Practice Questions

Q79.The equation of the plane passing through the point 1, 2, - 3 and perpendicular to the planes 3𝑥+ 𝑦- 2𝑧= 5 and 2𝑥- 5𝑦- 𝑧= 7, is (1) 11𝑥+ 𝑦+ 17𝑧+ 38 = 0 (2) 3𝑥- 10𝑦- 2𝑧+ 11 = 0 (3) 6𝑥- 5𝑦+ 2𝑧+ 10 = 0 (4) 6𝑥- 5𝑦- 2𝑧- 2 = 0

Q79.Equation of a plane at a distance √221 planes x −y −z −1 = 0 and 2x + y −3 z + 4 = 0, is (1) −x + 2y + 2z −3 = 0 (2) 3x −4z + 3 = 0 (3) 3x −1y −5z + 2 = 0 (4) 4x −y −5z + 2 = 0

Q79.If →a = 2, →b = 5 and →a×→b = 8, then →a⋅→b is equal to: (1) 6 (2) 4 (3) 3 (4) 5

Q79.Consider the line L given by the equation x−3 2 = y−11 = z−21 . Let Q be the mirror image of the point (2, 3, −1) with respect to L. Let a plane P be such that it passes through Q, and the line L is perpendicular to P. Then which of the following points is on the plane P? (1) (−1, 1, 2) (2) (1, 1, 1) (3) (1, 1, 2) (4) (1, 2, 2)

Q79.Let →a = 2ˆi + ˆj −2ˆk and b = ˆi + ˆj. If →cis a vector such that →a⋅→c= →c, →c−→a = 2√2 and the angle between π , then the value of is: and →cis × × 6 (→a → → b) (→a b) ×→c (1) 2 (2) 4 3 (3) 3 (4) 32

Q79.The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is: (1) x + 3z = 10 (2) x + 3z = 0 (3) 3x + z = 6 (4) 3x −z = 0

Q79.The angle between the straight lines, whose direction cosines l, m, n are given by the equations 2l + 2 m −n = 0 and mn + nl+ lm= 0, is: (1) π (2) π 3 2 (3) cos−1( 89 ) (4) π −cos−1( 94 )

Q79.Let the plane passing through the point (−1, 0, −2) and perpendicular to each of the planes 2x + y −z = 2 and x −y −z = 3 be ax + by + cz + 8 = 0. Then the value of a + b + c is equal to: (1) 3 (2) 8 (3) 5 (4) 4

Q79.For real numbers α and β ≠0, if the point of intersection of the straight lines x−α1 = y−12 = z−13 and x−4 β = y−63 = z−73 lies on the plane x + 2y −z = 8, then α −β is equal to : (1) 5 (2) 9 (3) 3 (4) 7

Q79.Let a, b ∈R. If the mirror image of the point P(a, 6, 9) with respect to the line x−37 = y−25 = z−1−9 is (20, b, −a −9), then |a + b| is equal to: (1) 86 (2) 90 (3) 84 (4) 88

Q79.Let →a and b be two non-zero vectors perpendicular to each other and →a = b , If →a× b = →a , then the angle between the vectors and →a is equal to : + b + × (→a → → (→a b)) JEE Main 2021 (18 Mar Shift 2) JEE Main Previous Year Paper (1) sin−1( √31 ) (2) cos−1( √31 ) (3) cos−1( √21 ) (4) sin−1( √61 )

Q79.If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line x+1 2 = y−31 = z+2−1 and containing the line x−23 = 1−y2 = z+11 is αx + βy + γz = 24 then α + β + γ is equal to: (1) 20 (2) 19 (3) 18 (4) 21

Q79.The distance of the point ( - 1, 2, - 2 ) from the line of intersection of the planes 2𝑥+ 3𝑦+ 2𝑧= 0 and 𝑥- 2𝑦+ 𝑧= 0 is : 1 √42 (1) (2) √2 2 5 √34 (3) (4) 2 2

Q79.If the shortest distance between the straight lines 3(x −1) = 6(y −2) = 2(z −1) and 4(x −2) = 2(y −λ) = (z −3), λ ∈R is 1 , then the integral value of λ is equal to: √38 (1) 3 (2) 2 (3) 5 (4) −1

Q79.Let the foot of perpendicular from a point 𝑃( 1, 2, - 1 ) to the straight line 𝐿: 𝑥 = 𝑦 = 𝑧 be 𝑁. Let a line be 1 0 -1 drawn from 𝑃 parallel to the plane 𝑥+ 𝑦+ 2𝑧= 0 which meets 𝐿 at point 𝑄. If 𝛼 is the acute angle between the lines 𝑃𝑁 and 𝑃𝑄, then cos𝛼 is equal to . 1 √3 (1) (2) √5 2 1 1 (3) (4) √3 2√3

Q79.Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes = 6. Then which of the following points does NOT lie on P ? →r⋅(ˆi + ˆj + 4ˆk) = 16 & →r⋅(−ˆi + ˆj + ˆk) JEE Main 2021 (26 Aug Shift 2) JEE Main Previous Year Paper (1) (4, 2, 2) (2) (6, −6, 2) (3) (−8, 8, 6) (4) (3, 3, 2)

Q79.Let →a and b be two vectors such that 2→a+ 3b = 3→a+ b and the angle between →a and b is 60°. If 8→a → vector, then b is equal to : (1) 8 (2) 4 (3) 6 (4) 5

Q79.In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is : (1) 14 (2) 7 45 45 (3) 8 (4) 28 45 45

Q79.Consider the three planes P1 : 3x + 15y + 21z = 9 P2 : x −3y −z = 5, and P3 : 2x + 10y + 14z = 5 Then, which one of the following is true? (1) P2 and P3 are parallel. (2) P1, P2 and P3 all are parallel. (3) P1 and P2 are parallel. (4) P1 and P3 are parallel.

Q79.Let P be a plane lx + my + nz = 0 containing the line, 1−x1 = y+42 = z+23 . If plane segment AB joining points A(−3, −6, 1) and B(2, 4, −3) in ratio k : 1 then the value of k is equal to : (1) 1. 5 (2) 3 (3) 2 (4) 4

Q79.The differential equation satisfied by the system of parabolas y2 = 4a(x + a) is (1) dy 2 dy (2) dy 2 dy −y = 0 + y = 0 y( dx ) −2x( dx ) y( dx ) −2x( dx ) + −y = 0 + −y = 0 (4) y( dxdy ) 2x( dxdy ) (3) y( dxdy ) 2 2x( dxdy )

Q79.If the foot of the perpendicular from point (4, 3, 8) on the line L1 : x−al = y−23 = z−b4 , l ≠0 is (3, 5, 7), then the shortest distance between the line L1 and line L2 : x−23 = y−44 = z−55 is equal to (1) 1 (2) 1 2 √6 (3) √23 (4) √31 JEE Main 2021 (16 Mar Shift 2) JEE Main Previous Year Paper

Q80.Let A denote the event that a 6 -digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3 . Then probability of event A is equal to : (1) 9 (2) 4 56 9 (3) 3 (4) 11 7 27

Q80.The probability that a randomly selected 2− digit number belongs to the set {n ∈N : (2n −2) is a multiple of 3} is equal to (1) 1 (2) 2 6 3 (3) 1 (4) 1 2 3

Q80.A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least n correct answers is less than 1 , is : 2 (1) 5 (2) 6 (3) 3 (4) 4