Practice Questions

Q82.If 1 + √3−√2 a + loge ( ab ), where a and b are + 49−20√6180 + … upto ∞= 2 + 2√3 + 5−2√618 + 9√3−11√236√3 (√b 1) integers with gcd(a, b) = 1, then 11a + 18 b is equal to ______

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

Q82.An arithmetic progression is written in the following way The sum of all the terms of the 10th row is_______

Q82.Let α, β be the roots of the equation x2 −√6x + 3 = 0 such that Im (α) >Im (β). Let a, b be integers not + i = √−1. Then n + a + b is divisible by 3 and n be a natural number such that α99β + α98 = 3n(a ib), equal to ___________.

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.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

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 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.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.The number of solutions of sin2 x + (2 + 2x −x2) sin x −3(x −1)2 = 0, where −π ≤x ≤π, 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.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.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 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.Remainder when 643232 is divided by 9 is equal to _____.

Q83.If 11C1 2 + 3 + … . . + 10 = mn with gcd (n, m) = 1, then n + m 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 α = ∑nr=0 (4r2 + 2r + 1)nCr and β = (∑nr=0 r+1nCr ) _______

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