Practice Questions

Q83.The integral ∫ sin2 x cos2 x dx, is equal to (sin5 x+cos3 x sin2 x+sin3 x cos2 x+cos5 x)2 (where C is the constant of integration). (1) −1 + C (2) 1 + C 1+cot3 x 3(1+tan3 x) (3) −1 + C (4) 1 + C 3(1+tan3 x) 1+cot3 x π 2 sin2x dx is

Q83.If f( x+2x−4 ) = 2x + 1, (x ∈R −{1, −2}), then ∫f(x)dx is equal to (1) 12 ln|1 −x| −3x + C (2) −12 ln|1 −x| −3x + C (3) 12 ln|1 −x| + 3x + C (4) −12 ln|1 −x| + 3x + C π 2 2+sin x is

Q83.If f(x) = ∫x0 t(sin (1) f ′′′(x) −f ′′(x) = cos x −2x sin x (2) f ′′′(x) + f ′′(x) −f ′(x) = cos x (3) f ′′′(x) + f ′′(x) = sin x (4) f ′′′(x) + f ′(x) = cos x −2x sin x

Q83. dx = A√7 −6x −x2 + B sin−1 + C ( 4 ) ∫ √7 2x−6x+ 5−x2 x + 3 (where C is a constant of integration), then the ordered pair (A, B) is equal to (1) (−2, −1) (2) (2, −1) (3) (−2, 1) (4) (2, 1) 3π dx is

Q83.If f ( x−4x+2 ) = 2x + 1, (x ∈R = {1, −2}), then ∫f ( x ) dx is equal to (where C is a constant of integration) (1) 12 loge |1 −x| −3x + c (2) −12 loge |1 −x| −3x + c (3) −12 loge |1 −x| + 3x + c (4) 12 loge |1 −x| + 3x + c JEE Main 2018 (15 Apr Shift 1 Online) JEE Main Previous Year Paper

Q84.The value of the integral ∫ sin4 x(1 + ln( 2−sin x ))dx −π2 (1) 3 (2) 3 π 4 8 (3) 0 (4) 163 π

Q84.The value of integral ∫ π 4 x 4 1+sin x (1) π 2 (√2 + 1) (2) π(√2 −1) (3) 2π(√2 −1) (4) π√2

Q84.The value of the integral π 2 2 + sin x sin4 x + log is 2 −sin x ))dx ∫ −π2 (1 ( (1) 3 π (2) 0 16 (3) 3 π (4) 3 8 4

Q84.The values of ∫ 1+2x −π2 (1) π (2) π 4 8 (3) π (4) 4π 2 JEE Main 2018 (08 Apr) JEE Main Previous Year Paper

Q84.If the area of the region bounded by the curves, y = x2, y = x1 and the lines y = 0 and x = t(t > 1) is 1 sq. unit, then t is equal to (1) e 23 (2) e 32 (3) 3 (4) 4 2 3

Q85.The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, center at the origin and passing through the point (0, 3) is (1) xyy′ −y2 + 9 = 0 (2) xyy′′ + x(y′)2 −yy′ = 0 (3) xyy′ + y2 −9 = 0 (4) x + yy′′ = 0 → → → →

Q85.Let g(x) = cos x2, f(x) = √x, and α, β(α < β) be the roots of the quadratic equation 18x2 −9πx + π2 = 0. Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines x = α, x = β and y = 0, is (1) 1 (2) 1 2 (√2 −1) 2 (√3 −1) (3) + 2 1 (√3 1) (4) 21 (√3 −√2)

Q85.The area (in sq. units) of the region {x ∈R : x ≥0 , y ≥0 , y ≥x −2 and y ≤√x} is (1) 13 (2) 8 3 3 (3) 10 (4) 5 3 3 . If dy + 2y = f(x), where f(x) =

Q85.If I1 = ∫10 e−x cos2 xdx; I2 = ∫10 e−x2 cos2 xdx and I3 = ∫10 e−x3dx; then (1) I2 > I3 > I1 (2) I3 > I1 > I2 (3) I2 > I1 > I3 (4) I3 > I2 > I1

Q85.The area (in sq. units) of the region {x ∈R : x ≥0, y ≥0, y ≥x −2 and y ≤√x}, is (1) 13 (2) 10 3 3 (3) 5 (4) 8 3 3

Q86.Let y = y(x) be the solution of the differential equation dxdy + 2y = f(x), where x ∈[0, 1] f(x) = {1,0, otherwise If y(0) = 0, then y ( 23 ) is (1) e2−1 (2) e2−1 2e3 e3 (3) 1 (4) e2+1 2e 2e4 →

Q86.Let y = y(x) be the solution of the differential equation sin x dxdy + y cos x = 4x, x ∈(0, π). If y( π2 ) = 0, then y( π6 ) is equal to (1) −49 π2 (2) 9√34 π2 (3) −8 π2 (4) −89 π2 9√3 → → →

Q86.Let →a = ˆi + ˆj + ˆk, →c= ˆj −ˆk and a vector b be such that →a× b =→cand →a⋅ b = 3. Then b equals (1) 11 (2) 11 3 √3 (3) √113 (4) √113

Q86.Let y = y(x) be the solution of the differential equation dx {1,0, otherwisex ∈[0, 1] y(0) = 0, then y ( 23 ) is JEE Main 2018 (15 Apr) JEE Main Previous Year Paper (1) e2−1 (2) 1 e3 2e (3) e2+1 (4) e2−1 2e4 2e3 → →

Q86.The curve satisfying the differential equation, (x2 −y2)dx + 2xydy = 0 and passing through the point (1, 1) is (1) a circle of radius two (2) a circle of radius one (3) a hyperbola (4) an ellipse

Q87.Let u be a vector coplanar with the vectors →a = 2ˆi + 3ˆj −ˆk and b = ˆj + ˆk . If u is perpendicular to →a and → → → 2 u ⋅ b = 24, then u is equal to: (1) 84 (2) 336 (3) 315 (4) 256

Q87.The sum of the intercepts on the coordinate axes of the plane passing through the point (–2, –2, 2) and containing the line joining the points (1, –1, 2) and (1, 1, 1) is JEE Main 2018 (16 Apr Online) JEE Main Previous Year Paper (1) 4 (2) 12 (3) −8 (4) −4

Q87.If →a, b, →care unit vectors such that →a+ 2 b + 2→c=→0, then →a×→c is equal to : (1) 1 (2) 15 4 16 (3) √15 (4) √15 4 16

Q87.If →a,→b, and→care unit vectors such that →a + 2→b + 2→c = 0 , then |→a ×→c| is equal to (1) 1 (2) √15 4 4 (3) 15 (4) √15 16 16

Q87.If the position vectors of the vertices A, B and C of a △ABC are respectively 4^i + 7^j + 8^k, 2^i + 3^j + 4^k and 2^i + 5^j + 7^k , then the position vector of the point, where the bisector of ∠A meets BC is (1) 1 2 (4^i + 8^j + 11^k) (2) 13 (6^i + 13^j + 18^k) (3) 1 4 (8^i + 14^j + 9^k) (4) 13 (6^i + 11^j + 15^k)