Practice Questions

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.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.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 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.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.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 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.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.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 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 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.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

Q66.The area (in sq. units) of the region {(x, y) ∈R2 : x2 ≤y ≤3 −2x}, is. (1) 32 (2) 34 3 3 (3) 29 (4) 31 3 3 JEE Main 2020 (08 Jan Shift 2) JEE Main Previous Year Paper

Q66.If ∫(e2x + 2ex −e−x −1)e(ex+e−x)dx = g(x)e(ex+e−x) (1) e (2) e2 (3) 1 (4) 2 1 2 x dx is :

Q66.Let f : (−1, ∞) →R be defined by f(0) = 1 and f(x) = x1 loge(1 + x), x ≠0 . Then the function f (1) Decreases in (−1, 0) and increases in (0, ∞) (2) Increases in (−1, ∞) (3) Increases in (−1, 0) and decreases in (0, ∞) (4) Decreases in (−1, ∞)

Q66.Which of the following points lies on the tangent to the curve x4ey + 2√y + 1 = 3 at the point (1, 0)? JEE Main 2020 (05 Sep Shift 2) JEE Main Previous Year Paper (1) (2, 2) (2) (2, 6) (3) (–2, 6) (4) (−2, 4) + C, where C is a constant of integration, then B(θ)A can be:

Q66.The area of the region (in sq. units), enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = x , is (1) 1 6 (24π −1) (2) 13 (6π −1) (3) 1 3 (12π −1) (4) 16 (12π −1) JEE Main 2020 (07 Jan Shift 1) JEE Main Previous Year Paper = ex such that y(0) = 0, then y(1) is

Q66.The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x2 −1 below the x-axis, is : (1) 2 (2) 1 3√3 3√3 (3) 4 (4) 4 3 3√3 π

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

Q66.The area (in sq. units) of the region {(x, y) : 0 ≤y ≤x2 + 1, 0 ≤y ≤x + 1, 21 ≤x ≤2} is (1) 23 (2) 79 16 24 (3) 79 (4) 23 16 6

Q66.The integral ∫( x sin x+cosx x ) 2dx (1) tan x − x sinx x+cossec x x + C (2) sec x + x sinx tanx+cosx x + C (3) sec x − x sinx tanx+cosx x + C (4) tan x + x sinx x+cossec x x + C

Q66.If for all real triplets (a, b, c), f(x) = a + bx + cx2; then ∫1 f(x)dx is equal to: 0 JEE Main 2020 (09 Jan Shift 1) JEE Main Previous Year Paper (1) 2{3f(1) + 2f( 12 )} (2) 12 {f(1) + 3f( 12 )} (3) 1 3 {f(0) + f( 12 )} (4) 16 {f(0) + f(1) + 4f( 12 )} dx is equal to:

Q66.The integral ∫21 ex. xx (2 + loge x) dx equals : (1) e(4e + 1) (2) 4e2 −1 (3) e(4e −1) (4) e(2e −1)

Q66.If ∫ 1 2 = f(x)(1 + sin6 x) λ + c, where c is a constant of integration, then λf( π3 ) is equal to sin3 x(1+sin6 x) 3 JEE Main 2020 (08 Jan Shift 1) JEE Main Previous Year Paper (1) −98 (2) 2 (3) 9 (4) −2 8

Q66.If ∫ cos2 θ(tandθ2θ+sec 2θ) = λ tan θ + 2 loge|f(θ)| + C where C is a constant of integration, then the ordered pair (λ, f(θ)) is equal to: (1) (1, 1 −tan θ) (2) (−1, 1 −tan θ) (3) (−1, 1 + tan θ) (4) (1, 1 + tan θ) JEE Main 2020 (09 Jan Shift 2) JEE Main Previous Year Paper