Practice Questions

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

Q83.If the area of the region bounded by the curves y2 −2y = −x and x + y = 0 is A , then 8A =

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

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.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.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 _____ .

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 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.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

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 ________

Q84.Let y = y1(x) and y = y2(x) be the solution curves the differential equation dxdy = y + 7 with initial conditions y1(0) = 0 and y2(0) = 1 respectively. Then the curves y = y1(x) and y = y2(x) intersect at (1) no point (2) two points (3) one point (4) infinite number of points → → → → → →

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.If the solution curve f(x, y) = 0 of the differential equation (1 + loge x) dxdy −x loge x = ey, x > 0, passes through the points (1, 0) and (a, 2), then aa is equal to (1) e2e2 (2) ee2 (3) e√2e2 (4) e2e√2 →

Q84.Let y = y(x) be the solution curve of the differential equation dxdy = xy (1 + x2(1 + loge x)), x > 0, y(1) = 3. y2(x) Then is equal to : 9 (1) x2 (2) x2 5−2x3(2+loge x3) 2x3(2+loge x3)−3 (3) x2 (4) x2 3x3(1+loge x2)−2 7−3x3(2+loge x2) JEE Main 2023 (25 Jan Shift 1) JEE Main Previous Year Paper be a vector such that = 2 . If →d

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.The number of integral terms in the expansion of 3 2 + 5

Q84.If the solution curve of the differential equation (y −2 loge x)dx + (x loge x2)dy = 0, x > 1 passes through the points (e, 34 ) and (e4, α) , then α is equal to _______

Q84.Let y = y(x) be the solution of the differential equation (x2– 3y2)dx + 3 xy dy = 0, y(1) = 1 . Then 6y2(e) is equal to (1) 3e2 (2) e2 (3) 2e2 (4) 3e22 → → → → → →