12 |

411 | If ( f(x)=x^{3 / 2}(3 x-10), x geq-0 ) ), then ( f(x) ) is decreasing in B. ( (2, infty) ) c. ( (-infty,-1] cup[1, infty) ) | 12 |

412 | The point for the curve ( y=x e^{x} ) is A. ( x=-1 ) is minimum B. ( x=0 ) is minimum c. ( x=-1 ) is maximum D. ( x=0 ) is maximum | 12 |

413 | Find the minimum value of a, such that function ( f(x)=x^{2}+a x+5, ) is increasing in interval [1,2] | 12 |

414 | Mark the correct alternative of the following. The sum of two non-zero numbers is 8 the minimum value of the sum of their reciprocals is? A ( cdot frac{1}{4} ) B. ( frac{1}{2} ) ( c cdot frac{1}{8} ) D. None of these | 12 |

415 | ( Delta A B C ) is not right-angled and is inscribed in a fixed circle. If ( a, A, b, B ) be slightly varied keeping ( c, C ) fixed, then ( frac{d a}{cos A}+frac{d b}{cos B}=? ) A ( .2 R ) в. ( pi ) ( c cdot 0 ) D. none of these | 12 |

416 | The function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}+boldsymbol{b} ) is strictly increasing for all real ( boldsymbol{x} ), if ( mathbf{A} cdot a>0 ) B. ( a<0 ) ( mathbf{c} cdot a=0 ) D. ( a leq 0 ) | 12 |

417 | The point of intersection of the tangents drawn to the curve ( x^{2} y=1-y ) at the point where it is intersected by the curve ( x y=1-y, ) is given by A ( .(0,1) ) в. (1,1) c. (0,-1) D. none of these | 12 |

418 | Find the maximum value of ( boldsymbol{y}= ) ( 2 sin ^{2} x-3 sin x+1 forall x in R ) | 12 |

419 | 36. Twenty metres of wire is available for fencing off a flower- bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is: (JEEM 2017] (a 30 (b) 12.5 (c) 10 (d) 25 | 12 |

420 | 6. The function defined by f(x) = (x + 2) exis (1994) (a) decreasing for all x (b) decreasing in (-0, -1) and increasing in (-1,..) (c) increasing for all x (d) decreasing in (-1,0) and increasing in (-00,-1) | 12 |

421 | The value of ( b ) for which the function ( f(x)=sin x-b x+c ) is a strictly decreasing function ( forall x in R ) A. ( b in(-1,1) ) в. ( b in(-infty, 1) ) C. ( b in(1, infty) ) D. ( b in[1, infty) ) | 12 |

422 | 17. Iff(x)= xetll-*), then f (x) is (20015) (a) increasing on [-1/2, 1] (b) decreasing on R (c) increasing on R (d) decreasing on [-1/2,1] | 12 |

423 | The rate of growth of population of a city at any time is proportional to the size of the population at that time. For a certain city, the consumer of proportionality is ( 0.04 . ) The population of the city after 25 years, if the initia population is 10,000 is ( (e=2.7182) ) ( mathbf{A} cdot 27182 ) B. 27164 c. 27000 D. 27272 | 12 |

424 | Mark the correct alternative of the following. ( f(x)=1+2 sin x+3 cos ^{2} x, 0 leq x leq ) ( frac{2 pi}{3} ) is? A. Minimum at ( x=pi / 2 ) B . Maximum at ( x=sin ^{-1}(1 / sqrt{3}) ) c. Minimum at ( x=pi / 6 ) D. Maximum at ( sin ^{-1}(1 / 6) ) | 12 |

425 | 17. Let f: R → (0,0) and g: R → R be twice differentiable functions such that f” and g” are continuous functions on R. Suppose f'(2) = g(2)= 0, f'(2) #0 and g'(2) + 0. If lim-81_) =1 then . f(x)g(x) (JEE Adv. 2016 x+2 f (x)g'(x) (a) f has a local minimum at x=2 (b) f has a local maximum at x=2 (c) f”(2)>f(2) (d) f(x)-f”(x)=0 for at least one x e R | 12 |

426 | The equation of the normal to the curve ( boldsymbol{y}=(1+boldsymbol{x})^{y}+sin ^{-1}left(sin ^{2} boldsymbol{x}right) boldsymbol{a t} quad boldsymbol{x}=mathbf{0} ) is A. ( x+y=1 ) B. x-y+1=0 c. ( 2 x+y=2 ) D. 2x-y+ | 12 |

427 | The approximate change in the volume of a cube of side ( x ) metres caused by increasing the side by ( 3 % ) is : B. ( 0.6 x^{3} m^{3} ) D. ( 0.9 x^{3} m^{3} ) | 12 |

428 | Find the slope of the tangent at (1,2) on the curve ( y=x^{2}-4 x+5 ) | 12 |

429 | 0 120, 0 The maximum value of sin x(1 + cos x) will be at the (b) x = * (c) x = 7 (d) x = 1 | 12 |

430 | If ( boldsymbol{P}=boldsymbol{x}^{3}-frac{mathbf{1}}{boldsymbol{x}^{3}} ) and ( boldsymbol{Q}=boldsymbol{x}-frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x} in ) ( (1, infty) ) then minimum value of ( frac{P}{Q^{2}} ) is: ( A cdot ) is ( 2 sqrt{3} ) B. is ( 4 sqrt{3} ) c. does not exist D. None of these | 12 |

431 | Which of the following functions is always increasing? ( mathbf{A} cdot x+sin 2 x ) B. ( x-sin 2 x ) c. ( 2 x+sin 3 x ) ( mathbf{D} cdot 2 x-sin x ) | 12 |

432 | The radius and height of a cylinder are equal. If the radius of the sphere is equal to the height of the cylinder, then the ratio of the rates of increase of the volume of the sphere and the volume of the cylinder is A .4: 3 в. 3: 4 ( mathbf{c} cdot 4: 3 pi ) ( mathbf{D} cdot 3: 4 pi ) | 12 |

433 | 14. If y = sec(tan-x), thena r- atx=1 is equal to : (JEE M200 | 12 |

434 | Time period ( T ) of a simple pendulum of length ( l ) is given by ( T=2 pi sqrt{frac{l}{g}} . ) If the length is increased by ( 2 % ), then an approximate change in the time period is A . ( 2 % ) B . ( 1 % ) c. ( frac{1}{2} % ) D. None of these | 12 |

435 | If the line ( a x+b y+c=0 ) is a norma to the curve ( boldsymbol{x} boldsymbol{y}=mathbf{1}, ) then This question has multiple correct options A ( . a>0, b>0 ) В. ( a>0, b<0 ) c. ( a0 ) D. ( a<0, b<0 ) | 12 |

436 | A man ( 160 mathrm{cm} ) tall, walks away from a source of light situated at the top of a pole ( 6 mathrm{m} ) high, at the rate of ( 1.1 mathrm{m} / mathrm{sec} ) How fast is the length of his shadow increasing when he is ( 1 mathrm{m} ) away from the pole? | 12 |

437 | 7. If f(x)=-, , for every real number x, then the minimum x +1° value off (1998 – 2 Marks) (a) does not exist because f is unbounded (b) is not attained even though f is bounded (c) is equal to 1 (d) is equal to -1 | 12 |

438 | The interval a for which the local minimum value of the function ( boldsymbol{f}(boldsymbol{x})= ) ( 2 x^{3}-21 x^{2}+60 x+a ) is positive. is в. ( (-infty,-25) ) c. ( (25, infty) ) D. ( (-25, infty) ) | 12 |

439 | If ( a_{1}, a_{2}, a_{3}, dots a_{n} in R ) then ( (x- ) ( left.a_{1}right)^{2}+left(x-a_{2}right)^{2}+ldotsleft(x-a_{n}right)^{2} ) assumes least value at ( boldsymbol{x}= ) A ( cdot a_{1}+a_{2}+a_{3}+ldots+a_{n} ) В . ( a_{n} ) ( mathbf{c} cdot nleft(a_{1}+a_{2}+ldots+a_{n}right) ) D. ( frac{left(a_{1}+a_{2}+ldots+a_{n}right)}{n} ) | 12 |

440 | The value of ( a_{1}+a_{2} ) is equal to A . 30 B. -30 c. 27 D. -27 | 12 |

441 | 16. Ifp and q are positive real numbers such that p2 + q2 = 1, then the maximum value of (p+q) is [2007] (a) (©) V (d) 2. | 12 |

442 | If ( f(x)=x^{2}+2 b x+2 c^{2} ) and ( g(x)= ) ( -x^{2}-2 c x+b^{2} ) are such that min ( f(x)>max g(x), ) then the relation between ( b ) and ( c ) is A. no relation в. ( 0<c<b / 2 ) C ( cdot|c||b| sqrt{2} ) | 12 |

443 | An object stars from rest at ( t=0 ) and accelerates at a rate given by ( a=6 t ) What i) its velocity and ii) its displacement at any time ( t ? ) A ( cdot t^{3}, 3 t^{2} ) B. ( 3 t^{2}, t^{3} ) ( mathbf{c} cdot t^{2}, t^{3} ) D. ( 2 t^{2}, 3 t^{3} ) | 12 |

444 | ( f(x)=x^{3}-6 x^{2}+12 x-16 ) is strictly decreasing for ( mathbf{A} cdot x in R ) в. ( x in R-{1} ) c. ( x in R^{+} ) D. ( x in phi ) | 12 |

445 | f ( a, b, c ) are real number, then find the intervals in which ( boldsymbol{f}(boldsymbol{x})= ) [ begin{array}{ccc} boldsymbol{x}+boldsymbol{a}^{2} & boldsymbol{a} boldsymbol{b} & boldsymbol{a c} \ boldsymbol{a b} & boldsymbol{x}+boldsymbol{b}^{2} & boldsymbol{b c} \ boldsymbol{a c} & boldsymbol{b c} & boldsymbol{x}+boldsymbol{c}^{2} end{array} mid ] increasing or decreasing | 12 |

446 | Let h(x)=f(x)-([(x))2 + (F(x))for every real number x. Then (1998 – 2 Marks (a) h is increasing whenever fis increasing (b) h is increasing whenever fis decreasing (c) h is decreasing whenever fis decreasing (d) nothing can be said in general. | 12 |

447 | The approximate value of ( sqrt[5]{33} ) correct to 4 decimal places is A . 2.0000 B. 2.1001 c. 2.0125 D. 2.0500 | 12 |

448 | Find the rate of change of the area of a circular disc with respect to its circumference, when the radius is 3 ( mathrm{cm} .left(text { incm }^{2}right) ) | 12 |

449 | Maximum value of ( sin theta+cos theta ) in ( left[0, frac{pi}{2}right] ) is A ( cdot sqrt{2} ) B. 2 c. 0 D. ( -sqrt{2} ) | 12 |

450 | Let tangent at a point ( mathrm{P} ) on the curve ( x^{2 m} y^{frac{n}{2}}=a^{frac{4 m+n}{2}} ) meets the x-axis and y-axis at A and B respectively if AP: ( mathrm{PB} ) is ( frac{n}{lambda m} ) where ( mathrm{P} ) lies between ( mathrm{A} ) and ( mathrm{B} ) then find the value of ( lambda ) | 12 |

451 | How fast is the farther end of the shadow moving on the pavement? A. ( 2 k m / h r ) в. ( 4 k m / h r ) ( mathrm{c} .6 mathrm{km} / mathrm{hr} ) D. ( 8 k m / h r ) | 12 |

452 | Suppose that ( f(0)=-3 ) and ( f^{prime}(x) leq 5 ) for all values of ( x ). Then the largest value which ( f(2) ) can attain is A. 7 B. -7 c. 13 D. | 12 |

453 | If ( a>b ) maximum value of ( a sin ^{2} x+ ) ( b cos ^{2} x ) A . a B. ( c cdot a+b ) D. ( sqrt{a^{2}+b^{2}} ) | 12 |

454 | The greatest value of the function ( f(x)=sin ^{2} x-20 cos x+1 ) is A . 20 B. 1 c. 21 D. | 12 |

455 | The maximum value of ( left(frac{1}{x}right)^{x} ) is A ( cdot(1 / e)^{e} ) B . ( e^{1 / c} ) ( c ) D. none of these | 12 |

456 | Function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}^{2}-mathbf{2}}{sqrt{mathbf{1}+boldsymbol{x}^{2}}} ) A. is always increasing B. is always decreasing C. has exactly one point of minima D. has exactly one point of maxima | 12 |

457 | The absolute minimum & maximum values of ( f(x)=frac{x^{2}-3 x+4}{x^{2}+3 x+4} ) are respectively A ( cdot frac{1}{7} ) and 7 B. ( 5 & 7 ) c. ( frac{1}{7} & 1 ) D. None of these | 12 |

458 | The radius of a circle is increasing uniformly at the rate of ( 3 mathrm{cm} / mathrm{s} ). Find the rate at which the area of the circle is increasing when the radius is ( 10 mathrm{cm} ) | 12 |

459 | 9. The function f(x)= t (e – 1) (t – 1) (t 2)} (t – 3) dt has -1 a local minimum atx= (1999 – 3 Marks) (a) o (6) 1 (c) 2 (d)3 1 .1 .21. 10. 191 – 1 Alan | 12 |

460 | The area of a triangle is computed using the formula ( S=frac{1}{2} b c sin A . ) If the relative errors made in measuring b, ( c ) and calculating ( mathrm{S} ) are respectively 0.02 0.01 and 0.13 the approximate error in ( A ) when ( A=pi / 6 ) is A . 0.05 radians B. 0.01 radians c. 0.05 degree D. 0.01 degree | 12 |

461 | The local maximum value of ( x(1-x)^{2}, 0 leq x leq 2 ) is ( A cdot 2 ) в. ( frac{4}{27} ) ( c .5 ) D. ( 2, frac{4}{27} ) | 12 |

462 | Let a function ( boldsymbol{f}:[mathbf{0}, mathbf{5}] rightarrow boldsymbol{R} ) be continuous, ( boldsymbol{f}(1)=boldsymbol{3} ) and ( boldsymbol{F} ) be defined as: ( boldsymbol{F}(boldsymbol{x})=int_{1}^{x} boldsymbol{t}^{2} boldsymbol{g}(boldsymbol{t}) boldsymbol{d} t, ) where ( boldsymbol{g}(boldsymbol{t})= ) ( int_{1}^{t} boldsymbol{f}(boldsymbol{u}) boldsymbol{d} boldsymbol{u} ) Then for the function ( F ) the point ( x=1 ) is A. a point of local minima B. a point of local maxima c. not a critical point D. a point of inflection | 12 |

463 | A particle moves along the curve ( y= ) ( x^{3 / 2} ) in the first quadrant in such a way that its distance from the origin increases at the rate of 11 units per second. The value of when ( x=3 ) is A . 4 B. ( frac{9}{2} ) c. ( frac{3 sqrt{3}}{2} ) D. none of these | 12 |

464 | ( operatorname{Let} g(x)=2 fleft(frac{x}{2}right)+f(x-2) ) and ( boldsymbol{f}^{prime prime}(boldsymbol{x})<mathbf{0} forall boldsymbol{x} in(mathbf{0}, mathbf{2}), ) then ( boldsymbol{g}(boldsymbol{x}) ) increases in A ( cdotleft(frac{1}{2}, 2right) ) в. ( left(frac{4}{3}, 2right) ) c. (0,2) (年. ( 0,2,2) ) D. ( left(0, frac{4}{3}right) ) | 12 |

465 | A stone is dropped into a quiet lake and waves move in a circle at a speed of ( 3.5 c m / )sec. At the instant when the radius of the circular wave is ( 7.5 mathrm{cm} ) The enclosed area increases as fastly as. A ( cdot 52.5 pi mathrm{cm}^{2} / mathrm{sec} ) в. ( 50.5 pi mathrm{cm}^{2} / mathrm{sec} ) ( mathbf{c} cdot 57.5 pi mathrm{cm}^{2} / mathrm{sec} ) D. ( 62.5 pi mathrm{cm}^{2} / mathrm{sec} ) | 12 |

466 | In the following increasing function is A ( cdot e^{x^{2}}^{2} ) B cdot ( e^{x^{3}} ) ( c cdot e^{0} ) D. all the above | 12 |

467 | The volume of metal in a hollow sphere is constant.If the inner radius is increasing at the rate of ( 1 mathrm{cm} / mathrm{sec} ), then the rate of increase of the outer radius when the radii are ( 4 c m ) and ( 8 c m ) respectively is. A. ( 0.75 mathrm{cm} / mathrm{sec} ) B. ( 0.25 mathrm{cm} / mathrm{sec} ) ( mathrm{c} cdot 1 mathrm{cm} / mathrm{sec} ) D. ( 0.50 mathrm{cm} / mathrm{sec} ) | 12 |

468 | The function ( f(x)=sqrt{3} cos x+sin x ) has an amplitude of A . 1.37 в. 1.73 ( c cdot 2 ) D. 2.73 E . 3.46 | 12 |

469 | Using differentials, find the approximate value of ( left(frac{17}{81}right)^{frac{1}{4}} ) | 12 |

470 | The greatest value of ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}+ ) 1) ( ^{1 / 3}-(x-1)^{1 / 3} ) on [0,1] is ( mathbf{A} cdot mathbf{1} ) B. ( c cdot 3 ) D. ( 1 / 3 ) | 12 |

471 | Let ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-boldsymbol{p})^{2}+(boldsymbol{x}-boldsymbol{q})^{2}+(boldsymbol{x}- ) ( boldsymbol{r})^{2} . ) Then ( boldsymbol{f}(boldsymbol{x}) ) has a minimum at ( boldsymbol{x}=boldsymbol{lambda} ) where ( lambda ) is equal to A. ( frac{p+q+r}{3} ) B. ( sqrt[3]{p q r} ) c. ( frac{3}{frac{1}{p}+frac{1}{q}+frac{1}{r}} ) D. none of these | 12 |

472 | Illustration 2.37 The particle’s position as a function of time is given as x = 5t2 – 9t + 3. Find out the maximum value of position co-ordinate? Also, plot the graph. | 12 |

473 | Normal to the curve ( x^{2}=4 y ) which passes through the point ( (mathbf{1}, mathbf{2}) ) A. ( x+y=3 ) в. ( x-y=3 ) c. ( 2 x+y=4 ) D. ( x+2 y=5 ) | 12 |

474 | Find all points of local maxima and local minima of the function f given by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}+boldsymbol{3} ) | 12 |

475 | A tangent to the hyperbola ( boldsymbol{y}=frac{boldsymbol{x}+mathbf{9}}{boldsymbol{x}+mathbf{5}} ) passing though the origin is A . ( x+25 y=0 ) в. ( 5 x+y=0 ) c. ( 5 x-y=0 ) D. ( x-25 y=0 ) | 12 |

476 | The maximum value of ( f(x)=100- ) ( |45-x| ) is A . 100 в. 145 c. 55 D. 45 | 12 |

477 | Tangent to parabola ( boldsymbol{y}^{2}=mathbf{4} boldsymbol{x}+mathbf{5} ) which is parallel to ( boldsymbol{y}=2 boldsymbol{x}+mathbf{7} ) A. ( y-2 x-3=0 ) В. ( y=x+3 ) c. ( y-2 x+1=0 ) D. ( y=x+1 ) | 12 |

478 | A stone is dropped into a quiet lake and waves move in circles at the speed of 5 ( mathrm{cm} / mathrm{s} ) At the instant when the radius of the circular wave is ( 8 mathrm{cm} ) how fast is the enclosed area increasing? ( mathbf{A} cdot 84 pi mathrm{cm}^{2} / mathrm{s} ) B. ( 80 pi mathrm{cm}^{2} / mathrm{s} ) ( mathbf{c} cdot 90 pi mathrm{cm}^{2} / mathrm{s} ) D. ( 96 pi mathrm{cm}^{2} / mathrm{s} ) | 12 |

479 | Write the equation of tangent at (1,1) on the curve ( 2 x^{2}+3 y^{2}=5 ) | 12 |

