Practice Questions

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

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.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.The area, enclosed by the curves 𝑦= sin𝑥+ cos𝑥 and 𝑦= | cos𝑥- sin𝑥| and the lines 𝑥= 0, 𝑥= 2, is : (1) 2√2 ( √2 + 1 ) (2) 2√2 ( √2 - 1 ) (3) 4 ( √2 - 1 ) (4) 2 ( √2 + 1 )

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.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 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.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

Q77. n→∞[ (1) 1 (2) 1 2 4 (3) 1 (4) 1 3

Q77.Let α be the angle between the lines whose direction cosines satisfy the equations l + m −n = 0 and l2 + m2 −n2 = 0. Then the value of sin4 α + cos4 α is : (1) 5 (2) 1 8 2 (3) 3 (4) 3 8 4

Q77.The population 𝑃= 𝑃𝑡 at time 𝑡 of a certain species follows the differential equation 𝑑𝑃 0 . 5𝑃- 450. If 𝑑𝑡= 𝑃0 = 850, then the time at which population becomes zero is: (1) log𝑒9 (2) 2log𝑒18 1 (3) log𝑒18 (4) 2log𝑒18 𝑥- 3 𝑦- 4 𝑧- 5

Q77.Let y = y(x) be a solution curve of the differential equation (y + 1) tan2 xdx + tan xdy + ydx = 0, x ∈(0, π2 ). If lim xy(x) = 1, then the value of y( π4 ) is: x→0+ (1) π 4 + 1 (2) π4 −1 (3) π 4 (4) −π4 is equal b

Q77.Let f(x) be a differentiable function defined on [0, 2] such that f ′(x) = f ′(2 −x) for all x ∈(0, 2), f(0) = 1 and f(2) = e2. Then the value of ∫20 f(x)dx is (1) 2(1 + e2) (2) 1 + e2 (3) 1 −e2 (4) 2(1 −e2) = 1 and

Q77.Let y = y(x) be the solution of the differential equation xdy = (y + x3 cos x)dx with y(π) = 0, then y( π2 ) is equal to: (1) π2 4 + π2 (2) π22 + π4 (3) π2 2 −π4 (4) π24 −π2

Q77.Let y = y(x) be the solution of the differential equation x tan( xy )dy = (y tan( xy ) −x)dx, −1 ≤x ≤1, y( 12 ) = π6 . Then the area of the region bounded by the curves x = 0, x = √21 and y = y(x) in the upper half plane is: (1) 1 8 (π −1) (2) 121 (π −3) (3) 4 1 (π −2) (4) 16 (π −1)

Q77.Let 𝑦= 𝑦( 𝑥) be the solution of the differential equation 𝑑𝑦 1 + 𝑥𝑒𝑦- 𝑥, - √2 < 𝑥< √2, 𝑦0 = 0 𝑑𝑥= , then the minimum value of 𝑦𝑥, 𝑥∈-√2, √2 is equal to : (1) 2 - √3 - loge2 (2) 2 + √3 + loge2 (3) 1 + √3 - loge√3 - 1 (4) 1 - √3 - loge√3 - 1

Q77.If vectors →a1 = xˆi −ˆj + ˆk and →a2 = ˆi + yˆj + zˆk are collinear, then a possible unit vector parallel to the vector xˆi + yˆj + zˆk is: (1) + 1 (−ˆj √2 ˆk) (2) √31 (ˆi +ˆj −ˆk) (3) + ˆk) √2 1 (ˆi −ˆj) (4) √31 (ˆi −ˆj JEE Main 2021 (26 Feb Shift 2) JEE Main Previous Year Paper

Q77.In a triangle ABC, if BC→ = 3, CA→ = 5 and BA→ = 7, then the projection of the vector BA→ on BC→ is equal to JEE Main 2021 (20 Jul Shift 2) JEE Main Previous Year Paper (1) 19 (2) 13 2 2 (3) 11 (4) 15 2 2