Practice Questions

Q77.The plane, passing through the points ( 0, – 1, 2 ) and ( – 1, 2, 1 ) and parallel to the line passing through ( 5, 1, – 7 ) and ( 1, – 1, – 1 ) , also passes through the point (1) -2, 5, 0 (2) 1, - 2, 1 (3) 2, 0, 1 (4) 0, 5, - 2

Q78.Let the image of the point P ( 1, 2, 6 ) in the plane passing through the points A ( 1, 2, 0 ) and B ( 1, 4, 1 ) C ( 0, 5, 1 ) be Q ( α, β, γ ) . Then α2 + β2 + γ2 equal to JEE Main 2023 (10 Apr Shift 2) JEE Main Previous Year Paper (1) 65 (2) 62 (3) 76 (4) 70 𝑥 6 - 𝑦 𝑧+ 8 𝑥- 5 𝑦- 7 𝑧+ 2 𝑥+ 3 3 - 𝑦 𝑧- 6

Q78.The domain of the function f(x) = 1 is (where [x] denotes the greatest integer less than or equal to √[x]2−3[x]−10 x) (1) (−∞, −3] ∪(5, ∞) (2) (−∞, −2) ∪[6, ∞) (3) (−∞, −2) ∪(5, ∞) (4) (−∞, −3] ∪[6, ∞)

Q78.Let f : R −{0, 1} →R be a function such that f(x) + f( 1−x1 ) = 1 + x. Then f(2) is equal to : (1) 9 (2) 9 2 4 (3) 7 (4) 7 4 3

Q78.Let the image of the point 𝑃2, - 1, 3 in the plane 𝑥+ 2𝑦- 𝑧= 0 be 𝑄. Then the distance of the plane 3𝑥+ 2𝑦+ 𝑧+ 29 = 0 from the point 𝑄 is (1) 22√2 (2) 24√2 7 7 (3) 2√14 (4) 3√14 𝑥- 5 𝑦- 2 𝑧- 4 𝑥+ 3 𝑦+ 5 𝑧- 1

Q78.Let (a, b) ⊂(0, 2π) be the largest interval for which sin−1(sin θ) −cos−1(sin θ) > 0, θ ∈(0, 2π), holds . If αx2 + βx + sin−1(x2 −6x + 10) + cos−1(x2 −6x + 10) = 0 and α −β = b −a, then α is equal to; JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper (1) π (2) π 8 48 (3) π (4) π 16 12

Q78.Let →𝑎= 2 ^𝑖+ ^𝑗+ ^𝑘, and →𝑏 and →𝑐 be two nonzero vectors such that →𝑎+ →𝑏+ →𝑐= →𝑎+ →𝑏- →𝑐 and →𝑏· →𝑐= 0. Consider the following two statement: 𝐴 →𝑎+ 𝜆→𝑐≥→𝑎 for all 𝜆∈ℝ. 𝐵 →𝑎 and →𝑐 are always parallel (1) only (B) is correct (2) neither (A) nor (B) is correct (3) only (A) is correct (4) both (A) and (B) are correct. 5 𝑦- 𝜆 𝑧+ 𝜆

Q78.Let the foot of perpendicular of the point P(3, –2, –9) on the plane passing through the points (–1, –2, –3), (9, 3, 4), (9, –2, 1) be Q(α, β, γ). Then the distance Q from the origin is (1) √42 (2) √38 (3) √35 (4) √29

Q78.The distance of the point 7, - 3, - 4 from the plane containing the points 2, - 3, 1, -1, 1, - 2 and 3, - 4, 2 is equal to: (1) 4 (2) 5 (3) 5√2 (4) 4√2 JEE Main 2023 (24 Jan Shift 1) JEE Main Previous Year Paper

Q78.Let ( 𝛼, 𝛽, 𝛾) be the image of point 𝑃( 2, 3, 5 ) in the plane 2𝑥+ 𝑦- 3𝑧= 6. Then 𝛼+ 𝛽+ 𝛾 is equal to (1) 5 (2) 10 (3) 12 (4) 9

Q78.One vertex of a rectangular parallelopiped is at the origin 𝑂 and the lengths of its edges along 𝑥, 𝑦 and 𝑧 axes are 3, 4 and 5 units respectively. Let 𝑃 be the vertex ( 3, 4, 5 ) . Then the shortest distance between the diagonal 𝑂𝑃 and an edge parallel to 𝑧 axis, not passing through 𝑂 or 𝑃 is 12 (1) (2) 12√5 √5 12 12 (3) (4) 5√5 5

Q78.The shortest distance between the lines 𝑥+ 2 = 𝑦 = 𝑧- 5 and 𝑥- 4 = 𝑦- 1 = 𝑧+ 3 is 1 -2 2 1 2 0 (1) 8 (2) 6 (3) 7 (4) 9 𝑥+ 3 𝑦+ 2 1 - 𝑧

