Practice Questions

Q78.Let 𝑃 be the plane containing the straight line = = and perpendicular to the plane containing the 9 -1 -5 straight lines 𝑥 = 𝑦 = 𝑧 and 𝑥 = 𝑦 = 𝑧 If 𝑑 is the distance of 𝑃 from the point 2, - 5, 11, then 𝑑2 is equal to 2 3 5 3 7 8. 147 (1) (2) 96 2 32 (3) (4) 54 3

Q79.The shortest distance between the lines x−3 2 = y−23 = z−1−1 and x+32 = y−61 = z−53 is (1) 18 (2) 22 √5 3√5 (3) 46 (4) 6√3 3√5

Q79.Five numbers x1, x2, x3, x4, x5 are randomly selected from the numbers 1, 2, 3, … … , 18 and are arranged in the increasing order (x1 < x2 < x1 < x4 < x2). The probability that x2 = 7 and x4 = 11 is JEE Main 2022 (27 Jun Shift 1) JEE Main Previous Year Paper (1) 1 (2) 1 136 68 (3) 7 (4) 5 68 68

Q79.Let X have a binomial distribution B(n, p) such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If P(X > n −3) = 2nk , then k is equal to (1) 528 (2) 529 (3) 629 (4) 630

Q79.The foot of the perpendicular from a point on the circle 𝑥2 + 𝑦2 = 1, 𝑧= 0 to the plane 2𝑥+ 3𝑦+ 𝑧= 6 lies on which one of the following curves? (1) 6𝑥+ 5𝑦- 122 + 43𝑥+ 7𝑦- 82 = 1, 𝑧= 6 - 2𝑥- 3𝑦(2) 5𝑥+ 6𝑦- 122 + 43𝑥+ 5𝑦- 92 = 1, 𝑧= 6 - 2𝑥- 3𝑦 (3) 6𝑥+ 5𝑦- 142 + 93𝑥+ 5𝑦- 72 = 1, 𝑧= 6 - 2𝑥- 3𝑦(4) 5𝑥+ 6𝑦- 142 + 93𝑥+ 7𝑦- 82 = 1, 𝑧= 6 - 2𝑥- 3𝑦

Q79.If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky −5z = 1 and 3kx −ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane JEE Main 2022 (27 Jul Shift 1) JEE Main Previous Year Paper P on the y-axis is (1) 1 (2) 5 11 11 (3) 6 (4) 7

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.If the plane 2x + y −5z = 0 is rotated about its line of intersection with the plane 3x −y + 4z −7 = 0 by an angle of π , then the plane after the rotation passes through the point 2 (1) (2, −2, 0) (2) (−2, 2, 0) (3) (1, 0, 2) (4) (−1, 0, −2) + = +

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.Let →a = αˆi + 3ˆj −ˆk, b = 3ˆi −βˆj + 4ˆk and →c= ˆi + 2ˆj −2ˆk where α, β ∈R be three vectors. If the projection → 10 of →a on →cis and b ×→c= −6ˆi + 10ˆj + 7ˆk , then the value of α + β equal to 3 (1) 3 (2) 4 (3) 5 (4) 6

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.Let 𝑄 be the foot of perpendicular drawn from the point 𝑃1, 2, 3 to the plane 𝑥+ 2𝑦+ 𝑧= 14. If 𝑅 is a point on the plane such that ∠𝑃𝑅𝑄= 60°, then the area of ∆𝑃𝑄𝑅 is equal to (1) √3 (2) √3 2 (3) 2√3 (4) 3

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.If the foot of the perpendicular from the point A(−1, 4, 3) on the plane P : 2x + my + nz = 4, is (−2, 72 , 32 ), then the distance of the point A from the plane P , measured parallel to a line with direction ratios 3, −1, −4, is equal to (1) 1 (2) √26 (3) 2√2 (4) √14

Q79.Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16 . Let T be a λ ∈R. Then, which of the plane passing through the point Q and contains the line →r= −ˆk + λ(ˆi + ˆj + 2ˆk), following points lies on T ? (1) (2, 1, 0) (2) (1, 2, 1) (3) (1, 2, 2) (4) (1, 3, 2)

Q79.Let →a be a vector which is perpendicular to the vector 3ˆi + 2 1 ˆj + 2ˆk. If →a× (2ˆi ˆk) the projection of the vector →a on the vector 2ˆi + 2ˆj + ˆk is (1) 1 (2) 1 3 (3) 5 (4) 7 3 3

Q79.Let the plane P :→r⋅→a = d contain the line of intersection of two planes →r⋅(ˆi + 3ˆj −ˆk) 13→a 2 → = 7. If the plane P passes through the point (2, 3, 21 ), then the value of d2 is equal to r ⋅(−6ˆi + 5ˆj −ˆk) (1) 90 (2) 93 (3) 95 (4) 97

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.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

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

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 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.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.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