Practice Questions

Q74.If ∫1x √1−x1+x + π3 (1) loge( √3+1√3−1 ) + π3 (2) loge( √3+1√3−1 ) (3) loge( √3−1√3+1 ) −π3 (4) 13 loge( √3−1√3+1 ) −π6

Q74.Let f be a differentiable function in (0, π2 ). If ∫1cos x t2f(t)dt = sin3 x + cos x, then √31 f ′( √31 ) (1) 6 −9√2 (2) 6 + 9 √2 (3) 6 − 9 (4) 3 + √2 √2 dx, where [⋅] denotes the greatest integer function, is equal to

Q74.The value of the integral ∫2−2 (ex|x|+1)x3+x (1) 5e2 (2) 3e−2 (3) 4 (4) 6 dy ax−by+a

Q74.If the tangent at the point (x1, y1) on the curve y = x3 + 3x2 + 5 passes through the origin, then (x1, y1) does NOT lie on the curve (1) x2 + 81y2 = 2 (2) y29 −x2 = 8 (3) y = 4x2 + 5 (4) x3 −y2 = 2

Q74.If the line 𝑦= 4 + 𝑘𝑥, 𝑘> 0, is the tangent to the parabola 𝑦= 𝑥- 𝑥2 at the point 𝑃 and 𝑉 is the vertex of the parabola, then the slope of the line through 𝑃 and 𝑉 is (1) 3 (2) 26 2 9 5 23 (3) (4) 2 6

Q74.If the maximum value of 𝑎, for which the function 𝑓𝑎𝑥= tan-12𝑥- 3𝑎𝑥+ 7 is non-decreasing in -𝜋 𝜋 is ¯𝑎, 6, 6, 𝜋 then 𝑓¯𝑎 8 is equal to (1) 8 - 9𝜋 (2) 8 - 4𝜋 49 + 𝜋2 94 + 𝜋2 1 + 𝜋2 𝜋 (4) 8 - (3) 8 4 9 + 𝜋2 JEE Main 2022 (26 Jul Shift 2) JEE Main Previous Year Paper Q75. 1 - 1 √3cos𝑥- sin𝑥 The integral ∫ 2 is equal to 1 + √3sin2𝑥𝑑𝑥 𝜋 𝜋 tan𝑥 + tan𝑥 + 2 12 2 (1) 1 (2) 𝑥 𝜋 + 𝐶 2log𝑒 6 + 𝜋 + 𝐶 log𝑒 𝑥 + 2 6 2 3 𝜋 𝜋 tan𝑥 + tan𝑥 - 2 2 12 (3) 1 6 (4) 1 𝑥 𝜋 + 𝐶 2log𝑒 + 𝜋 + 𝐶 2log𝑒 tan𝑥 - 2 3 2 6 Q76. 20𝜋sin𝑥+ cos𝑥2𝑑𝑥 is equal to: ∫0 (1) 10𝜋+ 4 (2) 10𝜋+ 2 (3) 20𝜋- 2 (4) 20𝜋+ 2

Q74.Let f : R →R be continuous function satisfying f(x) + f(x + k) = n, for all x ∈R where k > 0 and n is a positive integer. If I1 = ∫4nk0 f(x)dx and I2 = ∫3k−k f(x)dx, then (1) I1 + 2I2 = 4nk (2) I1 + 2I2 = 2nk (3) I1 + nI2 = 4n2 K (4) I1 + nI2 = 6n2k

Q74.Let 𝑔: 0, ∞→𝑅 be a differentiable function such that ∫ + d𝑥= + 𝐶, for all 𝑥> 0 e𝑥+ 1 e𝑥+ 12 e𝑥+ 1 , where 𝐶 is an arbitrary constant. Then 𝜋 𝜋 (1) 𝑔 is decreasing in 0, (2) 𝑔- 𝑔' is increasing in 0, 4 2 (3) 𝑔' is increasing in 0, 𝜋 (4) 𝑔+ 𝑔' is increasing in 0, 𝜋 4 2 𝜋 ecos𝑥sin𝑥

Q74.If 𝑡 denotes the greatest integer ≤t, then the value of ∫0 2𝑥- 3𝑥2 - 5𝑥+ 2 + 1𝑑𝑥 is JEE Main 2022 (29 Jul Shift 2) JEE Main Previous Year Paper (1) √37 + √13 - 4 (2) √37 - √13 - 4 6 6 (3) -√37 - √13 + 4 (4) -√37 + √13 + 4 6 6

Q75.The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is 3643 , equal to: (1) 3 (2) 5 (3) 7 (4) 9

Q75.Let y = y(x) be the solution curve of the differential equation dx 1 1 y = ( x−1x+1 ) 2 , x > 1 passing through x2−1 the point . Then √7y(8) is equal to 3 (2, √1 ) (1) 11 + 6 loge 3 (2) 19 (3) 12 −2 loge 3 (4) 19 −6 loge 3

