Practice Questions

Q78.The distance of the point 1, 1, 9 from the point of intersection of the line = = and the plane 1 2 2 𝑥+ 𝑦+ 𝑧= 17 is: (1) 19√2 (2) 2√19 (3) √38 (4) 38

Q78.The equation of the line through the point (0, 1, 2) and perpendicular to the line x−12 = y+13 = z−1−2 is : y−1 (1) x 3 = −4 = z−23 (2) x3 = y−14 = z−23 (3) −3x = y−14 = z−23 (4) x3 = y−14 = z−2−3

Q78.If (x, y, z) be an arbitrary point lying on a plane P which passes through the point (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of expression 3 + x−11 + y−19 + z−12 − 14(x−11)(y−19)(z−12)x+y+z is (y−19)2(z−12)2 (x−11)2(z−12)2 (x−11)2(y−19)2 (1) 0 (2) 3 (3) 39 (4) −45

Q78.The integral ∫ (2x−1) cos √(2x−1)2+5 dx is equal to (where c is a constant of integration) √4x2−4x+6 (1) 2 1 sin √(2x −1)2 + 5 + c (2) 21 cos √(2x + 1)2 + 5 + c (3) 1 2 cos √(2x −1)2 + 5 + c (4) 12 sin √(2x + 1)2 + 5 + c

Q78.The lines x = ay −1 = z −2 and x = 3y −2 = bz −2, (ab ≠0) are coplanar, if: (1) b = 1, a ∈R −{0} (2) a = 1, b ∈R −{0} (3) a = 2, b = 2 (4) a = 2, b = 3

Q78.In a triangle ABC , if BC→ = 8, CA→ = 7, AB→ = 10 , then the projection of the vector AB→ on AC→ is equal to : (1) 25 (2) 85 4 14 (3) 127 (4) 115 20 16 → → →

Q78.Let →a,→b and →cbe three vectors such that →a =→b × (→ → ×→c). → π 2 respectively and the angle between b and →cis θ(0 < θ < 2 ), then the value of 1 + tan θ is equal to : (1) √3 + 1 (2) 2 (3) 1 (4) √3+1 √3 JEE Main 2021 (27 Jul Shift 2) JEE Main Previous Year Paper

Q78.The distance of line 3𝑦- 2𝑧- 1 = 0 = 3𝑥- 𝑧+ 4 from the point ( 2, - 1, 6 ) is : (1) 2√5 (2) 2√6 (3) √26 (4) 4√2

Q78.The equation of the plane passing through the line of intersection of the planes →r⋅(ˆi + ˆj + ˆk) + 4 = 0 and parallel to the x-axis, is →r⋅(2ˆi + 3ˆj −ˆk) + + 6 = 0 (1) →r⋅(ˆi 3ˆk) + 6 = 0 (2) →r⋅(ˆi −3ˆk) + 6 = 0 (3) →r⋅(ˆj −3ˆk) −6 = 0 (4) →r⋅(ˆj −3ˆk)

Q78.Let →a = 2ˆi −3ˆj + 4ˆk and b = 7ˆi + ˆj −6ˆk If →r×→a =→r× b,→r⋅(ˆi ˆk) equal to: (1) 12 (2) 8 (3) 13 (4) 10

Q78.Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4 . If point P(α, β, γ) is the foot of perpendicular from (3, 2, 1) on L, then the value of 21(α + β + γ) equals: (1) 102 (2) 142 (3) 68 (4) 136

Q78.Let →𝑎, →𝑏, →𝑐 be three vectors mutually perpendicular to each other and have same magnitude. If a vector →𝑟 satisfies →𝑎× {→𝑟- →𝑏× →𝑎} + →𝑏× {→𝑟- →𝑐× →𝑏} + →𝑐× {→𝑟- →𝑎× →𝑐} = →0, then →𝑟 is equal to: (1) 1 (→𝑎+ →𝑏+ →𝑐) (2) 1 (2→𝑎+ →𝑏- →𝑐) 3 3 (3) 1 (→𝑎+ →𝑏+ →𝑐) (4) 1 ( →𝑎+ →𝑏+ 2 →𝑐) 2 2

Q78.Let →a = ˆi + ˆj + 2ˆk and b = −ˆi + 2ˆj + 3ˆk. Then the vector product × × is equal to : (→a+→b) ((→a ((→a−→b) ×→b)) ×→b) + + (1) 5(34ˆi −5ˆj 3ˆk) (2) 7(34ˆi −5ˆj 3ˆk) + + (3) 7(30ˆi −5ˆj 7ˆk) (4) 5(30ˆi −5ˆj 7ˆk)

Q78.Let L be the line of intersection of planes →r⋅(ˆi −ˆj + 2ˆk) = 2 and →r⋅(2ˆi + ˆj −ˆk) foot of perpendicular on L from the point (1, 2, 0), then the value of 35(α + β + γ) is equal to: (1) 101 (2) 119 (3) 143 (4) 134

Q78.A hall has a square floor of dimension 10 m × 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is cos−1 15 , then the height of the hall (in meters) is: (1) 5√2 (2) 5√3 (3) 5√10 (4) 5

Q78.Let the vectors 2 + 𝑎+ 𝑏 ^𝑖+ 𝑎+ 2𝑏+ 𝑐 ^𝑗- 𝑏+ 𝑐 ^𝑘, 1 + 𝑏 ^𝑖+ 2𝑏 ^𝑗- 𝑏 ^𝑘 and 2 + 𝑏 ^𝑖+ 2𝑏 ^𝑗+ 1 - 𝑏 ^𝑘, ∀𝑎, 𝑏, 𝑐∈𝑅 be co-planar. Then which of the following is true? (1) 2𝑏= 𝑎+ 𝑐 (2) 3𝑐= 𝑎+ 𝑏 (3) 𝑎= 𝑏+ 2𝑐 (4) 2𝑎= 𝑏+ 𝑐

Q78.If dy dx = 2y , y(0) = 1, then y(1) is equal to : (1) log2(1 + e2) (2) log2(2e) (3) log2(2 + e) (4) log2(1 + e) → → → → 1 is a unit

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

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.Let the acute angle bisector of the two planes 𝑥- 2𝑦- 2𝑧+ 1 = 0 and 2𝑥- 3𝑦- 6𝑧+ 1 = 0 be the plane 𝑃. Then which of the following points lies on 𝑃 ? 1 (1) ( 0, 2, - 4 ) (2) -2, 0, - 2 (3) ( 4, 0, - 2 ) (4) 3, 1, - 1 2

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.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.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.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.The coefficients a, b and c of the quadratic equation, ax2 + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is: (1) 1 (2) 1 72 36 (3) 1 (4) 5 54 216