Practice Questions

Q87.Let 𝑓( 𝑥) be a polynomial of degree 3 such that 𝑓𝑘= - for 𝑘= 2, 3, 4, 5 . Then the value of 𝑘 52 - 10 𝑓( 10 ) is equal to _____ .

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

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

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 ∫ 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 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.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.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 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 ________.

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.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. 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 ________.

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 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.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.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.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 _________.

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

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.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 Q be the foot of the perpendicular from the point P(7, −2, 13) on the plane containing the lines y−1 x+1 6 = 7 = z−38 and x−13 = y−25 = z−37 Then (PQ)2, is equal to ______. JEE Main 2021 (26 Aug 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

Q90.Let there be three independent events E1, E2 and E3. The probability that only E1 occurs is α only E2 occurs is β and only E3 occurs is γ. Let ′p′ denote the probability of none of events occurs that satisfies the equations (α −2β)p = αβ and (β −3γ)p = 2βγ. All the given probabilities are assumed to lie in the interval (0, 1). Then, Probability of occurrence of E1 is equal to ________. Probability of occurrence of E3 JEE Main 2021 (17 Mar Shift 1) JEE Main Previous Year Paper

Q90.If Im,n = ∫10 xm−1(1 −x)n−1dx, for m, n ⩾1, and ∫10 xm−1+xn−1(1+x)m+n ________. JEE Main 2021 (26 Feb Shift 2) JEE Main Previous Year Paper