Practice Questions

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.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 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 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 : 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.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 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 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

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 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

Q80.A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at 𝑘 is equal to least 4 successes is 311,𝑘 then (1) 82 (2) 75 (3) 164 (4) 123

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 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.A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is (1) 1 (2) 11 4 50 (3) 1 (4) 9 5 50

Q80.Let 𝑁 denote the sum of the numbers obtained when two dice are rolled. If the probability that 2𝑁< 𝑁! is 𝑛 where 𝑚 and 𝑛 are coprime, then 4𝑚- 3𝑛 is equal to (1) 6 (2) 12 (3) 10 (4) 8

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.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.The absolute minimum value, of the function f(x) = x2 −x + 1 + [x2 −x + 1], where [t] denotes the greatest integer function, in the interval [−1, 2], is (1) 3 (2) 1 2 4 (3) 5 (4) 3 4 4 dx = 16+20√215 then α is equal to :

Q80.Let 𝑆= 𝑀= 𝑎𝑖𝑗, 𝑎𝑖𝑗∈0, 1, 2, 1 ≤𝑖, 𝑗≤2 be a sample space and 𝐴𝑀∈𝑆: 𝑀 is invertible be an even. Then 𝑃𝐴 is equal to 16 47 (1) (2) 27 81 49 50 (3) (4) 81 81 + 𝑎17 + 𝑏17 is equal to

Q80.Let f and g be two functions defined by f(x) = {x|x+−1|,1, xx≥0< 0 {x1, + 1, xx≥0< 0 (gof)(x) is (1) Continuous everywhere but not differentiable (2) Continuous everywhere but not differentiable at exactly at one point x = 1 (3) Differentiable everywhere (4) Not continuous at x = 1

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.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 f(x) = x + a sin x + b cos x, x ∈R be a function which satisfies π2−4 π2−4 f(x) = x + ∫π/20 sin(x + y)f(y)dy. Then (a + b) is equal to (1) −π(π + 2) (2) −2π(π + 2) (3) −2π(π −2) (4) −π(π −2)

Q80.The sum of the abosolute maximum and minimum values of the function f(x) = x2 −5x + 6 −3x + 2 in the interval [−1, 3] is equal to : (1) 10 (2) 12 (3) 13 (4) 24 π 4 x+ π4 dx is :