Practice Questions

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.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 ______

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 𝑧= 1 + 𝑖 and 𝑧1 = 1 · Then 𝜋 arg𝑧1 is equal to ¯𝑧(1 - 𝑧) + 𝑧

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.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 value of the integral ∫ −π4 2−cos 2x (1) π2 (2) π2 6 12√3 (3) π2 (4) π2 3√3 6√3 kπ , then k is equal to _____ . 16

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

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 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 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is _____ . 1 1 1

Q81.The integral ∫(( x2 ) x + ( x2 ) x) log2 C (1) ( x2 ) x + ( x2 ) x + C (2) ( x2 ) x −( x2 ) x + C (3) ( x2 ) x log2( x2 ) + C (4) ( x2 ) x log2( x2 ) +

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

Q82.The area of the region enclosed by the curve y = x3 and its tangent at the point (–1, –1) is (1) 19 (2) 23 4 4 (3) 31 (4) 27 4 4

Q82.Let T and C respectively, be the transverse and conjugate axes of the hyperbola 16x2 −y2 + 64x + 4y + 44 = 0 . Then the area of the region above the parabola x2 = y + 4 , below the transverse axis T and on the right of the conjugate axis C is: (1) 4√6 + 443 (2) 4√6 + 283 (3) 4√6 −443 (4) 4√6 −283

Q82.If ϕ(x) 1 π −3ϕ′(t))dt, 4 (4√2 (1) 4 (2) 8 6+√π 6+√π (3) 8 (4) 4 √π 6−√π

Q82.The value of the integral ∫21/2 tan−1x x (1) π loge 2 (2) 21 loge 2 (3) π 4 loge 2 (4) π2 loge 2

Q82.Let 𝛼 denote the greatest integer ≤𝛼. Then √1 + √2 + √3 + . . . . . . . . . . . . . + √120 is equal to

Q82.The minimum value of the function f(x) = ∫20 e|x−t|dt is (1) 2(e −1) (2) 2e −1 (3) 2 (4) e(e −1)

Q82.Let A = {(x, . Then the ratio of the area of A to the area of B −(x B = y) ∈R × R : 0 ≤y −1)2}} {(x, ≤min{2x, √4 is (1) π−1 (2) π π+1 π−1 (3) π (4) π+1 π+1 π−1 −21 sin−1 2 ) is

Q82.If f : R →R be a continuous function satisfying ∫ 0π π 2 f(sin 2x) sin x dx + α ∫ 04 f(cos 2x) cos x dx = 0 , then the value of α is JEE Main 2023 (11 Apr Shift 2) JEE Main Previous Year Paper (1) √2 (2) −√3 (3) √3 (4) −√2

Q82.Let q be the maximum integral value of p in [0, 10] for which the roots of the equation x2 −px + 45 p = 0 are rational. Then the area of the region {(x, y) : 0 ≤y ≤(x −q)2, 0 ≤x ≤q} is (1) 243 (2) 25 (3) 125 (4) 164 3