Practice Questions

Q76.The area (in sq. units) of the region, given by the set 𝑥, 𝑦∈𝑅× 𝑅∣𝑥≥0, 2𝑥2 ≤𝑦≤4 - 2𝑥 is : JEE Main 2021 (25 Jul Shift 1) JEE Main Previous Year Paper 8 17 (1) (2) 3 3 (3) 13 (4) 7 3 3

Q76.The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of 2 k bacteria is 2000 after hours, then ( logek 2 ) is equal to: 6 ) loge( 5 (1) 8 (2) 4 (3) 16 (4) 2 is equal to: × × ×

Q76.The value of the integral ∫1−1 loge(√1 x)dx is equal to: (1) 2 1 loge 2 + π4 −32 (2) 2 loge 2 + π4 −1 (3) loge 2 + π2 −1 (4) 2 loge 2 + π2 −12

Q76.If a curve y = f(x) passes through the point (1, 2) and satisfies x dydx + y = bx4, then for what value of b, ∫21 f(x)dx = 625 ? (1) 31 (2) 10 5 (3) 5 (4) 625

Q76.The value of ∫ −11 1 ) √2 (( x−1x+1 + ( x−1x+1 ) 2 −2) 2 √2 (1) loge 4 (2) 2 loge 16 + (3) loge 16 (4) 4 loge(3 2√2)

Q76.If the area of the bounded region R = {(x, y) : max{0, loge x} ≤y ≤2x, 21 ≤x ≤2} is, α(loge 2)−1 + β(loge 2) + γ then the value of (α + β −2γ)2 is equal to: (1) 8 (2) 2 (3) 4 (4) 1 = 3x + 4y, with y(0) = 0. If

Q76.Let slope of the tangent line to a curve at any point P(x, y) be given by xy2+yx x + 2y = 4 at x = −2, then the value of y, for which the point (3, y) lies on the curve, is : (1) −43 (2) 3518 (3) −1819 (4) −1811 −−

Q76.If the value of the integral ∫50 x+[x]ex−[x] greatest integer less than or equal to x; then the value of (α + β)2 is equal to : (1) 25 (2) 100 (3) 36 (4) 16

Q76.Let y = y(x) be the solution of the differential equation cos sin x + cos x + = + y sin sin x + cos x + 0 ≤x ≤π2 , y(0) = 0. Then, y( π3 ) is x(3 3)dy (1 x(3 3))dx, equal to: JEE Main 2021 (17 Mar Shift 2) JEE Main Previous Year Paper 2 loge( 2√3+1011 ) (1) 2 loge( 2√3+96 ) (2) 2 loge( 3√3−84 ) (3) 2 loge( √3+72 ) (4)

Q76.If the solution curve of the differential equation (2x −10y3)dy + ydx = 0 , passes through the points (0, 1) and (2, β), then β is a root of the equation? (1) y5 −2y −2 = 0 (2) y5 −y2 −1 = 0 (3) 2y5 −y2 −2 = 0 (4) 2y5 −2y −1 = 0

Q76.If the functions are defined as f(x) = √x and g(x) following functions: f + g, f −g, f/g, g/f, g −f , where (f ± g)(x) = f(x) ± g(x), (f/g)(x) = f(x) g(x) (1) 0 ≤x ≤1 (2) 0 ≤x < 1 (3) 0 < x < 1 (4) 0 < x ≤1 1 ; |x| ≥1 |x| is differentiable at every point of the domain, then the values of a and b are

Q76.If 𝑦d𝑦 𝜙𝑦2 d𝑥= 𝑥2 𝑦2 , 𝑥> 0, 𝜙> 0, and 𝑦( 1 ) = - 1, then 𝜙𝑦24 𝜙' 𝑥2 (1) 2𝜙1 (2) 𝜙1 (3) 4𝜙2 (4) 4𝜙1 𝑑𝑦 2𝑥𝑦+ 2𝑦· 2𝑥

