Practice Questions

Q87.Three points 𝑂0, 0, 𝑃𝑎, 𝑎2, 𝑄−𝑏, 𝑏2, 𝑎> 0, 𝑏> 0, are on the parabola 𝑦= 𝑥2. Let 𝑆1 be the area of the region bounded by the line 𝑃𝑄 and the parabola, and 𝑆2 be the area of the triangle 𝑂𝑃𝑄. If the minimum value 𝑆1 𝑚 of is 𝑛, gcd𝑚, 𝑛= 1, then 𝑚+ 𝑛 is equal to: 𝑆2 JEE Main 2024 (01 Feb Shift 2) JEE Main Previous Year Paper

Q87.Let [t] denote the largest integer less than or equal to t. If + = a + b√2 −√3 −√5 + c√6 −√7, where a, b, c ∈Z, then a + b + c is equal ∫30 ([x2] [ x22 ])dx to_______

Q88.If ∫ B 1 dx = A( αx−1βx+3 ) 5√(x−1)4(x+3)6 α + β + 20AB is__________

Q88.For a differentiable function f : R →R, suppose f ′(x) = 3f(x) + α, where α ∈R, f(0) = 1 and limx→−∞f(x) = 7. Then 9f (−loge 3) is equal to_________

Q88.The value 9 ∫90 [√10x ] ,

Q88.If the solution of the differential equation (2x + 3y −2)dx + (4x + 6y −7)dy = 0, y(0) = 3, is αx + βy + 3 loge |2x + 3y −γ| = 6, then α + 2β + 3γ is equal to ______.

Q88.Let the area of the region enclosed by the curve y = min{sin x, cos x} and the x axis between x = −π to x = π be A . Then A2 is equal to ___________

Q88.The area of the region enclosed by the parabola ( 𝑦- 2 ) 2 = 𝑥- 1, the line 𝑥- 2 𝑦+ 4 = 0 and the positive coordinate axes is __________.

Q88.If ∫ π3 √1 −sin 2xdx = α + β√2 + γ√3, where α, β and γ are rational numbers, then 3α + 4β −γ is equal 6 to _____.

Q88.If the area of the region ( x, y ) : 0 ≤y ≤min2x, 6x - x2 is A, then 12 A is equal to _______.

Q88.Let 𝑦= 𝑦𝑥 be the solution of the differential equation sec2𝑥𝑑𝑥+ 𝑒2𝑦tan2𝑥+ tan𝑥𝑑𝑦= 0, 0 < 𝑥< 𝜋 𝑦𝜋 = 0. 2, 4 𝜋 If 𝑦 = 𝛼, then 𝑒8𝛼 is equal to ______. 6

Q88.The area (in sq. units) of the part of circle x2 + y2 = 169 which is below the line 5x −y = 13 is πα 2β −652 + αβ sin−1( 1312 ) where α, β are coprime numbers. Then α + β is equal to

Q88.If the solution y(x) of the given differential equation (ey + 1) cos x dx + ey sin x dy = 0 passes through the 6 ) is equal to_________ point ( π2 , 0), then the value of ey( π

Q88.If ∫−𝜋/𝜋/ 2 2 1 +8√2cos𝑥𝑑𝑥𝑒sin𝑥1 + sin4𝑥=

Q88.The sum of squares of all possible values of 𝑘, for which area of the region bounded by the parabolas 2𝑦2 = 𝑘𝑥 and 𝑘𝑦2 = 2𝑦−𝑥 is maximum, is equal to:

Q88.Let the solution y = y(x) of the differential equation dydx −y = 1 + 4 sin x satisfy y(π) = 1. Then y ( π2 ) + 10 is equal to ______ −−

Q88.Let y = y(x) be the solution of the differential equation (x + y + 2)2dx = dy, y(0) = −2. Let the maximum and minimum values of the function y = y(x) in [0, π3 ] be α and β , respectively. If (3α + π)2 + β2 = γ + δ√3, γ, δ ∈Z , then γ + δ equals ______

Q88.If S = {a ∈R : |2a −1| = 3[a] + 2{a}} , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t , then 72 ∑a∈S a is equal to ______

Q88.If f(t) = ∫π0 1−cos22x dxt sin2 x , 0 < t < π, then the value of ∫ 0 2 π2dtf(t) equals_________ 1 dy 2x (1+x2) y = xe ; y(0) = 0.

Q88.Let rk = , k ∈N. Then the value of ∑10k=1 7(rk−1)1 is equal to________ (1−x7)k+1dx ∫1 0

Q89.If 𝑥= 𝑥𝑡 is the solution of the differential equation 𝑡+ 1𝑑𝑥= 2𝑥+ 𝑡+ 14𝑑𝑡, 𝑥0 = 2, then 𝑥1 equals ________

Q89.Let →a = 2^i −3^j + 4^k,→b = 3^i + 4^j −5^k and a vector →c be such that →a × (→b + →c) + →b × →c = ^i + 8^j + 13^k . If →a ⋅→c = 13 , then (24 −→b ⋅→c) is equal to_______

Q89.If the solution curve y = y(x) of the differential equation (1 + y2)(1 + loge x)dx + xdy = 0, x > 0 passes α−tan( 23 ) through the point (1, 1) and y(e) = 3 , then α + 2β is β+tan( 2 )

Q89.Let 𝑌= 𝑌( 𝑋) be a curve lying in the first quadrant such that the area enclosed by the line 𝑌- 𝑦= 𝑌' (𝑥) (𝑋- 𝑥) and the co-ordinate axes, where ( 𝑥, 𝑦) is any point on the curve, is always -𝑦2 + 1, 𝑌'𝑥≠0. If 𝑌( 1 ) = 1, then 12 𝑌( 2 ) equals ________. 2𝑌' (𝑥)

Q89.Consider a line L passing through the points P(1, 2, 1) and Q(2, 1, −1). If the mirror image of the point A(2, 2, 2) in the line L is (α, β, γ), then α + β + 6γ is equal to _______