Practice Questions

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 ___ .

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

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

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

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

Q82.The equation of a circle is Re (z2) + 2(Im(z))2 + 2 Re (z) = 0, where z = x + iy. A line which passes through the centre of the given circle and the vertex of the parabola, x2 −6x −y + 13 = 0, has y-intercept 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.If 𝑆= + + + + … , then 160 𝑆 is equal to . 5 52 53 54

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

Q83.Let m, n ∈N and gcd(2, n) = 1 . If 30(300 ) 30 30 30 29( 1 ) +2(28 ) 1(29 ) n equal to _______. (Here = nCk) (k )

Q83.The term independent of x in the expansion of 10 [ x2/3−x1/3+1x+1 − x−x1/2x−1 ] , x ≠1 , is equal to ___.

Q83.Let n be a non-negative integer. Then the number of divisors of the form 4n + 1 of the number (10)10 ⋅(11)11 ⋅(13)13 is equal to _____.

Q83.The locus of the point of intersection of the lines (√3)kx + ky −4√3 = 0 and √3x −y −4(√3)k conic, whose eccentricity is a −b −tan( 2θ )