Practice Questions

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 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.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.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.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.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 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.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

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

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 f(a + b + 1 −x) = f(x), for all x, where a and b are fixed positive real numbers, then b 1 ∫ x(f(x) + f(x + 1))dx is equal to a+b a (1) b−1 (2) b−1 ∫ f(x + 1)dx ∫ f(x)dx a−1 a−1 (3) b+1 (4) b+1 ∫ f(x)dx ∫ f(x + 1)dx a+1 a+1

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.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.If I1 = ∫10 (1 −x50)100dx and I2 = ∫10 (1 −x50)101dx such that I2 = αI1 then (1) 5049 (2) 5050 5050 5049 (3) 5050 (4) 5051 5051 5050 Q66. ∫(x−1)20 t cos t2dt lim (x−1) sin(x−1) x→1( ) (1) is equal to 1 . (2) is equal to 1. 2 (3) is equal to −12 . (4) is equal to 0.

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.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

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 be a twice differentiable function on (1, 6), If f(2) = 8, f ′(2) = 5, f ′(x) ≥1 and f′′(x) ≥4, for all x ∈(1, 6), then : (1) f(5) + f ′(5) ≤26 (2) f(5) + f ′(5) ≥28 (3) f ′(5) + f′′(5) ≤20 (4) f(5) ≤10 is equal to, (where C is a constant of integration):

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 the function f(x) = {k1(xk2−π)2cos x,−1, xx ≤π> π to: (1) ( 21 , 1) (2) (1, 0) (3) ( 21 , −1) (4) (1, 1) + c, where c is a constant of integration, then g(0) is

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

Q66.Let P(h, k) be a point on the curve y = x2 + 7x + 2 , nearest to the line, y = 3x −3 . Then the equation of the normal to the curve at P is (1) x + 3y + 26 = 0 (2) x + 3y −62 = 0 (3) x −3y −11 = 0 (4) x −3y + 22 = 0