Practice Questions

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.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 ___________.

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

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.Let 𝑓𝑥= ∑𝑘=10 1 𝑘· 𝑥𝑘, 𝑥∈ℝ, if 2𝑓2 + 𝑓'2 = 1192𝑛+ 1 then 𝑛 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.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 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.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.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.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 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.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.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.The remainder, when 7103 is divided by 17, is

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

Q84.The number of integral terms in the expansion of 3 2 + 5

Q84.Let 𝑆 be the set of values of λ, for which the system of equations 6𝜆𝑥- 3𝑦+ 3𝑧= 4𝜆2, 2𝑥+ 6𝜆𝑦+ 4𝑧= 1 and 3𝑥+ 2𝑦+ 3𝜆𝑧= 𝜆 has no solution. Then,12 ∑𝜆∈𝑆𝜆 is equal to _______. 2𝑥

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 −−−