Practice Questions

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 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.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.Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6} . Then the number of functions f : A →B satisfying f(1) + f(2) = f(4) −1 is equal to........ .Then and g(x) =

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.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.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 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.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 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 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 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

Q80.The number of points, where the curve y = x5 −20x3 + 50x + 2 crosses the x-axis, is _____. x dx is equal to

Q80.If an unbiased die, marked with -2, - 1, 0, 1, 2, 3 on its faces is thrown five times, then the probability that the product of the outcomes is positive, is : 881 521 (1) (2) 2592 2592 (3) 440 (4) 27 2592 288 1 + i ¯𝑧 12

Q80.If aα is the greatest term in the sequence an = n3 , n = 1, 2, 3. . . . , then α is equal to ______ n4+147

Q80.The random variable 𝑋 follows binomial distribution 𝐵( 𝑛, 𝑝) , for which the difference of the mean and the variance is 1. If 2 𝑃( 𝑋= 2 ) = 3 𝑃( 𝑋= 1 ) , then 𝑛2𝑃( 𝑋> 1 ) is equal to (1) 15 (2) 11 (3) 12 (4) 16

Q80.Let x = 2 be a local minima of the function f(x) = 2x4 −18x2 + 8x + 12, x ∈(−4, 4). If M is local maximum value of the function f in (−4, 4), then M = (1) 12√6 −332 (2) 12√6 −312 (3) 18√6 −332 (4) 18√6 −312

Q80.Let a die be rolled n times. Let the probability of getting odd numbers seven times be equal to the probability 𝑘 of getting odd numbers nine times. If the probability of getting even numbers twice is 215, then 𝑘 is equal to (1) 60 (2) 15 (3) 90 (4) 30

Q80.If the equation of the normal to the curve y = (x+b)(x−2)x−a at the point (1, −3) is x −4y = 13 then the value of a + b is equal to ______

Q80.The integral 16 ∫21 x3(x2+2)2dx is equal to JEE Main 2023 (25 Jan Shift 2) JEE Main Previous Year Paper (1) 11 6 + loge 4 (2) 1211 + loge 4 (3) 12 11 −loge 4 (4) 116 −loge 4 m and n are coprime natural numbers, then m2 + n2 −5 is equal to

Q80.Let 𝛺 be the sample space and 𝐴⊆𝛺 be an event. Given below are two statements: (S1): If 𝑃( 𝐴) = 0, then 𝐴= 𝜙 (S2): If 𝑃( 𝐴) = , then 𝐴= 𝛺 Then (1) only (S1) is true (2) only (S2) is true (3) both (S1) and (S2) are true (4) both (S1) and (S2) are false

Q80.A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is (1) 5 (2) 2 7 7 3 5 (3) (4) 7 6

Q80.If ∫√sec 2x −1dx = α loge cos 2x + β + √cos 2x(1 ______.

Q80.If f(x) = x3 −x2f ′(1) + xf ′′(2) −f ′′′(3), x ∈R, then (1) 3f(1) + f(2) = f(3) (2) f(3) −f(2) = f(1) (3) 2f(0) −f(1) + f(3) = f(2) (4) f(1) + f(2) + f(3) = f(0) Q81. 3√34 48 ∫ 3√2 dx is equal to 4 √9−4x2 JEE Main 2023 (24 Jan Shift 2) JEE Main Previous Year Paper (1) π (2) π 3 2 (3) π (4) 2π 6 such that f(x) > 0 and