Practice Questions

Q82.If S(x) = (1 + x) + 2(1 + x)2 + 3(1 + x)3 + ⋯+ 60(1 + x)60, x ≠0, and (60)2 S(60) = a(b)b + b, where a, b ∈N , then (a + b) equal to ______

Q82.The coefficient of x2012 in the expansion of 1 - x20081 + x + x22007 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.If three successive terms of a G.P. with common ratio 𝑟𝑟> 1 are the length of the sides of a triangle and 𝑟 denotes the greatest integer less than or equal to r, then 3𝑟+ −𝑟 is equal to: 2𝜋

Q82.If 8 = 3 + 14 (3 + p) + 421 (3 + 2p) + 431 (3 + 3p) + … ∞, then the value of p is JEE Main 2024 (27 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

Q83.Remainder when 643232 is divided by 9 is equal to _____.

Q83.If the coefficient of 𝑥30 in the expansion of 1 + 1 + 𝑥271 −𝑥38; 𝑥≠0 is 𝛼, then 𝛼 equals _________. 𝑥 JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper

Q83.If 11C1 2 + 3 + … . . + 10 = mn with gcd (n, m) = 1, then n + m is equal to

Q83.Let the centre of a circle, passing through the points (0, 0), (1, 0) and touching the circle x2 + y2 = 9, be (h, k) . Then for all possible values of the coordinates of the centre (h, k), 4 (h2 + k2) is equal to_________

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 𝑆𝑛 be the sum to n-terms of an arithmetic progression 3, 7, 11, … … , if 40 < 𝑛( 𝑛+ 1 ) ∑𝑘= 1 𝑆𝑘< 42, then 𝑛 equals ____________. 𝑛C𝑘 𝑛C𝑘+ 1 𝑛 𝑛C𝑘 2

Q83.Let A, B and C be three points on the parabola y2 = 6x and let the line segment AB meet the line L through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and AM⋅BN 2 B on L. Then ( CD ) is equal to _________

Q83.Let a ray of light passing through the point (3, 10) reflects on the line 2x + y = 6 and the reflected ray passes through the point (7, 2). If the equation of the incident ray is ax + by + 1 = 0, then a2 + b2 + 3ab is equal to_________ , on the positive x-axis. Let C be the circle with its centre at

Q83.In the expansion of 1 + 𝑥1 −𝑥21 + + , 𝑥≠0, the sum of the coefficient of 𝑥3 and 𝑥-13 is equal to 𝑥+ 𝑥2 𝑥3 ______

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.If the constant term in the expansion of (1 + 2x −3x3)( 32 x2 − 3x1 ) 9

Q83.If the second, third and fourth terms in the expansion of (x + y)n are 135,30 and 103 , respectively, then 6 (n3 + x2 + y) is equal to _______

Q83.If the sum of squares of all real values of α, for which the lines 2x - y + 3 = 0, 6x + 3y + 1 = 0 and αx + 2y - 2 = 0 do not form a triangle is p, then the greatest integer less than or equal to p is ________.

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.Number of integral terms in the expansion of 1 1 824 is equal to ______. 2 ) + 11( )} {7(

Q83.Let 𝐴−2, −1, 𝐵1, 0, 𝐶𝛼, 𝛽 and 𝐷𝛾, 𝛿 be the vertices of a parallelogram 𝐴𝐵𝐶𝐷. If the point 𝐶 lies on 2𝑥−𝑦= 5 and the point 𝐷 lies on 3𝑥−2𝑦= 6, then the value of 𝛼+ 𝛽+ 𝛾+ 𝛿 is equal to ______.

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

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 ,