Practice Questions

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

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.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.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 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

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 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

Q77.Let y = y(x) be solution of the differential equation loge( dxdy ) y(−23 loge 2) = α loge 2 , then the value of α is equal to: JEE Main 2021 (27 Jul Shift 1) JEE Main Previous Year Paper (1) −14 (2) 41 (3) 2 (4) −12 →

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

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.Let three vectors →a, b and →cbe such that →a× b =→c, b ×→c=→a and →a = 2. Then which one of the following is not true? b b b × is 2 (1) →a× ((→ → → (2) → +→c) ( −→c)) = 0 Projection of →a on ( ×→c) + = 8 (4) 3→a+→b −2→c 2 = 51 (3) [→a →b →c] [→c →a →b ] JEE Main 2021 (22 Jul Shift 1) JEE Main Previous Year Paper = 2. If P(α, β, γ) is the

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