480 | Consider ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} ) Parameters ( a, b, c ) are chosen, respectively, by throwing a die three times. Then the probability that ( f(x) ) is an increasing function is A ( .5 / 12 ) в. ( 2 / 9 ) c. ( 4 / 9 ) D. ( 1 / 3 ) | 12 |

481 | If the points of local extremum of ( f(x)=x^{3}-3 a x^{2}+3left(a^{2}-1right) x+1 ) lies between ( -2 & 4, ) then ( ^{prime} a^{prime} ) belongs to A. (-2,2) B ( cdot(-infty,-1) cup(3, infty) ) c. (-1,3) ( D cdot(3, infty) ) | 12 |

482 | If a point is moving in a line so that its velocity at time ( t ) is proportional to the square of the distance covered, then its acceleration at time t varies as A. cube of the distance B. the distance c. square of the distance D. none of these | 12 |

483 | Find the equation of the tangent and the normal to the following curves at the indicated points. ( boldsymbol{y}=boldsymbol{x}^{4}-boldsymbol{6} boldsymbol{x}^{3}+1 mathbf{3} boldsymbol{x}^{2}-mathbf{1 0} boldsymbol{x}+mathbf{5} ) at ( boldsymbol{x}= ) ( mathbf{1} ) | 12 |

484 | Find the equation of the tangent line to the curve ( y=x^{2}-2 x+7 ) which is perpendicular to the line ( 5 y-15 x= ) ( mathbf{1 3} ) | 12 |

485 | A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius be ( 3 m m ) and 1 hour later it reduces to 2 mm, find an expression for the radius of the rain drop at any time ( t ) | 12 |

486 | Which of the following function has extreme point? ( mathbf{A} cdot 2^{x} ) B . ( log _{10} x ) c. ( x-[x] ) D. all of these | 12 |

487 | I et P(x)= a, + a,x2+aart + …… +a x2n be a polynomial in a real variable x with Oxao < a <a2 <….. < an.. The function P(x) has (a) neither a maximum nor a minimum (1986-2 Marks) (b) only one maximum (c) only one minimum (d) only one maximum and only one minimum (e) none of these. | 12 |

488 | If ( f(x)=frac{x}{1+x tan x}, x inleft(0, frac{pi}{2}right) ) then This question has multiple correct options A. ( f(x) ) has exactly one point of minima B. ( f(x) ) has exactly one point of maxima C ( f(x) ) is many one in ( left(0, frac{pi}{2}right) ) D. ( f(x) ) has maximum at ( x_{0} ) where ( x_{0}=cos x_{0} ) | 12 |

489 | A ladder ( 10 mathrm{m} ) long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of ( 3 c m / s ) The height of the upper end while it is descending at the rate of ( 4 mathrm{cm} / mathrm{s} ), is A. ( 4 sqrt{3} m ) В. ( 5 sqrt{3} m ) c. ( 5 sqrt{2} m ) D. ( 6 m ) | 12 |

490 | Find Stationary points of ( f(x)=sin x ) where ( 0<x<2 pi ) | 12 |

491 | Two pipes running together can fill a cistern in ( 3 frac{1}{13} ) minutes. If one pipe take 3 minutes more than other to fill it, find the time in which each pipe can fill the tank. | 12 |

492 | 28. A line is drawn through the point (1,2) to meet the coordinate axes at Pand Q such that it forms a triangle OPQ, where O is the origin. If the area of the triangle OPQ is least, then the slope of the line PQ is : [2012] (a) · (b) – 4 (C) -2 (d) | 12 |

493 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{2}-boldsymbol{b}|boldsymbol{x}|, ) where ( boldsymbol{a} ) and ( boldsymbol{b} ) are constants. Then at ( x=0, f(x) ) has This question has multiple correct options A. A maxima whenever ( a>0, b>0 ) B. A maxima whenever ( a>0, b0, b0, b<0 ) | 12 |

494 | 17. A window of perimeter P (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass transmits three times as much light per square meter as the coloured glass does. What is the ratio for the sides of the rectangle so that the window transmits the maximum light? (1991 – 4 Marks) | 12 |

495 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sqrt{mathbf{4} boldsymbol{a} boldsymbol{x}-boldsymbol{x}^{2}},(boldsymbol{a}>mathbf{0}) . . ) Then ( f(x) ) is decreasing in: This question has multiple correct options A. ( (5 a, infty) ) ) в. ( (-infty, 0) U(4 a, infty) ) c. Always increasing D. None of the above | 12 |

496 | The radius of a circular plate is increased at ( 0.01 mathrm{cm} / ) sec. If the area is increased at the rate of ( frac{pi}{10} . ) Then its radius is ( A cdot 5 mathrm{cm} ) B. ( 10 mathrm{cm} ) c. ( 15 mathrm{cm} ) D. ( 20 mathrm{cm} ) | 12 |

497 | 14. The perimeter of a sector is p. The area of the sector is maximum when its radius is (b) = | 12 |

498 | π π Let the function g:(-0,00) → be given by g(u) = 2 tan-(e”) . Then, g is (2008) (a) even and is strictly increasing in (0,0) n odd and is strictly decreasing in (-0,0) odd and is strictly increasing in (-00,00) (d) neither even nor odd, but is strictly increasing in (-00,00) | 12 |

499 | If ( mathbf{f}(mathbf{x}) ) is minimum at ( mathbf{x}=mathbf{a} ) then A. There exists ( delta>0 ) such that ( a-delta<x<a Rightarrow f(x)0 ) such that ( a<xf(a) ) C. There exists ( delta>0 ) such that ( a-delta<x0 ) such that ( a-delta<x<a+delta Rightarrow ) ( f(x) leq f(a) ) | 12 |

500 | If ( 2 x-7-5 x^{2} ) has maximum value at ( x=a, ) then ( a=dots dots ) A. ( -1 / 5 ) в. ( 1 / 5 ) ( c cdot 34 / 5 ) D. ( -34 / 5 ) | 12 |

501 | 23. Tangent is drawn to ellipse * + y2 = 1 at (3-13 cos 0, sin e) (where 0 € (0,7/2)). 27 Then the value of such that sum of intercepts on axes made by this tangent is minimum, is (2003) (a) T3 (6) Tu6 (C) T8 (d) T4 Teco 32 | 12 |

502 | Let ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{1})^{boldsymbol{m}}(boldsymbol{2}-boldsymbol{x})^{n} ; boldsymbol{m}, boldsymbol{n} boldsymbol{epsilon} boldsymbol{N} ) and ( boldsymbol{m}, boldsymbol{n}>2 ) | 12 |

503 | The equation of the common normal at the point of contact of the curves ( x^{2}=y ) and ( x^{2}+y^{2}-8 y=0 ) A ( . x=y ) B. ( x=0 ) ( mathbf{c} cdot y=0 ) D. ( x+y=0 ) | 12 |

504 | Using differentials, find the approximate value of ( (3.968)^{frac{3}{2}} ) | 12 |

505 | Mark the correct alternative of the following. The least and greatest values of ( f(x)= ) ( x^{3}-6 x^{2}+9 x ) in ( [0,6], ) are? A .3,4 в. 0,6 c. 0,3 D. 3,6 | 12 |

506 | The function ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{mathbf{1}+boldsymbol{x}^{2}} ) is decreasing in the interval ( mathbf{A} cdot(-infty,-1] ) в. ( (-infty, 0] ) c. ( [1, infty) ) (i) D. ( (0, infty) ) | 12 |

507 | Observe the following lists Let ( f(x) ) be any function List – I A) ( f^{prime}(mathrm{a})=0 ) and I) ( f(x) ) is increasing at ( x=a ) [ f^{prime prime}(mathrm{a})0 text { then } quad text { at } x=a ] 3) ( f(x) ) has C) ( f^{prime}(a) neq 0 ) then neither maximum nor minimum D) ( f^{prime}(a)>0 ) 4) ( f(x) ) has minimum value at ( x=a ) 5) ( f(x) ) is decreasing at ( x=a ) ( A cdot A-4, B-2, C-3, D-5 ) B. A -2,B -4, C -3, D -1 C. ( A-2, B-4, C-3, D-5 ) D. A -2,B -4, C -5, D-1 | 12 |

508 | If two curves ( boldsymbol{y}=boldsymbol{a}^{x} ) and ( boldsymbol{y}=boldsymbol{b}^{x} ) intersect at an angle ( alpha ) then find the value of tana A ( cdotleft|frac{ln a-ln b}{1+ln a ln b}right| ) в. ( left|frac{ln a-ln b}{1-ln a ln b}right| ) c. ( mid frac{ln a+ln b}{1+ln a ln b} ) D. ( mid frac{ln a+ln b}{1-ln a ln b} ) | 12 |

509 | The function ( boldsymbol{f}(boldsymbol{x})=log _{e}left[boldsymbol{x}^{3}+sqrt{boldsymbol{x}^{6}+mathbf{1}}right] ) is an This question has multiple correct options A. even function B. odd function c. increasing function D. decreasing function | 12 |

510 | If the area of circle increases at a uniform rate, then prove that the perimeter varies inversely as the radius. | 12 |

511 | The maximum value of the function ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-boldsymbol{4}| ) exists at ( mathbf{A} cdot x=0 ) B. ( x=2 ) c. ( x=4 ) D. ( x=-4 ) | 12 |

512 | Find the stationary point of ( boldsymbol{y}=boldsymbol{x}^{2}+ ) ( 5 x-6 ) | 12 |

513 | Investigate the behaviour of the function ( boldsymbol{y}=left(boldsymbol{x}^{3}+boldsymbol{4}right)(boldsymbol{x}+mathbf{1})^{3} ) and construct its graph. How many solutions does the equation ( left(x^{3}+right. ) 4)( (x+1)^{3}=c ) possess? | 12 |

514 | Find the points at which the function ( boldsymbol{f} ) ( operatorname{given} operatorname{by} f(x)=(x-2)^{4}(x+1)^{3} ) has local maxima | 12 |

515 | The function ( f(x)=sin x-k x-c ) where ( k ) and ( c ) are constants, decreases always when ( mathbf{A} cdot k>1 ) в. ( k geq 1 ) c. ( k<1 ) D. ( k leq 1 ) E . ( k<-1 ) | 12 |

516 | Assertion If ( f(x)=(x-2)^{3} ) then ( f(x) ) has neither maximum nor minimum at ( boldsymbol{x}=mathbf{2} ) Reason ( boldsymbol{f}^{prime}(boldsymbol{x})=mathbf{0}=boldsymbol{f}^{prime prime}(boldsymbol{x}) ) when ( boldsymbol{x}=mathbf{2} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |

517 | Illustration 3.24 Find the maximum value of 13 sin x + cos x and x for which a maximum value occurs. | 12 |

518 | If metallic circular plate of radius ( 50 mathrm{cm} ) is heated so that its radius increases at the rate of ( 1 mathrm{mm} ) per hour, then the rate at which the area of the plate increases ( left(operatorname{in} c m^{2} / h rright) ) is A . ( 5 pi ) в. ( 10 pi ) c. ( 100 pi ) D. ( 50 pi ) | 12 |

519 | The two curves ( x=y^{2}, x y=a^{3} ) cut orthogonally at a point. Then ( a^{2} ) is equal to A ( cdot frac{1}{3} ) B. 3 ( c cdot 2 ) D. | 12 |

520 | If an edge of a cube measure ( 2 mathrm{m} ) with a possible error of ( 0.5 mathrm{cm} . ) Find the corresponding error in the calculated volume of the cube. ( mathbf{A} cdot 0.6 m^{3} ) В. ( 0.06 m^{3} ) c. ( 0.006 m^{3} ) D. ( 0.0006 mathrm{m}^{3} ) | 12 |

521 | Find the least value of ( k ) for which the function ( x^{2}+k x+1 ) is an increasing function in the interval ( 1<x<2 ) A . 1 B. – ( c cdot 2 ) D. – | 12 |

522 | Let for a function ( boldsymbol{f}(boldsymbol{x}), boldsymbol{h}(boldsymbol{x})= ) ( (f(x))^{2}+(f(x))^{3} ) for every real number ( boldsymbol{x} ). Then This question has multiple correct options | 12 |

523 | Find the interval of increase and decrease of the following functions. ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+ln (1-boldsymbol{4} boldsymbol{x}) ) | 12 |

524 | In a culture, the bacteria count is ( 1,00,000 . ) The number is increased by ( 10 % ) in 2 hours. In how many hours will the count reach ( 2,00,000, ) if the rate of growth of bacteria is proportional to the number present? | 12 |

525 | 22. Letf:R → R be defined by f(x) = k-2x, if xs-1 J(x) = 2x+3, if x >-1 Iff has a local minimum at x=-1, then a possible value of k is [2010] (a) O (b) (C) -1 (d) 1 | 12 |

526 | If ( x>0, ) then find greatest value of the expression ( frac{boldsymbol{x}^{100}}{1+boldsymbol{x}+boldsymbol{x}^{2}+boldsymbol{x}^{3}+ldots .+boldsymbol{x}^{200}} ) | 12 |

527 | Find the derivative of ( f(x)=3 x ) at ( x= ) 2 | 12 |

528 | If the radius of a sphere is measured as ( 9 mathrm{cm} ) with an error of ( 0.02 mathrm{cm}, ) then find the approximate error in calculating its volume | 12 |

529 | The length of the longest interval in which the function ( 3 sin x-4 sin ^{3} x ) is increasing is A. в. c. ( frac{3 pi}{2} ) D. | 12 |

530 | Find the values of ( a ) for which the function ( f(x)=sin x-a x+4 ) is increasing function on ( boldsymbol{R} ) | 12 |

531 | The absolute maximum of ( boldsymbol{y}=boldsymbol{x}^{3}- ) ( 3 x+2 ) in ( 0 leq x leq 2 ) is: A . 4 B. 6 ( c cdot 3 ) D. | 12 |

532 | On the interval [0,1] the function ( f(x)=x^{1005}(1-x)^{1002} ) assumes maximum value equal to. A ( cdot frac{(1005)^{1002}}{(2007)^{2007}} ) в. ( frac{(2007)^{2007}}{(1005)^{12055}(1002)^{1002}} ) c. ( frac{(2007)^{2007}}{(1005)^{1202}(1002)^{1005}} ) D. ( frac{(1005)^{1005}(1002)^{1002}}{(2007)^{2007}} ) | 12 |

533 | Assertion Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function such that ( f(x)=x^{3}+x^{2}+3 x+sin x . ) Then, ( f ) is one-one. Reason ( f(x) ) is decreasing function A. Assertion and Reason are correct and the reason is the correct explanation for the assertion B. Assertion and Reason are correct and the reason is Not the correct explanation for the assertion. c. Assertion is correct while the Reason is incorrect D. Assertion is incorrect while the Reason is correct | 12 |

534 | The greatest value of ( boldsymbol{f}(boldsymbol{x})= ) ( (x+1)^{1 / 3}-(x-1)^{1 / 3} ) on [0,1] is ( A ) B. 2 ( c cdot 3 ) D. ( 1 / 3 ) | 12 |

535 | The function which has neither maximum nor minimum at ( x=0=0 ) is A ( cdot f(x)=x^{2} ) B. ( f(x)=cos x ) ( c cdot f(x)=x^{3}-8 ) D. ( f(x)=cos h x ) | 12 |

536 | A particle’s velocity ( v ) at time ( t ) is given by ( v=2 e^{2 t} cos frac{pi t}{3} . ) The least value of ( t ) at which the acceleration becomes zero is ( mathbf{A} cdot mathbf{0} ) в. ( frac{3}{2} ) ( ^{mathrm{c}} cdot frac{3}{pi} tan ^{-1}left(frac{6}{pi}right) ) D. ( frac{3}{pi} cot ^{-1}left(frac{6}{pi}right) ) | 12 |

537 | () 1.JWs 3. If by dropping a stone in a quiet lake a wave moves in circle at a speed of 3.5 cm/sec, then the rate of increase of the enclosed circular region when the radius of the circular 22 wave is 10 cm, is (a) 220 sq. cm/sec (c) 35 sq. cm/sec (b) 110 sq. cm/sec (d) 350 sq. cm/sec | 12 |

538 | If ( f(x)=frac{80}{3 x^{4}+8 x^{3}-18 x^{2}+60}, ) then the points of local maxima for the function ( boldsymbol{f}(boldsymbol{x}) ) are A . 1,3 в. -3,1 c. -1,3 D. -1,-3 | 12 |

539 | Show that the function ( f ) given by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}^{2}+boldsymbol{4} boldsymbol{x}, boldsymbol{x} in boldsymbol{R} ) is strictly increasing. | 12 |

540 | If 1 degree ( =0.017 ) radians, then the approximate value of ( sin 46 ) degrees is A .0 .7194 B. ( frac{0.017}{sqrt{2}} ) c. ( frac{1.017}{sqrt{2}} ) D. none of these | 12 |

541 | 40. The maximum volume (in cu.m) of the right circular como having slant height 3 m is: [JEE M 2019-9 Jan (M) (a) 61 (b) 3131 (d) 213 T | 12 |

542 | Function ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}+mathbf{2}) boldsymbol{e}^{-boldsymbol{x}} ) is A. decreasing B. decreasing in ( (-infty,-1) ) and increasing in ( (-1, infty) ) c. increasing D. decreasing in ( (-1, infty) ) and increasing in ( (-infty,-1) ) | 12 |

543 | A balloon is pumped at the rate of a cm ( ^{3} ) /minute. The rate of increase of its surface area when the radius is ( b mathrm{cm} ), is A ( cdot frac{2 a^{2}}{b^{4}} mathrm{cm}^{2} / mathrm{min} ) B. ( frac{a}{2 b} mathrm{cm}^{2} / mathrm{min} ) c. ( frac{2 a}{b} mathrm{cm}^{2} / mathrm{min} ) D. none of these | 12 |

544 | A stone is dropped into a quiet lake. If the waves moves in circle at the rate of ( 30 mathrm{cm} / mathrm{sec} ) when the radius is ( 50 mathrm{m} ), the rate of increase of enclosed area is A ( .30 pi m^{2} / ) sec B. ( 30 m^{2} / )sec c. ( 3 pi m^{2} / )sec D. none of these | 12 |

545 | Find the sum of minimum and maximum values of ( y ) in ( 4 x^{2}+12 x y+ ) ( mathbf{1 0 y}^{2}-mathbf{4 y}+mathbf{3}=mathbf{0} ) | 12 |

546 | 10. f(x) is cubic polynomial with f(2)= 18 and f(1) = -1. Also f(x) has local maxima at x=-1 and f'(x) has local minima at x=0, then (2006 – 5M, -1) (a) the distance between (-1,2) and (a f(a)), where x=a is the point of local minima is 275 (b) f(x) is increasing for x = [1,2 V5] (c) f(x) has local minima at x=1 (d) the value of f(0) = 15 | 12 |

547 | At what time ( t ) will the volume of the sphere be 27 times its volume at ( t=0 ) | 12 |

548 | The position vector of a particle at time ( t^{prime} ) is given by ( vec{r}=t^{2} hat{t}+t^{3} hat{j} ). The velocity makes an angle ( theta ) with positive ( x ) -axis. Find the of ( frac{boldsymbol{d} boldsymbol{theta}}{boldsymbol{d} boldsymbol{t}} ) at ( boldsymbol{t}=frac{sqrt{mathbf{2}}}{mathbf{3}} ) | 12 |

549 | For what value of ‘a’ the function ( f(x)=x+cos x-a ) increases ( A cdot 0 ) B. ( c .-1 ) D. Any value | 12 |

550 | Let ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{1})^{4} cdot(boldsymbol{x}-boldsymbol{2})^{n}, boldsymbol{n} boldsymbol{epsilon} boldsymbol{N} ) Then ( f(x) ) has This question has multiple correct options A. a maximum at ( x=1 ) if ( n ) is odd B. a maximum at ( x=1 ) if ( n ) is even c. a minimum at ( x=2 ) if ( n ) is even D. a maximum at ( x=2 ) if ( n ) is odd | 12 |

