Practice Questions

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.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 _______

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

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

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

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.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.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 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.The value 9 ∫90 [√10x ] ,

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 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.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.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.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 _____.

Q89.The area of the region enclosed by the parabolas y = x2 −5x and y = 7x −x2 is → →

Q89.Let →𝑎 and 𝑏 be two vectors such that →𝑎= 1, 𝑏= 4 and →𝑎⋅ 𝑏= 2. If →𝑐= 2 →𝑎× 𝑏−3 𝑏 and the angle between →𝑏 and→𝑐 is 𝛼, then 192sin2𝛼 is equal to _________

Q89.Let →a = 9^i −13^j + 25^k,→b = 3^i + 7^j −13^k and →c = 17^i −2^j + ^k be three given vectors. If →r is a vector such |593→r+67→a|2 is equal to___________ that →r × →a = (→b + →c) × →a and →r ⋅(→b −→c) = 0 , then (593)2

Q89.Let the set of all positive values of λ , for which the point of local minimum of the function (1 + x (λ2 −x2)) satisfies x2+x+2 < 0, be (α, β). Then α2 + β2 is equal to _________ x2+5x+6

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 )