Practice Questions

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 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 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 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.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 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.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.Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x −3y + 5z = 8 . If the mirror image of the point (2, −12 , 2) in the rotated plane is B(a, b, c), then (1) a 8 = 5b = −4c (2) a4 = 5b = −2c (3) a 8 = −5b = 4c (4) a4 = 5b = 2c JEE Main 2022 (26 Jun Shift 1) JEE Main Previous Year Paper

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 →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.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 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.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 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.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

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.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.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.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 ¯

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.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.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.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.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