551 | The rate of change of surface area of a sphere of radius ( r ) when the radius is increasing at the rate of ( 2 mathrm{cm} / ) sec is proportional to A ( cdot frac{1}{r^{2}} ) B. ( frac{1}{r} ) ( c cdot r^{2} ) D. | 12 |

552 | The total number of local maxima and local minima of the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}(2+x)^{3}, & -3<x leq-1 \ x^{2 / 3}, & -1<x<2end{array}right. ) ( A cdot O ) B. ( c cdot 2 ) ( D ) | 12 |

553 | 2. A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of 1.5 m/sec. The length of the highest point of the ladder when the foot of the ladder 4.0 m away from the wall decreases at the rate of (a) 2 m/sec (b) 3 m/sec (c) 2.5 m/sec d) 1.5 m/sec | 12 |

554 | If ( A, B, C ) are angles of a triangle, then the minimum value of ( tan ^{2} frac{A}{2}+ ) ( tan ^{2} frac{B}{2}+tan ^{2} frac{C}{2} ) is | 12 |

555 | Consider the function ( boldsymbol{f}:(-infty, infty) rightarrow ) ( (-infty, infty) ) defined by ( boldsymbol{f}(boldsymbol{x})= ) ( frac{x^{2}-a x+1}{x^{2}+a x+1}, 0<a<2 ) Which of the following is true? A. ( f(x) ) is decreasing on (1,1) and has a local minimum at ( x=1 ) B. ( f(x) ) is increasing on (1,1) and has a local maximum at ( x=1 ) c. ( f(x) ) is increasing on (1,1) but has neither a local maximum nor a local minimum at ( x=1 ) maximum nor a local minimum at ( x=1 ) | 12 |

556 | A point is moving on ( y=4-2 x^{2} ). The ( x ) coordinate of the point is decreasing at the rate of 5 units per second. Then the rate at which y-coordinate of the point is changing when the point is at (1,2) is. A . 5 units/sec B. 10 units/sec c. 15 units/sec D. 20 units/sec | 12 |

557 | Set up an equation of a tangent to the graph of the following function. ( y=4 x-x^{2} ) at the points of its intersections with the ( 0 x ) axis. | 12 |

558 | Number of critical points of the function ( f(x)=(x-2)^{frac{2}{3}}(2 x+1) ) is equal to | 12 |

559 | Let, ( boldsymbol{f}: boldsymbol{A} rightarrow boldsymbol{B} ) be an invertible function If ( f(x)=2 x^{3}+3 x^{2}+x-1, ) then ( f^{prime-1}(5)= ) A ( cdot frac{1}{13} ) B. ( c cdot 6 ) D. can not be determined | 12 |

560 | Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( (0.999)^{frac{1}{10}} ) | 12 |

561 | 3×2+12x-1, -1<x<2, 5. { 37-x 2<x<3 then: (1993 – 2 Marks) (a) f(x) is increasing on [-1,2] (b) f(x) is continues on [-1,3] (c) f'(2) does not exist d) f(x) has the maximum value at x=2 | 12 |

562 | The minimum value of ( x log x ) is equal to. ( A ) B. ( frac{1}{e} ) ( c cdot-frac{1}{e} ) ( D cdot 2 ) | 12 |

563 | What is minimum value of ( sec ^{2} theta+ ) ( cos ^{2} theta ? ) | 12 |

564 | Find the approximate value of ( (0.009)^{1 / 3} ) ( mathbf{A} cdot 0.208 ) B. 0.108 c. 0.205 D. 0.204 | 12 |

565 | Find the intervals in which the function ( f(x) ) is (i) increasing, (ii) decreasing ( f(x)=2 x^{3}-9 x^{2}+12 x+15 ) | 12 |

566 | The equation of the normal to the curve ( x^{4}=4 y ) through the point (2,4) is A. ( x+8 y=34 ) B. ( x-8 y+30=0 ) c. ( 8 x-2 y=0 ) D. ( 8 x+y=20 ) | 12 |

567 | 20. Find the equation of the normal to the curve y = (1+x)” + sin (sin? x) at x =0 (1993 – 3 Marks) | 12 |

568 | 33. Find a point on the curve x2 + 2y2 = 6 whose distance from the line x+y=7, is minimum. (2003 – 2 Marks) | 12 |

569 | The length of the longest interval, in which ( f(x)=3 sin x-4 sin ^{2} x ) is increasing, is A. в. ( c cdot frac{3 pi}{2} ) D. | 12 |

570 | Write the set of values of ( a ) for which ( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x}+boldsymbol{a}^{2} boldsymbol{x}+boldsymbol{b} ) is strictly increasing on ( boldsymbol{R} ) | 12 |

571 | Show that the function ( x^{2}-x+1 ) is neither increasing nor decreasing on (0,1) | 12 |

572 | Consider the following statements: 1. The function ( f(x)=x^{2}+2 cos x ) is increasing in the interval ( (0, pi) ) 2. The function ( f(x)=ln (sqrt{1+x^{2}}-x) ) is decreasing in the interval ( (-infty, infty) ) Which of the above statements is lare correct? A. 1 only B. 2 only c. Both 1 and 2 D. Neither 1 nor 2 | 12 |

573 | Assertion The equation ( boldsymbol{f}(boldsymbol{x})left(boldsymbol{f}^{prime prime}(boldsymbol{x})right)^{2}+ ) ( boldsymbol{f}(boldsymbol{x}) boldsymbol{f}^{prime}(boldsymbol{x}) boldsymbol{f}^{prime prime prime}(boldsymbol{x})+left(boldsymbol{f}^{prime}(boldsymbol{x})right)^{2} boldsymbol{f}^{prime prime}(boldsymbol{x})= ) 0 has atleast 5 real roots Reason The equation ( f(x)=0 ) has atleast 3 real distinct roots & if ( f(x)=0 ) has ( k ) real distinct roots, then ( f^{prime}(x)=0 ) has atleast k-1 distinct roots. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |

574 | Find the slope of the tangent and the normal to the curve ( x^{2}-2 y-y^{2}=1 ) at (-1,-2) | 12 |

575 | The equation ( x+e^{x}=0 ) has A. only one real root B. only two real roots c. no real root D. none of these | 12 |

576 | If the rate of change of area of a square plate is equal to that of the rate of change of its perimeter, then length of the side is? A . 1 unit B. 2 unit c. 3 unit D. 4 unit | 12 |

577 | The greatest value of ( f(x)=x-2 ln x ) in ( [1, e] ) is attained at ( x= ) ( mathbf{A} cdot mathbf{1} ) B. ( sqrt{e} ) ( c cdot 2 ) D. | 12 |

578 | Find the slope of tangent of the curve ( boldsymbol{x}=boldsymbol{a} sin ^{3} boldsymbol{t}, boldsymbol{y}=boldsymbol{b} cos ^{3} boldsymbol{t} ) at ( boldsymbol{t}=frac{pi}{2} ) A . cott B. ( – ) tant c. ( – ) cott D. not defined at ( frac{pi}{2} ) | 12 |

579 | The maximum slope of the curve ( mathbf{y}= ) ( -x^{3}+3 x^{2}+9 x-27 ) is ( A ) B. 12 ( c cdot 6 ) D. | 12 |

580 | 24. Iff(x)= x3 + bx2 + cx + d and 0<b2 <c, then in (-0,00) (a) f(x) is a strictly increasing function (2004) (b) f(x) has a local maxima (c) f(x) is a strictly decreasing function (d) f (x) is bounded | 12 |

581 | The normal to the curve ( y(x-2)(x- ) ( mathbf{3})=boldsymbol{x}+boldsymbol{6} ) at the point where the curve intersects the y-axis passes through the point. ( ^{A} cdotleft(-frac{1}{2},-frac{1}{2}right) ) в. ( left(frac{1}{2}, frac{1}{2}right) ) ( ^{mathrm{c}} cdotleft(frac{1}{2},-frac{1}{3}right) ) D ( cdotleft(frac{1}{2}, frac{1}{3}right) ) | 12 |

582 | 5. The sum of two numbers is fixed. Then its multiplication is maximum, when (a) Each number is half of the sum 13 (b) Each number is espectively of the sum (c) Each number is respectively of the sum (d) None of these | 12 |

583 | The values of ( a ) and ( b ) for which the function ( boldsymbol{y}=boldsymbol{a} log _{e} boldsymbol{x}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{x}, ) has extremum at the points ( x_{1}=1 ) and ( boldsymbol{x}_{2}=2 ) are ( ^{A} cdot_{a}=frac{2}{3}, b=-frac{1}{6} ) B. ( quad a=-frac{2}{3}, b=-frac{1}{6} ) c. ( _{a=-frac{2}{3}, b=frac{1}{6}} ) D. ( _{a=-frac{1}{3}, b=-frac{1}{6}} ) | 12 |

584 | Which one of the following is correct in respect of the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(boldsymbol{x}-mathbf{1})(boldsymbol{x}+mathbf{1}) ? ) A. The local maximum value is larger than local minimum value B. The local maximum value is smaller than local minimum value c. The function has no local maximum D. The function has no local minimum | 12 |

585 | The slope of the tangent to the curve ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}} cos boldsymbol{x} ) is minimum at ( boldsymbol{x}=boldsymbol{alpha}, boldsymbol{0} leq ) ( a leq 2 pi, ) then the value of ( alpha ) is A . 0 в. ( pi ) c. ( 2 pi ) D. ( frac{3 pi}{2} ) | 12 |

586 | If ( f(x)=sin x+a^{2} x+b ) is an increasing function for all values of ( x ) then A ( . a epsilon(-infty,-1) ) B. ( a in R ) ( mathbf{c} cdot a epsilon(-1,1) ) D. none of these | 12 |

587 | 2. The adjacent sides of a rectangle with given perimeter as 100 cm and enclosing maximum area are (a) 10 cm and 40 cm (b) 20 cm and 30 cm (c) 25 cm and 25 cm (d) 15 cm and 35 cm 1 .11 1 . | 12 |

588 | The angle made by the tangent at any point on the curve ( boldsymbol{x}=boldsymbol{a}(boldsymbol{t}+ ) ( sin t cos t), y=a(1+sin t)^{2} ) with ( x ) -axis is A ( cdot frac{pi}{2} ) B. ( frac{pi}{4} ) c. ( _{pi+frac{t}{2}} ) D. ( frac{pi}{4}+frac{t}{2} ) | 12 |

589 | Function ( f(x)=a^{x} ) is increasing on ( R ) if ( mathbf{A} cdot a>0 ) B. ( a<0 ) c. ( 0<a1 ) | 12 |

590 | Let ( f ) be function defined on ( [a, b] ) such that ( f^{prime}(x)>0, ) for all ( x in(a, b) . ) Then prove that ( f ) is an increasing function on ( (a, b) ) | 12 |

591 | The equation of the tangent to the curve ( y=sqrt{9-2 x^{2}} ) at the point where the ordinate and abscises are equal is? A. ( 2 x+y-3 sqrt{3}=0 ) В. ( 2 x+y+3 sqrt{3}=0 ) c. ( 2 x-y-3 sqrt{3}=0 ) D. ( 2 x-y+3 sqrt{3}=0 ) | 12 |

592 | The values of ( x ) at which ( f(x)=sin x ) is stationary are given by A. ( mathrm{n} pi, forall n in Z ) в. ( (2 mathrm{n}+1) frac{pi}{2}, forall n in Z ) c. ( frac{mathrm{n} pi}{4}, forall n in Z ) D. ( frac{mathrm{n} pi}{2}, forall n in Z ) | 12 |

593 | The critical points of the function ( f(x)=(x-2)^{2 / 3}(2 x+1) ) are A. 1 and 2 B. 1 and ( -frac{1}{2} ) c. -1 and 2 ( D ) | 12 |

594 | Sand is pouring from a pipe at the rate of ( 12 mathrm{cm}^{3} / mathrm{s} ). The falling sand forms a cone on the ground in such a way that the height of the cone is always onesixth of the radius of the base. How fast is the height of the sand cone increasing when the height is ( 4 mathrm{cm} ) | 12 |

595 | The nearest point on the line ( 3 x-4 y= ) 25 from the origin is : A ( cdot(-4,5) ) В ( cdot(3,-4) ) ( mathbf{c} cdot(3,4) ) D ( cdot(3,5) ) | 12 |

596 | The total revenue received from the sale of ( x ) units is given by ( R(x)=10 x^{2}+ ) ( 20 x+1500 . ) The marginal revenue when ( x=2015, ) is A. 4032 B. 40320 ( c .403 ) D. 40300 | 12 |

597 | 1. The function f(x)=sin4 x + cos4 x increases if (1999 – 2 Marks) (a) 0<x< 1/8 (b) /4<x<31/8 (c) 37/8<x< 571/8 (d) 51/8<x<31/4 | 12 |

598 | If ( boldsymbol{x}, boldsymbol{y}, boldsymbol{z}>mathbf{0} ) and ( boldsymbol{x} boldsymbol{y} boldsymbol{z}=mathbf{1} ) Then This question has multiple correct options ( mathbf{A} cdot x^{2}+y^{2}+z^{2} leq x^{3}+y^{3}+z^{3} ) B . ( x^{2}+y^{2}+z^{2} geq x^{3}+y^{3}+z^{3} ) ( mathbf{c} cdotleft(x^{4}+y^{4}+z^{4}right) geqleft(x^{3}+y^{3}+z^{3}right) ) D. ( left(x^{4}+y^{4}+z^{4}right) leqleft(x^{3}+y^{3}+z^{3}right) ) | 12 |

599 | A rectangular tank is ( 80 m ) long and ( 25 m ) broad. Water flows into it through a pipe whose cross-section is ( 25 mathrm{cm}^{2}, ) at the rate of ( 16 k m ) per hour. How much the level of the water rises in the tank in 45 minutes? | 12 |

600 | ( mathbf{f}(mathbf{x})=frac{mathbf{a x}+mathbf{b}}{mathbf{c x}+mathbf{d}}(boldsymbol{a d}-boldsymbol{b c} neq mathbf{0}) ) A. Has a maximum B. Has a minimum c. Neither max nor min is true D. Both max and min are true | 12 |

601 | For all ( a, b in R ) the function ( f(x)= ) ( 3 x^{4}-4 x^{3}+6 x^{2}+a x+b ) has A. no extremumm B. exactly one extremum c. exactly two extremum D. three extremum | 12 |

602 | Two sides of a triangle are given. If the area of the triangle is maximum then the angle between the given sides is A ( cdot 45^{circ} ) B. ( 30^{circ} ) ( c cdot 60^{circ} ) D. ( 90^{circ} ) | 12 |

603 | Let ( boldsymbol{E}=boldsymbol{x}^{3}left(boldsymbol{x}^{3}+mathbf{1}right)left(boldsymbol{x}^{mathbf{3}}+mathbf{2}right)left(boldsymbol{x}^{mathbf{3}}+right. ) ( mathbf{3}) ; boldsymbol{x} epsilon boldsymbol{R} ). Then minimum value of ( boldsymbol{E} ) be ( A ) B. – – ( c cdot-1 ) D. Minimum value is not attained | 12 |

604 | 09 19. – cos(2x) cos(2x) sin(2x) If f(x) = -COS X COS X -sin x , then sin x sin x cOS X (JEE Adv. 2017) (a f'(x)=0 at exactly three points in (-T, T) (b) f'(x) = 0 at more than three points in (-1, T) (C f(x) attains its maximum at x=0 (d) f(x) attains its minimum at x=0 | 12 |

605 | If ( f(x)=x^{2} e^{-x^{2} / a^{2}} ) is an increasing function then (for ( a>0 ) ), x lies in the interval ( mathbf{A} cdot[a, 2 a] ) в. ( (-infty,-a) cup(0, a) ) c ( .(-a, 0) ) D. None of these | 12 |

606 | 2. Let x and y be two real variables such that x>0 and xy=1. Find the minimum value of x+y. (1981 – 2 Marks) 1 . : . : . C ….. . …. .. | 12 |

607 | A particle moves on a line according to the law ( s=a t^{2}+b t+c . ) If the displacement after 1 sec is ( 16 mathrm{cm}, ) the velocity after 2 sec is ( 24 mathrm{cm} / mathrm{sec} ) and acceleration is ( 8 mathrm{cm} / mathrm{sec}^{2}, ) then A. ( a=4, b=8, c=4 ) В. ( a=4, b=4, c=8 ) c. ( a=8, b=4, c=4 ) D. none of these | 12 |

608 | 13. Investigate for maxima and minima the function (1988 – 5 Marks) f(x)= [2(t – 1)(t – 2)2 + 3(t – 1)(t – 2)?]dt | 12 |

609 | Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( (0.0037)^{frac{1}{2}} ) | 12 |

610 | 18. A cubic f (x) vanishes at x = 2 and has relative minimum/ 14. maximum at x = -1 and x = = if . Ix = , find the cubic f(x). (1992 – 4 Marks) | 12 |

611 | The function ( f(x)=4-3 x+3 x^{2}-x^{3} ) is A. decreasing on ( R ) B. increasing on ( R ) c. strictly decreasing on ( R ) D. strictly increasing on ( R ) | 12 |

612 | Find the set of values of ( b ) for which ( boldsymbol{f}(boldsymbol{x})=boldsymbol{b}(boldsymbol{x}+cos boldsymbol{x})+boldsymbol{4} ) is decreasing on ( boldsymbol{R} ) | 12 |

613 | For ( mathbf{0} leq boldsymbol{x} leq mathbf{1}, ) the function ( boldsymbol{f}(boldsymbol{x})= ) ( |boldsymbol{x}|+|boldsymbol{x}-mathbf{1}| ) is A. Monotonically increasing B. Monotonically decreasing c. constant function D. Identity function | 12 |

614 | 26. A spherical balloon is filled with 45001 cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72īt cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is: [2012] (a) = (b) | 12 |

615 | Show that the function ( x^{x} ) is minimum at ( boldsymbol{x}=frac{mathbf{1}}{boldsymbol{e}} ) | 12 |

616 | Which one of the following statements is correct? A ( cdot e^{x} ) is a increasing function B. ( e^{x} ) is a decreasing function. ( mathrm{c} cdot e^{x} ) is neither increasing nor decreasing function D. ( e^{x} ) is a constant function | 12 |

617 | If ( f(x)=frac{x^{2}-1}{x^{2}+1}, ) for every real ( x, ) then the minimum value of ( boldsymbol{f} ) is A. does not exist because fis unbounded B. is not attained even through f is bounded c. is equal to 1 D. is equal to -1 | 12 |

618 | A spherical balloon is being inflated so that its volume increases uniformly at the rate of ( 40 mathrm{cm}^{3} / ) min. When ( r=8 ) then the increase in radius in the next ( 1 / 2 m i n ) is A ( .0 .025 mathrm{cm} ) B . ( 0.050 mathrm{cm} ) c. ( 0.075 mathrm{cm} ) D. ( 0.01 mathrm{cm} ) | 12 |

619 | 13. Let interval (a) (-00,-2) (c) (1,2) – 1)(x – 2)dx . Then f decreases in the (2000) (b) (-2,-1) (d) (2,700) | 12 |

620 | To find the equation of tangent and normal to the circle ( x^{2}+y^{2}-3 x+ ) ( 4 y-31=0 ) at the point (2,3) | 12 |

621 | Construct the graphs of the following functions and carry out a complete investigation ( y=sin ^{4} x+cos ^{4} x ) | 12 |

622 | Maximum value of ( f(x)=frac{x^{2}-x+1}{x^{2}+x+1} ) is A. B. 3 ( c cdot frac{3}{7} ) D. | 12 |

623 | If ( x=a(cos 2 t+2 t sin 2 t) ) and ( y= ) ( a(sin 2 t-2 t cos 2 t) ) then find ( frac{d^{2} y}{d x^{2}} ) | 12 |

