Practice Questions

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.The coefficient of 𝑥18 in the expansion of 𝑥4 - is ____________ 𝑥3

Q82.Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5 , and are divisible by 15 , is equal to Q83. ∞ 𝑛3 ( (2𝑛)!) + (2𝑛- 1) (𝑛!) 𝑏 ∞ 1 ∑𝑛= 0 ( 𝑛! ) ( ( 2𝑛) ! ) = 𝑎𝑒+ 𝑒+ 𝑐 where 𝑎, 𝑏, 𝑐 ∈ℤ and 𝑒= ∑𝑛= 0 𝑛! Then 𝑎2 - 𝑏+ 𝑐 is equal to _______ JEE Main 2023 (30 Jan Shift 1) JEE Main Previous Year Paper

Q82.Let 𝑎1, 𝑎2, … … , 𝑎𝑛 be in A.P. If 𝑎5 = 2𝑎7 and 𝑎11 = 18, then 12 + + + + … . . + √𝑎10 √𝑎11 √𝑎11 √𝑎12 √𝑎17 √𝑎18 is equal to _____ .

Q82.A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

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.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.Suppose 𝑎1, 𝑎2, 2, 𝑎3, 𝑎4 be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is 49 , then 𝑎4 is 2 equal to ______________

Q83.Let A be the area of the region {(x, y) : y ≥x2, y ≥(1 −x)2, y ≤2x(1 −x)}. Then 540A is equal to y(1) = 0 is

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

Q83.Let the area enclosed by the lines x + y = 2, y = 0 , x = 0 and the curve f(x) = min{x2 + 43 , 1 + [x]} where [x] denotes the greatest integer ≤x, be A . Then the value of 12A 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 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.The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ______ 1

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.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 _____ .

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.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.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.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.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.If the area of the region bounded by the curves y2 −2y = −x and x + y = 0 is A , then 8A =

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.Let Δ be the area of the region {(x, y) ∈R2 : x2 + y2 ≤21, y2 ≤4x, x ≥1}. Then 21 (Δ √7 equal to (1) 2√3 −13 (2) √3 −23 (3) 2√3 −23 (4) √3 −43

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 ___________.