Practice Questions

Q81.Let α, β ∈ be roots of equation x2 −70x + λ = 0, where λ2 , λ3 ∉ . If λ assumes the minimum possible value, (√α−1+√β−1)(λ+35) then is equal to : |α−β|

Q81.The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is ______.

Q81.The number of real solutions of the equation x|x + 5| + 2|x + 7| −2 = 0 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.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

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.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.If α satisfies the equation x2 + x + 1 = 0 and (1 + α)7 = A + Bα + Cα2, A, B, C ≥0 , then 5(3 A −2 B −C) is equal to

Q81.Let a = 1 + 2C23! + 3C24! + 4C25! + … 1! + 2! + 3! + … Then 2b is equal to a2

Q81.The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________

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

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.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.The remainder when 4282024 is divided by 21 is__________

Q82.All the letters of the word GTWENTY are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word GTWENTY IS 11C2 11C9

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 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 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.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 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 ( α+11 + α+21 + … … + α+10121 ) −( 2⋅11 + 4⋅31 + 6⋅51 + … . . + 2024⋅20231 ) = 20241 , then α 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.An arithmetic progression is written in the following way The sum of all the terms of the 10th row is_______