Practice Questions

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.Let the points on the plane P be equidistant from the points (−4, 2, 1) and (2, −2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is (1) π (2) π 6 4 (3) π (4) 5π 3 12

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+7 −6 = 7 = z and 7−x2 = y −2 = z −6 is (1) 2√29 (2) 1 2 (3) √3729 (4) √29

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 𝑃 be the plane passing through the intersection of the planes →𝑟· ^𝑖+ 3 ^𝑗- ^𝑘= 5 and →𝑟· 2 ^𝑖- ^𝑗+ ^𝑘= 3, and the point 2, 1, - 2. Let the position vectors of the points 𝑋 and 𝑌 be ^𝑖- 2 ^𝑗+ 4 ^𝑘 and 5 ^𝑖- ^𝑗+ 2 ^𝑘 respectively. Then the points (1) 𝑋 and 𝑋+ 𝑌 are on the same side of 𝑃 (2) 𝑌 and 𝑌- 𝑋 are on the opposite sides of 𝑃 (3) 𝑋 and 𝑌 are on the opposite sides of 𝑃 (4) 𝑋+ 𝑌 and 𝑋- 𝑌 are on the same side of 𝑃

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.A vector →𝑎 is parallel to the line of intersection of the plane determined by the vectors ^𝑖, ^𝑖+ ^𝑗 and the plane determined by the vectors ^𝑖- ^𝑗, ^𝑖+ ^𝑘. The obtuse angle between →𝑎 and the vector →𝑏= ^𝑖- 2 ^𝑗+ 2 ^𝑘 is (1) 3𝜋 (2) 2𝜋 4 3 4𝜋 5𝜋 (3) (4) 5 6 4

Q80.Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is draw from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is 4 5 (1) (2) 9 18 (3) 1 (4) 3 6 10

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.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.A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with 1 mark 𝑛 is 𝑛. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is (1) 7 (2) 7 211 212 3 13 (3) (4) 210 212

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.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.If the numbers appeared on the two throws of a fair six faced die are 𝛼 and 𝛽, then the probability that 𝑥2 + 𝛼𝑥+ 𝛽> 0, for all 𝑥∈𝑅, is 17 4 (1) (2) 36 9 (3) 1 (4) 19 2 36

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.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 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.A random variable X has the following probability distribution: X 0 1 2 3 4 P(X) k 2k 4k 6k 8k The value of P( 1<x<4x≤2 )is equal to (1) 4 (2) 2 7 3 (3) 3 (4) 4 7 5 ¯

Q80.Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is 2 11 (1) (2) 3 16 23 13 (3) (4) 32 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.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.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.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.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