624 | The equation of one of the tangents to the curve ( boldsymbol{y}=cos (boldsymbol{x}+boldsymbol{y}),-2 boldsymbol{pi} leq boldsymbol{x} leq ) ( 2 pi ; ) that is parallel to the line ( x+2 y= ) ( 0, ) is A. ( x+2 y=1 ) B. ( x+2 y=frac{pi}{2} ) c. ( x+2 y=frac{pi}{4} ) D. None of these | 12 |

625 | The function ( frac{sin (x+alpha)}{sin (x+beta)} ) has no maximum or minimum if (k an integer) A. ( beta-alpha=k pi ) B. ( beta-alpha neq k pi ) c. ( beta-alpha=2 k pi ) D. none of the above | 12 |

626 | Find a point of inflation for the curve ( y=frac{x+1}{x^{2}+1} ) | 12 |

627 | if ( f^{prime}(x)<0 forall x in R ) and ( g(x)=fleft(x^{2}-right. ) 2) ( +boldsymbol{f}left(boldsymbol{6}-boldsymbol{x}^{2}right) ) then ( A cdot g(x) ) is an increasing in ( [2, infty) ) B. ( g(x) ) is an increasing in [-2,0] c. ( g(x) ) has a local minima ( a t x=-2 ) D. ( g(x) ) has a local maxima at ( x=2 ) | 12 |

628 | An edge of a variable cube is increasing at the rate of ( 5 mathrm{cm} / mathrm{s} ). How fast is the volume of the cube increasing when the edge is ( 10 mathrm{cm} ) long? | 12 |

629 | Find the slopes of the tangent and the normal to the following curves at the indicated points. ( boldsymbol{x}^{2}+mathbf{3} boldsymbol{y}+boldsymbol{y}^{2}=mathbf{5} ) at ( (mathbf{1}, mathbf{1}) ) | 12 |

630 | Find the slope of a line. Which bisects the first quadrant angle. | 12 |

631 | syualt * 1933 y = 10 then the making point of us is tu Nile ul uitst 4. If x + y = 10, then the maximum value of xy is (a) 5 (b) 20 (c) 25 (d) None of these | 12 |

632 | If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is a differentiable function such that ( f^{prime}(x)>2 f(x) ) for all ( x in R ) and ( f(0)=1, ) then This question has multiple correct options A ( cdot f(x) ) is decreasing in ( (0, infty) ) B ( cdot f^{prime}(x)e^{2 x} ) in ( (0, infty) ) | 12 |

633 | 25. For xe ( 0.574), define $(w) = vi sint dt. Thenſ has [2011] (a) local minimum at n and 21 (b) local minimum at n and local maximum at 21 (c) local maximum at n and local minimum at 21 (d) local maximum at n and 21 | 12 |

634 | A particle starts moving from rest from a fixed point in a fixed direction. The distance ( s ) from the fixed point at a time ( t ) is given by ( s=t^{2}+a t-b+17, ) where ( a, b ) are real numbers. If the particle comes to rest after 5 sec at a distance of ( s=25 ) units from the fixed point then values of ( a ) and ( b ) are respectively. A .10,-33 B. -10,-33 c. -8,33 D. -10,33 | 12 |

635 | total number of local maxima and local minima of the 29. The total numbe is (2008) function f(x) = (2+x)}, -3, [(2+x), -3<xs-1 7×2/3,-1<x<2 (6) 1 (c) 2 (+2/3 (a) 0 . (d) 3 | 12 |

636 | Let ( f ) be a decreasing function in ( (a, b] ) then which of the following must be true? A. ( f ) is continuous at ( b ) B ( cdot f^{prime}(b)<0 ) c. ( lim _{x rightarrow b} f(x) leq f(b) ) D. ( lim _{x rightarrow b} f(x) geq f(b) ) | 12 |

637 | The function ( boldsymbol{f}(boldsymbol{x})=int_{0}^{x} boldsymbol{e}^{-boldsymbol{x}^{2} / 2}left(boldsymbol{x}^{2}-right. ) ( 3 x+2) d x ) is maximum at ( x= ) ( A ) B. 2 ( c cdot 3 ) D. 4 | 12 |

638 | If ( mathbf{f}(boldsymbol{theta})=sin ^{99} boldsymbol{theta} cos ^{94} boldsymbol{theta} ; boldsymbol{theta} inleft(-frac{boldsymbol{pi}}{2}, frac{boldsymbol{pi}}{2}right) ) attains a maximum at ( boldsymbol{theta} ) equals A ( cdot tan ^{-1} sqrt{frac{94}{99}} ) B. ( tan ^{-1} sqrt{frac{99}{94}} ) c. D. | 12 |

639 | II (X)= and e(r) * where 0<x s 1, the tan x this interval (1997 – 2 Marks) (a) both f(x) and g(x) are increasing functions (b) both f(x) and g(x) are decreasing functions (C) f(x) is an increasing function (d) g(x) is an increasing function. TLC | 12 |

640 | Find the co-ordinates of the points on the ellipse ( x^{2}+2 y^{2}=9 ) at which tangent has slope ( frac{1}{4} . ) Also find the equation of normal. | 12 |

641 | While measuring the side of an equilateral triangle an error of ( k % ) is marked, the percentage error in its area is A. ( k % ) в. ( 2 k % ) ( c cdot frac{k}{2} % ) D. ( 3 k % ) | 12 |

642 | Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{lll}boldsymbol{k}-boldsymbol{2} boldsymbol{x}, & boldsymbol{i} boldsymbol{f} & boldsymbol{x} leq-1 \ 2 boldsymbol{x}+boldsymbol{3}, & boldsymbol{i f} & boldsymbol{x}>-1end{array}right. ) If ( f ) has a local minimum at ( x=-1 ) then the possible value of ( k ) is A. ( -1 / 2 ) B. – ( c cdot 1 ) ( D ) | 12 |

643 | If the ratio of base radius and height of a cone is 1: 2 and percentage error in radius is ( lambda %, ) then the error in its volume is A . ( lambda % ) B . 2lambda% c. ( 3 lambda % ) D. none of these | 12 |

644 | If ( boldsymbol{f}(boldsymbol{x})=sum_{boldsymbol{r}=mathbf{0}}^{boldsymbol{n}} boldsymbol{a}_{boldsymbol{r}} boldsymbol{x}^{boldsymbol{r}} ) for ( boldsymbol{a}_{boldsymbol{r}} boldsymbol{epsilon} boldsymbol{R} ; boldsymbol{r} boldsymbol{epsilon} boldsymbol{N} ; boldsymbol{n} geq mathbf{3} ) If ( boldsymbol{f}(boldsymbol{x}) neq mathbf{0} ) for ( boldsymbol{x} boldsymbol{epsilon}(boldsymbol{alpha}, boldsymbol{beta}) ) Then, if ( (boldsymbol{alpha}<boldsymbol{t}<boldsymbol{beta}) ) This question has multiple correct options A. ( f(x) ) is continuous and Differentiable over ( (alpha, beta) ) atleast B. ( (x-alpha)(x-beta) f(x) ) is continuous and Differentiable over ( (alpha, beta) ) atleast C ( .(x-alpha)(x-beta) f(x) ) is continuous, but Not differentiable over ( (alpha, beta) ) ( frac{f^{prime}(t)}{f(t)}=frac{1}{alpha-t}+frac{1}{beta-t}, ) for atleast one ( ^{prime} t^{prime} ) in ( (alpha, beta) ) | 12 |

645 | a, b>c, x>-cis (Va-c+Vb-c)2. (1979) Tet x and y be two real variables such that x > 0 and xy = 1. Find the minimum value of x+y. (1981 – 2 Marks) | 12 |

646 | 13. A triangular park is enclosed on two side park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sid fence are of same length x. The maximum area enclosed the park is [2006] (a) 3 (c) 1 x² (d) rx² | 12 |

647 | Example 2.2 From point A located on a highway as shown in Fig. 2.41, one has to get by car as soon as possible to point B located in the field at a distance I from the highway. It is known that the car moves in the field time slower on the highway. At what distance from point D one must turn off the highway? X Fig. 2.41 | 12 |

648 | The length ( x ) of an rectangle is decreasing at the rate ( 2 mathrm{cm} / ) sec and width y is increasing at the rate of ( 2 c m / )sec. When ( x=12 c m ) and ( y= ) ( 12 c m, ) the rate of change of the area of the rectangle is ( k c m^{2} / s e c, ) then ( k-9 ) is | 12 |

649 | 3. The normal to the curve x = a (cos + O sin o), y = a (sin 0-0 cos O) at any point ‘O’ is such that (1983 – 1 Mark) (a) it makes a constant angle with the x-axis (b) it passes through the origin (c) it is at a constant distance from the origin (d) none of these | 12 |

650 | Find the values of ( x, ) for which the function ( f(x)=x^{3}+12 x^{2}+36 x+6 ) is increasing. | 12 |

651 | sin tax 12. Let f(x) = 2 ,x>0 Let x, <x, <x,<…….. <x<……. be all the points of local maximum off and y, <y, <y,< ……. <y, 2 (b) x, el 21,2n + „ }for every n (©) |xn – yul>1 for every n (d) x,<y, | 12 |

652 | f ( boldsymbol{y}=cos ^{2}left(45^{circ}+xright)(sin x-cos x)^{2} ) then the maximum value of y is: | 12 |

653 | The slope of the curve ( y=sin x+cos ^{2} x ) is zero at the point where – A ( cdot x=frac{pi}{4} ) B. ( x=frac{pi}{2} ) c. ( x=pi ) D. No where | 12 |

654 | ( operatorname{Let} frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=ln sqrt{frac{1+x}{1-x}}, ) then find ( x ) | 12 |

655 | where Il 15 UISLUMUUUU 13. Let h(x)=min {x, r2), for every real number of x, (1998 – 2 Marks) (a) h is continuous for all x (b) h is differentiable for all x (c) h'(x)=1, for all x >1 (d) h is not differentiable at two values of x. | 12 |

656 | 38. Let f(x) = x² + + and g(x) = x , XER-{-1,0,1}. If x h(x)=-***, then the local minimum value of h(x) is : g (a) -3 (b) -2/2 (c) 2/2 [JEE M 2018] (d) 3 | 12 |

657 | Prove that the function given by ( boldsymbol{f}(boldsymbol{x})= ) ( cos x ) is strictly increasing in ( (pi, 2 pi) ) | 12 |

658 | If the percentage error in the edge of a cube is ( 1, ) then error in its volume is A. ( 1 % ) B. 2% ( c .3 % ) D. none of these | 12 |

659 | A cylindrical tank of radius ( 10 m ) is being filled with wheat at the rate of 314 cubic metre per hour. then the depth of the wheat is increasing at the rate of A ( cdot 1 m^{3} / h ) B. ( 0.1 m^{3} / h ) ( mathbf{c} cdot 1.1 m^{3} / h ) D. ( 0.5 m^{3} / h ) E. None of these | 12 |

660 | 22. For every pair of continuous functions f, g: [0, 1] → R such that max f(x): x €[0,1]} = max {g(x): x €[0,11), the correct statement(s) is (are): (JEE Adv. 2014) (a) f(c))2 +3f (c)= (g(c))2 + 3g(c) for some ce [0, 1] (b) (c)2 +f(c)=(g(c))2 +3g(c) for some c € [0, 1] (c) (c))2 + 3f (c)= (g(c))2+g(c) for some c e [0, 1] (d) (C))2 = (g(c))2 for some c e [0, 1] | 12 |

661 | The approximate value of ( f(x)=x^{3}+ ) ( 5 x^{2}-7 x+9=0 ) at ( x=1.1 ) is A . 8.6 B. 8.5 ( c .8 .4 ) D. 8.3 | 12 |

662 | Let ( f(x)=(x-2)^{33}(x-3)^{44} ) then which of the following is true A. ( x=2 ) is point of inflexion B. ( x=3 ) is point of minima c. ( _{x}=frac{17}{7} ) is point of maxima D. All of these | 12 |

663 | 4. Use the function f(x) = x1/x , x>0. to determine the bigger of the two numbers er and me (1981 – 4 Marks) | 12 |

664 | The function ( f(x)=x^{2}+frac{lambda}{x} ) then: A. minimum at ( x=2 ) if ( lambda=16 ) B. maximum at ( x=2 ) if ( lambda=16 ) c. maximum for no real value of ( lambda ) D. point of inflection at ( x=1 ) if ( lambda=-1 ) | 12 |

665 | 21. The equation of the tangent to the curve y=x+ 2 , that is parallel to the x-axis, is (a) y=1 (b) y=2 [2010] (d) y=0 (c) y=3 | 12 |

666 | 14. Find all maxima and minima of the function y = x(x – 1)2,0 < x < 2 (1989 – 5 Marks) Also determine the area bounded by the curvey=x (x – 1)2, the y-axis and the line y=2. | 12 |

667 | Let the absolute maxima/minima value of ( boldsymbol{f}(boldsymbol{x})=mathbf{3} boldsymbol{x}^{4}-boldsymbol{8} boldsymbol{x}^{3}+mathbf{1} boldsymbol{2} boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{8} boldsymbol{x}+ ) ( mathbf{2 5} ; boldsymbol{x} in[mathbf{0}, mathbf{3}] ) be max, min. Find ( 2(max )+min ? ) | 12 |

668 | Find the greatest and the least values of the following functions. ( f(x)=left(2^{x}+2^{-x}right) / I n 2 ) on the interval [-1,2] | 12 |

669 | Minimum value of the polynomial ( boldsymbol{p}(boldsymbol{x})=mathbf{4} boldsymbol{x}^{2}-boldsymbol{6} boldsymbol{x}+mathbf{1} ) A. ( -frac{3}{4} ) B. ( -frac{5}{4} ) ( c cdot-frac{5}{16} ) D. – œ | 12 |

670 | A sphere of radius ( 100 mathrm{mm} ) shrinks to radius ( 98 mathrm{mm} ), then the approximate decrease in its volume is A. ( 12000 pi m m^{3} ) В. ( 800 pi m m^{3} ) c. ( 80000 pi m m^{3} ) D. ( 120 pi m m^{3} ) | 12 |

671 | ( 6 то п опало – (1995) 12. The function flx) = max (1-x)(1+x), 2}, xe(-0, 0) is (a) continuous at all points (6) differentiable at all points (© differentiable at all points except at x=landx=-1 (d) continuous at all points except at x = 1 and x = -1. where it is discontinuous Loof, Then 7 | 12 |

672 | Assertion ( (A): ) If ( f(x)=|x|, ) then ( f ) has minimum value at ( x=0 ) Reason (R): A function f(x) has minimum value at ( x=a ) if ( f^{prime}(a)=0 ) and ( mathbf{f}^{prime prime}(mathbf{a})>mathbf{0} ) A. Both A and R are true and R is the correct explanation of A B. Both A and R are true and R is not the correct explanation of A C. A is true and R is false D. A is false and R is true | 12 |

673 | The function ( x^{5}-5 x^{4}+5 x^{3}-10 ) has a maximum, when ( boldsymbol{x}= ) ( A cdot 3 ) B . 2 ( c .1 ) D. | 12 |

674 | Find limits of the error when ( frac{mathbf{9 1 6}}{mathbf{1 9 1}} ) is taken for ( sqrt{mathbf{2 3}} ) | 12 |

675 | Find intervals in which ( f(x)=frac{4 x^{2}+1}{x} ) is increasing and decreasing. | 12 |

676 | The equation of normal to the curve ( boldsymbol{y}=tan boldsymbol{x} ) at the point ( (boldsymbol{0}, boldsymbol{0}) ) is A. ( x+y=0 ) В. ( x-y=0 ) c. ( x+2 y=0 ) D. None of these | 12 |

677 | A particle is moving in a straight line such that its distance at any time ( t ) is given by ( s=frac{t^{4}}{4}-2 t^{3}+4 t^{2}-7 . ) The acceleration of the particle is minimum when ( t= ) ( A cdot 1 ) B. 2 ( c .3 ) ( D ) | 12 |

678 | 23. Let f: R R be a function defined by f(x)=min (x +1,|x+1),Then which of the following is true ? (a) f(x) is differentiable everywhere [2007] (6) f() is not differentiable at x=0 © f(x) > 1 for all X ER (d) f(x) is not differentiable at x=1 | 12 |

679 | Find the rate of change of the area of a circle with respect to its radius ( r ) when (i) ( r=3 mathrm{cm} ) (ii) ( r=4 mathrm{cm} ) ( mathbf{A} cdot 6 pi, 8 pi ) B . ( 5 pi, 8 pi ) ( mathbf{c} .4 pi, 10 pi ) D . ( 2 pi, 8 pi ) | 12 |

680 | Assertion(A): If the tangent at any point ( P ) on the curve ( x y=a^{2} ) meets the axes at ( A ) and ( B ) then ( A P: P B=1: 1 ) Reason(R): The tangent at ( P(x, y) ) on the curve ( boldsymbol{X}^{boldsymbol{m}} cdot boldsymbol{Y}^{boldsymbol{n}}=boldsymbol{a}^{boldsymbol{m}+boldsymbol{n}} ) meets the axes at ( A ) and ( B ). Then the ratio of ( P ) divides ( overline{A B} ) is ( n: m ) A. Both A and R are true R is the correct explanation of B. Both A and R are true but R is not correct explanation of A c. A is true but R is false D. A is false but R is true | 12 |

681 | The point on the curve ( y^{2}=8 x ) for which the abscissa and ordinate change at the same rate is. A ( cdot(4,2) ) в. (-4,2) ( c cdot(2,4) ) D. (-2,-4) | 12 |

682 | Find the critical points of the function ( f(x)=(x-2)^{2 / 3}(2 x+1) ) ( A .-1 ) and 2 B. c. 1 and -2 D. 1 and 2 | 12 |

683 | the radius of a sphere increases at a rate of ( 2 mathrm{cm} / mathrm{sec} . ) Find the rate at which its volume and area increases when radius is ( 4 mathrm{cm} ) | 12 |

684 | Find the values of ( x ) for which ( y= ) ( [x(x-2)]^{2} ) is an increasing function. | 12 |

685 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{5} sin ^{2} boldsymbol{x} ) be an increasing function on the set ( R ) Then, ( a ) and ( b ) satisfy A ( cdot a^{2}-3 b-15>0 ) B . ( a^{2}-3 b+15>0 ) c. ( a^{2}-3 b+150 ) and ( b>0 ) 0 | 12 |

686 | ff ( f(x)=x^{2}-4 x+5 ) on [0,3] then the absolute maximum value is: ( A cdot 2 ) B. 3 ( c cdot 4 ) D. 5 | 12 |

687 | A point on the parabola ( y^{2}=18 x ) at which the ordinate increases at twice the rate of the abscissa is A. (2,6) (年) (2,6) в. (2,-6) ( ^{mathrm{c}} cdotleft(frac{9}{8},-frac{9}{2}right) ) D ( cdotleft(frac{9}{8}, frac{9}{2}right) ) | 12 |

688 | If ( f ) is an increasing and ( g ) is a decreasing function and fog is defined, then fog will be A. increasing function B. decreasing funtion c. neither increasing nor decreasing D. None of these | 12 |

689 | The side of a square sheet is increasing at the rate of ( 4 mathrm{cm} ) per minute. The rate by which the area increasing when the side is ( 8 mathrm{cm} ) long is. ( mathbf{A} cdot 60 mathrm{cm}^{2} / mathrm{minute} ) B. ( 66 mathrm{cm}^{2} / ) minute c. ( 62 mathrm{cm}^{2} / mathrm{minute} ) D. ( 64 mathrm{cm}^{2} / mathrm{minute} ) | 12 |

690 | If the rate of change in the circumference of a circle of ( 0.3 mathrm{cm} / mathrm{s} ) then the rate of change in the area of the circle when the radius is ( 5 mathrm{cm}, ) is: A. ( 1.5 mathrm{sq} mathrm{cm} / mathrm{s} ) B. ( 0.5 mathrm{sq} mathrm{cm} / mathrm{s} ) c. ( 5 mathrm{sq} mathrm{cm} / mathrm{s} ) D. 3 sq cm/s | 12 |