Q75.The area of the region {(x, y) : |x −1| ≤y ≤√5 −x2} (1) 5 2 sin−1( 53 ) −12 (2) 5π4 −32 (3) 3π 4 + 23 (4) 5π4 −12 + = 1 pass through the point

Q75.A wire of length 22m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is (1) 22 (2) 66 9+4√3 9+4√3 (3) 22 (4) 66 4+9√3 4+9√3 t, is equal toQ76. ∫50 cos(π(x −[ x2 ]))dx, where [t] denotes greatest integer less than or equal to (1) 0 (2) 2 (3) −3 (4) 4 JEE Main 2022 (29 Jun Shift 1) JEE Main Previous Year Paper

Q75.The sum of absolute maximum and absolute minimum values of the function f(x) = 2x2 + 3x −2 + sin x cos x in the interval [0, 1] is 1 sin(1) cos2( (1) 2 ) (2) 3 + 12 (1 + 2 cos(1)) sin(1) 3 + 2 (3) 5 + 12 (sin(1) + sin(2)) (4) 2 + sin( 21 ) cos( 12 )

Q75.Let f(x) = 2 cos−1 x + 4 cot−1 x −3x2 −2x + 10, x ∈[−1, 1]. If [a, b] is the range of the function, then 4a −b is equal to (1) 11 (2) 11 −π (3) 11 + π (4) 15 −π

Q75.If ∫20 (√2x −√2x −x2)dx + I , then I equal to + ∫21 (2 −y22 )dy ∫10 (1 −√1 −y2 −y22 )dy −y2 + + √1 −y2)dy (2) ∫10 ( y22 −√1 1)dy (1) ∫10 (1 + √1 −y2 + 1)dy (3) ∫10 (1 −√1 −y2)dy (4) ∫10 ( y22

Q75.The area of the smaller region enclosed by the curves y2 = 8x + 4 and x2 + y2 + 4√3x −4 = 0 is equal to (1) 1 + + 3 (2 −12√3 8π) (2) 13 (2 −12√3 6π) (3) 1 −12√3 + −12√3 + 3 (4 8π) (4) 13 (4 6π)

Q75.The area of the region bounded by y2 = 8x and y2 = 16(3 −x) is equal to (1) 32 (2) 40 3 3 (3) 16 (4) 9

Q75. n→∞(lim (n2+1)(n+1)n2 + (n2+4)(n+2)n2 + (n2+9)(n+3)n2 + … + (n2+n2)(n+n)n2 ) is equal to (1) π 8 + 14 ln 2 (2) π4 + 18 ln 2 (3) π 4 −18 ln 2 (4) π8 + ln √2

Q75.Let the solution curve of the differential equation 𝑥𝑑𝑦= √𝑥2 + 𝑦2 + 𝑦𝑑𝑥, 𝑥> 0, intersect the line x = 1 at 𝑦= 0 and the line 𝑥= 2 at 𝑦= 𝛼. Then the value of 𝛼 is (1) 1 (2) 3 2 2 3 5 (3) - (4) 2 2

Q75.The area of the region enclosed by y ≤4x2, x2 ≤9y and y ≤4 , is equal to (1) 40 (2) 56 3 3 (3) 112 (4) 80 3 3

Q75.The slope of the tangent to a curve 𝐶: 𝑦= 𝑦𝑥 at any point [𝑥, 𝑦) on it is 2e2x - 6e-x + 9 . If 𝐶 passes through the 2 + 9e-2x 1 𝜋 1 points 0, + and 𝛼, then 𝑒𝛼 is equal to 2 2√2 2e2𝛼 (1) 3 + √2 (2) 3 3 + √2 3 - √2 √2 3 - √2 (3) 1 √2 + 1 (4) √2 + 1 √2 √2 - 1 √2 - 1

Q75.Let [t] denote the greatest integer less than or equal to t. Then the value of the integral ∫101−3 ([sin(πx)] + e[cos(2πx)])dx is equal to (1) 52(1−e) (2) 52 e e (3) 52(2+e) (4) 104 e e

Q75.If the solution curve of the differential equation 𝑑𝑦 𝑥+ 𝑦- 2 passes through the point 2, 1 and 𝑘+ 1, 2, k > 0, 𝑑𝑥= 𝑥- 𝑦 then (1) 2tan-11 + 1 𝑘= loge𝑘2 + 1 (2) tan-11𝑘= loge𝑘2 1 𝑘2 + 1 (3) 2tan-1 = loge𝑘2 + 2𝑘+ 2 (4) 2tan-11 𝑘+ 1 𝑘= loge 𝑘2

Q75.The value of ∫0 1 + cos2𝑥ecos𝑥+ e-cos𝑥d𝑥 is equal to (1) 𝜋2 (2) 𝜋 4 4 (3) 𝜋 (4) 𝜋2 6 2