Practice Questions

Q87.Let A = {(x, y) : 2x + 3y = 23, x, y ∈N} and B = {x : (x, y) ∈A}. Then the number of one-one functions from A to B is equal to _______

Q87.Let 𝑆= −1, ∞ and 𝑓: 𝑆→ℝ be defined as 𝑓𝑥= ∫ 𝑒𝑡−1112𝑡−15𝑡−27𝑡−3122𝑡−1061𝑑𝑡. Let 𝑝= Sum −1 of square of the values of 𝑥, where 𝑓𝑥 attains local maxima on 𝑆. and 𝑞= Sum of the values of 𝑥, where 𝑓𝑥 attains local minima on 𝑆. Then, the value of 𝑝2 + 2𝑞 is ________ 𝜋 1 Q88. 2 11 5 If the integral 525 ∫ sin2𝑥 cos 2 𝑥1 + cos 2𝑥 2𝑑𝑥 is equal to 𝑛√2 −64, then 𝑛 is equal to ________ 0 → → → → →

Q87.Let 𝐴= 1, 2, 3, . ..20 . Let 𝑅1 and 𝑅2 two relation on 𝐴 such that 𝑅1 = {𝑎, 𝑏: 𝑏 is divisible by 𝑎} 𝑅2 = {𝑎, 𝑏: 𝑎 is an integral multiple of 𝑏} Then, number of elements in 𝑅1 −𝑅2 is equal to __________. 𝛼𝜋+ 𝛽log𝑒3 + 2√2, where 𝛼, 𝛽 are integers, then 𝛼2 + 𝛽2 equals __________

Q87.Let f(x) = √limr→x{ 2r2[(f(r))2−f(x)f(r)] f(r) the value of ae , such that f(a) = 0, is equal to ______. π

Q87.Let f(x) = 2x −x2, x ∈R. If m and n are respectively the number of points at which the curves y = f(x) and y = f ′(x) intersects the x−axis, then the value of m + n is

Q87.Let the area of the region {(x, y) : x −2y + 4 ≥0, x + 2y2 ≥0, x + 4y2 ≤8, y ≥0} be mn , where m and n are coprime numbers. Then m + n is equal to ______.

Q87.Let the maximum and minimum values of −x2 −12 2 + (x −7)2, x ∈R be M and m , (√8x −4) respectively. Then M2 −m2 is equal to _________ π

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 A be the region enclosed by the parabola y2 = 2x and the line x = 24. Then the maximum area of the rectangle inscribed in the region A is________ + C, where C is the constant of integration, then the value of

Q87.If ∫ 0 4 1+sinsin2x xcos x dx = 1a loge ( 3a ) + b√3π , where

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

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.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.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.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 value 9 ∫90 [√10x ] ,

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.If the area of the region ( x, y ) : 0 ≤y ≤min2x, 6x - x2 is A, then 12 A 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.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.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 ∫ B 1 dx = A( αx−1βx+3 ) 5√(x−1)4(x+3)6 α + β + 20AB is__________

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