Practice Questions

Q74.The area enclosed by the curves y = loge(x + e2), x = loge( 2y ) and (1) 2 + e −loge 2 (2) 1 + e −loge 2 (3) e −loge 2 (4) 1 + loge 2 dy +

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.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 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.Let the solution curve y = y(x) of the differential equation, [ √x2−y2x [ √x2−y2x through the points (1, 0) and (2α, α), α > 0 . Then α is equal to (1) 2 1 exp( π6 + √e −1) (2) 12 exp( π3 + √e −1) (3) exp( π6 + √e + 1) (4) 2 exp( π3 + √e −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.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.The value of ∫0 1 + cos2𝑥ecos𝑥+ e-cos𝑥d𝑥 is equal to (1) 𝜋2 (2) 𝜋 4 4 (3) 𝜋 (4) 𝜋2 6 2

Q75.The integral ∫10 [11x ] 7 (1) 1 −6 ln( 76 ) (2) 1 + 6 ln( 76 ) (3) 1 −7 ln( 76 ) (4) 1 + 7 ln( 76 )

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.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.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.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. 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.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 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.Let = , where a, b, c are constants. represent a circle passing through the point (2, 5). Then the dx bx+cy+a shortest distance of the point (11, 6) from this circle is (1) 10 (2) 8 (3) 7 (4) 5 dy 2x−y(2y−1)

Q75.The area of the bounded region enclosed by the curve y = 3 −x −12 −|x + 1| and the x-axis is (1) 9 (2) 45 4 16 (3) 278 (4) 1663 x x −4xe y2 = 0 such that x(1) = 0.

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.If the angle made by the tangent at the point 𝑥0, 𝑦0 on the curve 𝑥= 12𝑡+ sin𝑡cos𝑡, 𝜋 𝜋 𝑦= 121 + sin𝑡2, 0 < 𝑡< 2, with the positive 𝑥-axis is 3, then 𝑦0 is equal to (1) 63 + 2√2 (2) 37 + 4√3 (3) 27 (4) 48 𝜋 𝑛∈ℕ, then

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

Q76.Let y = y1(x) and y = y2(x) be two distinct solutions of the differential equation dxdy = x + y, with y1(0) = 0 and y2(0) = 1 respectively. Then, the number of points of intersection of y = y1(x) and y = y2(x) is (1) 0 (2) 1 (3) 2 (4) 3 → →

Q76.Let 𝑦= 𝑦𝑥 be the solution curve of the differential equation 𝑑𝑦 2𝑥2 + 11𝑥+ 13 𝑥+ 3 𝑥> - 1, which 𝑑𝑥+ 𝑥3 + 6𝑥2 + 11𝑥+ 6𝑦= 𝑥+ 1, passes through the point 0, 1. Then 𝑦1 is equal to 1 3 (1) (2) 2 2 5 7 (3) (4) 2 2

Q76.The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is (1) 9 (2) 7 (3) 5 (4) 3