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 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 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.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.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.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.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 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 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.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.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

Q81.If p and q are real number such that p + q = 3, p4 + q4 = 369 , then the value of −2 ( p1 + 1q ) is equal to is equal to _____.

Q81.In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, −2 marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets 5 marks is _____ JEE Main 2022 (24 Jun Shift 1) JEE Main Previous Year Paper

Q81.Sum of squares of modulus of all the complex numbers z satisfying z = iz2 + z2 −z is equal to

Q81.Let z = a + ib, b ≠0 be complex numbers satisfying z2 = ¯z ⋅21−|z| . Then the least value of n ∈N , such that zn = (z + 1)n , is equal to _____ .

Q81.Let 𝛼, 𝛽𝛼> 𝛽 be the roots of the quadratic equation 𝑥2 - 𝑥- 4 = 0. If 𝑃𝑛= 𝛼𝑛- 𝛽𝑛, 𝑛∈ℕ, then 𝑃15𝑃16 - 𝑃14𝑃16 - 𝑃152 + 𝑃14𝑃15 is equal to _____. 𝑃13𝑃14

Q81.Let S ={ z ∈C : |z −3| ≤1 and z(4 + 3i) + z(4 −3i) ≤24}. If α + iβ is the point in S which is closest to 4i , then 25(α + β) is equal to ______.

Q81.Let S = {4, 6, 9} and T = {9, 10, 11, … , 1000}. If A = {a1 + a2 + … + ak : k ∈N, a1, a2, a3, … , ak ∈S} then the sum of all the elements in the set T −A is equal to _______.

Q81.Let 𝑎, 𝑏 be two non-zero real numbers. If 𝑝 and 𝑟 are the roots of the equation 𝑥2 - 8𝑎𝑥+ 2𝑎= 0 and 𝑞 and 𝑠 1 1 1 1 are the roots of the equation 𝑥2 + 12𝑏𝑥+ 6𝑏= 0, such that 𝑝, 𝑞, 𝑟, 𝑠 are in A.P., then 𝑎-1 - 𝑏-1 is equal to _____ .

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,

Q81.Let f(x) be a quadratic polynomial with leading coefficient 1 such that f(0) = p, p ≠0 , and f(1) = 31 . If the equations f(x) = 0 and fofofof(x) = 0 have a common real root, then f(−3) is equal to ______. JEE Main 2022 (25 Jul Shift 2) JEE Main Previous Year Paper + = k + 6√3 + 8√6 ,

Q81.The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

Q81.Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is _______.