Practice Questions

Q78.If the line of intersection of the planes ax + by = 3 and ax + by + cz = 0, a > 0 makes an angle 30° with the plane y −z + 2 = 0 , then the direction cosines of the line are (1) 1 , 1 , 0 (2) 1 , −1 , 0 √2 √2 √2 √2 (3) 1 , −2 , 0 (4) 1 2 , −√32 , 0 √5 √5

Q78.Let ˆa,ˆb be unit vectors. If →cbe a vector such that the angle between ˆa and →cis 12 π , and ˆb =→c+ 2(→c ˆa), then 6→c 2 is equal to: + (1) 6(3 −√3) (2) 6(3 √3) + (3) 3 + √3 (4) 6(√3 1)

Q78.If two straight lines whose direction cosines are given by the relations l + m −n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is (1) 6 (2) 4 (3) 3 (4) 2

Q78.Let x−2 3 = −2 = z+3−1 lie on the plane px −qy + z = 5, for some p, q ∈R. The shortest distance of the plane from the origin is: (1) √ 1093 (2) √ 1425 (3) √571 (4) √ 1421

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.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 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.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

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.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.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.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.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.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.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.A plane P is parallel to two lines whose direction ratios are −2, 1, −3, and −1, 2, −2 and it contains the point (2, 2, −2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts α, β, γ . If V is the volume of the tetrahedron OABC , where O is the origin and p = α + β + γ , then the ordered pair (V , p) is equal to (1) (48, −13) (2) (24, −13) (3) (48, 11) (4) (24, −5)

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.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

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.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 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.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 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.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