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.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 the roots of the equation x2 −x + 2 = 0 with Im (α) >Im (β). Then α6 + α4 + β4 −5α2 is equal to

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 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.Let 𝑎, 𝑏, 𝑐 be the length of three sides of a triangle satisfying the condition 𝑎2 + 𝑏2𝑥2 −2𝑏𝑎+ 𝑐 𝑥+ 𝑏2 + 𝑐2 = 0. If the set of all possible values of 𝑥 is in the interval 𝛼, 𝛽, then 12𝛼2 + 𝛽2 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.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.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.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|x + 5| + 2|x + 7| −2 = 0 is_________

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

Q81.The sum of the square of the modulus of the elements in the set {z = a + ib : a, b ∈Z, z ∈C, |z −1| ≤1, |z −5| ≤|z −5i|} is ________

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 = 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.The coefficient of x2012 in the expansion of 1 - x20081 + x + x22007 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.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.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.The remainder when 4282024 is divided by 21 is__________

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 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 ___________.