Practice Questions

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.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 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.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 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 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.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.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 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.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 ¯

Q81.The number of real solutions of the equation e4x + 4e3x −58e2x + 4ex + 1 = 0 is _____.

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 _______.

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.The total number of three-digit numbers, with one digit repeated exactly two times, is ______. JEE Main 2022 (25 Jun Shift 2) JEE Main Previous Year Paper 3 10

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.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 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.For a natural number 𝑛, let 𝛼𝑛= 19𝑛- 12𝑛. Then, the value of is ______ 57𝛼8

Q81.The sum of the cubes of all the roots of the equation x4 −3x3 −2x2 + 3x + 1 = 0 is _____.

Q81.The sum of all real values of 𝑥 for which 3𝑥2 - 9𝑥+ 17 = 5𝑥2 - 7𝑥+ 19 is equal to 𝑥2 + 3𝑥+ 10 3𝑥2 + 5𝑥+ 12

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 ______.

Q82.Let a1, a2, a3, … be an A.P. If ∑∞r=1 ar2r = 4, then 4a2 is equal to ______.

Q82.Let 𝑆 be the set of all passwords which are six to eight characters long, where each character is either an alphabet from 𝐴, 𝐵, 𝐶, 𝐷, 𝐸 or a number from 1, 2, 3, 4, 5 with the repetition of characters allowed. If the number of passwords in 𝑆 whose at least one character is a number from 1, 2, 3, 4, 5 is 𝛼× 56, then 𝛼 is equal to