691 | (2006 – 5M, -1) 15. Iff(x)=min {1, x2, x3, then (a) f(x) is continuous VXER (b) f(x) is continuous and differentiable everywhere. © f(x) is not differentiable at two points (d) f(x) is not differentiable at one point | 12 |

692 | 19. Let f(x)=(1 + b2)x2 + 2bx + 1 and let m(b) be the minimum value of f(x). As b varies, the range of m(b) is (20015) (a) [0,1] (b) (0,1/2] (c) [1/2,1] (d) (0,1] | 12 |

693 | ( boldsymbol{f}(boldsymbol{x})left{begin{array}{l}=mathbf{2} boldsymbol{x}^{2}+frac{2}{x^{2}} text { for }-mathbf{2} leq boldsymbol{x}<mathbf{0} boldsymbol{a} \ =mathbf{1} quad text { for } boldsymbol{x}=mathbf{0}end{array}right. ) Determine the greatest and least values. What is the minimum value of the function? A. 0 B. 1 ( c cdot 4 ) ( D ) | 12 |

694 | The maximum value of ( frac{ln x}{x} ) is A ( . e ) B. ( frac{1}{e} ) ( c cdot frac{2}{e} ) D. | 12 |

695 | 4. Let f and g be increasing and decreasing functions, respectively from [0, 0 ) to [0, 0). Let h(x)=f(g(x)). If h(0)=0, then h(x)-h(1) is (1987 – 2 Marks) (a) always zero b) always negative (©) always positive d) strictly increasing (e) None of these. | 12 |

696 | The family of curves represented by ( frac{d y_{1}}{d x}=frac{x^{2}+x+1}{y^{2}+y+1} ) and the family represented by ( frac{boldsymbol{d} boldsymbol{y}_{2}}{boldsymbol{d} boldsymbol{x}}+frac{boldsymbol{y}^{2}+boldsymbol{y}+mathbf{1}}{boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1}}=mathbf{0} ) A. touch each other B. orthogonal to each other c. identical D. intersect at an angle of ( frac{pi}{4} ) | 12 |

697 | The values of ( a ) for which ( f(x)= ) ( frac{a^{2} x^{3}}{3}+frac{3 a x^{2}}{2}+2 x+1 ) is strictly decreasing at ( boldsymbol{x}=mathbf{1} ) A. ( a in(-2,-1) ) в. ( a in(-1,0) ) c. ( a in(1,2) ) D. ( a in(-2,1) ) | 12 |

698 | 6. If sum of two numbers is 3, then maximum value of the product of first and the square of second is (a) 4 (6) 3 (c) 2 (d) 1 Tecn ie footh 24 .1. | 12 |

699 | If the percentage error in measuring the surface area of a sphere is ( alpha % ), then the error in its volume is A ( cdot frac{3}{2} alpha % ) в. ( frac{2}{3} alpha % ) ( c .3 alpha % ) D. none of these | 12 |

700 | The normal to the curve ( sqrt{x}+sqrt{y}=sqrt{a} ) is perpendicular to ( x ) axis at the point в. ( (a, 0) ) c. ( left(frac{a}{4}, frac{a}{4}right) ) D. No where | 12 |

701 | 38. Ify=(sinx + cosec x)2 + (cosx + sec x)?, then the minimum value of y, Vxe R, is b. 3 o levo ! c. 9 d. 0 a. 7 000 | 12 |

702 | If the distance ‘s’ metres transversed by a particle in ( t ) seconds is given by ( s= ) ( t^{3}-3 t^{2}, ) then the velocity of the particle when the acceleration is zero, in metre/sec is ( A cdot 3 ) B. – – ( c cdot-3 ) D. | 12 |

703 | ff ( x y(y-x)=2 a^{3}, ) at what point does ( y ) have a minimum value A . a, 2a B. a,-a c. २а,२а D. None of these | 12 |

704 | ( f(x)=x+sin x ) is decreasing when ( x ) lies in the interval A ( cdot[-1,1] ) B. [2,3] c. [3,4] D. No value of ( x ) | 12 |

705 | The function ( f(x)=frac{lambda sin x+2 cos x}{sin x+cos x} ) is increasing, if ( A cdot lambda1 ) c. ( lambda2 ) | 12 |

706 | The function ( f(x)=2-3 x+3 x^{2}- ) ( x^{3}, x varepsilon R ) is A. neither increasing nor decreasing B. increasing c. decreasing D. none of these | 12 |

707 | The distance between the origin and the tangent to the curve ( y=e^{2 x}+x^{2} ) drawn at the point ( x=0 ) is A ( cdot frac{1}{sqrt{5}} ) в. ( frac{2}{sqrt{5}} ) c. ( frac{-1}{sqrt{5}} ) D. ( frac{2}{sqrt{3}} ) | 12 |

708 | A point particle moves along a straight line such that ( x=sqrt{t}, ) where ( t ) is time. Then, ratio of acceleration to cube of the velocity is A . -1 в. -0.5 c. -3 D. – | 12 |

709 | The slope of the normal to the curve ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}-sin theta), boldsymbol{y}=boldsymbol{a}(1-cos boldsymbol{theta}) ) at point ( boldsymbol{theta}=frac{boldsymbol{pi}}{2} ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot-1 ) D. ( frac{1}{sqrt{2}} ) | 12 |

710 | 12. For the function x x x f (x) = x COS-, X21, (2009) (a) for at least onex in the interval [1,00 ), f(x + 2)-f(x)00 (c) for all x in the interval [1,0), f (x + 2)-f(x)>2 (d) f'(x) is strictly decreasing in the interval [1,0 ) | 12 |

711 | 1. A man 2 metre high walks at a uniform speed 5 metre/hour away from a lamp post 6 metre high. The rate at which the length of his shadow increases is (a) 5 m/h (b) m/h (0 m/h (a) m/h | 12 |

712 | Area of the greatest rectangle that can be inscribed in the ellipse + =lis [20051 2 64 (a) zab (b) ab (c) Jab (d) 1 2 | 12 |

713 | In a bank, principal increases continuously at the rate of ( 5 % ) per year. An amount of ( R s .1000 ) is deposited with this bank, how much will it worth after 10 years( left(e^{0.5}=1.648right) ) | 12 |

714 | 1 ui 10. Define the collections {E, E., Ez, ……} of ellipses and {R1, R2, Rz, …..} of rectangles as follows: x2 y = 1; E: 74 R : rectangle of largest area, with sides parallel to the axes, inscribed in Ej; 12 E: ellipse – + 5=1 of largest area inscribed in R . a b- n>1; R, : rectangle oflargest area, with sides parallel to the axes, inscribed in E n >1. Then which of the following options is/are correct? (JEE Adv. 2019) (a) The eccentricities of E, and E, are NOT equal (b) The length of latus rectum of E, is – (C) (area of R.)< 24, for each positive integer N n=1 (d) The distance of a focus from the centre in E, is را در | 12 |

715 | Let ( f(x)=x^{3}+3 x^{2}-9 x+2 . ) Then A. ( f(x) ) has a maximum at ( x=1 ) B. ( f(x) ) has neither a minimum nor a maximum at ( x= ) -3 c. ( f(x) ) has a minimum at ( x=1 ) D. none of these | 12 |

716 | If ( f^{prime}(x)=g(x)(x-a)^{2}, ) where ( g(a) neq 0 ) and ( g ) is continuous at ( x=a ) then This question has multiple correct options A. ( f ) is increasing near a if ( g(a)>0 ) B. ( f ) is increasing near a if ( g(a)0 ) D. ( f ) is decreasing near a if ( g(a)<0 ) | 12 |

717 | The set of values of ( p ) for which the points of extremum of the function, ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{p} boldsymbol{x}^{2}+boldsymbol{3}left(boldsymbol{p}^{2}-mathbf{1}right) boldsymbol{x}+mathbf{1} ) lie in the interval (-2,4) is A. (-3,5) в. (-3,3) c. (-1,3) D. (-1,5) | 12 |

718 | Consider the function ( f(x)=frac{x^{2}-1}{x^{2}+1} ) where ( boldsymbol{x} boldsymbol{epsilon} boldsymbol{R} ) At what value of ( x operatorname{does} f(x) ) attain minimum value? A . -1 B. c. 1 D. | 12 |

719 | The distance (in metre) travelled by a vehicle in time ( t ) (in seconds) is given by the equation ( s=t^{3}+2 t^{2}+t+1 . ) The difference in the acceleration between ( boldsymbol{t}=boldsymbol{2} ) and ( boldsymbol{t}=boldsymbol{4} ) is A ( cdot 12 m / s^{2} ) B . ( 18 m / s^{2} ) c. ( 16 m / s^{2} ) D. ( 14 m / s^{2} ) | 12 |

720 | The focal length of a mirror is given by ( frac{1}{v}-frac{1}{u}=frac{2}{f} . ) If equal errors ( (alpha) ) are made in measuring ( u ) and ( v ), then the relative error in ( boldsymbol{f} ) is A ( cdot frac{2}{alpha} ) B ( cdot alphaleft(frac{1}{u}+frac{1}{v}right) ) ( ^{c} cdot alphaleft(frac{1}{u}-frac{1}{v}right) ) D. none of these | 12 |

721 | Find the derivative of ( f(x)=3 x ) at ( x= ) 2 | 12 |

722 | 18. Iff:RRis a differentiable function such that f'(x) >2f(x) for all x e R, and f(0) = 1, then (JEE Adv. 2017) (a) f(x) is increasing in (0,00) (b) f(x) is decreasing in (0,0) (c) f(x)>e2x in (0,00) (d) f'(x) <e2x in (0,0) | 12 |

723 | The points on the curve ( 12 y=x^{3} ) whose ordinate and abscissa change at the same rate, are A ( cdot(-2,-2 / 3),(2,2 / 3) ) в. ( (-2,2 / 3),(2 / 3,2) ) c. ( (-2,-2 / 3) ) only D. ( (2 / 3,2) ) only | 12 |

724 | If the volume of spherical ball is increasing at the rate of ( 4 pi c c / ) sec then the rate of change of its surface area when the volume is ( 288 pi c c ) is A ( cdot frac{4}{3} pi c m^{2} / ) sec в. ( frac{2}{3} pi c m^{2} / ) sec ( mathrm{c} cdot 4 pi mathrm{cm}^{2} / mathrm{sec} ) D. ( 2 pi c m^{2} / ) sec | 12 |

725 | ( f(x)=x+2 cos x ) is increasing in A ( cdotleft(0, frac{pi}{2}right) ) В ( cdotleft(frac{-pi}{2}, frac{pi}{6}right) ) c. ( left(frac{pi}{2}, piright) ) D. ( left(frac{-pi}{2} frac{pi}{2}right) ) | 12 |

726 | Using differentials, find the approximate value of each of the following up to 3 places of decimal. (i) ( sqrt{25.3} ) (ii) ( sqrt{49.5} ) (iii) ( sqrt{mathbf{0 . 6}} ) ( (i v)(0.009)^{frac{1}{3}} ) ( (v)(0.999)^{frac{1}{10}} ) ( (v i)(15)^{frac{1}{4}} ) (vii) ( (26)^{frac{1}{3}} ) ( (text { viii })(255)^{frac{1}{4}} ) ( (i x)(82)^{frac{1}{4}} ) ( (x)(401)^{frac{1}{2}} ) ( (x i)(0.0037)^{frac{1}{2}} ) | 12 |

727 | From a variable point of an ellipse ( frac{x^{2}}{d^{2}}+ ) ( frac{boldsymbol{y}^{2}}{boldsymbol{b}^{2}}=1 ) normal is drawn to the ellipse. Find the maximum distance of the normal from the centre of the ellipse. ( mathbf{A} cdot a+b ) B. ( a-b ) c. ( a^{2}-b^{2} ) D. ( -a+b ) | 12 |

728 | A particle moving on a curve has the position given by ( boldsymbol{x}=boldsymbol{f}^{prime}(boldsymbol{t}) sin boldsymbol{t}+ ) ( f^{prime prime}(t) cos t, y=f^{prime}(t) cos t-f^{prime prime}(t) sin t ) at time ( t ) where ( f ) is a thrice-differentiable function.Then the velocity of the particle at time ( t ) is ( mathbf{A} cdot f^{prime prime prime}(t) ) B . ( f^{prime}(t)+f^{prime prime prime}(t) ) c. ( f^{prime}(t)+f^{prime prime}(t) ) D. ( f^{prime}(t)-f^{prime prime prime}(t) ) | 12 |

729 | A particular point moves on the parabola ( y^{2}=4 a x ) in such a way that its projection on ( y ) -axis has a constant velocity. Then its projection on ( x ) -axis moves with This question has multiple correct options A. constant velocity B. constant acceleration c. variable velocity D. variable acceleration | 12 |

730 | 9. x and y be two variables such that x > 0 and xy = 1. Then the minimum value of x + y is (a) 2 (6) 3 (c) 4 (d) o | 12 |

731 | If ( frac{x^{2}}{f(4 a)}=frac{y^{2}}{fleft(a^{2}-5right)} ) respresents and ellipse with major axis as y-axis and ( boldsymbol{f} ) is a decreasing function, then A ( . a in(-infty, 1) ) B . ( a in(5, infty) ) c. ( a in(1,4) ) D. ( a in(-1,5) ) | 12 |

732 | What is the value of ( x, ) when ( f(x)= ) ( 6+(x-2)^{2} ) is at its minimum? A . -6 B. – c. 0 D. 2 E . 5 | 12 |

733 | For the function ( f(x)=x cos frac{1}{x}, x geq 1 ) which of the following is/are true?? This question has multiple correct options A. There is at least one ( x ) in the interval ( [1, infty) ) for which ( f(x+2)-f(x)2 ) for all ( x ) in the interval ( [1, infty) ) D. ( f^{prime}(x) ) is strictly decreasing in the interval ( [1, infty) ) | 12 |

734 | If displacement ( s ) at time ( t ) is ( s=t^{3}- ) ( 3 t^{2}-15 t+12, ) then acceleration at time ( t=1 ) sec is A ( cdot ) 6units ( / ) sec ( ^{2} ) B. – 6units/sec ( ^{2} ) ( c cdot 0 ) D. 4units ( / )sec( ^{2} ) | 12 |

735 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{5} boldsymbol{s} boldsymbol{i} boldsymbol{n}^{2} boldsymbol{x} ) be an increasing function on the set ( R ) Then, a and b satisfy A ( cdot a^{2}-3 b-15>0 ) B . ( a^{2}-3 b+15>0 ) c. ( a^{2}-3 b+150 ) and ( b>0 ) | 12 |

736 | A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. The rate at which the radius of the balloon is increasing when the radius is ( 15 mathrm{cm} ) is. A ( cdot frac{1}{pi} c m / s e c ) B ( cdot frac{2}{pi} c m / ) sec c. ( pi c m / s e c ) D. ( frac{pi}{2} mathrm{cm} / mathrm{sec} ) | 12 |

737 | 12. Consider the following statments in S and R (2000S) S: Both sin x and cos x are decreasing functions in the interval , R: If a differentiable function decreases in an interval (a, b), then its derivative also decreases in (a, b). Which of the following is true ? (a) Both S and R are wrong (b) Both S and R are correct, but R is not the correct explanation of S Sis correct and Ris the correct explanation for S (d) Sis correct and R is wrong | 12 |

738 | The surface area of a spherical balloon is increasing at the rate of ( 2 mathrm{cm}^{2} / mathrm{sec} ) At what rate is the volume of the balloon is increasing when the radius of the balloon is ( 6 mathrm{cm} ? ) | 12 |

739 | Let ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}^{2}+mathbf{2}}{[boldsymbol{x}]}, mathbf{1} leq boldsymbol{x} leq mathbf{3}, ) where [ represents greatest integer function, then A ( . f(x) ) is increasing in [1,3] B. Least value of ( f(x) ) is 3 c. Greatest value of ( f(x) ) is ( frac{11}{2} ) D. ( f(x) ) has no greatest value | 12 |

740 | Find the point of local maxima & local minima of the function ( boldsymbol{f}(boldsymbol{x})=sin ^{4} boldsymbol{x}+cos ^{4} boldsymbol{x} boldsymbol{i n}[mathbf{0}, boldsymbol{pi}] ) | 12 |

741 | Find the set of values of ( a ) for which ( f(x)=x+cos x+a x+b ) is increasing on ( boldsymbol{R} ) | 12 |

742 | Prove that the following functions are increasing on ( boldsymbol{R} ) (i) ( f(x)=3 x^{5}+40 x^{3}+240 x ) (ii) ( f(x)=4 x^{3}-18 x^{2}+27 x-27 ) | 12 |

743 | A particle moves along the ( x ) -axis obeying the equation ( boldsymbol{x}=boldsymbol{t}(boldsymbol{t}-mathbf{1})(boldsymbol{t}- ) 2), where ( x ) is in meter and ( t ) is in second Find the acceleration of the particle when its velocity is zero. | 12 |

744 | For all x in [0, 1], let the second derivative f” (x) of a function f(x) exist and satisfy f” (x) < 1. Iff(0)=f(1), then show that f(x)<1 for all x in [0, 1]. (1981 – 4 Marks) | 12 |

745 | Write the set of values of ( k ) for which ( f(x)=k x-sin x ) is increasing on ( R ) | 12 |

746 | A stone is dropped into a quiet lake and waves move in a circle at a speed of 3.5 ( mathrm{cm} / mathrm{sec} . ) At the instant when the radius of the circular wave is ( 7.5 mathrm{cm}, ) how fast is the enclosed area increasing? | 12 |

747 | Find the greatest and the least values of the following function: ( f(x)=cos 3 x-15 cos x+8 ) where ( boldsymbol{x} epsilonleft[frac{boldsymbol{pi}}{mathbf{3}}, frac{boldsymbol{3} boldsymbol{pi}}{boldsymbol{2}}right] ) | 12 |

748 | 27. Let a, b e R be such that the function f given byf(x)= In x+ bx2 + ax, x = 0 has extreme values at x=-1 and x=2 Statement-1: f has local maximum at x = -1 and at x=2. Statement-2 : a = 1 and b= -1 [2012] . 4 (a) Statement-1 is false, Statement-2 is true. (b) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1. Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1. (d) Statement-1 is true, statement-2 is false. | 12 |

749 | Illustration 2.36 Find the minimum and maximum values of the function y = x – 3x + 6. Also find the values of x at which these occur. | 12 |

750 | The displacement of a body varies with the time as ( S=t^{3}+3 t^{2}+2 t-1 . ) If the velocity at ( t=4 sec ) is ( 2+12 K m / s ) then find ( k ) ( A cdot 6 ) B. 3 ( c cdot 2 ) D. | 12 |

751 | Find the maximum value of ( f(x)=left(frac{1}{x}right)^{x} ) | 12 |

752 | The function ( boldsymbol{y}=boldsymbol{a} log |boldsymbol{x}|+boldsymbol{b x}^{2}+boldsymbol{x} ) has its extremum values at ( x=-1 ) and ( x=2 ) then A ( a=2, b=-1 ) – ( 1, a=2, b=2=2 ) В. ( a=2, b=-1 / 2 ) c. ( a=-2, b=1 / 2 ) D. None of these | 12 |

753 | Arrange ( A, B, C, D ) in ascending order A) Maximum value of ( sin 5 x ) B) Minimum value of ( x+frac{1}{x}(x>0) ) C) Minimum value of ( 4 times 2^{left(x^{2}-3right)^{3}+27} ) D) Minimum value of ( 5 sin ^{2} x+3 cos ^{2} x ) ( A cdot A, B, D, C ) B. A, C, B, D c. ( A, D, B, C ) D. B, D, A, C | 12 |

754 | Coffee is coming out from a conical filter, with height and diameter both ( 25 mathrm{cm} ) into a cylindrical coffee pot with diameter ( 15 mathrm{cm} . ) The constant rate at which coffee comes out from the filter into the pot is ( 100 mathrm{cm}^{3} / ) min. The rate in cm / min at which the level in the pot is rising at the instance when the coffee in the pot is ( 10 mathrm{cm} ), is A ( cdot frac{9}{16 pi} ) В ( cdot frac{25}{9 pi} ) c. ( frac{5}{3 pi} ) D. ( frac{16}{9 pi} ) | 12 |

