Practice Questions

Q81.Let S = {z ∈C : |z −2| ≤1, z(1 + i) + z(1 −i) ≤2} . Let |z −4 i| attains minimum and maximum values, + = α + β√5 , where α and β are integers, then the value respectively, at z1 ∈S and z2 ∈S . If 5(|z1|2 |z2|2) of α + β is equal to ______.

Q81.For a natural number 𝑛, let 𝛼𝑛= 19𝑛- 12𝑛. Then, the value of is ______ 57𝛼8

Q81.The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ______.

Q81.The sum of all real values of 𝑥 for which 3𝑥2 - 9𝑥+ 17 = 5𝑥2 - 7𝑥+ 19 is equal to 𝑥2 + 3𝑥+ 10 3𝑥2 + 5𝑥+ 12

Q81.If for some p, q, r ∈R, all have positive sign, one of the roots of the equation q2+r2 (p2 + q2)x2 −2q(p + r)x + q2 + r2 = 0 is also a root of the equation x2 + 2x −8 = 0 , then p2 is equal to-

Q81.The total number of three-digit numbers, with one digit repeated exactly two times, is ______. JEE Main 2022 (25 Jun Shift 2) JEE Main Previous Year Paper 3 10

Q81.The sum of the cubes of all the roots of the equation x4 −3x3 −2x2 + 3x + 1 = 0 is _____.

Q81.Let S = {z ∈C : z2 + z = 0}. Then ∑z∈S(Re (z)+ Im (z)) is equal to _______.

Q81.The number of real solutions of the equation e4x + 4e3x −58e2x + 4ex + 1 = 0 is _____.

Q82.The number of 5 -digit natural numbers, such that the product of their digits is 36 , is

Q82.The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____ .

Q82.The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____.

Q82.If the sum of the co-efficients of all the positive even powers of 𝑥 in the binomial expansion of 2𝑥3 + is 𝑥 510 - 𝛽· 39, then 𝛽 is equal to _____.

Q82.If z2 + z + 1 = 0, z ∈C , then ∑15n=1 (zn + (−1)a zn1 ) 2

Q82.The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is _____.

Q82.A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168 , then b + 3g is equal to

Q82.Let for the 9th term in the binomial expansion of (3 + 6x)n , in the increasing powers of 6x, to be the greatest for x = 23 , the least value of n is n0 . If k is the ratio of the coefficient of x6 to the coefficient of x3 , then k + n0 is equal to does not pass through the fourth

Q82.Let 𝑆 be the set of all passwords which are six to eight characters long, where each character is either an alphabet from 𝐴, 𝐵, 𝐶, 𝐷, 𝐸 or a number from 1, 2, 3, 4, 5 with the repetition of characters allowed. If the number of passwords in 𝑆 whose at least one character is a number from 1, 2, 3, 4, 5 is 𝛼× 56, then 𝛼 is equal to

Q82.The number of elements in the set { z = a + ib ∈C : a, b ∈Z and 1 < |z −3 + 2i| < 4 } is _____.

Q82.Let for n = 1, 2, … … , 50, Sn be the sum of the infinite geometric progression whose first term is n2 and whose common ratio is 1 . Then the value of 26 1 + ∑50n=1(Sn + n+12 −n −1) is equal to (n+1)2

Q82.Let 3, 6, 9, 12, … upto 78 terms and 5, 9, 13, 17, … upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ______. 15 1 1

Q82.Let a1, a2, a3, … be an A.P. If ∑∞r=1 ar2r = 4, then 4a2 is equal to ______.

Q82.The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____.

Q82.If the circles x2 + y2 + 6x + 8y + 16 = 0 and x2 + y2 + 2(3 −√3)x 2(4 −√6)y 2 2 + + + k > 0 , touch internally at the point P(α, β), then (α √3) (β √6) is equal to _______.

Q82.Let f(x) = 2x2 −x −1 and S = {n ∈Z : |f(n)| ≤800} . Then, the value of ∑n∈S f(n) is equal to _______.