Practice Questions

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 [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 =

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

Q83.Let 𝑓𝑥= ∑𝑘=10 1 𝑘· 𝑥𝑘, 𝑥∈ℝ, if 2𝑓2 + 𝑓'2 = 1192𝑛+ 1 then 𝑛 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 of the region {(x, y) : x2 ≤y ≤8 −x2, y ≤7} is (1) 27 (2) 18 (3) 20 (4) 21

Q83.Let 𝑆= 109 + 108 + 107 + … . + 2 + 1 Then the value of 16𝑆- ( 25 -54 is equal to 5 52 5107 5108. ) 1 1 680 4 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.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.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.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 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 _____ .

Q84.If the four points, whose position vectors are 3ˆi −4ˆj + 2ˆk,ˆi + 2ˆj −ˆk, −2ˆi −ˆj + 3ˆk and 5ˆi −2αˆj + 4ˆk are coplanar, then α is equal to (1) 7317 (2) −10717 (3) −7317 (4) 10717 → → →

Q84.Let a, b, c be three distinct real numbers, none equal to one. If the vectors aˆi + ˆj + ˆk, ˆi + bˆj + ˆk and ˆi + ˆj + cˆk are coplanar, then 1−a1 + 1−b1 + 1−c1 is equal to (1) 2 (2) −1 (3) −2 (4) 1 →

Q84.Let the solution curve x = x(y), 0 < y < π2 , of the differential equation (loge(cos y))2 cos y dx −(1 + 3x loge(cos y)) sin y dy = 0 satisfy x( π3 ) = 2 loge1 2 . If x( π6 ) = loge m−loge1 n , where m and n are coprime, then mn is equal to −−−

Q84.Let an ellipse with centre (1, 0) and latus rectum of length 21 have its major axis along x-axis. If its minor axis subtends an angle 60∘ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.

Q84.The sum of all those terms, of the arithmetic progression 3, 8, 13, . . . , 373, which are not divisible by 3, is equal to ________. JEE Main 2023 (10 Apr Shift 1) JEE Main Previous Year Paper

Q84.The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted, 𝑎 and 𝑏 are respectively mean and variance of remaining 6 observation, then 𝑎+ 3 𝑏- 5 is equal to ________