Practice Questions

Q82.Let S = {sin2 2θ : (sin4 θ + cos4 θ)x2 + (sin 2θ)x + (sin6 θ + cos6 θ) = 0 has real roots }. If α and β be the smallest and largest elements of the set S , respectively, then 3 ((α −2)2 + (β −1)2) equals _________

Q82.Let the first term of a series be T1 = 6 and its rth term Tr = 3Tr−1 + 6r, r = 2, 3, n. If the sum of the first n terms of this series is 1 (n2 −12n + 39) (4 ⋅6n −5 ⋅3n + 1), then n is equal to______ 5 JEE Main 2024 (06 Apr Shift 1) JEE Main Previous Year Paper

Q82.Let a1, a2, a3, … be in an arithmetic progression of positive terms. Let Ak = a21 −a22 + a23 −a24 + … + a22k−1 −a22k . If A3 = −153, A5 = −435 and a21 + a22 + a23 = 66 , then a17 −A7 is equal to______ is p , then 108p is equal to

Q83.Consider a triangle ABC having the vertices A(1, 2), B(α, β) and C(γ, δ) and angles ∠ABC = π6 and ∠BAC = 2π3 . If the points B and C lie on the line y = x + 4, then α2 + γ 2 is equal to ________ = and the determinant of A be 1 . If A−1 = αA + βI ,

Q83.Let 𝐴𝐵𝐶 be an isosceles triangle in which 𝐴 is at −1, 0, ∠𝐴= , 𝐴𝐵= 𝐴𝐶 and 𝐵 is on the positive 𝑥- 3 𝛽4 axis. If 𝐵𝐶= 4√3 and the line 𝐵𝐶 intersects the line 𝑦= 𝑥+ 3 at 𝛼, 𝛽, then is: 𝛼2

Q83.Let the set of all a ∈R such that the equation cos 2x + a sin x = 2a −7 has a solution be [p, q] and r = tan 9°−tan 27°− cot163° + tan 81°, then pqr is equal to ________. Q84. ⎡ 2 0 1⎤ ⎡ 1 ⎤ Let A = 1 1 0 , B = [B1 B2 B3 ], where B1 , B2, B3 are column matrices, and AB1 = 0 , ⎣ 1 0 1⎦ ⎣ 0 ⎦ ⎡2 ⎤ ⎡ 3 ⎤ AB2 = 3 , AB3 = 2 ⎣0 ⎦ ⎣ 1 ⎦ If α = |B| and β is the sum of all the diagonal elements of B , then α3 + β3 is equal to

Q83.Let α = ∑nr=0 (4r2 + 2r + 1)nCr and β = (∑nr=0 r+1nCr ) _______

Q84.Let the latus rectum of the hyperbola x2 = 1 subtend an angle of π3 at the centre of the hyperbola. If b2 9 −y2b2 is equal to l (1 + √n), where l and m are co-prime numbers, then l2 + m2 + n2 is equal to __________. m

Q84.Consider the circle C : x2 + y2 = 4 and the parabola P : y2 = 8x. If the set of all values of α, for which three chords of the circle C on three distinct lines passing through the point (α, 0) are bisected by the parabola P is the interval (p, q), then (2q −p)2 is equal to ________ JEE Main 2024 (09 Apr Shift 2) JEE Main Previous Year Paper

Q84.Let the line 𝐿: √2𝑥+ 𝑦= 𝛼 pass through the point of the intersection 𝑃(in the first quadrant)of the circle 𝑥2 + 𝑦2 = 3 and the parabola 𝑥2 = 2𝑦. Let the line 𝐿 touch two circles 𝐶1 and 𝐶2 of equal radius 2√3. If the centres 𝑄1 and 𝑄2 of the circles 𝐶1 and 𝐶2 lie on the 𝑦- axis, then the square of the area of the triangle 𝑃𝑄1𝑄2 is equal to _________.