755 | 12. The function f(x) = * +2 has a local minimum at [2006] (a) x=2 (C) x=0 (b) x= -2 (d) x=1 | 12 |

756 | Find the local maxima and local minima for the given function and also find the local maximum and local minimum values ( f(x)=sin x- ) ( cos x, 0<x<2 pi ) | 12 |

757 | The function ( y=sqrt{2 x-x^{2}} ) A. increases in (0,1) but decreases in (1,2) B. decreases in (0,2) C. increases in (1,2) but decreases in (0,1) D. increases in (0,2) | 12 |

758 | ( y=[x(x-3)]^{2} ) is increasing when A ( cdot 0<x<frac{3}{2} ) B. ( 0<x<infty ) c. ( -infty<x<0 ) D. ( 1<x<3 ) | 12 |

759 | If ( x ) and ( y ) are sides of two squares such that ( y=x-x^{2} ). Find the rate of change of area of second square (side ( y ) ) with respect to area of the first square (side ( x) ) when ( x=1 mathrm{cm} ) ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 3 | 12 |

760 | The function has ( boldsymbol{f}(boldsymbol{x})= ) ( (log (x-1))^{2}(x-1)^{2} ) has A. local extremum at ( x=1 ) B. point of inflection at ( x=1 ) c. local extremum at ( x=2 ) D. point of inflection at ( x=2 ) | 12 |

761 | Find the values of ( a ) and ( b ), if the slope of the tangent to the curve ( boldsymbol{x} boldsymbol{y}+boldsymbol{a} boldsymbol{x}+ ) ( b y=2 ) at (1,1) is 2 | 12 |

762 | Bacteria multiply at a rate proportional to the number present. If the original number ( N ) double in 3 hours, the number of the bacteria will be ( 4 N ) is (in hours) ( mathbf{A} cdot mathbf{6} ) B. 4 ( c .5 ) D. | 12 |

763 | A computer solved several problems in succession. The time it took the computer to solve each successive problem was the same number of times smaller than the time it took it to solve the preceding problem. How many problems were suggested to the computer if it spent ( 63.5 mathrm{mm} ) to solve al the problems except for the first, 127 ( mathrm{mm} ) to solve all the problems except for the last one, and ( 31.5 mathrm{mm} ) to solve all the problems except for the first two? | 12 |

764 | Mark the correct alternative of the following. For the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}} ) A. ( x=1 ) is a point of maximum B. ( x=-1 ) is a point of minimum c. Maximum value> minimum value D. Maximum value < minimum value | 12 |

765 | Consider the function ( f(x)=0.75 x^{4}- ) ( x^{3}-9 x^{2}+7 ) Consider the following statements: 1. The function attains local minima at ( x=-2 ) and ( x=3 ) 2. The function increases in the interval (-2,0) Which of the above statements is/are correct? A. 1 only B. 2 only c. Both 1 and 2 D. Neither 1 nor | 12 |

766 | The radius of a spherical soap bubble is increasing at the rate of ( 0.2 mathrm{cm} / mathrm{sec} ) Find the rate of increase of its surface area, when the radius is ( 7 mathrm{cm} ) | 12 |

767 | The function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{1+boldsymbol{x} tan boldsymbol{x}} ) has A. one point of minimum in the interval ( (0, pi / 2) ) B. one point of maximum in the interval ( (0, pi / 2) ) C. no point of maximum,no point of minumum in the interval ( (0, pi / 2) ) D. two points of maximum in the interval ( (0, pi / 2) ) | 12 |

768 | Let the function ( f(x)=sin x+cos x ) be defined in ( [0,2 pi], ) then ( f(x) ) A ( cdot ) increases in ( left(frac{pi}{4}, frac{pi}{2}right) ) B. decreases in ( left[frac{pi}{4}, frac{5 pi}{4}right] ) C . increases in ( left[0, frac{pi}{4}right] cup[pi, 2 pi] ) D. decreases in ( left[0, frac{pi}{4}right) cupleft(frac{pi}{2}, 2 piright] ) | 12 |

769 | Let the function ( f(x) ) be defined as follows: ( f(x)= ) ( left{begin{array}{cc}x^{3}+x^{2}-10 x & -1 leq x<0 \ cos x & 0 leq x<frac{pi}{2} \ 1+sin x & frac{pi}{2} leq x leq piend{array}, ) then right. which of the following statement(s) is/are correct This question has multiple correct options A. Local maximum at ( x=0 ) B . Local maximum at ( x=frac{pi}{2} ) c. Absolute maxima at ( x=-1 ) D. Absolute minima at ( x=pi ) | 12 |

770 | Find the derivative of ( f(x)=tan x ) at ( boldsymbol{x}=mathbf{0} ) | 12 |

771 | Two roads ( O A ) and ( O B ) intersect at an angle fo ( 60^{circ} . ) A car driver approaches 0 from ( A, ) where ( A O=800 ) metres, at a uniform speed of 20 metres per second. Simultaneously a runner starts running from ( boldsymbol{O} ) towards ( boldsymbol{B} ) at uniform speed of ( mathbf{5} ) metres per second. Find the time when the car and the runner are closest. A ( cdot frac{240}{7} ) sec B. ( frac{260}{7} ) sec c. ( frac{250}{7} mathrm{sec} ) D. ( frac{210}{7} mathrm{sec} ) | 12 |

772 | The radius of the balloon is variable. Find the rate of change of its volume, when the radius is ( 5 mathrm{cm} ) ? | 12 |

773 | The intercept on x-axis made by tangent to the curve, ( y=int_{0}^{x}|t| d t, x in R, ) which are parallel to the line ( y=2 x, ) are equal to A . ±1 B. ±2 ( c .pm 3 ) ( mathrm{D} cdot pm 4 ) | 12 |

774 | ( operatorname{Let} y=left(x+frac{4}{x^{2}}right) ) and ( x in+R ) The minimum value of ( y ) is A . -4 B. 3 c. 8 D. 7 | 12 |

775 | ( f(x)=x^{3}+a x^{2}+b x+c ) has a max. at ( x=-1 ) and ( min . ) at ( x=3 . ) Determine the constants ( a, b, c ) ( mathbf{A} ldots a=-3, b=-3, c=0 ) В ( ldots a=-3, b=-9, c=3 ) ( mathbf{c} ldots a=-3, b=9, c=9 ) D ( ldots a=3, b=9, c=-3 ) | 12 |

776 | The real number ( x ) when added to its inverse gives the minimum value of the ( operatorname{sum} operatorname{at} x= ) ( (boldsymbol{x} in+boldsymbol{R}) ) ( mathbf{A} cdot mathbf{1} ) B. 3 ( c cdot 2 ) D. | 12 |

777 | ( f(x) ) is cubic polynomial which has local maximum at ( mathbf{x}=-1 . ) If ( mathbf{f}(mathbf{2})= ) ( mathbf{1 8}, mathbf{f}(mathbf{1})=-mathbf{1} ) and ( mathbf{f}(mathbf{x}) ) has local minima at ( mathbf{x}=mathbf{0}, ) then This question has multiple correct options A. the distance between (-1,2) and ( (a, f(a)), ) where ( x=a ) is the point of local minima is ( 2 sqrt{5} ) B. f(x) is increasing for ( x in[1,2 sqrt{5} ) ( mathrm{c} . mathrm{f}(mathrm{x}) ) has local minima at ( mathrm{x}=1 ) D. the value of ( f(0)=5 ) | 12 |

778 | Given ( boldsymbol{f}(boldsymbol{x})=cos ^{2} boldsymbol{x}, ) find whether ( boldsymbol{f}(boldsymbol{x}) ) is increasing or decreasing in the range ( left[0, frac{pi}{2}right] ) | 12 |

779 | The sides of an equilateral triangle are increasing at the rate of ( 2 mathrm{cm} / mathrm{s} ). The rate at which the area increases when the side is ( 10 mathrm{cm} ), is ( mathbf{A} cdot sqrt{3} c m^{2} / s ) B. ( 10 mathrm{cm}^{2} / mathrm{s} ) c. ( 10 sqrt{3} mathrm{cm}^{2} / mathrm{s} ) D. ( frac{10}{sqrt{3}} c m^{2} / s ) | 12 |

780 | A particle moves along a straight line according to the law ( s=16-2 t+3 t^{3} ) where ( s ) metres is the distance of the particle from a fixed point at the end of ( t ) second. The acceleration of the particle at the end of 2 s is A ( cdot 3.6 m / s^{2} ) B. ( 36 m / s^{2} ) c. ( 36 k m / s^{2} ) D. ( 360 m / s^{2} ) | 12 |

781 | The maximum value of ( left(frac{log x}{x}right) ) is A ( cdotleft(frac{1}{e}right. ) B. ( frac{2}{e} ) ( c cdot e ) D. | 12 |

782 | If there is an error of ( 2 % ) in measuring the length of a simple pendulum, then percentage error in its period is A . ( 1 % ) B. 2% ( c .3 % ) D. ( 4 % ) | 12 |

783 | : The function ( f(x)=2 x^{3}-3 x^{2}- ) ( 12 x+8 ) attains minimum value at ( x= ) 2 II: The function of ( f(x)=x^{4}-6 x^{2}+ ) ( 8 x+11 ) attains minimum value at ( x= ) 2 which of the above statements are true A. onlyı B. only II c. both I and II D. neither I nor II | 12 |

784 | *** v y u taurus 7. The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when the side is 10 cm is (a) V3 sq. unit/sec (b) 10 sq. unit/sec 10 (c) 103 sq. unit/sec (d) to sq. unit/sec | 12 |

785 | The total revenue in Rupees received from the sale of ( x ) units of a product is given by ( boldsymbol{R}(boldsymbol{x})=mathbf{3} boldsymbol{x}^{2}+mathbf{3 6} boldsymbol{x}+mathbf{5} . ) The marginal revenue, when ( boldsymbol{x}=mathbf{1 5} ) is. A . 116 B. 96 ( c cdot 90 ) D. 126 | 12 |

786 | Consider the following statements: Statement I ( x>sin x ) for all ( x>0 ) Statement II: ( f(x)=x-sin x ) is an increasing function for all ( x>0 ) Which one of the following is correct in respect of the above statements? A. Both Statements I and II are true and Statement II is the correct explanation of statement B. Both Statements I and II are true and Statement II is the not correct explanation of Statement c. statement lis true but Statement II is false D. Statement I is true but Statement II is true | 12 |

787 | Find the minimum distance of any point on the curve ( x^{2}+y^{2}+2 x y=8 ) from the origin. | 12 |

788 | Find intervals in which the function ( operatorname{given} operatorname{by} f(x)=sin 3 x, x inleft[0, frac{pi}{2}right] ) is decreasing. | 12 |

789 | If ( y=7 x-x^{3} ) and ( x ) increases at the rate of 4 units per second, how fast is the slope of the curve changing when ( x=2 ? ) | 12 |

790 | If ( y=6 x-x^{3} ) and ( x ) increases at the rate of 5 units per second, the rate of change of slope when ( x=3 ) is A. -90 units/sec B. 90 units/ sec c. 180 units/sec D. -180 units/sec | 12 |

791 | Find the value of ( theta ) for attaining a max value of ( sin ^{p} theta cos ^{q} theta ) is A ( cdot tan ^{-1} frac{p}{q} ) B. ( tan ^{-1} frac{sqrt{p}}{sqrt{q}} ) c. ( tan ^{-1}(p+q) ) ( mathbf{D} cdot tan ^{-1} q ) | 12 |

792 | cosine of the angle of intersection of curves ( y=3^{x-1} ln x ) and ( y=x^{x}-1 ) is ( mathbf{A} cdot mathbf{1} ) B. ( 1 / 2 ) ( c cdot 0 ) D. ( 1 / 3 ) | 12 |

793 | A stone dropped into a pond of still water sends out concentric circular waves from the point of disturbance of water at the rate of ( 4 mathrm{cm} / ) sec. Find the rate of change of disturbed area at the instant when the radius of wave ring is ( mathbf{1 5} c m ) | 12 |

794 | 9. A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase of the surface area of the balloon when its diameter is 14 cm is (a) 7 sq. cm/min (b) 10 sq. cm/min (c) 17.5 sq. cm/min (d) 28 sq. cm/min . | 12 |

795 | The rate of change of the area of a circle with respect to its radius ( r ) at ( r=6 c m ) is. ( mathbf{A} cdot 10 pi ) в. ( 12 pi ) ( c cdot 8 pi ) D. ( 11 pi ) | 12 |

796 | The equation normal to the curve ( boldsymbol{x}^{2 / 3}+boldsymbol{y}^{2 / 3}=boldsymbol{a}^{2 / 3} ) at the point ( (boldsymbol{a}, boldsymbol{0}) ) is ( mathbf{A} cdot x=a ) B. ( x=-a ) ( mathbf{c} cdot y=a ) D. ( y=-a ) | 12 |

797 | The largest term of the sequence ( a_{n}= ) ( frac{n}{left(n^{2}+10right)} ) is A ( cdot frac{3}{19} ) B. ( frac{2}{13} ) c. 1 D. | 12 |

798 | If the line ( a x+b y+c=0 ) is a normal to the rectangular hyperbola ( boldsymbol{x} boldsymbol{y}=mathbf{1} ) then This question has multiple correct options A ( . a>0, b>0 ) в. ( a>0, b<0 ) c. ( a0 ) D. ( a<0, b<0 ) | 12 |

799 | Example 2.4 Two bodies start moving in the same straight line at the same instant of time from the same origin. The first body moves with a constant velocity of 40 ms, and the second starts from rest with a constant acceleration of 4 ms. Find the time that elapses before the second catches the first body. Find also the greatest distance between them prior to it and time at which this occurs. | 12 |

800 | A point on the parabola ( y^{2}=18 x ) at which the ordinate increases at twice the rate of the abscissa is в. (2,-4) ( ^{mathrm{c}} cdotleft(-frac{9}{8}, frac{9}{2}right) ) D ( cdotleft(frac{9}{8}, frac{9}{2}right) ) | 12 |

801 | If the length of the diagonal of a square is increasing at the rate of ( 0.2 mathrm{cm} / mathrm{sec} ) then rate of increase of its area when its side is ( 30 / sqrt{2} mathrm{cm}, ) is ( mathbf{A} cdot 3 mathrm{cm}^{2} / mathrm{sec} ) в. ( frac{6}{sqrt{2}} mathrm{cm}^{2} / mathrm{sec} ) ( c cdot sqrt[3]{2} mathrm{cm}^{2} / mathrm{sec} ) D. ( 6 mathrm{cm}^{2} / mathrm{sec} ) | 12 |

802 | The function ( f(x)=frac{x}{1+x tan x} ) has a point of minimum in the interval ( left(0, frac{pi}{2}right) ) Bne point of maximum in the interval ( left(0, frac{pi}{2}right) ) C. No point of maximum, no point of minimum in the interval ( left(0, frac{pi}{2}right) ) Two points of maxima in the interval ( left(0, frac{pi}{2}right) ) | 12 |

803 | The maximum value of ( frac{log x}{x} ) in ( (2, infty) ) is A. 5 B. ( frac{5}{e} ) ( mathbf{c} cdot e^{e} ) D. | 12 |

804 | A particle is moving in a straight line such that its distance at any time ( t ) is given by ( s=frac{1}{4} t^{4}-2 t^{3}+4 t^{2}-7 . ) Find twhen its velocity is maximum( ( t_{v} ) ) and acceleration minimum(t ( _{a} ) ). A ( cdot t_{v}=2-2 / sqrt{3}, t_{a}=4 ) В ( cdot t_{v}=2, t_{a}=1 ) C ( cdot t_{v}=2-2 / sqrt{3}, t_{a}=2 ) D. ( t_{v}=-2 / sqrt{3}, t_{a}=2 ) | 12 |

805 | Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( (26)^{frac{1}{3}} ) | 12 |

806 | Solve: ( boldsymbol{x}^{4}-boldsymbol{x}^{mathbf{3}}+boldsymbol{x}^{2}-boldsymbol{x}+mathbf{1}=mathbf{0} ) | 12 |

807 | ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) has A. minimum at ( x=0 ) B. maximum at ( x=0 ) c. neither a maximum nor a minimum at ( x=0 ) D. none of these | 12 |

808 | Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=[boldsymbol{x}+sqrt{boldsymbol{x}+} sqrt{boldsymbol{x}}]^{1 / 2}, ) at ( boldsymbol{x}= ) ( mathbf{1} ) A. ( frac{3+4 sqrt{2}}{8 sqrt{2}(sqrt{1+sqrt{2}})} ) B. Not defined ( c cdot 0 ) D. | 12 |

809 | A spherical balloon is being inflated so that its volume increase uniformly at the rate of ( 40 mathrm{cm}^{3} / ) minute. The rate of increase in its surface area when the radius is ( 8 mathrm{cm} ) is ( mathbf{A} cdot 10 mathrm{cm}^{2} / mathrm{minute} ) B . ( 20 mathrm{cm}^{2} / ) minute ( mathbf{c} cdot 40 mathrm{cm}^{2} / mathrm{minute} ) D. none of these | 12 |

810 | If the sum of the lengths of the hypothesis and another side of a right angle is given, show that the area of the triangle is maximum when the angle between these sides is ( frac{pi}{3} ) | 12 |

811 | If ( f ) and ( g ) are two decreasing function such that ( f o g ) is defined, then fog will be A. increasing function B. decreasing function c. neither increasing nor decreasing D. None of these | 12 |

812 | In a ( Delta A B C ) the sides b and ( c ) are given. If there is an error ( Delta A ) in measuring angle ( A, ) then the error ( Delta a ) in side a is given by ( ^{text {A }} cdot frac{S}{2 a} Delta A ) в. ( frac{2 S}{a} Delta A ) ( c cdot b c sin A Delta A ) D. none of these | 12 |

813 | Using differential, find the approximate value of the following: ( sqrt{401} ) | 12 |

814 | Find the absolute maximum and minimum values of a function ( f ) given by ( f(x)=2 x^{3}-15 x^{2}+36 x+1 ) on the interval ( [mathbf{1}, mathbf{5}] ) | 12 |

815 | The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is ( 8 m / s^{2} . ) The time taken by the particle to move the second metre is A ( cdot frac{sqrt{2}-1}{2} s ) B. ( frac{sqrt{2}+1}{2} s ) ( mathbf{c} cdot(1+sqrt{2}) s ) D. ( (sqrt{2}-1) s ) | 12 |

816 | defined by ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{l}k-2 x, text { if } x leq 1 \ 2 x+3, text { if } x>-1end{array}right} ), if has a loca minimum at ( x=-1, ) then a pair | 12 |

817 | The displacement ( x ) of a particle moving in one dimension under the action of a constant force is related to time ( t ) by the equation ( t=sqrt{x}+3, ) where ( x ) is in meter and ( t ) is in second. Find the displacement of the particle when its velocity is zero. | 12 |

818 | 21. The point(s) on the curve y3 + 3×2 = 12y where the tange is vertical, is (are) (2002) (c) (0,0) | 12 |

819 | 27. The tangent to the le tangent to the curve y = et drawn at the point (c, e) intersects the line joining the points (C – 1, e ) and (c +1, ec+1) (2007 -3 marks) (a) on the left of x = c . (b) on the right of x = 0 (C) at no point (d) at all points | 12 |

820 | Find the point on the parabola ( y^{2}=18 x ) at which the ordinate increases at twice the rate of the abscissa. | 12 |

821 | A man ( 1.5 mathrm{m} ) tall walks away from a lamp post ( 4.5 mathrm{m} ) high at the rate of 4 km/hr. How fast is the farther end of shadow moving on the pavement A. ( 4 mathrm{km} / mathrm{hr} ) B. ( 2 mathrm{km} / mathrm{hr} ) c. ( 6 mathrm{km} / mathrm{hr} ) D. None of these | 12 |

