Practice Questions

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 :

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.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.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.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 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 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 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.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.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

Q81.If ∫0.15−0.15 100x2 −1

Q81.Let [x] denote the greatest integer ≤x. Consider the function f(x) = max{x2, 1 + [x]}. Then the value of the integral ∫20 f(x)dx is : (1) 5+4√2 (2) 8+4√2 3 3 (3) 1+5√2 (4) 4+5√2 3 3 and y) ∈R2 : y ≥0, 2x ≤y ≤√4 −(x −1)2}

Q81.Let f(x) be a function satisfying f(x) + f(π −x) = π2, ∀x ∈R. Then ∫π0 f(x) sin (1) π2 (2) 2π2 4 (3) π2 (4) π2 2

Q81.Let f(x) = ∫ (x2+1)(x2+3)2x dx. If f(3) = 21 (loge 5 −loge 6), then f(4) is equal to (1) 1 2 (loge 17 −logc 19) (2) loge 17 −loge 18 (3) 1 2 (logc 19 −logc 17) (4) logc 19 −logc 20

Q81.If ∫3 m n2 1 |loge x|dx = n loge( e ), where 3 _____ .

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.Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________

Q81.Let α > 0 . If ∫α0 √x+α−√xx (1) 2 (2) 2√2 (3) 4 (4) √2 = sin t ∫xπ x > 0 then ϕ′( 4 ) is equal to √x

Q81.Let I(x) = ∫ x+1 dx, x > 0. If lim = 0 then I(1) is equal to x(1+xex)2 x→∞I(x) (1) e+1 e+2 −loge(e + 1) (2) e+1e+2 + loge(e + 1) (3) e+2 e+1 −loge(e + 1) (4) e+2e+1 + loge(e + 1) 6 (8[cosec x] −5[cot x])dx is equal to _______ 2 ∫ π

Q81. lim n3 {4 + (2 + n1 )2 + (2 + n2 )2 + … + (3 −1n )2} is equal to n→∞ (1) 12 (2) 193 (3) 0 (4) 19 JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper

Q81.The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is _____ 1 15

Q81.Let the function f : [0, 2] →R be defined as f(x) = {emin{x2,x−[x]},e[x−loge x], xx ∈[0,∈[1, 1)2] , where [t] denotes the greatest integer less than or equal to t. Then the value of the integral ∫20 xf(x)dx is (1) 1 + 3e2 (2) (e −1)(e2 + 12 ) (3) 2e −1 (4) 2e −12

Q81.The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to ____.

Q81.The value of the integral ∫21 ( t4+1t6+1 )dt is : (1) tan−1 12 + 31 tan−1 8 −π3 (2) tan−1 2 −13 tan−1 8 + π3 (3) tan−1 2 + 13 tan−1 8 −π3 (4) tan−1 21 −13 tan−1 8 + π3 dx is equal to