Practice Questions

Q82.Let for x ∈R, S0(x) = x, Sk(x) = Ckx + k ∫x0 Sk−1(t)dt where k = 1, 2, 3, … Then S2(3) + 6C3 is equal to _______. C0 = 1, Ck = 1 −∫10 Sk−1(x)dx,

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 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 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.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.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 _______

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 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 7103 is divided by 17, is

Q85.Suppose ∑𝑟=20230 𝑟2 · 2023𝐶𝑟= 2023 × 𝛼× 22022, then the value of 𝛼 is

Q85.Let 𝑆= {1, 2, 3, 4, 5, 6}. Then the number of oneone functions 𝑓: 𝑆→𝑃( 𝑆) , where 𝑃( 𝑆) denote the power set of 𝑆, such that 𝑓( 𝑛) ⊂𝑓( 𝑚) where 𝑛< 𝑚 is

Q85.The foci of a hyperbola are ( ± 2, 0 ) and its eccentricity is 32. A tangent, perpendicular to the line 2𝑥+ 3𝑦= 6, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the 𝑥- and 𝑦-axes are 𝑎 and 𝑏 respectively, then |6𝑎| + | 5𝑏| is equal to

Q86.Let 𝑎∈ℤ and 𝑡 be the greatest integer ≤𝑡, then the number of points, where the function 𝑓𝑥= 𝑎+ 13 sin𝑥, 𝑥∈0, 𝜋 is not differentiable, is ____________

Q86.Let a common tangent to the curves 𝑦2 = 4𝑥 and 𝑥- 42 + 𝑦2 = 16 touch the curves at the points 𝑃 and 𝑄. Then 𝑃𝑄2 is equal to ________.

Q86.Let →a = ˆi + 2ˆj + 3ˆk and b = ˆi + ˆj −ˆk. If →cis a vector such that →a⋅→c= 11, b ⋅(→a×→c) 2 is equal to −√3→b , then →a×→c

Q86.In the figure, θ1 + θ2 = π2 and √3BE θ1 then the perimeter (in unit) of ∆CED is equal to

Q86.If 1 1 / 1 𝑛 𝑥21 + 𝑥14 + 𝑥72𝑥14 + 3𝑥7 + 6 7𝑑𝑥= where 𝑙, 𝑚, 𝑛∈𝑁, 𝑚 and 𝑛 are co-prime then 𝑙+ 𝑚+ 𝑛 ∫0 𝑙11𝑚/ is equal to _____ .

Q87.Let 𝐶 be the largest circle centred at 2, 0 and inscribed in the ellipse 𝑥2 + 𝑦2 = 1. If 1, 𝛼 lies on 𝐶, then 10𝛼2 is 36 16 equal to ______ 𝜋 2 cos𝑥2023

Q87.Let f(x) = ∫ 2 . If f(0) = 0 and f(1) = αβ1 tan−1( αβ ), √3 (3+4x2)√4−3x2 equal to _______.

Q87.Let the quadratic curve passing through the point -1, 0 and touching the line 𝑦= 𝑥 at 1, 1 be 𝑦= 𝑓𝑥. Then the 𝑥-intercept of the normal to the curve at the point 𝛼, 𝛼+ 1 in the first quadrant is

Q88.Let the co-ordinates of one vertex of ΔABC be A(0, 2, α) and the other two vertices lie on the line x+α 5 = y−12 = z+43 . For α ∈Z , if the area of ΔABC is 21 sq. units and the line segment BC has length 2√21 units, then α2 is equal to _______.

Q88.If the area of the region 𝑆= ( 𝑥, 𝑦) : 2𝑦- 𝑦2 ≤𝑥2 ≤2𝑦, 𝑥≥𝑦 is equal to 𝑛+ 2 - 𝜋 then the natural number 𝑛+ 1 𝑛- 1, 𝑛 is equal to _______ JEE Main 2023 (06 Apr Shift 1) JEE Main Previous Year Paper

Q88.Let 𝑓: ℝ→ℝ be a differentiable function such that 𝑓'𝑥+ 𝑓𝑥= ∫0 𝑓𝑡𝑑𝑡. If 𝑓0 = 𝑒-2, then 2𝑓0 - 𝑓2 is equal to _____ .

Q89.Let P1 be the plane 3x −y −7z = 11 and P2 be the plane passing through the points (2, −1, 0), (2, 0, −1), and (5, 1, 1). If the foot of the perpendicular drawn from the point (7, 4, −1) on the line of intersection of the JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper planes P1 and P2 is (α, β, γ), then α + β + γ is equal to

Q89.Let 𝑦= 𝑦𝑥 be a solution of the differential equation 𝜋 𝜋 𝜋 𝜋 𝜋 is equal to (𝑥 cos𝑥)𝑑𝑦+ (𝑥𝑦 sin𝑥+ 𝑦 cos𝑥- 1)𝑑𝑥= 0, 0 < 𝑥< 2 . If 3𝑦 3 = √3, then 6𝑦" 6 + 2𝑦'𝜋6 _______ .