Practice Questions

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

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 )

Q76.Let y = y(x) be the solution of the differential equation x(1 −x2) dxdy + (3x2y −y −4x3) = 0, x > 1 with y(2) = −2. Then y(3) is equal to (1) −18 (2) −12 (3) −6 (4) −3

Q76.The general solution of the differential equation 𝑥- 𝑦2𝑑𝑥+ 𝑦5𝑥+ 𝑦2𝑑𝑦= 0 is 4 3 4 3 (1) 𝑦2 + 𝑥 = 𝐶𝑦2 + 2𝑥 (2) 𝑦2 + 2𝑥 = 𝐶𝑦2 + 𝑥 3 4 3 4 (3) 𝑦2 + 𝑥 = 𝐶2𝑦2 + 𝑥 (4) 𝑦2 + 2𝑥 = 𝐶2𝑦2 + 𝑥 → → → → → →

Q76.If y = y(x) is the solution of the differential equation (1 + e2x) dxdy + 2(1 + y2)ex = 0 and y(0) = 0, then 2 + (y(logc √3)) is equal to: 6(y′(0) ) (1) 2 (2) −2 (3) −4 (4) −1

Q76.Let a smooth curve y = f(x) be such that the slope of the tangent at any point (x, y) on it is directly proportional to ( −yx ). If the curve passes through the points (1, 2) and (8, 1), then y( 81 ) is equal to (1) 2 loge 2 (2) 4 (3) 1 (4) 4 loge 2 → → → →

Q76.If the solution curve of the differential equation ((tan−1 y) −x)dy = (1 + y2)dx passes through the point (1, 0) then the abscissa of the point on the curve whose ordinate is tan(1) is (1) 2 (2) 2e (3) 3 (4) 2e e →

Q76.Let x = x(y) be the solution of the differential equation 2ye y2 dx + (y2 )dy Then, x(e) is equal to (1) e loge(2) (2) −e loge(2) (3) e2 loge(2) (4) −e2 loge(2)

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.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

Q76.If dx dy + 2y tan x = sin x, 0 < x < π2 and y( π3 ) = 0 , then the maximum value of y(x) is JEE Main 2022 (26 Jul Shift 1) JEE Main Previous Year Paper (1) 1 (2) 3 8 4 (3) 1 (4) 3 4 8 → →

Q76.If dx + 2x−1 = 0, x, y > 0, y(1) = 1 , then y(2) is equal to (1) 2 + log2 3 (2) 2 + log2 2 (3) 2 −log−2 3 (4) 2 −log2 3 → →

Q76.If 𝑏𝑛= ∫02 cos2𝑛𝑥sin𝑥𝑑𝑥, 1 1 1 (1) 𝑏3 - 𝑏2, 𝑏4 - 𝑏3, 𝑏5 - 𝑏4 are in an A.P. with (2) 𝑏3 - 𝑏2, 𝑏4 - 𝑏3, 𝑏5 - 𝑏4 are in an A.P. with common common difference-2 difference 2 (3) 𝑏3 - 𝑏2, 𝑏4 - 𝑏3, 𝑏5 - 𝑏4 are in a G.P. (4) 1 1 1 are in an A.P. with common 𝑏3 - 𝑏2, 𝑏4 - 𝑏3, 𝑏5 - 𝑏4 difference -2

Q76.The area bounded by the curve y = x2 −9 and the line y = 3 is (1) 8√6 −16√12 −72 (2) 8√6 + 8√12 −72 (3) 16√6 + 16√12 −72 (4) 16√6 −16√12 −64 → → → → → is b b × b × × (→c×→a) →c

Q76.The differential equation of the family of circles passing through the points (0, 2) and (0, −2) is (1) 2xy dxdy + (x2 −y2 + 4) = 0 (2) 2xy dxdy + (x2 + y2 −4) = 0 (3) 2xy dxdy + (y2 −x2 + 4) = 0 (4) 2xy dxdy −(x2 −y2 + 4) = 0 →

Q76.Let the solution curve y = y(x) of the differential equation (1 + e2x)( dxdy y) (0, π2 ). Then, x→∞exy(x)lim is equal to JEE Main 2022 (29 Jul Shift 1) JEE Main Previous Year Paper (1) π (2) 3π 4 4 (3) π (4) 3π 2 2 → b = b + λ→c. If→b and →care non-

Q76.The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by x2 . If the curve xy−x2y2−1 passes through the point (1, 1), then e ⋅y(e) is equal to (1) 1−tan(1) (2) tan(1) 1+tan(1) (3) 1 (4) 1+tan(1) 1−tan(1)

Q77.Let 𝐴𝐵𝐶 be a triangle such that 𝐵𝐶= →𝑎, 𝐶𝐴= 𝑏, 𝐴𝐵= →𝑐, →𝑎= 6√2, 𝑏= 2√3 and 𝑏· →𝑐= 12 Consider the statements : 𝑆1: →𝑎× →𝑏+ →𝑐× →𝑏- →𝑐= 62√2 - 1 𝑆2: ∠𝐴𝐵𝐶= cos-1√ 23. Then (1) both 𝑆1 and 𝑆2are true (2) only 𝑆1 is true (3) only 𝑆2 is true (4) both 𝑆1 and 𝑆2 are false 𝑥- 3 𝑦+ 4 𝑧- 7

Q77.Let A, B, C be three points whose position vectors respectively are: →a = ˆi + 4ˆj + 3ˆk → b = 2ˆi + αˆj + 4ˆk, α ∈R →c= 3ˆi −2ˆj + 5ˆk → If α is the smallest positive integer for which →a, b, →care non-collinear, then the length of the median, △ABC , through A is: (1) √82 (2) √62 2 2 (3) √69 (4) √66 2 2 y+1