Practice Questions

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

Q83.If the length of the latus rectum of the ellipse x2 + 4y2 + 2x + 8y −λ = 0 is 4 , and l is the length of its major axis, then λ + l is equal to _____. . Let the major

Q83.For 𝑝, 𝑞∈𝑅, consider the real valued function 𝑓𝑥= 𝑥- 𝑝2 - 𝑞, 𝑥∈𝑅 and 𝑞> 0. Let 𝑎1, 𝑎2, 𝑎3 and 𝑎4 be in an arithmetic progression with mean 𝑝 and positive common difference. If 𝑓𝑎𝑖= 500 for all 𝑖= 1, 2, 3, 4, then the absolute difference between the roots of 𝑓𝑥= 0 is

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

Q83.The total number of 3 -digit numbers, whose greatest common divisor with 36 is 2 , is ______.

Q83.If one of the diameters of the circle x2 + y2 −2√2x −6√2y + 14 = 0 is a chord of the circle 2 (x −2√2) 2 = r2 , then the value of r2 is equal to +(y −2√2)

Q83.The remainder on dividing 1 + 3 + 32 + 33 + … + 32021 by 50 is _____.

Q83.If ∑𝑘=10 1 𝐾210𝐶𝐾 2 = 22000 𝐿, then 𝐿 is equal to _____.

Q83.The number of positive integers k such that the constant term in the binomial expansion of 12 (2x3 + xk3 ) , x ≠0 is 28 ⋅l, where l is an odd integer, is ______.

Q83.Let the coefficients of x−1 and x−3 in the expansion of (2x 5 − x 51 ) , x > 0, be mand n respectively. If r is a positive integer such mn2 = 15Cr.2r , then the value of r is equal to ______. JEE Main 2022 (29 Jun Shift 2) JEE Main Previous Year Paper

Q83.Let the eccentricity of the hyperbola 𝑥2 - 𝑦2 = 1 be 5 If the equation of the normal at the point 8 12 on the 𝑎2 𝑏2 4. √5, 5 hyperbola is 8√5𝑥+ 𝛽𝑦= 𝜆, then 𝜆- 𝛽 is equal to _____. 5𝑛+ 1

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, ______