Practice Questions

Q79.For real numbers α and β ≠0, if the point of intersection of the straight lines x−α1 = y−12 = z−13 and x−4 β = y−63 = z−73 lies on the plane x + 2y −z = 8, then α −β is equal to : (1) 5 (2) 9 (3) 3 (4) 7

Q79.Let the foot of perpendicular from a point 𝑃( 1, 2, - 1 ) to the straight line 𝐿: 𝑥 = 𝑦 = 𝑧 be 𝑁. Let a line be 1 0 -1 drawn from 𝑃 parallel to the plane 𝑥+ 𝑦+ 2𝑧= 0 which meets 𝐿 at point 𝑄. If 𝛼 is the acute angle between the lines 𝑃𝑁 and 𝑃𝑄, then cos𝛼 is equal to . 1 √3 (1) (2) √5 2 1 1 (3) (4) √3 2√3

Q79.If →a = 2, →b = 5 and →a×→b = 8, then →a⋅→b is equal to: (1) 6 (2) 4 (3) 3 (4) 5

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.A fair die is tossed until six is obtained on it. Let X be the number of required tosses, then the conditional probability P(X ⩾5 ∣X > 2) is : (1) 25 (2) 5 36 6 (3) 11 (4) 125 36 216

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 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.Two dices are rolled. If both dices have six faces numbered 1, 2, 3, 5, 7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is: (1) 4 (2) 17 9 36 (3) 5 (4) 1 12 2

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.Let 𝑆= {1, 2, 3, 4, 5, 6} . Then the probability that a randomly chosen onto function 𝑔 from 𝑆 to 𝑆 satisfies 𝑔3 = 2 𝑔1 is : 1 1 (1) (2) 15 5 (3) 1 (4) 1 30 10

Q80.The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is: (1) 65 (2) 65 28 27 (3) 35 (4) 135 27 29

Q80.Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is: (1) 5 (2) 1 8 8 (3) 5 (4) 1 16

Q80.Let A be a set of all 4 -digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is: (1) 1 (2) 122 5 297 (3) 97 (4) 2 297 9

Q80.Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be 21 and probability of occurrence of 0 at the odd place be 31 . Then the probability that 10 is followed by 01 is equal to : (1) 1 (2) 1 18 3 (3) 1 (4) 1 6 9

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.A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is (1) 15 (2) 15 213 214 (3) 15 (4) 15 212 28

Q80.An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is: (1) 1 (2) 5 32 16 3 1 (3) (4) 16 2

Q80.When a certain biased die is rolled, a particular face occurs with probability 16 −x and its opposite face occurs with probability 61 + x. All other faces occur with probability 16 . Note that opposite faces sum to 7 in any die. If 0 < x < 61 , and the probability of obtaining total sum = 7, when such a die is rolled twice, is 9613 , then the value of x is JEE Main 2021 (27 Aug Shift 1) JEE Main Previous Year Paper (1) 161 (2) 121 (3) 81 (4) 19 Z is the set of ∈R : x −2 > B = C = ∈R : x −4

Q80.A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least n correct answers is less than 1 , is : 2 (1) 5 (2) 6 (3) 3 (4) 4

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, 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.Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 × 2 matrices. The probability that such formed matrices have all different entries and are non-singular, is: (1) 45 (2) 23 162 81 (3) 22 (4) 43 81 162

Q80.Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0. 4096 and 0. 2048 respectively. Then the probability of getting exactly 3 successes is equal to : (1) 32 (2) 80 625 243 (3) 40 (4) 128 243 625

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.A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is : (1) 3 (2) 52 4 867 (3) 39 (4) 22 50 425 ¯