Practice Questions

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.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.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.Let f : R →R be a differentiable function that satisfies the relation f(x + y) = f(x) + f(y) −1, ∀ x, y ∈R. If f ′(0) = 2 , then |f(−2)| is equal to

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.Let the sets A and B denote the domain and range respectively of the function f(x) = 1 , where [x] √[x]−x denotes the smallest integer greater than or equal to x. Then among the statements (S1) : A ∩B = (1, ∞) −N and (S2) : A ∪B = (1, ∞) (1) Only (S2) is true (2) Only (S1) is true (3) Neither (S1) nor (S2) is true (4) Both (S1) and (S2) are true

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.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.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 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.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 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 [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 )

Q79.If the equation of the plane passing through the line of intersection of the planes 𝑥+ 1 𝑦+ 3 𝑧- 2 2𝑥- 𝑦+ 𝑧= 3, 4𝑥- 3𝑦+ 5𝑧+ 9 = 0 and parallel to the line = = is 𝑎𝑥+ 𝑏𝑦+ 𝑐𝑧+ 6 = 0, -2 4 5 then 𝑎+ 𝑏+ 𝑐 is equal to (1) 12 (2) 14 (3) 16 (4) 13

Q79.Let the shortest distance between the lines L: 𝑥- = = , 𝜆≥0 and L1: 𝑥+ 1 = 𝑦- 1 = 4 - 𝑧 be 2√6. -2 0 1 If ( 𝛼, 𝛽, 𝛾) lies on L, then which of the following is NOT possible? (1) 𝛼+ 2𝛾= 24 (2) 2𝛼+ 𝛾= 7 (3) 2𝛼- 𝛾= 9 (4) 𝛼- 2𝛾= 19

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 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 curve y = f(x), x ∈(0, ∞) pass through the points P(1, 32 ) and Q(a, 12 ). If the tangent at any point R(b, f(b)) to the given curve cuts the y-axis at the point S(0, c) such that bc = 3, then (PQ)2 is equal to JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper _____.

Q79.Let f : R −{2, 6} →R be real valued function defined as f(x) = x+2x+1 . Then range of f is x2−8x+12 (1) (−∞, −214 ] ∪[ 214 , ∞) (2) (−∞, −214 ] ∪[0, ∞) (3) (−∞, −214 ) ∪(0, ∞) (4) (−∞, −214 ] ∪[1, ∞)

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.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.Let the line = = intersect the lines = = and = = at the points A and B 1 2 5 4 3 1 6 3 1 respectively. Then the distance of the mid-point of the line segment 𝐴𝐵 from the plane 2𝑥- 2𝑦+ 𝑧= 14 is (1) 3 (2) 11 3 10 (3) 4 (4) 3

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