Practice Questions

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 the positive integers be written in the form : If the kth row contains exactly k numbers for every natural number k, then the row in which the number 5310 will be, is _______ + n+11 . If 140 < 2αβ < 281, then the value of n is

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 length of the focal chord PQ of the parabola y2 = 12x be 15 units. If the distance of PQ from the origin is p, then 10p2 is equal to _______ JEE Main 2024 (04 Apr Shift 1) JEE Main Previous Year Paper

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.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 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.In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is _________. 6 𝑛

Q82.The total number of words (with or without meaning) that can be formed out of the letters of the word "DISTRIBUTION" taken four at a time, is equal to ______. 3 3 1 5

Q82.Let the coefficient of 𝑥𝑟 in the expansion of 𝑥+ 3𝑛−1 + 𝑥+ 3𝑛−2𝑥+ 2 + 𝑥+ 3𝑛−3𝑥+ 22 + . ... + 𝑥+ 2𝑛−1 be 𝛼𝑟. If ∑𝑟=𝑛 0 𝛼𝑟= 𝛽𝑛−𝛾𝑛, 𝛽, 𝛾∈𝑁, then the value of 𝛽2 + 𝛾2 equals _______.

Q82.If ( α+11 + α+21 + … … + α+10121 ) −( 2⋅11 + 4⋅31 + 6⋅51 + … . . + 2024⋅20231 ) = 20241 , then α is equal to________

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 3, 7, 11, 15, . . , 403 and 2, 5, 8, 11, . . . , 404 be two arithmetic progressions. Then the sum, of the common terms in them, is equal to_________ 1 6

Q82.The coefficient of x2012 in the expansion of 1 - x20081 + x + x22007 is equal to _____.

Q82.The remainder when 4282024 is divided by 21 is__________

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

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

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.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.The number of solutions of sin2 x + (2 + 2x −x2) sin x −3(x −1)2 = 0, where −π ≤x ≤π, is________