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 _____ .

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

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

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

Q82.Let b1b2b3b4 be a 4-element permutation with bi ∈{1, 2, 3, … … … , 100} for 1 ≤i ≤4 and bi ≠bj for i ≠j , such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then the number of such permutations b1b2b3b4 is equal to ______.

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

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.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.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.Let A( √a3 , √a), a > 0 , be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C . If D(3 cos θ, a sin θ), is a point in the fourth quadrant such that the maximum area of ΔACD is 12 square units, then a 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 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

Q83.The greatest integer less than or equal to the sum of first 100 terms of the sequence 1 5 19 65 … is equal to 3, 9, 27, 81, ______

Q83.If 6 + 10 + 20 + 40 + … . . + 102403 = 2n ⋅m, where m is odd, then m. n is equal to _____ . 312 311 310 39