Practice Questions

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.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.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.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.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

Q84.Let 𝛼> 0, be the smallest number such that the expansion of 𝑥 3 + 2 has a term 𝛽𝑥-𝛼, 𝛽∈𝑁. Then 𝛼 is 𝑥3 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 the point 𝑝, 𝑝+ 1 lie inside the region 𝐸= 𝑥, 𝑦: 3 - 𝑥≤𝑦≤√9 - 𝑥2 , 0 ≤𝑥≤3 . If the set of all values of 𝑝 is the interval 𝑎, 𝑏, then 𝑏2 + 𝑏- 𝑎2 is equal to ________ .

Q84.The remainder when 19200 + 23200 is divided by 49, is _____ .

Q84.The remainder, when 7103 is divided by 17, is

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 → → → → → →

Q84.Let y = y(x) be the solution of the differential equation (3y2 −5x2)ydx + 2x(x2 −y2)dy = 0 such that y(1) = 1. Then (y(2))3 −12y(2) is equal to : (1) 64 (2) 32√2 (3) 32 (4) 16√2 →

Q84.Let αx = exp(xβyγ) be the solution of the differential equation 2x2ydy −(1 −xy2)dx = 0 , x > 0, y(2) = √loge 2 . Then α + β −γ equals : (1) 1 (2) −1 (3) 0 (4) 3 →

Q84.Let y = y(x) be the solution of the differential equation dxdy + x(x5+1)5 y(2) is equal to (1) 637 (2) 679 128 128 (3) 693 (4) 697 128 128 is equal to

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

Q84.Let y = y(x) be the solution of the differential equation x loge x dxdy + y = x2 loge x, (x 1). If then y(e) is equal to (1) 4+e2 (2) 1+e2 4 4 (3) 2+e2 (4) 1+e2 2 2

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 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.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 y = f(x) be the solution of the differential equation y(x + 1)dx −x2dy = 0, y(1) = e. Then lim x→0+ f(x) is equal to (1) 0 (2) 1e (3) e2 (4) 1 e2 →