Practice Questions

Q82.Let f(x) = x , x ∈R −{−1}, n ∈N, n > 2 . If f n(x) = (fofof. . . . upto n times) (x), then (1+xn) 1n lim 0 xn−2(f n(x))dx is equal to n→∞∫1

Q82.Let [t] denote the greatest integer ≤t. Then π 5π 6

Q82.Let [t] denote the greatest integer function. If α + β√2 + γ√3 + δ√5, then α + β + γ + δ is equal to ∫2.40 [x2]dx =

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.Let f be a differentiable function defined on [0, π2 ] 2 e ∀x f(x) + ∫x0 f(t)√1 −(loge(f(t)))2dt = ∈[0, π2 ], then {6 loge(f( π6 ))} 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.The number of permutations, of the digits 1, 2, 3, … , 7 without repetition, which neither contain the string 153 nor the string 2467, 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

Q83.The area of the region A = {(x, y) : |cos x −sin x| ≤y ≤sin x, 0 ≤x ≤π2 } (1) 1 − 3 + 4 (2) √5 + 2√2 −4. 5 √2 √5 (3) 3 − 3 + 1 (4) √5 −2√2 + 1 √5 √2 > y(2) = 2,

Q83.Let y = y(x), y > 0, be a solution curve of the differential equation (1 + x2)dy = y(x −y)dx. If y(0) = 1 = β, then and y(2√2) = + + 2√2) (2) e3β−1 e(5 √2) (1) e3β−1 = e(3 = + + 2√2) (4) eβ−1 e−2(5 √2) (3) eβ−1 = e−2(3

Q83.The area of the region given by {(x, y) : xy ≤8, 1 ≤y ≤x2} is : (1) 8 loge 2 −133 (2) 16 loge 2 −143 (3) 8 loge 2 + 76 (4) 16 loge 2 + 37

Q83.The area of the region {(x, y) : x2 ≤y ≤8 −x2, y ≤7} is (1) 27 (2) 18 (3) 20 (4) 21

Q83.A circle passing through the point 𝑃𝛼, 𝛽 in the first quadrant touches the two coordinate axes at the points 𝐴 and 𝐵. The point 𝑃 is above the line 𝐴𝐵. The point 𝑄 on the line segment 𝐴𝐵 is the foot of perpendicular from 𝑃 on 𝐴𝐵. If 𝑃𝑄 is equal to 11 units, then the value of 𝛼𝛽 is _______

Q83.Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is ________.

Q83.Let the equations of two adjacent sides of a parallelogram 𝐴𝐵𝐶𝐷 be 2𝑥- 3𝑦= - 23 and 5𝑥+ 4𝑦= 23. If the equation of its one diagonal 𝐴𝐶 is 3𝑥+ 7𝑦= 23 and the distance of 𝐴 from the other diagonal is 𝑑, then 50𝑑2 is equal to ______________

Q83.Let 𝑓𝑥= ∑𝑘=10 1 𝑘· 𝑥𝑘, 𝑥∈ℝ, if 2𝑓2 + 𝑓'2 = 1192𝑛+ 1 then 𝑛 is equal to ______.

Q83.Let y = y(t) be a solution of the differential equation dydt + αy = γe−βt Where, α > 0, β > 0 and γ > 0 . Then Limt→∞ y(t) (1) is 0 (2) does not exist (3) is 1 (4) is −1

Q83.If A is the area in the first quadrant enclosed by the curve C : 2x2 −y + 1 = 0 , the tangent to C at the point (1, 3) and the line x + y = 1 , then the value of 60A is................ (x5+1)2 y = , x > 0 . If y(1) = 2, then x7

Q83.Consider the triangles with vertices A(2, 1), B(0, 0) and C(t, 4), t = [0, 4]. If the maximum and the minimum perimeters of such triangles are obtained at t = α and t = β respectively, then 6α + 21β is equal to ___________.

Q83.Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to _____ . JEE Main 2023 (31 Jan Shift 1) JEE Main Previous Year Paper 2 30

Q83.The area bounded by the curves y = |x −1| + |x −2| and y = 3 is equal to (1) 4 (2) 6 (3) 3 (4) 5

Q83.The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is _____ .

Q83.The area of the region enclosed by the curve f(x) = max{sin x, cos x}, −π ≤x ≤π and the x−axis is + (1) 2√2(√2 1) (2) 4 + 1) (3) 4(√2) (4) 2(√2

Q83.Let the area of the region {(x, y) : |2x −1| ≤y ≤x2 −x , 0 ≤x ≤1} be A . Then (6A + 11)2 is equal to _____ .

Q83.If the area enclosed by the parabolas P1 : 2y = 5x2 and P2 : x2 −y + 6 = 0 is equal to the area enclosed by P1 and y = αx, α > 0, then α3 is equal to _____ .