Practice Questions

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.For some a, b, c ∈N, let f(x) = ax −3 and g(x) = xb + c, x ∈R. If (fog)−1 (x) = ( 1 2 ) 3 , then (f ∘g)(ac) + (g ∘f)(b) is equal to _____ .

Q78.Let [x] be the greatest integer ≤x . Then the number of points in the interval (–2, 1) where the function f(x) = |[x]| + √x −[x] is discontinuous, is _____. sin2 x √3e is , x ∈(0, π2 ), is ke , then ( ke ) 8 + k8e5 + k8 sin x )

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.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

Q78.Consider a function f : N →R, satisfying f(1) + 2f(2) + 3f(3) + … + xf(x) = x(x + 1)f(x) ; x ≥2 with f(1) = 1 . Then f(2022)1 + f(2028)1 is equal to JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper (1) 8200 (2) 8000 (3) 8400 (4) 8100

Q78.If domain of the function loge( 6x2+5x+12x−1 ) cos−1( 2x2−3x+43x−5 ) is is equal to JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper

Q78.Let A = {1, 2, 3, 5, 8, 9} . Then the number of possible functions f : A →A such that f(m ⋅n) = f(m) ⋅f(n) for every m, n ∈A with m ⋅n ∈A is equal to ax + bx2, a ≠2b have a common extreme point,

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.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 (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.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.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

Q79.Let f(x) = sinsinx+cos−√2x−cos x , x ∈[0, π] −{ π4 }, then f( 7π12 )f ′′( 7π12 ) is equal to JEE Main 2023 (08 Apr Shift 1) JEE Main Previous Year Paper (1) 2 (2) −2 9 3 (3) −1 (4) 2 3√3 3√3

Q79.Suppose f is a function satisfying f(x + y) = f(x) + f(y) for all x, y ∈N and f(1) = 51 . If ∑mn=1 n(n+1)(n+2)f(n) = 121 then m is equal to ______.

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

Q79.Let R = {a, b, c, d, e} and S = {1, 2, 3, 4} . Total number of onto functions f : R →S such that f(a) ≠1, is equal to ________.

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 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 y(x) = (1 + x)(1 + x2)(1 + x4)(1 + x8)(1 + x16) . Then y′ −y′′ at x = −1 is equal to (1) 976 (2) 464 (3) 496 (4) 944

Q79.Let the function f(x) = 2x3 + (2p −7)x2 + 3(2p −9)x −6 have a maxima for some value of x < 0 and a minima for some value of x > 0 . Then, the set of all values of p is (1) ( 92 , ∞) (2) (0, 29 ) (3) (−∞, 92 ) (4) (−92 , 92 )

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 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 +