Practice Questions

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.Let the plane ax + by + cz = d pass through (2, 3, −5) and is perpendicular to the planes 2x + y −5z = 10 and 3x + 5y −7z = 12 If a, b, c, d are integers d > 0 and gcd(|a|, |b|, |c|, d) = 1 then the value of a + 7b + c + 20d is equal to JEE Main 2022 (28 Jun Shift 2) JEE Main Previous Year Paper (1) 18 (2) 20 (3) 24 (4) 22 ¯

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 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.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.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.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.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.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 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 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.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.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 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 the lines →r= (ˆi −ˆj + ˆk) λ(3ˆj −ˆk) and →r (αˆi −ˆj) μ(2ˆi −3ˆk) are co-planar, the the distance of the plane containing these two lines from the point (α, 0, 0) is (1) 2 (2) 2 9 11 (3) 4 (4) 2 11

Q80.Let S be the sample space of all five digit numbers. If p is the probability that a randomly selected number from S , is a multiple of 7 but not divisible by 5 , then 9p is equal to (1) 1. 0146 (2) 1. 2085 (3) 1. 0285 (4) 1. 1521 ¯

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

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

Q81.Let α, β be the roots of the equation x2 −4λx + 5 = 0 and α, γ be the roots of the equation + + 7 + 3λ√3 = 0. If β + γ = 3√2, then (α + 2β + γ)2 is equal to x2 −(3√2 2√3)x is 939,

Q87.Let 𝐴= 1 -1 and 𝐵= 𝛽1 , 𝛼, 𝛽∈𝑅. Let 𝛼1 be the value of 𝛼 which satisfies 𝐴+ 𝐵2 = 𝐴2 + 2 2 and 2 𝛼 1 0 2 2 𝛼2 be the value of 𝛼 which satisfies 𝐴+ 𝐵2 = 𝐵2. Then 𝛼1 - 𝛼2 is equal to

Q88.Let 𝑓: 0, 1 →𝑅 be a twice differentiable function in 0, 1 such that 𝑓0 = 3 and 𝑓1 = 5. If the line 𝑦= 2𝑥+ 3 intersects the graph of 𝑓 at only two distinct points in 0, 1, then the least number of points 𝑥∈0, 1, at which 𝑓''𝑥= 0, is √3 15𝑥3

Q61.The equation arg( z+1z−1 ) = π4 represents a circle with: (1) centre at (0, 0) and radius √2 (2) centre at (0, 1) and radius 2 (3) centre at (0, −1) and radius √2 (4) centre at (0, 1) and radius √2 22

Q61.The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is: (1) 77 (2) 82 (3) 42 (4) 35

Q61.The sum of 10 terms of the series 3 + 5 + 7 + … is : 12×22 22×32 32×42 (1) 143 (2) 99 144 100 (3) 1 (4) 120121

Q61.Let n denote the number of solutions of the equation z2 + 3z = 0, where z is a complex number. Then the value of ∑∞k=0 nk1 is equal to (1) 1 (2) 34 (3) 32 (4) 2