Q84.Suppose AB is a focal chord of the parabola y2 = 12x of length l and slope m < √3 . If the distance of the chord AB from the origin is d , then l d2 is equal to _______ for

Q84.If lim 𝑎𝑥2𝑒𝑥−𝑏log𝑒1 + 𝑥+ 𝑐𝑥𝑒−𝑥 = 1, then 16𝑎2 + 𝑏2 + 𝑐2 is equal to ______. 𝑥→0 𝑥2sin𝑥 JEE Main 2024 (31 Jan Shift 2) JEE Main Previous Year Paper

Q84.Let n − 2n + n − 8n + … + n − 2n⋅n2 be πk , limn→∞( √n4+1 (n2+1)√n4+1 √n4+16 (n2+4)√n4+16 √n4+n4 (n2+n2)√n4+n4 ) using only the principal values of the inverse trigonometric functions. Then k2 is equal to ________

Q84.Let a conic C pass through the point (4, −2) and P(x, y), x ≥3, be any point on C . Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and (3, −5). If the focal distance of the point (7, 1) on C is d, then 12d equals ______

Q84.If the orthocentre of the triangle formed by the lines 2x + 3y −1 = 0, x + 2y −1 = 0 and ax + by −1 = 0, is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3, 4) and (−6, −8), then the value of |a −b| is_______ is

Q85.Let a > 0 be a root of the equation 2x2 + x −2 = 0. If limx→1a 16(1−cos(2+x−2x2))(1−ax)2 α, β ∈Z , then α + β is equal to_______

Q85.Let f be a differentiable function in the interval (0, ∞) such that f(1) = 1 and limt→x t2f(x)−x2f(t)t−x = 1 each x > 0 . Then 2f(2) + 3f(3) is equal to _______

Q85.The value of limx→0 2 ( 1−cos x√cos 2x3√cosx2 3x……10√cos 10x )

Q85.Let 𝑥 denote the fractional part of 𝑥 and 𝑓𝑥= cos−11 −𝑥2sin−11 −𝑥 , 𝑥≠0. If 𝐿 and 𝑅 respectively denotes the 𝑥−𝑥3 32 left hand limit and the right hand limit of 𝑓𝑥 at 𝑥= 0, then 𝜋2𝐿2 + 𝑅2 is equal to __________.

Q85.Consider two circles 𝐶1: 𝑥2 + 𝑦2 = 25 and 𝐶2: ( 𝑥- 𝛼) 2 + 𝑦2 = 16, where 𝛼∈( 5, 9 ) . Let the angle between . If the the two radii (one to each circle) drawn from one of the intersection points of C1and C2 be sin-1√638 length of common chord of 𝐶1 and 𝐶2 is 𝛽, then the value of ( 𝛼𝛽) 2 equals _________. JEE Main 2024 (30 Jan Shift 2) JEE Main Previous Year Paper

Q85.Let the slope of the line 45x + 5y + 3 = 0 be 27r1 + 9r22 for some r1, r2 ∈R. Then 8t2 is equal to ______. lim 3 3r2x x→3(∫x 2 −r2x2−r1x3−3x dt)

Q85.If α = limx→0+ e√tan x−e√x and β = limx→0(1 + sin x) 1 ( √tan x−√x ) 2 cot x are the roots of the quadratic equation ax2 + bx −√e = 0, then 12 loge(a + b) is equal to__________

Q85.Let f(x) = x3 + x2f ′(1) + xf ′′(2) + f ′′′(3), x ∈R. Then f ′(10) is equal to + x −y, ∀x, y ∈(0, ∞). Then

Q85.If the points of intersection of two distinct conics x2 + y2 = 4b and x216 + y2b2 then 3√3 times the area of the rectangle formed by the intersection points is _______.

Q85.Let L1, L2 be the lines passing through the point P(0, 1) and touching the parabola 9x2 + 12x + 18y −14 = 0. Let Q and R be the points on the lines L1 and L2 such that the △PQR is an isosceles triangle with base QR. If the slopes of the lines QR are m1 and m2 , then 16 (m21 + m22) is equal to _______