Practice Questions

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

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 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,

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

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

Q84.The 4th term of GP is 500 and its common ratio is 𝑚∈𝑁. Let 𝑆𝑛 denote the sum of the first 𝑛 terms of 𝑚, 𝑚 is ______ this GP. If 𝑆6 > 𝑆5 + 1 and 𝑆7 < 𝑆6 + 12, then the number of possible values of

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

Q84.The solution of the differential equation dxdy = −( x2+3y23x2+y2 ), (1) loge|x + y| − xy = 0 (2) loge|x + y| + xy = 0 (x+y)2 (x+y)2 (3) loge|x + y| + (x+y)2 2xy = 0 (4) loge|x + y| − (x+y)22xy = 0 + × × × − = 8ˆi −40ˆj −24ˆk then

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.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.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 _______