Practice Questions

Q63.Let f(x) = (sin(tan−1 x) + sin(cot−1 x))2 −1 , |x| > 1. If dxdy = 12 dxd (sin−1(f(x))) and y(√3) y(−√3) is equal to: (1) 2π 3 (2) −π6 (3) 5π (4) π 6 3 [3, 4], where

Q63.The length of the perpendicular from the origin, on normal to the curve, x2 + 2xy −3y2 = 0, at the point (2, 2), is. (1) √2 (2) 4√2 (3) 2 (4) 2√2 ∫x0 tsin(10t)dt , is equal to

Q63.If f(x + y) = f(x) f(y) and x=1f(x) of f(4) is f(2) (1) 2 (2) 1 3 9 (3) 1 (4) 4 3 9

Q64.For all twice differentiable functions f : R →R, with f(0) = f(1) = f′(0) = 0 , (1) f′′(x) ≠0 at every point xε(0, 1) (2) f′′(x) = 0, for some x ε (0, 1) (3) f′′(0) = 0 (4) f′′(x) = 0, at every point x ε(0, 1) JEE Main 2020 (06 Sep Shift 2) JEE Main Previous Year Paper

Q64. lim x x→0 (1) 0 (2) 101 (3) −15 (4) −110 2 dx

Q64.If c is a point at which Rolle’s theorem holds for the function, f(x) = loge( x2+α7x ) in the interval α ∈R, then f ''(c) is equal to (1) −112 (2) 121 (3) −124 (4) √37

Q64.If the surface area of a cube is increasing at a rate of 3. 6cm2/sec, retaining its shape; then the rate of change of its volume (in cm3/sec), when the length of a side of the cube is 10cm, is: (1) 20 (2) 10 (3) 18 (4) 9 = A(x) tan−1(√x) + B(x) + C , where C is a constant of integration, then the

Q64.If S is the sum of the first 10 terms of the series, tan−1( 13 ) + tan−1( 17 ) + tan−1( 131 ) + tan−1( 211 ) + … … then tan(S) is equal to : (1) 65 (2) 115 (3) −56 (4) 1011 is twice differentiable, then the ordered pair (k1, k2) is equal

Q64.If (a + √2b cos x)(a −√2b y) (1) a−2b (2) a−b a+2b a+b (3) a+b (4) 2a+b a−b 2a−b JEE Main 2020 (04 Sep Shift 1) JEE Main Previous Year Paper

Q64.Let the function , f : [−7, 0] →R be continuous on [−7, 0] and differentiable on (−7, 0). If f(−7) = −3 and f '(x) ≤2 for all x ∈(−7, 0), then for all such functions f, f(−1) + f(0) lies in the interval (1) (−∞, 20] (2) [−3, 11] (3) (−∞, 11] (4) [−6, 20]

Q64.A spherical iron ball of 10cm radius is coated with a layer of ice of uniform thickness that melts at a rate of 50cm3/min . When the thickness of ice is 5cm , then the rate (in cm/min .) at which of the thickness of ice decreases, is: (1) 5 (2) 1 6π 54π (3) 1 (4) 1 36π 18π

Q64.The function, f(x) = (3x −7)x 32 , x ∈R, is increasing for all x lying in (1) (−∞, 0) ∪( 1514 , ∞) (2) (−∞, 0) ∪( 73 , ∞) (3) (−∞, 1514 ) (4) (−∞, −1415 ) ∪(0, ∞) Q65. ∫π−π|π −|x||dx is equal to (1) √2π2 (2) 2π2 (3) π2 (4) π2 2 JEE Main 2020 (03 Sep Shift 1) JEE Main Previous Year Paper

Q64.The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t1, t2] is attained at the point: (1) (t2−t1) (2) a(t2 −t1) + b 2 (3) (t1+t2) (4) 2a(t1 + t2) + b 2 α equals to :

Q64.Let f : R →R be a function which satisfies f(x + y) = f(x) + f(y), ∀x, y ∈R . If f(1) = 2 and g(n) = ∑(n−1)k=1 f(k), n ∈N then the value of n, for which g(n) = 20, is (1) 5 (2) 20 (3) 4 (4) 9

Q64.The function f(x) = π 1 (|x| −1), |x| > 1 { 2 JEE Main 2020 (04 Sep Shift 2) JEE Main Previous Year Paper (1) continuous on R −{1} and differentiable on (2) both continuous and differentiable on R −{1} R −{−1, 1}. (3) continuous on R −{−1}and differentiable on (4) both continuous and differentiable on R −{−1} R −{−1, 1}

Q64.The derivative of tan−1( √1+x2−1x ) with respect to tan−1( 2x√1−x21−2x2 ) (1) 2√3 (2) √3 5 12 (3) 2√3 (4) √3 3 10

Q65.If I = ∫ , then √2x3−9x2+12x+4 1 (1) 8 1 < I 2 < 41 (2) 91 < I 2 < 81 (3) 16 1 < I 2 < 19 (4) 16 < I 2 < 21

Q65.The value of α for which 4α ∫2 e−α|x|dx = 5 , is −1 (1) loge 2 (2) loge( 23 ) (3) loge √2 (4) loge( 34 )

Q65.Let a function f : [0, 5] →R be continuous, f(1) = 3 and F be defined as: F(x) = ∫x1 t2g(t)dt, where g(t) = ∫t1 f(u)du. Then for the function F(x), the point x = 1 is: (1) a point of local minima (2) not a critical point (3) a point of local maxima (4) a point of inflection

Q65.If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2 then p(0) is equal to (1) 6 (2) −12 (3) 24 (4) 12

Q65.Let f : (0, ∞) →(0, ∞) be a differentiable function such that f(1) = e and lim t2f 2(x)−x2f 2(t) = 0. If t→x t−x f(x) = 1, then x is equal to: (1) 1 (2) 2e e (3) 1 (4) e 2e

Q65.If ∫sin−1( 1+x√x )dx ordered pair (A(x), B(x)) can be : (1) (x −1, √x) (2) (x −1, −√x) (3) (x + 1, √x) (4) (x + 1, −√x) 2 x2

Q65.Let f(x) = xcos−1(−sin|x|), x ∈[−π2 , π2 ], then which of the following is true? (1) f' is increasing in (−π2 , 0) and decreasing in (2) f '(0) = −π2 (0, π2 ) (3) f is not differentiable at x = 0 (4) f' is decreasing in (−π2 , 0) and increasing in (0, π2 ) cos xdx

Q65.If x = 1 is a critical point of the function f(x) = (3x2 + ax −2 −a)ex, then (1) x = 1 and x = −23 are local minima of f (2) x = 1 and x = −23 is a local maxima of f (3) x = 1 is a local maxima and x = −22 is a local (4) x = 1 is a local minima and x = −23 are local minima of f maxima of f

Q65.If the tangent to the curve, y = f(x) = x loge x, (x > 0) at a point (c, f(c)) is parallel to the line-segment joining the points (1, 0) and (e, e),then c is equal to : 1 ) e−1 (1) e−1 (2) e( e 1 1−e 1 ) (4) (3) e( e−1