Practice Questions

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.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 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.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.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.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 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 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 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 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 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.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

Q77.If 𝑦0 = 0, then for 𝑦= 1, the value of 𝑥 lies in the interval : 𝑑𝑥= 2𝑥+ 2𝑥+ 𝑦log𝑒2, 1 (1) 1, 2 (2) 2, 1 (3) 2, 3 (4) 0, 1 2

Q77.Let y = y(x) be the solution of the differential equation dydx = 2(y + 2 sin x −5)x −2 cos x such that y(0) = 7. Then y(π) is equal to (1) 7eπ2 + 5 (2) eπ2 + 5 (3) 2eπ2 + 5 (4) 3eπ2 + 5

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 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.If →a and→b are perpendicular, then →a× (→a (→a (→a →b))) 4→ (1) →a b (2) →0 → 4→ 1 (3) →a× b (4) 2 →a b

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

Q77.Which of the following is true for y(x) that satisfies the differential equation dy = xy −1 + x −y; y(0) = 0 dx (1) y(1) = e−12 −1 (2) y(1) = e 12 −e−12 (3) y(1) = 1 (4) y(1) = e 21 −1 → → + 2ˆj + = −3, then →r⋅(2ˆi −3ˆj + ˆk) is

Q77.A differential equation representing the family of parabolas with axis parallel to y−axis and whose length of latus rectum is the distance of the point (2, −3) from the line 3x + 4y = 5, is given by: (1) 11 d2x dy2 = 10 (2) 11 dx2d2y = 10 d2y (3) 10 = 11 (4) 10 d2xdy2 = 11 dx2 = 1 and

Q77.If for a > 0, the feet of perpendiculars from the points A(a, −2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, −a, −1) and D respectively, then the length of line segment CD is equal to : (1) √31 (2) √41 (3) √55 (4) √66

Q77.If 𝑦= 𝑦( 𝑥) is the solution curve of the differential equation 𝑥2 d𝑦+ 𝑦- 1 0; 𝑥> 0 and 𝑦( 1 ) = 1, 𝑥d𝑥= 1 then 𝑦 is equal to : 2 (1) 3 + e (2) 3 - e 3 1 1 (3) - (4) 3 + 2 √e √e