Practice Questions

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 y2 + loge(cos2 x) = y, x ∈(−π2 , π2 ) then : (1) y′′(0) = 0 (2) |y′(0)| + |y′′(0)| = 1 (3) |y′′(0)| = 2 (4) |y′(0)| + |y′′(0)| = 3

Q63.The set of all real values λ for which the function f(x) = (1 −cos2 x). (λ + sin x), xε (−π2 2 ), has exactly one maxima and exactly one minima, is : (1) (−12 , 12 ) −{0} (2) (−32 , 32 ) (3) (−12 , 12 ) (4) (−32 , 32 ) −{0}

Q63.Let f be any function continuous on [a, b] and twice differentiable on (a, b) . If all x ∈(a, b), f '(x) > 0 and f ''(x) < 0 , then for any c ∈(a, b), f(c)−f(a)f(b)−f(c) (1) b+a (2) 1 b−a (3) b−c (4) c−a c−a b−c

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

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

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.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.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.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.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

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) = (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.If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joining (0, 23 ) and ( 21 , 2) , then (1) b = a (2) |b −a| = 1 (3) |a + b| = 1 (4) b = π2 + a JEE Main 2020 (02 Sep Shift 1) JEE Main Previous Year Paper

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.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 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.Let f(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If x→0(2lim + f(x)x3 ) = 4, then which one of the following is not true? (1) f is an odd function (2) f(1) −4f(−1) = 4 . x = 1 is a point of maximum and x = −1 (3) x = 1is a point of local minimum and x = −1 is (4) x = 1 is a point of local maxima of f a point of local maximum JEE Main 2020 (07 Jan Shift 2) JEE Main Previous Year Paper

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

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.Let f and g be differentiable functions on R such that fog is the identity function. If for some a, b ∈R, g'(a) = 5 and g(a) = b, then f '(b) is equal to: (1) 1 (2) 1 5 (3) 5 (4) 52

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.The integral ∫ 8dx 6 is equal to: (where C is a constant of integration) (x+4) 7 (x−3) 7 (1) x−3 71 (2) x−3 −17 ( x+4 ) + C ( x+4 ) + C (3) 1 x−3 73 (4) x−3 −137 2 ( x+4 ) + C −113 ( x+4 ) + C

Q65.The equation of the normal to the curve y = (1 + x)2y + cos2(sin−1 x) , at x = 0 is (1) y + 4x = 2 (2) y = 4x + 2 (3) x + 4y = 8 (4) 2y + x = 4