Practice Questions

Q80.A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5 . The probability, that number so formed is divisible by 2 , is (1) 4 (2) 3 7 7 (3) 1 (4) 6 7 7

Q80.Let A, B and C be three events such that the probability that exactly one of A and B occurs is (1 −k), the probability that exactly one of B and C occurs is (1 −2k), the probability that exactly one of C and A occurs is (1 −k) and the probability of all A, B and C occur simultaneously is k2, where 0 < k < 1. Then the probability that at least one of A, B and C occur is: (1) greater than 1 but less than 1 (2) greater than 1 8 4 2 (3) greater than 1 but less than 1 (4) exactly equal to 1 4 2 2 + −4 = 0, x > 0, is

Q80.A vector →a has components 3p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, →a has components p + 1 and √10, then a value of p is equal to: (1) 1 (2) −54 (3) 4 (4) −1 5 JEE Main 2021 (18 Mar Shift 1) JEE Main Previous Year Paper

Q80.The probability that a randomly selected 2− digit number belongs to the set {n ∈N : (2n −2) is a multiple of 3} is equal to (1) 1 (2) 2 6 3 (3) 1 (4) 1 2 3

Q80.Let 9 distinct balls be distributed among 4 boxes, 𝐵1, 𝐵2, 𝐵3 and 𝐵4. If the probability that 𝐵3 contains 9 exactly 3 balls is 𝑘3 then 𝑘 lies in the set : 4 (1) {𝑥∈𝑅: | 𝑥- 3 | < 1} (2) {𝑥∈𝑅: | 𝑥- 2 | ≤1} (3) {𝑥∈𝑅: | 𝑥- 1 | < 1} (4) {𝑥∈𝑅: | 𝑥- 5 | ≤1}

Q80.When a missile is fired from a ship, the probability that it is intercepted is 1 and the probability that the 3 missile hits the target, given that it is not intercepted, is 3 . If three missiles are fired independently from the 4 ship, then the probability that all three hit the target, is: (1) 3 (2) 1 8 27 (3) 1 (4) 3 8 4 JEE Main 2021 (25 Feb Shift 1) JEE Main Previous Year Paper

Q80.Let the equation of the plane, that passes through the point (1, 4, −3) and contains the line of intersection of the planes 3 x −2 y + 4 z −7 = 0 and x + 5 y −2 z + 9 = 0, be αx + βy + γz + 3 = 0, then α + β + γ is equal to : (1) −15 (2) 15 (3) −23 (4) 23

Q80.Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is : JEE Main 2021 (01 Sep Shift 2) JEE Main Previous Year Paper 1 1 (1) (2) 9 7 (3) 2 (4) 1 7 18

Q80.Let X be a random variable such that the probability function of a distribution is given by P(X = 0) = 21 , P(X = j) = 3j1 (j = 1, 2, 3, … , ∞). Then the mean of the distribution and P(X is positive and even ) respectively, are: (1) 3 and 1 (2) 3 and 1 8 8 4 8 (3) 3 and 1 (4) 3 and 1 4 9 4 16

Q80.Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = 59 , is: (1) 4 (2) 1 9 5 (3) 5 (4) 2 12 9

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.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.The number of solutions of the equation log4(x −1) = log2(x −3) 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.The total number of two digit numbers ′n′, such that 3n + 7n is a multiple of 10 , is ___ .

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

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

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.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.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 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 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.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