Practice Questions

Q81.Let 𝑎, 𝑏 be two non-zero real numbers. If 𝑝 and 𝑟 are the roots of the equation 𝑥2 - 8𝑎𝑥+ 2𝑎= 0 and 𝑞 and 𝑠 1 1 1 1 are the roots of the equation 𝑥2 + 12𝑏𝑥+ 6𝑏= 0, such that 𝑝, 𝑞, 𝑟, 𝑠 are in A.P., then 𝑎-1 - 𝑏-1 is equal to _____ .

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

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

Q81.If p and q are real number such that p + q = 3, p4 + q4 = 369 , then the value of −2 ( p1 + 1q ) is equal to 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.The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

Q81.Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is _______.

Q81.Let 𝛼, 𝛽𝛼> 𝛽 be the roots of the quadratic equation 𝑥2 - 𝑥- 4 = 0. If 𝑃𝑛= 𝛼𝑛- 𝛽𝑛, 𝑛∈ℕ, then 𝑃15𝑃16 - 𝑃14𝑃16 - 𝑃152 + 𝑃14𝑃15 is equal to _____. 𝑃13𝑃14

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

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

Q82.There are ten boys B1, B2, … . , B10 and five girls G1, G2, … . G5 in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B1 and B2 together should not be the members of a group, is _____.

Q82.The number of elements in the set { z = a + ib ∈C : a, b ∈Z and 1 < |z −3 + 2i| < 4 } 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.If the sum of the first ten terms of the series 5 1 + 652 + 3253 + 10254 + 25015 + … . is mn , where m and n are co- prime numbers, then m + n is equal to ______. 60 √x √5

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.If 10 𝑘 = 𝑚 where 𝑚 and 𝑛 are co-prime, then 𝑚+ 𝑛 is equal to ∑𝑘= 1 𝑘4 + 𝑘2 + 1 𝑛,

Q82.The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, 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.If the sum of the coefficients of all the positive powers of x, in the binomial expansion of (xn + x52 ) 7 then the sum of all the possible integral values of n is JEE Main 2022 (27 Jun Shift 2) JEE Main Previous Year Paper

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