Practice Questions

Q87.If ∫π0 (sin3 x)e−sin2 xdx =

Q88.Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through a −2√2 b = 3 , then (a2 + b2 + ab) is equal to _____. (3, −3) and (4, −2√2), given that

Q88.For real numbers α, β, γ and δ, if (x2−1)+tan−1( x2+1x ) x2+1 γ(x2−1) x2+1 + β + δ + C where C is ∫ x2+1 dx = α loge(tan−1( x )) tan−1( x ) tan−1( x ) (x4+3x2+1) tan−1( x ) an arbitrary constant, then the value of 10(α + βγ + δ) is equal to ______ . → = 8 , then

Q88.Let f : [−3, 1] →R be given as f(x) = {max{√x,min{(x + 6),x2},x2}, −30 ≤x≤x≤1≤0 .If the area bounded by y = f(x) and x-axis is A sq units, then the value of 6A is equal to → → →

Q88.If xϕ(x) = ∫x5 (3t2 −2ϕ′(t))dt, JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper

Q88.Let 𝑆= 𝑛∈𝑁, 𝑏, 𝑐, 𝑑∈𝑅, where 𝑖= √-1 . Then the number of 2 - digit 1 0 𝑐 𝑑= 𝑐 𝑑∀𝑎, numbers in the set 𝑆 is

Q88.If ∫ sin𝑥 d𝑥= | 1 + tan𝑥| + - tan𝑥+ tan2𝑥+ 𝛾tan-1 2tan𝑥- 1 + 𝐶, when 𝐶 is constant sin3𝑥+ cos3𝑥 𝛼loge 𝛽loge1 √3 of integration, then the value of 18𝛼+ 𝛽+ 𝛾2 is 3

Q88.Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = −1 and x = 1 . If lim f(x) = 1, then 5 ⋅f(2) is equal to x→0 x3

Q88.If a + α = 1, b + β = 2 and af(x) + αf( x1 ) = bx + xβ , x ≠0, then the value of the expression f(x)+f(x+ x ___________.

Q88. if |x| ≤2 2 ) . Let f : R →R be a function defined as f(x) = { 3(1 −|x|0 if |x| > 2 Let g : R →R be given by g(x) = f(x + 2) −f(x −2). If n and m denote the number of points in R where g is not continuous and not differentiable, respectively, then n + m is equal to ________.

Q88.Let [t] denote the greatest integer ≤t . The number of points where the function 𝑓(𝑥) = [𝑥]𝑥2 - 1 + sin 𝜋 - [𝑥+ 1], 𝑥∈( - 2, 2) is not continuous is _____ . [𝑥] + 3

Q88.Let y = y(x) be the solution of the differential equation xdy −ydx = √(x2 −y2)dx, x ≥1 , with y(1) = 0. If the area bounded by the line x = 1, x = eπ, y = 0 and y = y(x) is αe2π + β, then the value of 10(α + β) is equal to ___ . −b = 0 be (−3, 5, 2).

Q88.If 𝑎𝑥+ 𝑥- 2𝑑𝑥= 22, 𝑎> 2 and 𝑥 denotes the greatest integer ≤𝑥, then -𝑎𝑥+ 𝑥𝑑𝑥 is equal to ∫-𝑎 ∫𝑎

Q88.If the normal to the curve y(x) = ∫x0 (2t2 −15t + 10)dt at a point (a, b) is parallel to the line x + 3y = −5, a > 1 , then the value of |a + 6b| is equal to ________.

Q89.Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ______. dx = αIm,n, α ∈R, then α equals

Q89.Let a curve y = y(x) be given by the solution of the differential equation y-axis at y = −1, and the intersection point of the cos( 12 cos−1(e−x))dx = (√e2x −1)dy. If it intersects curve with x− axis is (α, 0), then eα is equal to

Q89.Let P be a plane passing through the points (1, 0, 1), (1, −2, 1) and (0, 1, −2). Let a vector →a = αˆi + βˆj + γˆk = 2 , then be such that →a is parallel to the plane P , perpendicular to (ˆi + 2ˆj + 3ˆk) and →a⋅(ˆi + ˆj + 2ˆk) (α −β + γ)2 equals______. → + λ ∈R, α > 0 and

Q89.Let 𝑦= 𝑦( 𝑥) be solution of the following differential equation 𝑒𝑦𝑑𝑦 2𝑒𝑦sin𝑥+ sin𝑥cos2𝑥= 0, 𝑦 𝜋 = 0. 𝑑𝑥- 2 If 𝑦0 = loge𝛼+ 𝛽e-2, then 4 ( 𝛼+ 𝛽) is equal to .

Q89.If y = y(x) is the solution of the equation esin y cos y dxdy + esin y cos x = cos x, y(0) = 0; then 1 + y( π6 ) + √32 y( π3 ) + √21 y( π4 ) is equal to _______.

Q89.The area (in sq. units) of the region bounded by the curves x2 + 2y −1 = 0, y2 + 4x −4 = 0 and y2 −4x −4 = 0 in the upper half plane is _________.

Q90.A line l passing through origin is perpendicular to the lines l1 :→r= (3 + t)ˆi + (−1 + 2t)ˆj + (4 + 2t)ˆk l2 :→r= (3 + 2s)ˆi + (3 + 2s)ˆj + (2 + s)ˆk If the co-ordinates of the point in the first octant on l2 at a distance of √17 from the point of intersection of l and l1 are (a, b, c), then 18(a + b + c) is equal to ___ . JEE Main 2021 (25 Feb Shift 2) JEE Main Previous Year Paper

Q90.Let 𝐵𝑖𝑖= 1, 2, 3 be three independent events in a sample space. The probability that only 𝐵1 occur is 𝛼, only 𝐵2 occurs is 𝛽 and only 𝐵3 occurs is 𝛾. Let 𝑝 be the probability that none of the events 𝐵𝑖 occurs and these 4 probabilities satisfy the equations 𝛼- 2𝛽𝑝= 𝛼𝛽 and 𝛽- 3𝛾𝑝= 2𝛽𝛾 (All the probabilities are assumed to lie in 𝑃𝐵1 the interval 0, 1 Then is equal to______. 𝑃𝐵3 JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper

Q90.Let y = y(x) be the solution of the differential equation x+2 ) + (y + = (x + 2)dy, y(1) = 1. If the domain of y = y(x) is an open interval (α, β), + 2)e( 1))dx ((x y+1 then |α + β| is equal to ___________. JEE Main 2021 (22 Jul Shift 1) JEE Main Previous Year Paper

Q90.For p > 0, a vector →v2 = 2ˆi + (p + 1)ˆj is obtained by rotating the vector →v1 = √3pˆi + ˆj by an angle θ about (α√3−2) origin in counter clockwise direction. If tan θ = , then the value of α is equal to (4√3+3) JEE Main 2021 (20 Jul Shift 2) JEE Main Previous Year Paper

Q90.Let the line L be the projection of the line x−1 2 = y−31 = z−42 in the plane x −2y −z = 3. If d is the distance of the point (0, 0, 6) from L, then d2 is equal to JEE Main 2021 (26 Aug Shift 1) JEE Main Previous Year Paper