Practice Questions

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 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.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 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

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

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

Q81.For a natural number 𝑛, let 𝛼𝑛= 19𝑛- 12𝑛. Then, the value of is ______ 57𝛼8

Q82.If the circles x2 + y2 + 6x + 8y + 16 = 0 and x2 + y2 + 2(3 −√3)x 2(4 −√6)y 2 2 + + + k > 0 , touch internally at the point P(α, β), then (α √3) (β √6) is equal to _______.

Q82.If the sum of the co-efficients of all the positive even powers of 𝑥 in the binomial expansion of 2𝑥3 + is 𝑥 510 - 𝛽· 39, then 𝛽 is equal to _____.

Q82.Let 3, 6, 9, 12, … upto 78 terms and 5, 9, 13, 17, … upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ______. 15 1 1

Q82.If the sum of the coefficients of all the positive powers of x, in the binomial expansion of (xn + x52 ) 7 then the sum of all the possible integral values of n is JEE Main 2022 (27 Jun Shift 2) JEE Main Previous Year Paper

Q82.If the sum of the first ten terms of the series 5 1 + 652 + 3253 + 10254 + 25015 + … . is mn , where m and n are co- prime numbers, then m + n is equal to ______. 60 √x √5

Q82.The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____.

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

Q82.The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____ .

Q82.Let for the 9th term in the binomial expansion of (3 + 6x)n , in the increasing powers of 6x, to be the greatest for x = 23 , the least value of n is n0 . If k is the ratio of the coefficient of x6 to the coefficient of x3 , then k + n0 is equal to does not pass through the fourth

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