Practice Questions

Q78.Let the lines x−1 λ = y−21 = z−32 and x+26−2 = y+183 = z+28λ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lies on P ? (1) (0, −2, −2) (2) (−5, 0, −1) (3) (3, −1, 0) (4) (0, 4, 5)

Q78.Let the solution curve 𝑦= 𝑓𝑥 of the differential equation 𝑑𝑦 𝑥𝑦 = 𝑥4 + 2𝑥 , 𝑥∈-1, 1 pass through the 𝑑𝑥+ 𝑥2 - 1 √1 - 𝑥2 √3 origin. Then ∫ 2 𝑓𝑥𝑑𝑥 is equal to -√3 2 𝜋 1 𝜋 √3 (1) - (2) - 3 4 3 4 (3) 𝜋 - √3 (4) 𝜋 - √3 6 4 6 2

Q78.If the two lines l1 : x−23 = y+1−2 , z = 2 and l2 : x−11 = 2y+3α = z+52 are perpendicular, then an angle between the lines l2 and l3 : 1−x3 = 2y−1−4 = 4z is (1) cos−1( 294 ) (2) sec−1( 294 ) (3) cos−1( 292 ) (4) cos−1( √292 )

Q78.Let a vector →𝑎 has a magnitude 9. Let a vector →𝑏 be such that for every 𝑥, 𝑦𝑅× 𝑅- 0, 0, the vector 𝑥→𝑎+ 𝑦 →𝑏 is → → perpendicular to the vector 6𝑦 →𝑎- 18𝑥 𝑏. Then the value of →𝑎× 𝑏 is equal to (1) 9√3 (2) 27√3 (3) 9 (4) 81

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.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.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.Let the foot of the perpendicular from the point (1, 2, 4) on the line x+24 = y−12 = z+13 be distance of P from the plane 3x + 4y + 12z + 23 = 0 is (1) 50 (2) 63 13 13 (3) 65 (4) 4 13

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 = 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.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 shortest distance between the lines x−1 2 = y−23 = z−3λ and x−21 = y−44 = z−55 is √31 , then the sum of all possible values of λ is: (1) 16 (2) 6 (3) 12 (4) 15

Q78.If 𝑦= 𝑦𝑥 is the solution of the differential equation 2𝑥2𝑑𝑦 2𝑥𝑦+ 3𝑦2 = 0 such that 𝑦𝑒= 𝑒 then 𝑦1 is equal 𝑑𝑥- 3, to (1) 1 (2) 2 3 3 3 (3) (4) 3 2

Q78.Let →a = 2ˆi −ˆj + 5ˆk and b = αˆi + βˆj + 2ˆk. If ((→a b) ׈i) (1) 4 (2) 5 (3) √21 (4) √17

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 the solution curve of the differential equation x dxdy −y = √y2 + 16x2, y(1) = 3 be y = y(x). Then y(2) is equal to (1) 15 (2) 11 (3) 14 (4) 17 →

Q78.Let →𝑎= 𝑎1 ^𝑖+ 𝑎2 ^𝑗+ 𝑎3 ^𝑘, 𝑎𝑖> 0, 𝑖= 1, 2, 3 be a vector which makes equal angles with the coordinate axes 𝑂𝑋, 𝑂𝑌 and 𝑂𝑍. Also, let the projection of →𝑎 on the vector 3 ^𝑖+ 4 ^𝑗 be 7 . Let →𝑏 be a vector obtained by rotating →𝑎 with 90°. If →𝑎, →𝑏 and 𝑥-axis are coplanar, then projection of a vector →𝑏 on 3 ^𝑖+ 4 ^𝑗 is equal to (1) √7 (2) √2 (3) 2 (4) 7

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 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 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.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 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 →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.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