Practice Questions

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 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.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.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

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

Q80.In a binomial distribution B ( 𝑛, 𝑝) , the sum and product of the mean & variance are 5 and 6 respectively, then find 6 ( 𝑛+ 𝑝- 𝑞) is equal to :- (1) 51 (2) 52 (3) 53 (4) 50

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

Q80.Let I(x) = ∫√x+7x dx and I(9) = 12 + 7 loge 7. If I(1) = α + 7 loge(1 2√2), then α4 is equal to _____. dx = 3000k , then k is equal to _____.

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 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.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 k and m be positive real numbers such that the function f(x) = {3x2mx2+ k√x+ k2,+ 1, 0 <x ≥1x < 1 8f ′(8) is differentiable for all x > 0 . Then 1 is equal to f ′( 8 ) x dx is equal to

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 :

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

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.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.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

Q81.Among (S1) : lim 1 + 4 + 6 + … + = 1 n→∞ n2 (2 2n) JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper (S2) : lim 1 (115 + 215 + 315 + … + n15) = 161 n16 n→∞ (1) Both (S1) and (S2) are true (2) Only (S1) is true (3) Both (S1) and (S2) are false (4) Only (S2) is true

Q81.Let 𝜆∈ℝ and let the equation 𝐸 be |𝑥| 2 - 2 | 𝑥| + | 𝜆- 3 | = 0. Then the largest element in the set 𝑆= {𝑥+ 𝜆: 𝑥 is an integer solution of 𝐸} is ______