Q78.The line, that is coplanar to the line 𝑥+ 3 = 𝑦- 1 = 𝑧- 5 , is -3 1 5 (1) 𝑥+ 1 = 𝑦- 2 = 𝑧- 5 (2) 𝑥+ 1 = 𝑦- 2 = 𝑧- 5 -1 2 4 -1 2 5 (3) 𝑥- 1 = 𝑦- 2 = 𝑧- 5 (4) 𝑥+ 1 = 𝑦- 2 = 𝑧- 5 -1 2 5 1 2 5

Q78.If f(x) = 22x , x ∈R, then f( 20231 ) + f( 20232 ) + f( 20233 ). . . . . . . . . f( 20222023 ) is equal to 22x+2 (1) 2011 (2) 1010 (3) 2010 (4) 1011

Q78.The line 𝑙1 passes through the point 2, 6, 2 and is perpendicular to the plane 2𝑥+ 𝑦- 2𝑧= 10. Then the 𝑥+ 1 𝑦+ 4 𝑧 shortest distance between the line 𝑙1 and the line 2 = -3 = 2 is: (1) 7 (2) 19 3 19 (3) (4) 9 2

Q79.Let S be the set of all values of λ, for which the shortest distance between the lines x−λ0 = y−34 = z+61 and x+λ 3 = −4y = z−60 is 13. Then 8 ∑λ∈S λ is equal to (1) 306 (2) 304 (3) 308 (4) 302

Q79.If the equation of the plane that contains the point ( - 2, 3, 5 ) and is perpendicular to each of the planes 2𝑥+ 4𝑦+ 5𝑧= 8 and 3𝑥- 2𝑦+ 3𝑧= 5 is 𝛼𝑥+ 𝛽𝑦+ 𝛾𝑧+ 97 = 0 then 𝛼+ 𝛽+ 𝛾= (1) 15 (2) 18 (3) 16 (4) 17

Q79.If the functions f(x) = x33 + 2bx + ax22 and g(x) = x33 + then a + 2b + 7 is equal to (1) 4 (2) 32 (3) 3 (4) 6 1 + constant, then β −α is equal to + cos β x)

Q79.Let a unit vector →𝑂𝑃 make angle 𝛼, 𝛽, 𝛾 with the positive directions of the co-ordinate axes OX, OY, OZ 𝜋 respectively, where 𝛽∈0, →𝑂𝑃 is perpendicular to the plane through points 1, 2, 3, 2, 3, 4 and 1, 5, 7, then 2. which one of the following is true ? (1) 𝛼∈𝜋 𝜋 and 𝛾∈𝜋 𝜋 (2) 𝛼∈0, 𝜋 and 𝛾∈0, 𝜋 2, 2, 2 2 𝜋 𝜋 𝜋 𝜋 (3) 𝛼∈ 2, 𝜋 and 𝛾∈0, 2 (4) 𝛼∈0, 2 and 𝛾∈ 2, 𝜋

Q79.If the total maximum value of the function f(x) = ( 2 equal to (1) e3 + e6 + e11 (2) e5 + e6 + e11 (3) e3 + e6 + e10 (4) e3 + e5 + e11 +

Q79.The set of all a ∈R for which the equation x|x −1| + |x + 2| + a = 0 has exactly one real root, is (1) (−7, ∞) (2) (−∞, ∞) (3) (−6, −3) (4) (−∞, −3) dx = Q80. ∫∞0 e3x+6e2x+11ex+66 (1) loge( 3227 ) (2) loge( 51281 ) (3) loge( 25681 ) (4) loge( 30227 )

Q79.Let 𝑁 be the foot of perpendicular from the point 𝑃( 1, - 2, 3 ) on the line passing through the points ( 4, 5, 8 ) and ( 1, - 7, 5 ) . Then the distance of 𝑁 from the plane 2𝑥- 2𝑦+ 𝑧+ 5 = 0 is (1) 8 (2) 6 (3) 9 (4) 7

Q79.Let f and g be twice differentiable functions on R such that f ′′(x) = g′′(x) + 6x f ′(1) = 4g′(1) −3 = 9 f(2) = 3 g(2) = 12 Then which of the following is NOT true ? (1) g(−2) −f(−2) = 20 (2) If −1 < x < 2 , then |f(x) −g(x)| < 8 (3) |f ′(x) −g′(x)| < 6 ⇒−1 < x < 1 (4) There exists x0 ∈(1, 23 ) such that f(x0) = g(x0)

Q79.The distance of the point -1, 9, - 16 from the plane 2𝑥+ 3𝑦- 𝑧= 5 measure parallel to the line 𝑥+ 4 2 - 𝑦 𝑧- 3 = = is 3 4 12 (1) 13√2 (2) 31 (3) 26 (4) 20√3

Q79.The shortest distance between the lines = = and = = is 1 2 -3 1 4 -5 (1) 7√3 (2) 5√3 (3) 6√3 (4) 4√3