Practice Questions

Q82.If the real part of the complex number z = 1−3i3+2i coscos θθ , θ ∈(0, π2 ) is zero, then the value of sin2 3θ + cos2 θ is equal to ______.

Q82.The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is k is equal to is the term, independent of x, in the binomial expansion of ( x4 −12x2 )12, then

Q82.The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

Q82.Let A1, A2, A3, … . . be squares such that for each n ⩾1, the length of the side of An equals the length of diagonal of An+1 . If the length of A1 is 12 cm, then the smallest value of n for which area of An is less than one, is = 0 is a

Q82.Let {an}∞n=1 be a sequence such that a1 = 1, a2 = 1 and an+2 = 2an+1 + an for all n ≥1. Then the value of 47 ∑∞n=1( 23nan ) is equal to ________.

Q82.If 𝑆= + + + + … , then 160 𝑆 is equal to . 5 52 53 54

Q82.The sum of 162th power of the roots of the equation x3 −2x2 + 2x −1 = 0 is ______. + + … . . + = n. 2m , then n + m is

Q82.There are 5 students in class 10, 6 students in class 11 and 8 students in class 12 . If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is 100𝑘, then 𝑘 is equal to + … . upto ∞ 2 + 2 6 10 log0 . 25 3 + 3 33

Q82.Let z be those complex numbers which satisfy |z + 5| ≤4 and z(1 + i) + z(1 −i) ⩾−10, i = √−1. If the maximum value of |z + 1|2 is α + β√2 , then the value of (α + β) is

Q82.Let i = √−1. If (−1+i√3) 21 + (1+i√3) 21 = k, and n = [|k|] be the greatest integral part of |k|. (1−i)24 (1+i)24 Then ∑n+5j=0 (j + 5)2 −∑n+5j=0 (j + 5) is equal to ________.

Q82.The minimum distance between any two points P1 and P2 while considering point P1 on one circle and point P2 on the other circle for the given circles' equations x2 + y2 −10x −10y + 41 = 0 x2 + y2 −24x −10y + 160 = 0 is ________ then the value of det (A4)+ det (A10 −(Adj (2 A))10) is equal to ________.

Q82.Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to _______.

Q82.For real numbers α and β, consider the following system of linear equations: x + y −z = 2, x + 2y + αz = 1 and 2x −y + z = β. If the system has infinite solutions, then α + β is equal to ______.

Q82.Let Sn(x) = loga1/2 x + loga1/3 x + loga1/6 x + loga1/11 x + loga1/18 x + loga1/27 x + … up to n-terms, where a > 1 . If S24(x) = 1093 and S12(2x) = 265, then value of a is equal to _____ . k )

Q82.The number of rational terms in the binomial expansion of 1 1 120 4 + 5 6 (4 ) is_______.

Q82.If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)2022 is divided by 8 is ___ .

Q82.The sum of all the elements in the set {n ∈{1, 2, … … , 100} ∣ H.C.F. of n and 2040 is 1} is equal to __________.

Q82.Let z = 1−i√32 , i = √−1. Then the value of 21 + (z + 1z ) 3 + (z2 + z21 ) 3 + (z3 + z31 ) 3 + … + (z21 + z211 ) 3 is______.

Q82.If one of the diameters of the circle 𝑥2 + 𝑦2 - 2𝑥- 6𝑦+ 6 = 0 is a chord of another circle '𝐶', whose center is at 2, 1, then its radius is_____.

Q82.A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is ________.

Q82.Let S = {1, 2, 3, 4, 5, 6, 9} . Then the number of elements in the set T = {A ⊆S : A ≠ϕ and the sum of all the elements of A is not a multiple of 3} is Q83. 3 × 722 + 2 × 1022 −44 when divided by 18 leaves the remainder

Q82.The least positive integer n such that , i = √−1, is a positive integer, is ______. (1−i)n−2

Q82.All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two 𝑅 appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is _____ .

Q82.Let the coefficients of third, fourth and fifth terms in the expansion of (x + x2a )n, 12 : 8 : 3. Then the term independent of x in the expansion, is equal to _______. n ∈N be the slopes of the three line segments OA, OB and

Q82.If ∑10r=1 r!(r3 + 6r2 + 2r + 5) = α(11!), then the value of α is equal to ___ .