Practice Questions

Q81.The number of distinct real roots of the equation |x + 1||x + 3| −4|x + 2| + 5 = 0, is JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper

Q81.The lines 𝐿1, 𝐿2, . .. , 𝐿20 are distinct. For 𝑛= 1, 2, 3, . .. , 10 all the lines 𝐿2𝑛−1 are parallel to each other and all the lines 𝐿2𝑛 pass through a given point 𝑃. The maximum number of points of intersection of pairs of lines from the set 𝐿1, 𝐿2, . .. , 𝐿20 is equal to:

Q81.There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____

Q81.Let the complex numbers α and lie on the circles z - z02 = 4 and z - z02 = 16 respectively, where z0 = 1 + i. ¯α Then, the value of 100 | α| 2 is__________.

Q81.Let α, β be roots of x2 + √2x −8 = 0. If Un = αn + βn , then U10+√2U9 is equal to______ 2U8

Q81.The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) 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.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 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.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.An arithmetic progression is written in the following way The sum of all the terms of the 10th row is_______

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

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 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 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 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.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 _______.

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

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 _________