822 | In which one of the following intervals is the function ( f(x)=x^{2}-5 x+6 ) decreasing? A ( cdot x<frac{5}{2} ) в. ( xfrac{2}{5}} ) D. none | 12 |

823 | Let ( f ) be a function defined on ( R ) (the set of all real numbers) such that ( mathbf{f}^{prime}(mathbf{x})= ) ( 2010(x-2009)(x-2010)^{2}(x- ) 2011)( ^{3}(x-2012)^{4}, ) for all ( x in R ). If ( g ) is a function defined on ( mathbf{R} ) with values in the interval ( (0, infty) ) such that ( f(x)= ) ( ln (mathrm{g}(mathrm{x})), ) for all ( mathrm{x} in mathrm{R}, ) then the number of points in ( mathbf{R} ) at which ( mathbf{g} ) has a local maximum is A . B. ( c cdot 2 ) D. 3 | 12 |

824 | The minimum value of ( 4 cos ^{2} x+ ) ( 5 sin ^{2} x ) is | 12 |

825 | Assertion ( f(x) ) is increasing with concavity upwards, then concavity of ( boldsymbol{f}^{-1}(boldsymbol{x}) ) is also upwards. Reason If ( boldsymbol{f}(boldsymbol{x}) ) is decreasing function with concavity upwards, then concavity of ( f^{-1}(x) ) is also upwards 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 and Reason are correct | 12 |

826 | (a) 1 (0 ) 2 The shortest distance between the line y – x = 1 and the curve x=y2 is: [2009] (a | 12 |

827 | Assertion Let ( f ) and ( g ) be increasing and decreasing functions respectively from ( [0, infty] ) to ( [0, infty] . operatorname{Let} h(x)=f(g(x)) . ) If ( h(0)=0, ) then ( h(x) ) is always zero Reason ( h(x) ) is an increasing function of ( x ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |

828 | Find the approximate error in the volume of a cube with edge ( x mathrm{cm}, ) when the edge is increased by ( 2 % ) A . ( 4 % ) в. ( 2 % ) ( c .6 % ) D. ( 8 % ) | 12 |

829 | Function ( x-sin x ) has A. a maxima B. a minima c. a maxima and a minima D. no maxima and no minima | 12 |

830 | Assertion The largest term in the sequence ( a_{n}= ) ( frac{n^{2}}{n^{3}+200}, n in N ) is the 7 th term. Reason The function ( f(x)=frac{x^{2}}{x^{3}+200} ) attains local maxima at ( x=7 ) 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 |

831 | Mark the correct alternative of the following. The maximum value of ( x^{1 / x}, x>0 ) is? A ( cdot e^{1 /} ) ( ^{mathrm{B}}left(frac{1}{e}right) ) c. 1 D. None of these | 12 |

832 | For ( boldsymbol{a} in[boldsymbol{pi}, boldsymbol{2} boldsymbol{pi}] ) and ( boldsymbol{n} in boldsymbol{Z}, ) the critical points of ( boldsymbol{f}(boldsymbol{x})=frac{1}{mathbf{3}} sin boldsymbol{a} tan ^{3} boldsymbol{x}+ ) ( (sin a-1) tan x+sqrt{frac{a-2}{8-a}} ) are A . ( x=n pi ) B. ( x=2 n pi ) c. ( x=(2 n+1) pi ) D. None of these | 12 |

833 | The top of a ladder 6 meters long is resting against a vertical wall.Suddenly the ladder begins to slide outwards. At the instant when the foot of the ladder is 4 meters from the wall, it is sliding away at the rate of ( 0.5 mathrm{m} / ) sec. How fast is the top sliding downward at this moment? | 12 |

834 | 43. If the tangent to the curve, y=x3 + ax – b at the point (1,-5) is perpendicular to the line, -x+y+4=0, then which one of the following points lies on the curve? JJEEM 2019-9 April (M) (a) (-2,1) (b) (-2,2) (c) (2,-1) (d) (2,-2) | 12 |

835 | Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. 1 ( (255)^{overline{4}} ) | 12 |

836 | If the radius of a circle is diminished by ( 10 %, ) then its area is diminished by: A . ( 10 % ) B. ( 19 % ) ( c cdot 20 % ) D. ( 36 % ) | 12 |

837 | If ( e^{d y / d x}=x+1 ) given that when ( x= ) ( mathbf{0}, boldsymbol{y}=mathbf{3} ) then minimum (local) value of ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) is | 12 |

838 | The approximate value of ( (1.0002)^{3000} ) is A . 1.2 B. 1.4 c. 1.6 D. 1.8 | 12 |

839 | The two tangents to the curve ( a x^{2}+ ) ( 2 h x y+b y^{2}=1, a>0 ) at the points where it crosses the X-axis, are A. parallel B. perpendicular C . inclined at an angle ( frac{pi}{4} ) D. None of these | 12 |

840 | If an error of ( 1^{circ} ) is made in measuring the angle of a sector of radius ( 30 mathrm{cm} ) then the approximate error in its area is A ( cdot 450 mathrm{cm}^{2} ) В ( cdot 25 pi c m^{2} ) ( mathbf{c} cdot 2.5 pi c m^{2} ) D. none of these | 12 |

841 | Find the derivative of the following function at the indicated points. ( 2 cos x ) at ( x=frac{pi}{2} ) | 12 |

842 | If a function ( f(x) ) has ( f^{prime}(a)=0 ) and ( f^{prime prime}(a)=0, ) then A. ( x=a ) is a maximum for ( f(x) ) B. ( x=a ) is minimum for ( f(x) ), c. It is difficult to say ( (a) ) and ( (b) ) D. ( f(x) ) is necessarily a constant function. | 12 |

843 | I. If the curve ( y=x^{2}+b x+c ) touches the straight line ( y=x ) at the point (1,1) then ( b ) and ( c ) are given by 1,1 Il. If the line ( boldsymbol{P x}+boldsymbol{m} boldsymbol{y}+boldsymbol{n}=boldsymbol{0} ) is a normal to the curve ( x y=1, ) then ( P> ) ( mathbf{0}, boldsymbol{m}<mathbf{0} ) Which of the above statements is correct A. onlyı B. only II c. both I and II D. Neither I nor II | 12 |

844 | Consider the function. ( f(x)=3 x^{4}-20 x^{3}-12 x^{2}+288 x+1 ) In which one of the following intervals is the function decreasing? ( mathbf{A} cdot(-1,0) ) в. (0,2) c. ( (2, infty) ) D. none of these | 12 |

845 | The function ( f(x)=2 x^{3}-15 x^{2}+ ) ( 36 x+6 ) is strictly decreasing in the interval A. (2,3) (年) (2,3,3) в. ( (-infty, 2) ) c. (3,4) (年. ( (3,4)) ) D. ( (-infty, 3) cup(4, infty) ) E ( .(-infty, 2) cup(3, infty) ) | 12 |

846 | 13. If 2)(t-3)dt for all x € (0,00), then (2012) (a) f has a local maximum at x=2 (b) f is decreasing on (2,3) (c) there exists somece (0,0), such that f'(c)=0 (d) f has a local minimum at x=3 | 12 |

847 | The value of ( x ) at which tangent to the curve ( boldsymbol{y}=boldsymbol{x}^{3}-boldsymbol{6} boldsymbol{x}^{2}+boldsymbol{9} boldsymbol{x}+boldsymbol{4}, boldsymbol{0} leq boldsymbol{x} leq ) 5 has maximum slope is ( mathbf{A} cdot mathbf{0} ) B. 2 ( c cdot frac{5}{2} ) ( D ) | 12 |

848 | The slope of the tangent to the curve ( boldsymbol{x}=mathbf{3} boldsymbol{t}^{2}+mathbf{1}, boldsymbol{y}=boldsymbol{t}^{3}-mathbf{1} ) at ( boldsymbol{x}=mathbf{1} ) is A ( cdot frac{1}{2} ) B. ( c cdot-2 ) ( D cdot infty ) | 12 |

849 | For which region is ( f(x)=3 x^{2}-2 x+ ) 1 strictly increasing? This question has multiple correct options ( mathbf{A} cdot(2,5) ) B. ( left(frac{1}{3}, inftyright) ) c. ( left(-1, frac{1}{3}right] ) D. ( left(-infty, frac{1}{3}right) ) | 12 |

850 | The difference between the greatest and the least value of ( boldsymbol{f}(boldsymbol{x})= ) ( cos ^{2} frac{x}{2} sin x, x in[0, pi] ) is ( A cdot frac{3 sqrt{3}}{8} ) в. ( frac{sqrt{3}}{8} ) ( c cdot frac{3}{8} ) D. ( frac{1}{2 sqrt{2}} ) | 12 |

851 | A particle starts with some initial velocity with an acceleration along the direction of motion. Draw a graph depicting the variation of velocity ( (v) ) along y-axis with the variation of displacement ( (s) ) along ( x ) -axis. | 12 |

852 | The minimum value of ( boldsymbol{f}(boldsymbol{x})= ) ( max {x, 1+x, 2-x} ) is A ( cdot frac{1}{2} ) B. ( frac{3}{2} ) c. D. 0 E . 2 | 12 |

853 | 10. Find all the tangents to the curve y = cos(x + y), – 21 Sxs 21, that are parallel to the line x+2y=0. (1985 – 5 Marks) | 12 |

854 | ff ( boldsymbol{y}=mathbf{8} boldsymbol{x}^{3}-boldsymbol{6} mathbf{0} boldsymbol{x}^{2}+mathbf{1 4 4} boldsymbol{x}+mathbf{2 7} ) is a strictly decreasing function in the interval A. (-5,6) В ( cdot(-infty, 2) ) c. (5,6) (年. ( 5,6,6) ) D. ( (3, infty) ) E ( .(2,3) ) | 12 |

855 | In the interval ( boldsymbol{x} epsilonleft(boldsymbol{e}^{2 r boldsymbol{pi}+boldsymbol{pi} / 4}, boldsymbol{e}^{2 boldsymbol{r} boldsymbol{pi}+boldsymbol{5} boldsymbol{pi} / boldsymbol{4}}right) ) in which the function ( f(x)= ) ( sin left(log _{e} xright)+cos left(log _{e} xright) ) is A. decreasing B. increasing c. const. D. no monotonicity | 12 |

856 | The maximum value of the function ( y=frac{1}{4 x^{2}+2 x+1} ) is A ( cdot frac{4}{3} ) в. ( frac{5}{2} ) c. ( frac{13}{4} ) D. None of these | 12 |

857 | A particle moves along the curve ( y= ) ( (2 / 3) x^{3}+1 . ) Find the points on the curve at which the y-coordinate is changing twice as fast as the ( x ) coordinate. | 12 |

858 | 50 COLUuus The function f(x)=max {(1-x), (1+x), 2), xe(-0, 0) is (a) continuous at all points (1995) (6) differentiable at all points c) differentiable at all points except at x = 1 and x=-1 (d) continuous at all points except at x = 1 and x = -1, where it is discontinuous | 12 |

859 | The equation of tangent to the curve ( y=b e^{-x / a} ) where it cuts the ( y ) -axis is A ( cdot frac{x}{a}+frac{y}{b}=1 ) B. ( frac{x}{a}+frac{y}{b}=-1 ) c. ( frac{x}{a}-frac{y}{b}=1 ) D. none of these | 12 |

860 | Assertion ( f(x)=frac{1}{x-7} ) is decreasing ( forall x epsilon R ) {7} Reason ( boldsymbol{f}^{prime}(boldsymbol{x})<mathbf{0} forall boldsymbol{x} neq mathbf{7} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |

861 | The minimum value of ( 2^{left(x^{2}-3right)^{3}+27} ) is ( A cdot 2^{27} ) B. ( c cdot 2 ) D. None of the above | 12 |

862 | For what values of a does the minimum value of the function ( y=x^{2}-4 a x- ) ( a^{4} ) assume the greatest value? | 12 |

863 | If the distance of the point on ( y=x^{4}+ ) ( 3 x^{2}+2 x ) which is nearest to the line ( boldsymbol{y}=2 boldsymbol{x}-mathbf{1} ) is ( mathbf{p}, ) Find ( mathbf{5} boldsymbol{p}^{2} ) | 12 |

864 | A swimming pool is to be drained for cleaning. If L represents the number of liters of water in the pool t seconds after the pool has been plugged off to drain and ( L=200(10-t)^{2} . ) What is the average rate at which the water flows out during the first 5 seconds ( left(frac{text {litres}}{text {sec}}right) ) 7 | 12 |

865 | It is desired to construct a cylindrical vessel of capacity 500 cubic metres open at the top.What should be the dimensions of the vessel so that the material need is minimum, given that the thickness of the material used is 2 ( mathrm{cm} ) ( ^{mathbf{A}} cdot quad r=left(frac{100}{pi}right)^{1 / 3}, h=5left(frac{100}{pi}right)^{1 / 3} ) B. ( _{r}=left(frac{500}{pi}right)^{1 / 3}=h ) ( r=left(frac{1000}{pi}right)^{1 / 3}, h=left(frac{125}{pi}right)^{1 / 3} ) ( r=left(frac{20}{pi}right)^{1 / 3}, h=left(frac{100}{pi}right)^{1 / 3} ) | 12 |

866 | Find the maximum and the minimum values, if any, without using derivatives of the following function: ( f(x)=4 x^{2}-4 x+4 ) on ( R ) | 12 |

867 | A window is in the shape of a rectangle surmounted by a semi circle. If the perimeter of the window is of fixed length I then the maximum area of the window is A ( cdot frac{I^{2}}{2 pi+4} ) B. ( frac{I^{2}}{pi+8} ) c. ( frac{I^{2}}{2 pi+8} ) D. ( frac{I^{2}}{8 pi+4} ) | 12 |

868 | ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+frac{1}{x}, boldsymbol{x} neq 0, ) then A. ( f(x) ) has no point of local maxima B. ( f(x) ) has no point local minima C ( . f(x) ) has exactly has one point of local minima D. ( f(x) ) has exactly has two point of local minima | 12 |

869 | 10. A population p(t) of 1000 bacteria introduced into nutrient medium grows according to the relation p(t) = 1000 + 1000t “. The maximum size of this bacterial population 100+ 2 is (a) 1100 (c) 1050 (b) 1250 (d) 5250 | 12 |

870 | Calculate the rate of flow of glycerin of density ( 1.25 times 10^{3} k g / m^{3} ) through the conical section of a pipe, if the radii of its ends are ( 0.1 m ) and 0.04 m and the pressure drop across its length is ( mathbf{1 0} N / boldsymbol{m}^{2} ) A ( cdot 6.43 times 10^{-4} m^{3} / s ) B. ( 5.43 times 10^{-4} mathrm{m}^{3} / mathrm{s} ) c. ( 5.44 times 10^{-3} mathrm{m}^{3} / mathrm{s} ) D. ( 6.43 times 10^{-3} mathrm{m}^{3} / mathrm{s} ) | 12 |

871 | 31. Let f(x) be a polynomial of degree four having extreme values [ f(x) at x= 1 and x=2. If lim 1+- then f(2) is equal to : x>0L (a) o (6) 4 (6 -8 [JEEM 2015] (d) -4 | 12 |

872 | A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is ( 15 mathrm{cm} ) | 12 |

873 | ( operatorname{Let} h(x)=f(x)-(f(x))^{2}+(f(x))^{3} ) for every real number ( x, ) then A. ( h ) is increasing whenever ( f ) is increasing and decreasing whenever ( f ) is decreasing B. ( h ) is increasing whenever ( f ) is decreasing c. ( h ) is decreasing whenever ( f ) is increasing D. Nothing can be said in general | 12 |

874 | The maximum value of ( x^{1 / x} ) is A ( cdot frac{1}{e^{e}} ) B. ( mathbf{c} cdot e^{1 / e} ) D. | 12 |

875 | Find an angle ( theta, 0<theta<frac{pi}{2}, ) which increases twice as fast as its sine. | 12 |

876 | 15. Let f(x)= 11 |xl, for 0 < x < 2 1 then at x=0, f has for x = 0 (a) a local maximum (c) a local minimum (2000) (b) no local maximum (d) no extremum | 12 |

877 | Which one of the following curves cut the parabola y2 = 4ax at right angles? (1994) (a) x2 + y2 = a? (b) y=e-x/2a (c) y = ax (d) x2 = 4ay | 12 |

878 | The real value of ( k ) for which ( f(x)= ) ( x^{2}+k x+1 ) is increasing on ( (1,2), ) is A . -2 B. – 1 ( c .1 ) D. 2 | 12 |

879 | Illustration 3.42 If a + B = 90°, find the maximum value of sin a sin ß. | 12 |

880 | The function ( f(x)=x^{2}+2 x-5 ) is strictly increasing in the interval B ( cdot(-infty,-1] ) ( c cdot[-1, infty) ) D. ( (-1, infty) ) | 12 |

881 | 9 10. Divide 20 into two parts such that the product of one part and the cube of the other is maximum. The two parts are (a) (10, 10) (b) (5, 15) (c) (13,7) (d) None of these | 12 |

882 | The function ( f(x)=x^{3}-3 x^{2}+6 ) is an increasing function for: A. ( 0<x<2 ) B. ( x2 ) or ( x<0 ) D. all ( x ) | 12 |

883 | 35. The normal to the curve y(x – 2)(x-3)=x+6 at the point where the curve intersects the y-axis passes through the JEE M 2017] point: | 12 |

884 | A small hole is made at the button of a symmetrical jar as shown in figure. liquid is filled into the jar upto a certain height. The rate of descension of liquid is independent of the level of liquid in the jar. Then the surface of jar is a surface of revolution of the curve ( mathbf{A} cdot y=k x^{4} ) ( mathbf{B} cdot y=k x^{2} ) ( mathbf{c} cdot y=k x^{3} ) D. ( y=k x^{5} ) | 12 |

885 | A stone is dropped into a quiet lake and waves move in circles at a speed of 4 ( mathrm{cm} / mathrm{sec} . ) At the instant when the radius of the circular wave is ( 10 mathrm{cm}, ) how fast is the enclosed area increasing? | 12 |

886 | Given that carbon ( 14left(C_{14}right) ) decays at a constant rate in such a way that it reduces to ( 50 % ) in 5568 years. Find the age of an old wooden piece in the carbon is only ( 12 frac{1}{2} % ) of the original. | 12 |

887 | Show that ( f(x)=sin x ) is an increasing function on ( (-pi / 2, pi / 2) ? ) | 12 |

888 | ( f(x)=sin x ) -ax is decreasing in ( R ) if ( mathbf{A} cdot mathbf{a}>1 ) B. ( afrac{1}{2}} ) D. ( a<frac{1}{2} ) | 12 |

889 | If ( x ) changes from 4 to 4.01 then find the approximate change in ( log x ) | 12 |

890 | OLLOIOL 21001 20. Given P(x)=x4 +ar3 + bx2 + cx + d such that x=0 is the only real root of P'(x) = 0. If P(-1)< P(1), then in the interval [-1,1]: [2009] (a) P(-1) is not minimum but P(1) is the maximum of P (b) P(-1is the minimum but P(1) is not the maximum of P (C) Neither P(-1) is the minimum nor P(1) is the maximum of P (d) P(-1) is the minimum and P(1) is the maximum ofP | 12 |

891 | The value of ( K ) in order that ( f(x)= ) ( sin x-cos x-K x+5 ) decreases for all positive real values of ( x ) is given by A ( . Ksqrt{2} ) D. ( K<sqrt{2} ) | 12 |

892 | The coordinates of a point ( boldsymbol{P} ) on the line ( 2 x-y+5=0 ) such that ( |P A-P B| ) is maximum where ( A ) is (4,-2) and ( B ) is (2,-4) will be A ( .(11,27) ) B. (-11,-17) c. (-11,17) D. (0.5) | 12 |

