Practice Questions

Q78.The acute angle between the planes P1 and P2 , when P1 and P2 are the planes passing through the intersection of the planes 5x + 8y + 13z −29 = 0 and 8x −7y + z −20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is (1) π (2) π 3 4 (3) π (4) π 6 12 JEE Main 2022 (28 Jun Shift 1) JEE Main Previous Year Paper = 6 and

Q78.A plane E is perpendicular to the two planes 2x −2y + z = 0 and x −y + 2z = 4 , and passes through the point P(1, −1, 1). If the distance of the plane E from the point Q(a, a, 2) is 3√2 , then (PQ)2 is equal to (1) 9 (2) 12 (3) 21 (4) 33 y−6

Q78.Let →𝑎, →𝑏, →𝑐 be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and →𝑎× →𝑏· →𝑏× →𝑐+ →𝑏× →𝑐· →𝑐× →𝑎+ →𝑐× →𝑎· →𝑎× →𝑏= 168 then →𝑎+ →𝑏+ →𝑐 is equal to (1) 10 (2) 14 (3) 16 (4) 18

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

Q78.The length of the perpendicular from the point (1, −2, 5) on the line passing through (1, 2, 4) and parallel to the line x + y −z = 0 = x −2y + 3z −5 is: (1) √212 (2) √92 (3) √732 (4) 1

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 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

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.Let →a = 2ˆi −ˆj + 5ˆk and b = αˆi + βˆj + 2ˆk. If ((→a b) ׈i) (1) 4 (2) 5 (3) √21 (4) √17

Q78.Let →a = ˆi + ˆj + 2ˆk, b = 2ˆi −3ˆj + ˆk and →c= ˆi −ˆj + ˆk be the three given vectors. Let →vbe a vector in the → plane of →a and b whose projection on →cis 2 . If →v,ˆj = 7 , then →v + is equal to √3 ⋅(ˆi ˆk) (1) 6 (2) 7 (3) 8 (4) 9

Q78.Let →a = αˆi + 2ˆj −ˆk and b = −2ˆi + αˆj + ˆk, where α ∈R. If the area of the parallelogram whose adjacent → 2 → → 2 b is equal to ⋅ sides are represented by the vectors →a and b is √15(α2 + 4), then the value of 2→a + (→a b) (1) 10 (2) 7 (3) 9 (4) 14 + = 2ˆi −13ˆj −4ˆk, then

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.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 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 →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.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 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 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.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.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.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.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.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.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.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)