Q76.The integral ∫ 1 dx is equal to : (where C is a constant of integration) 4√(x−1)3(x+2)5 (1) 5 1 4 + C 4 3 ( x−1x+2 ) 4 + C (2) 34 ( x+2x−1 ) (3) 4 x−1 54 (4) 3 x+2 14 3 ( x+2 ) + C 4 ( x−1 ) + C JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper

Q76.Which of the following statement is correct for the function g(α) for α ∈R such that π 3 sinα x dx g(α) = ∫ π 6 cosα x+sinα x (1) g(α) is a strictly increasing function (2) g(α) has an inflection point at α = −12 (3) g(α) is a strictly decreasing function (4) g(α) is an even function

Q76.The area (in sq. unit) bounded by the curve 4y2 = x2(4 −x)(x −2) is equal to (1) π8 (2) 3π8 (3) 3π (4) π 2 16 0 < x < 2. 1 , with

Q76.If In = ∫ π2 cotn xdx, then 4 (1) I2 + I4, (I3 + I5)2, I4 + I6 are in G. P. (2) I2 + I4, I3 + I5, I4 + I6 are in A. P. (3) 1 , 1 , 1 are in A. P. (4) 1 , 1 , 1 are in G. P. I2+I4 I3+I5 I4+I6 I2+I4 I3+I5 I4+I6 is equal to lim n1 + (n+1)2n + (n+2)2n + … + (2n−1)2n ]

Q76.The area of the region bounded by y −x = 2 and x2 = y is equal to :- (1) 16 (2) 2 3 3 (3) 9 (4) 4 2 3

Q76.Let a vector αˆi + βˆj be obtained by rotating the vector √3ˆi +ˆj by an angle 45° about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices (α, β), (0, β) and (0, 0) is equal to (1) 1 (2) 1 2 (3) 1 (4) 2√2 √2

Q76.Let a vector →a be coplanar with vectors b = 2ˆi + ˆj + ˆk and →c= ˆi −ˆj + ˆk. If →a is perpendicular to → → → → → d = 3ˆi + 2ˆj + 6ˆk, and →a = √10. Then a possible value of [→a b →c] + [→a b d ] + [→a →c d ] is equal to: (1) −42 (2) −40 (3) −29 (4) −38 → → →

Q76.Let us consider a curve, y = f(x) passing through the point (−2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf ′(x) = x2. Then (1) x3 −3xf(x) −4 = 0 (2) x2 + 2xf(x) −12 = 0 (3) x3 + xf(x) + 12 = 0 (4) x2 + 2xf(x) + 4 = 0

Q76. y sin x 1 dy ⎡ ⎤ Let y = y(x) satisfies the equation dx −|A| = 0, for all x > 0, where A = 0 −1 1 . If y(π) = π + 2, ⎣ 2 0 x1 ⎦ then the value of y( π2 ) is: (1) π 2 + π4 (2) π2 −1π (3) 3π 2 −1π (4) π2 −4π −−−−−

Q76.The area (in sq. units) of the part of the circle 𝑥2 + 𝑦2 = 36, which is outside the parabola 𝑦2 = 9𝑥, is equal to (1) 12𝜋+ 3√3 (2) 24𝜋+ 3√3 (3) 24𝜋- 3√3 (4) 12𝜋- 3√3

Q76.If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is x2−4x+y+8x−2 , curve also passes through the point: (1) (5, 4) (2) (4, 4) (3) (4, 5) (4) (5, 5)

Q76.The value of the integral ∫1−1 log(x + √x2 + 1)dx is: (1) 2 (2) 0 (3) −1 (4) 1

Q76.Let C1 be the curve obtained by the solution of differential equation 2xy dxdy = y2 −x2, x > 0 . Let the curve C2 be the solution of x2−y22xy = dxdy . If both the curves pass through (1, 1), then the area (in sq. units) enclosed by the curves C1 and C2 is equal to : (1) π −1 (2) π2 −1 (3) π + 1 (4) π4 + 1 → → = 3 and