Practice Questions

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 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.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 ( 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.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 [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.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 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 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 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.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.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

Q79.If y(x) = xx, x > 0 , then y′′(2) −2y′(2) is equal to : (1) 8 loge 2 −2 (2) 4 loge 2 + 2 (3) 4(loge 2)2 −2 (4) 4(loge 2)2 + 2

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.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 𝑃 be the point of intersection of the line = = and the plane 𝑥+ 𝑦+ 𝑧= 2. If the distance of 3 1 2 the point 𝑃 from the plane 3𝑥- 4𝑦+ 12𝑧= 32 is 𝑞, then 𝑞 and 2𝑞 are the roots of the equation (1) 𝑥2 - 18𝑥- 72 = 0 (2) 𝑥2 - 18𝑥+ 72 = 0 (3) 𝑥2 + 18𝑥+ 72 = 0 (4) 𝑥2 + 18𝑥- 72 = 0 𝑚

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 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.Let f(x) be a function such that f(x + y) = f(x) ⋅f(y) for all x, y ∈N , If f(1) = 3 and ∑nk=1 f(k) = 3279 , then the value of n is (1) 6 (2) 8 (3) 7 (4) 9

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 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 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 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.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