893 | If the global maximum value of ( boldsymbol{f}(boldsymbol{x})= ) ( -x^{2}+a x-a^{2}-2 a-2 ) for ( x epsilon[0,1] ) be -5′, then ‘a’ can take the value This question has multiple correct options ( ^{mathrm{A}} cdot frac{-4+2 sqrt{13}}{3} ) B. ( frac{-4-2 sqrt{13}}{3} ) c. 1 D. – 3 | 12 |

894 | N characters of information are held on magnetic tape, in batches of ( x ) characters each,the batch processing time is ( alpha+beta x^{2} ) seconds, ( alpha ) and ( beta ) are constants. The optical value of ( x ) for fast processing is, ( mathbf{A} cdot alpha / beta ) в. ( beta / alpha ) c. ( sqrt{alpha / beta} ) D. ( sqrt{beta / alpha} ) | 12 |

895 | If the radius of a sphere is measured as ( 9 m ) with an error of ( 0.03 m, ) then find the approximate error in calculating in surface area. | 12 |

896 | ( ln a Delta A B C ) if sides a and b remain constant such that ( alpha ) is the error in ( C ) then relative error in its area is A. ( alpha cot C ) B. ( alpha sin C ) ( c cdot alpha tan C ) D. ( alpha cos C ) | 12 |

897 | 32. Consider : X 1. A normal to y=f(x) at x = also passes through the point : [JEE M 2016] (a) (5.0) (0) (69) (C) (0,0)(a) (637 | 12 |

898 | The side of a square of metal is increasing at the rate of ( 5 mathrm{cm} / ) minute. At what are its area is increasing when the side is ( 20 mathrm{cm} ) long? | 12 |

899 | The pressure ( boldsymbol{P} ) and the volume ( boldsymbol{v} ) of ( mathbf{a} ) gas are connected by the relation ( p v^{1.4}= ) const. Find the percentage error in ( p ) corresponding to a decrease of ( frac{1}{2} % ) in ( boldsymbol{v} ) | 12 |

900 | A spherical balloon is filled with ( 4500 pi ) cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of ( 72 pi ) cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is: A ( cdot 6 / 7 ) B. ( 4 / 9 ) c. ( 2 / 9 ) D. None of these | 12 |

901 | The velocity of any particle at maximum height is equal to | 12 |

902 | Let ( f(x) ) and ( g(x) ) are two functions which are defined and differentiable for all ( boldsymbol{x} geq boldsymbol{x}_{0} . ) If ( boldsymbol{f}left(boldsymbol{x}_{0}right)=boldsymbol{g}left(boldsymbol{x}_{0}right) ) and ( boldsymbol{f}^{prime}(boldsymbol{x})> ) ( g^{prime}(x) ) for all ( x>x_{0} ) then A ( cdot f(x)x_{0} ) B. ( f(x)=g(x) ) for some ( x>x_{0} ) C. ( f(x)>g(x) ) only for some ( x>x_{0} ) D. ( f(x)>g(x) ) for all ( x>x_{0} ) | 12 |

903 | A spherical balloon is being inflated so that its volume increases uniformly at the rate of ( 40 mathrm{cm}^{3} / mathrm{min.} ) At ( r=8 mathrm{cm}, ) its surface area increases at the rate ( mathbf{A} cdot 8 mathrm{cm}^{2} / mathrm{min} ) B . ( 10 mathrm{cm}^{2} / mathrm{min} ) c. ( 20 mathrm{cm}^{2} / mathrm{min} ) D. none of these | 12 |

904 | The smallest value of ( x^{2}-3 x+3 ) in the interval ( left(-3, frac{3}{2}right) ) is A ( cdot frac{3}{4} ) B. 5 ( c .-15 ) D . -20 | 12 |

905 | Show that the function ( boldsymbol{f}(boldsymbol{x})= ) ( cot ^{-1}(sin x+cos x) ) is decreasing on ( left(0, frac{pi}{4}right) ) and increasing on ( left(frac{pi}{4}, frac{pi}{2}right) ) | 12 |

906 | 25. If f(x)=x2 + 2br + 2.2 and g(x) = -x2 – 2cx+bsuch that min f(x) > max g(x), then the relation between b and c, is (2003) (a) no real value of b&c (b) 0<c<bv2 (c) lcl|b|v2 11 | 12 |

907 | For what values of ( x ) is the rate of increase of ( x^{3}-5 x^{2}+5 x+8 ) is twice the rate of increase of ( x ? ) A ( cdot-3,-frac{1}{3} ) в. ( _{-3, frac{1}{3}} ) c. ( _{3,-frac{1}{3}} ) D. ( _{3, frac{1}{3}} ) | 12 |

908 | 31. The least value of a E R for which 4ax2 +-21, for allx>0, is (JEE Adv. 2016) | 12 |

909 | The rate of change of area of a circle with respect to its radius at ( r=2 mathrm{cm} ) is A .4 в. ( 2 pi ) ( c cdot 2 ) D. ( 4 pi ) | 12 |

910 | Find the greatest and the least values of the following functions. Fin the extrema of the function ( boldsymbol{f}(boldsymbol{x})= ) ( 2 x sin 2 x+cos 2 x-sqrt{3} ) on the interval ( [-boldsymbol{pi} / 2, mathbf{3} boldsymbol{pi} / mathbf{8}] ) | 12 |

911 | Find the point on the curve ( y^{2}=8 x ) for which the abscissa and ordinate change at the same rate. | 12 |

912 | Find the maximum and minimum values, if any of the following function given by ¡) ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}+mathbf{2}|-mathbf{1} ) ii) ( boldsymbol{g}(boldsymbol{x})=-|boldsymbol{x}+mathbf{1}|+mathbf{3} ) iii) ( boldsymbol{f}(boldsymbol{x})=|sin 4 boldsymbol{x}+boldsymbol{3}| ) | 12 |

913 | Find the maximum and minimum value of the function: ( f(x)=2 x^{3}-21 x^{2}+36 x-20 ) | 12 |

914 | 11. A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of 3 cm/sec. The height of the upper end while it is descending at the rate of 4 cm/sec is (a) 413m (b) 6 m (c) 572m (d) 8 m | 12 |

915 | The two curves ( x^{3}-3 x y^{2}+2=0 ) and ( mathbf{3} boldsymbol{x}^{2} boldsymbol{y}-boldsymbol{y}^{3}-boldsymbol{2}=mathbf{0} ) A. cut at right angles B. touch each other c. cut at an angle ( frac{pi}{3} ) D. cut at an angle ( frac{pi}{4} ) | 12 |

916 | A ( 13 f t . ) ladder is leaning against a wall when its base starts to slide away At the instant when the base is 12 ft. away from the wall, the base is moving away from the wall at the rate of 5 ft/sec. The rate at which the angle ( theta ) between the ladder and the ground is changing is A ( cdot-frac{12}{13} r a d / sec ) B. – 1 rad( / )sec. c. ( -frac{13}{12} r a d / s e c ) D. ( -frac{10}{13} ) rad/sec | 12 |

917 | At present a firm manufacturing 2000 items it is estimated that the rate of change of production (P) with respect to additional number of workers ( (x) ) is given by ( frac{d P}{d x}=100-12 sqrt{x} . ) If the firm employes 25 more workers then the new production is? A . 2500 в. 3000 c. 3500 D. 4500 | 12 |

918 | If ( boldsymbol{x}>mathbf{0} ) and ( boldsymbol{x} boldsymbol{y}=mathbf{1}, ) the minimum value of ( (x+y) ) is A . -2 B. ( c cdot 2 ) D. none of these | 12 |

919 | The value of ( (127)^{1 / 3} ) to four decimal places is A .5 .0267 B. 5.4267 c. 5.5267 D. 5.001 | 12 |

920 | A variable triangle is inscribed in a circle of radius ( mathrm{R} ). If the rate of change of side is ( R ) times the rate of change of the opposite angle, then that angle is ( mathbf{A} cdot pi / 6 ) в. ( pi / 4 ) c. ( pi / 3 ) D. ( pi / 2 ) | 12 |

921 | Let ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{4}}{mathbf{3}} boldsymbol{x}^{3}-mathbf{4} boldsymbol{x}, mathbf{0} leq boldsymbol{x} leq mathbf{2} . ) Then the global minimum value of the function is ( mathbf{A} cdot mathbf{0} ) B. ( -8 / 3 ) c. -4 D. none of these | 12 |

922 | Let ( boldsymbol{f}: boldsymbol{I} boldsymbol{R} rightarrow boldsymbol{I} boldsymbol{R} quad ) be defined as ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}|+left|boldsymbol{x}^{2}-1right| . ) The total number of points at which ( f ) attains either a local maximum or local minimum is | 12 |

923 | A point on the parabola ( y^{2}=18 x ) at which the ordinate increases at twice the rate of the absicca is ( ^{A} cdotleft(frac{9}{8}, frac{9}{2}right) ) в. (2,-4) ( ^{mathbf{c}} cdotleft(frac{-9}{8}, frac{9}{2}right) ) D. (2,4) | 12 |

924 | The slope of the tangent to the locus ( y=cos ^{-1}(cos x) ) at ( x=frac{pi}{4} ) is ( mathbf{A} cdot mathbf{1} ) B. ( c cdot 2 ) D. – | 12 |

925 | The focal length of a mirror is given by ( frac{2}{f}=frac{1}{v}-frac{1}{u} . ) In finding the values of and ( v, ) the errors are equal and equal to ‘p’. The the relative error in ( f ) is A ( cdot frac{p}{2}left(frac{1}{u}+frac{1}{v}right) ) в. ( pleft(frac{1}{u}+frac{1}{v}right) ) c. ( frac{p}{2}left(frac{1}{u}-frac{1}{v}right) ) D. ( _{p}left(frac{1}{u}-frac{1}{v}right) ) | 12 |

926 | The straight line which is parallel to ( x ) axis and crosses the curve ( y=sqrt{x} ) at an angle of ( 45^{0} ) is A ( cdot x=frac{1}{4} ) в. ( y=frac{1}{4} ) c. ( y=frac{1}{2} ) D. ( x=frac{1}{2} ) | 12 |

927 | The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is ( 1 mathrm{cm} ) the altitude is ( 6 mathrm{cm} . ) When the radius is ( 6 mathrm{cm} ), then volume is increasing at the rate of ( 1 mathrm{Cu} ) ( c m / )sec. When the radius is ( 36 mathrm{cm}, ) the volume is increasing at a rate of ( n ) cu. ( mathrm{cm} / ) sec. The value of ‘ ( n^{prime} ) is equal to: A .12 в. 22 ( c .30 ) D. 33 | 12 |

928 | Find the min. value of ( frac{boldsymbol{a}^{2}}{cos ^{2} boldsymbol{x}}+frac{boldsymbol{b}^{2}}{sin ^{2} boldsymbol{x}} ) A. ( (a+b)^{2} ) 2) ( a+b+b^{-2} b+b^{-2} ) B . ( (a-b)^{2} ) ( mathbf{c} cdot a^{2}+b^{2} ) D. ( a^{2}-b^{2} ) | 12 |

929 | You are given a rod of length ( L ). The linear mass density is ( lambda ) such that ( lambda= ) ( a+b x . ) Here ( a ) and ( b ) are constant and the mass of the rod increases as ( x ) decreases. Find the mass of the rod. | 12 |

930 | The number of Points of Inflexion in ( boldsymbol{y}= ) ( x^{3}-3 x^{2}+3 x ) are: ( A cdot 0 ) B. ( c cdot 2 ) D. 3 | 12 |

931 | The function ( f(x)=x^{3}-27 x+8 ) is increasing when A . ( |x|3 ) c. ( -3<x<3 ) D. none of these | 12 |

932 | If ( f(x)=x^{3}-10 x^{2}+200 x-10, ) then ( (x) ) is A. decreasing ( (-infty, 10] ) and increasing in ( (10, infty) ) B. increasing ( (-infty, 10] ) and decreasing in ( (10, infty) ) c. increasing for every value of ( x ) D. decreasing for every value of ( x ) | 12 |

933 | A flower bed is made in the shape of sector of a circle. ( 20 m ) of wire is available to make a fence for the flower bed. Find the radius of the circle so that area of the flower bed is maximum. | 12 |

934 | Show that the function ( f ) given by ( f(x)=10^{x} ) is increasing for all ( x ) | 12 |

935 | 5. A function y = f(x) has a second order derivative f”(x) = 6(x-1). If its graph passes through the point (2,1) and at that point the tangent to the graph is y=3x-5, then the function is [2004] (a) (x+1)2 (b) (x – 1)3() (x+1)3 (d) (x-1)2 | 12 |

936 | Find the points of the maxima or local minima of the following function, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be. ( f(x)=x^{3}(x-1)^{2} ) | 12 |

937 | The number of values of ( x ) where the function ( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x}+cos (sqrt{mathbf{2}} boldsymbol{x}) ) attains its maximum is A . в. ( c cdot 2 ) D. infinite | 12 |

938 | Let ( f ) and ( g ) be two functions defined on an interval I such that ( f(x) geq 0 ) and ( g(x) leq 0 ) for all ( x epsilon I ) and ( f ) is strictly decreasing on ( I ) while ( g ) is strictly increasing on ( I ) then This question has multiple correct options | 12 |

939 | The slope of the normal to the curve ( y^{3}-x y-8=0 ) at the point (0,2) is equal to: A . -3 в. -6 ( c cdot 3 ) D. 6 E . 8 | 12 |

940 | 39. If the curves y2 = 6x, 9x² + by2 = 16 intersect each other at right angles, then the value of bis: [JEEM 2018] (a) ₃ (1) 4 (c) 7 (d) 6 | 12 |

941 | 39. If]f(x,)-f(x))}<(x, – x,)”, for all x,, x, € R. Find the equation of tangent to the cuve y=f(x) at the point (1, 2). (2005 – 2 Marks) | 12 |

942 | – 40. Ifp(x) be a polynomial of degree 3 satisfying p(-1)= 10,p(1) =-6 and p(x) has maxima at x=-1 and p'(x) has minima atx = 1. Find the distance between the local maxima and local minima of the curve. (2005 – 4 Marks) | 12 |

943 | Find an angle ( theta, 0<theta<frac{pi}{2}, ) which increases twice as fast as its sine. | 12 |

944 | The side of a square sheet is increasing at the rate of ( 4 mathrm{cm} ) per minute. The rate by which the area increasing when the side is ( 8 mathrm{cm} ) long is- A ( cdot 60 mathrm{cm}^{2} / mathrm{sec} ) B. ( 66 mathrm{cm}^{2} / mathrm{sec} ) ( mathrm{c} cdot 62 mathrm{cm}^{2} / mathrm{sec} ) D. ( 64 mathrm{cm}^{2} / mathrm{sec} ) | 12 |

945 | The minimum value of ( mathbf{f}(mathbf{x})= ) ( 4 sec ^{2} x+9 operatorname{cosec}^{2} x ) is A . 5 B . 25 c. 13 ( D cdot 13^{2} ) | 12 |

946 | The ( boldsymbol{f}(boldsymbol{x})=(3-boldsymbol{x}) e^{2 x}-4 boldsymbol{x} e^{x}-boldsymbol{x} ) has ( mathbf{A} cdot ) a maximum at ( x=0 ) B. a minimum at ( x=0 ) C. neither of two at ( x=0 ) D. ( f(x) ) is not differentiable at ( x=0 ) | 12 |

947 | Show that the equation of normal at any point on the curve ( x=3 cos t-cos ^{3} t ) ( boldsymbol{y}=mathbf{3} sin t-sin ^{3} boldsymbol{t} ) is ( 4left(y cos ^{3} t-x sin ^{3} tright)=3 sin 4 t ) | 12 |

948 | The minimum value of the polynomial. ( p(x)=3 x^{2}-5 x+2 ) A ( cdot-frac{1}{6} ) B. ( c cdot frac{1}{12} ) D. ( -frac{1}{12} ) | 12 |

949 | Find the maximum and minimum values, if any, without using derivatives of the following function. ( f(x)=16 x^{2}-16 x+28 ) on ( R ) | 12 |

950 | The least value of ( boldsymbol{f}(boldsymbol{x})=left(boldsymbol{e}^{boldsymbol{x}}+boldsymbol{e}^{-boldsymbol{x}}right) ) is A . -2 B. ( c cdot 2 ) D. none of these | 12 |

951 | Using differentials, find the sum of digits approximate value of the following up to 3 places of decimal. ( sqrt{0.6} ) | 12 |

952 | The normal to the curve ( 2 x^{2}+y^{2}=12 ) at the point (2,2) cuts the curve again at ( ^{mathbf{A}} cdotleft(-frac{22}{9},-frac{2}{9}right) ) в. ( left(frac{22}{9}, frac{2}{9}right) ) c. (-2,-2) D. none of these | 12 |

953 | What normal to the curve ( y=x^{2} ) form the shortest chord? ( mathbf{A} cdot x+sqrt{2} y=sqrt{2} ) or ( x-sqrt{2} y=-sqrt{2} ) B ( . x+sqrt{2} y=sqrt{2} ) or ( x-sqrt{2} y=sqrt{2} ) c. ( x+sqrt{2} y=-sqrt{2} ) or ( x-sqrt{2} y=-sqrt{2} ) D. ( x+sqrt{2} y=-sqrt{2} ) or ( x-sqrt{2} y=sqrt{2} ) | 12 |

954 | 6. The normal to the curve x = a(1+cos), y = a sino at ‘o’ always passes through the fixed point [2004] (a) (a, a) (b) (0, a) (c) (0,0) (d) (a,0) | 12 |

955 | 2. AB is a diameter of a circle and C is any point on the circumference of the circle. Then (1983 – 1 Mark) (a) the area of A ABC is maximum when it is isosceles (b) the area of A ABC is minimum when it is isosceles (c) the perimeter of A ABC is minimum when it is isosceles (d) none of these | 12 |

956 | The side of an equilateral triangle is ‘a’ units and is increasing at the rate of ( lambda ) units /sec. The rate of increase of its area is A ( cdot frac{2}{sqrt{3}} lambda a ) в. ( sqrt{3} lambda a ) c. ( frac{sqrt{3}}{2} lambda a ) D. none of these | 12 |

957 | Find intervals in which the function ( operatorname{given} operatorname{by} f(x)=sin 3 x, x inleft[0, frac{pi}{2}right] ) is increasing function | 12 |

958 | If an error of ( k % ) is made in measuring the radius of a sphere, then percentage error in its volume is A. ( k % ) B. ( 3 k % ) c. ( 2 k % ) D. ( frac{2}{3} k % ) | 12 |

959 | Find the coordinates of the point on ( y ) axis which is nearest to the point (-2,5) | 12 |

960 | A particle moves along a curve so that its coordinates at time ( t ) are ( boldsymbol{x}=boldsymbol{t}, boldsymbol{y}= ) ( frac{1}{2} t^{2}, z=frac{1}{3} t^{3} ) acceleration at ( t=1 ) is ( mathbf{A} cdot j+2 k ) B. ( j+k ) ( c cdot 2 j+k ) D. none of these | 12 |

961 | The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate, then the ratio of the change of their areas is A . ( 2: pi ) в. ( pi: 1 ) ( c cdot 4: pi ) D. 1: 2 | 12 |

962 | 26. Determine the points of maxima and minima of the function In x-bx + x², x > 0, where b > O is a constant. | 12 |

963 | If ( I^{2}+mathbf{m}^{2}=1, ) then the max values of ( boldsymbol{I}+mathbf{m} ) is ( A cdot 1 ) B. ( sqrt{2} ) c. ( frac{1}{sqrt{2}} ) D. 2 | 12 |

964 | If ( f^{prime}(x) ) exists for all ( x in R ) and ( g(x)= ) ( boldsymbol{f}(boldsymbol{x})-(boldsymbol{f}(boldsymbol{x}))^{2}+(boldsymbol{f}(boldsymbol{x}))^{3} ) for all ( boldsymbol{x} in boldsymbol{R} ) then A. ( g(x) ) is increasing whenever ( f ) is increasing B. ( g(x) ) is increasing whenever ( f ) is decreasing ( mathrm{c} . g(x) ) is decreasing whenever ( f ) is increasing D. none of these | 12 |

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