Practice Questions

Q81.Let 𝑃= 𝑧∈ℂ: 𝑧+ 2 −3𝑖≤1 and 𝑄= 𝑧∈ℂ: 𝑧1 + 𝑖+ ¯𝑧1 −𝑖≤−8. Let in 𝑃∩𝑄, 𝑧−3 + 2𝑖 be maximum and minimum at 𝑧1 and 𝑧2 respectively. If 𝑧12 + 2𝑧2 = 𝛼+ 𝛽√2, where 𝛼, 𝛽 are integers, then 𝛼+ 𝛽 equals __________

Q81.Let α, β be the roots of the equation x2 −x + 2 = 0 with Im (α) >Im (β). Then α6 + α4 + β4 −5α2 is equal to

Q81.The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to__________ JEE Main 2024 (08 Apr Shift 1) JEE Main Previous Year Paper

Q81.Let x1, x2, x3, x4 be the solution of the equation 4x4 + 8x3 −17x2 −12x + 9 = 0 and (4 + x21) (4 + x22) (4 + x23) (4 + x24) = 12516 m. Then the value of m is

Q81.Let the set C = {(x, y) ∣x2 −2y = 2023, x, y ∈N}. Then ∑(x,y)∈C(x y)

Q81.If 𝛼 denotes the number of solutions of 1 −𝑖𝑥= 2𝑥 and 𝛽= 𝑧 where 𝑧= 𝜋 + 𝑖41 −√𝜋· 𝑖 √𝜋−𝑖 arg𝑧, 41 𝑖+ 1 + 𝑖, √𝜋+ √𝜋· 𝑖= √−1, then the distance of the point 𝛼, 𝛽 from the line 4𝑥−3𝑦= 7 is ______ JEE Main 2024 (31 Jan 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

Q82.Let α = 12 + 42 + 82 + 132 + 192 + 262 + … … . upto 10 terms and β = ∑10n=1 n4 . If 4α −β = 55k + 40, then k is equal to _______. 6

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 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.The remainder when 4282024 is divided by 21 is__________

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 A be a square matrix of order 2 such that |A| = 2 and the sum of its diagonal elements is -3 . If the points (x, y) satisfying A2 + x A + yI = O lie on a hyperbola, whose length of semi major axis is x and semi minor axis is y, eccentricity is e and the length of the latus rectum is l, then 81 (e4 + l2) is equal to

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.The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and x = ± 4 , √13 respectively. Let the line y −√3x + √3 = 0 touch this hyperbola at (x0, y0). If m is the product of the focal distances of the point (x0, y0), then 4e2 + m is equal to ________

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

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

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.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.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 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.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

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.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