Practice Questions

Q81.The sum of all integral values of k(k ≠0) for which the equation x−12 − x−21 = k2 in x has no real roots, is_____.

Q81.Let z and w be two complex numbers such that w = zz −2z + 2, z−3iz+i = 1 and Re (w) has minimum value. Then, the minimum value of n ∈N for which wn is real, is equal to _______.

Q81.Let 1 , a and b be in G.P. and a1 , 1b , 6 be in A.P., where a, b > 0 . Then 72(a + b) is equal to _______ . 16

Q81.If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to:

Q81.If A = {x 1}, {x ∈R : √x2 −3 > 1}, {x ⩾2} and all integers, then the number of subsets of the set (A ∩B ∩C)c ∩Z is _________.

Q81.A point z moves in the complex plane such that arg( z+2z−2 ) = π4 , then the minimum value of z −9√2 −2i 2 is equal to

Q81.There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is

Q81.If 𝛼, 𝛽 are roots of the equation 𝑥2 + 5√2𝑥+ 10 = 0, 𝛼> 𝛽 and 𝑃𝑛= 𝛼𝑛- 𝛽𝑛 for each positive integer 𝑛, then the value of 𝑃17𝑃20 + 5√2𝑃17𝑃192 is equal to 𝑃18𝑃19 + 5√2𝑃18

Q81.Let α and β be two real numbers such that α + β = 1 and αβ = −1. Let pn = (α)n + (β)n , pn−1 = 11 and pn+1 = 29 for some integer n ⩾1 . Then, the value of p2n is______. ¯

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.The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is

Q81.The number of solutions of the equation log(x+1)(2x2 + 7x + 5) + log(2x+5)(x 1)2

Q81.If log3 2, log3(2x −5), log3(2x −72 ) are in an arithmetic progression, then the value of x is equal to _____.

Q81.If for the complex numbers 𝑧 satisfying |𝑧- 2 - 2𝑖| ≤1, the maximum value of |3𝑖𝑧+ 6| is attained at 𝑎+ 𝑖𝑏, then 𝑎+ 𝑏 is equal to _____ .

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

Q81.The number of real roots of the equation e4x −e3x −4e2x −ex + 1 = 0 is equal to

Q81.The total number of two digit numbers ′n′, such that 3n + 7n is a multiple of 10 , is ___ .

Q81.Let z1 and z2 be two complex numbers such that arg(z1 −z2) = π4 and z1, z2 satisfy the equation |z −3| =Re (z). Then the imaginary part z1 + z2 is equal to

Q81.Let λ ≠0 be in R. If α and β are the roots of the equation x2 −x + 2λ = 0, and α and γ are the roots of the equation 3x2 −10x + 27λ = 0, then βγλ is equal to ________. (2i)n

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.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.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 ________.