Practice Questions

Q81.If 1, log10(4x −2) and log10(4x + 185 ) are in arithmetic progression for a real number x then the value of the 2(x −12 ) x −1 x2 determinant 1 0 x is equal to: x 1 0 x ≠0, be in the ratio

Q81.Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is

Q81.The number of solutions of the equation log4(x −1) = log2(x −3) is ______.

Q81.If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x3) + xg(x3) is divisible by x2 + x + 1, then P(1) is equal to ___ .

Q81.If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 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.If 𝑆= + + + + … , then 160 𝑆 is equal to . 5 52 53 54

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

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

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

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