Practice Questions

Q76.If the value of the integral ∫50 x+[x]ex−[x] greatest integer less than or equal to x; then the value of (α + β)2 is equal to : (1) 25 (2) 100 (3) 36 (4) 16

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.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.The number of the real roots of the equation (x + 1)2 + x −5 = 274 is ________.

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 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.If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10, 000 is equal to __________.

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

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

Q81.If (2021)3762 is divided by 17, then the remainder is _______.

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.The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is . JEE Main 2021 (31 Aug Shift 2) JEE Main Previous Year Paper 7 9 13 19

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

Q81.If the least and the largest real values of 𝛼, for which the equation 𝑧+ 𝛼𝑧- 1 + 2𝑖= 0 𝑧∈𝐶 and 𝑖= √-1 has a solution, are 𝑝 and 𝑞 respectively; then 4𝑝2 + 𝑞2 is equal to_______.

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

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