Practice Questions

Q79.The shortest distance between the lines x+7 −6 = 7 = z and 7−x2 = y −2 = z −6 is (1) 2√29 (2) 1 2 (3) √3729 (4) √29

Q79.Let →a = αˆi + 3ˆj −ˆk, b = 3ˆi −βˆj + 4ˆk and →c= ˆi + 2ˆj −2ˆk where α, β ∈R be three vectors. If the projection → 10 of →a on →cis and b ×→c= −6ˆi + 10ˆj + 7ˆk , then the value of α + β equal to 3 (1) 3 (2) 4 (3) 5 (4) 6

Q79.The mean and variance of a binomial distribution are α and α 3 respectively. If P(X = 1) = 2434 , then P(X = 4 or 5) is equal to: (1) 5 (2) 64 9 81 (3) 16 (4) 145 27 243

Q79.Let X have a binomial distribution B(n, p) such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If P(X > n −3) = 2nk , then k is equal to (1) 528 (2) 529 (3) 629 (4) 630

Q79.If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is : (1) 33 (2) 33 232 229 (3) 33 (4) 33 228 227

Q79.The shortest distance between the lines x−3 2 = y−23 = z−1−1 and x+32 = y−61 = z−53 is (1) 18 (2) 22 √5 3√5 (3) 46 (4) 6√3 3√5

Q79.Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random are found to be 1 red and 1 black. If the probability that both balls come from Bag A is 116 , then n is equal to _____ (1) 13 (2) 6 (3) 4 (4) 3

Q79.A plane P is parallel to two lines whose direction ratios are −2, 1, −3, and −1, 2, −2 and it contains the point (2, 2, −2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts α, β, γ . If V is the volume of the tetrahedron OABC , where O is the origin and p = α + β + γ , then the ordered pair (V , p) is equal to (1) (48, −13) (2) (24, −13) (3) (48, 11) (4) (24, −5)

Q79.Let 𝑄 be the mirror image of the point 𝑃1, 0, 1 with respect to the plane 𝑆: 𝑥+ 𝑦+ 𝑧= 5. If a line 𝐿 passing through 1, - 1, - 1, parallel to the line 𝑃𝑄 meets the plane 𝑆 at 𝑅, then 𝑄𝑅2 is equal to (1) 2 (2) 5 (3) 7 (4) 11 3 and 𝑃𝐸2 ∣𝐸1 =

Q79.Let the plane P :→r⋅→a = d contain the line of intersection of two planes →r⋅(ˆi + 3ˆj −ˆk) 13→a 2 → = 7. If the plane P passes through the point (2, 3, 21 ), then the value of d2 is equal to r ⋅(−6ˆi + 5ˆj −ˆk) (1) 90 (2) 93 (3) 95 (4) 97

Q79.If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky −5z = 1 and 3kx −ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane JEE Main 2022 (27 Jul Shift 1) JEE Main Previous Year Paper P on the y-axis is (1) 1 (2) 5 11 11 (3) 6 (4) 7

Q80.If the mirror image of the point (2, 4, 7) in the plane 3x −y + 4z = 2 is (a, b, c), the 2a + b + 2c is equal to (1) 54 (2) −6 (3) 50 (4) −42 ¯

Q80.Let 𝑋 be a binomially distributed random variable with mean 4 and variance 3. Then 54 𝑃𝑋≤2 is equal to (1) 73 (2) 146 27 27 146 126 (3) (4) 81 81

Q80.The probability, that in a randomly selected 3 -digit number at least two digits are odd, is (1) 19 (2) 16 36 36 (3) 19 (4) 13 33 36

Q80.Let E1, E2, E3 be three mutually exclusive events such that P(E1) = 2+3p6 , P(E2) = 2−p8 and P(E3) = 1−p2 . If the maximum and minimum values of p are p1 and p2 then (p1 + p2) is equal to: (1) 2 (2) 5 3 3 (3) 5 (4) 1 4

Q80.If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x −6y = 30, then the probability that y < 1 is (1) 16 (2) 56 (3) 2 (4) 6 3 7

Q80.Let S = {1, 2, 3, … , 2022}. Then the probability, that a randomly chosen number n from the set S such that HCF(n, 2022) = 1, is (1) 128 (2) 166 1011 1011 (3) 127 (4) 112 337 337

Q80.Let 𝐸1 and 𝐸2 be two events such that the conditional probabilities 𝑃𝐸1 ∣𝐸2 = 12, 4 1 𝑃𝐸1 ∩𝐸2 = 8. Then (1) 𝑃𝐸1 ∩𝐸2 = 𝑃𝐸1 · 𝑃𝐸2 (2) 𝑃𝐸1' ∩𝐸2' = 𝑃𝐸1' · 𝑃𝐸2 (3) 𝑃𝐸1 ∩𝐸2' = 𝑃𝐸1 · 𝑃𝐸2 (4) 𝑃𝐸1 ∪𝐸2 = 𝑃𝐸1𝑃𝐸2 31𝛼9 - 𝛼10

Q80.Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(X = 4), then the sum of the mean and the variance of X is (1) 105 (2) 77 16 36 (3) 3631 (4) 3536

Q80.If a random variable X follows the Binomial distribution B(33, p) such that 3P(X = 0) = P(X = 1), then the value of P(X=15) −P(X=16) is equal to P(X=18) P(X=17) (1) 1320 (2) 1088 (3) 1088 (4) 120 1089 1331

Q80.Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is (1) 46 (2) 275 64 65 (3) 41 (4) 36 55 54

Q80.A six faced die is biased such that 3 × P (a prime number) = 6 × P (a composite number) = 2 × P(1). Let X be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of X is (1) 3 (2) 5 11 11 (3) 7 (4) 8 11 11 43−33+23−13 63−53+43−33+23−13 303−293+283−273+…+23−13Q81. 23−13 is equal to ______. 1×7 + 2×11 + 3×15 + … . . + 15×63

Q80.The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to: (1) 5 (2) 9 16 16 (3) 11 (4) 13 16 16

Q80.If A and B are two events such that P(A) = 31 , P(B) = 15 and P(A ∪B) = 12 , then P(A B′) + P(B A′) is equal to (1) 3 (2) 5 4 8 (3) 5 (4) 7 4 8

Q80.Let A and B be two events such that P(B ∣A) = 25 , P(A ∣B) = 71 and P(A ∩B) = 19 . Consider (S1)P(A′ ∪B) = 65 , (S2)P(A′ ∩B′) = 181 . Then (1) Both (S1) and (S2) are true (2) Both (S1) and (S2) are false (3) Only (S1) is true (4) Only (S2) is true