Practice Questions

Q86.Let 𝐴= 1, 2, 3, 4, 5, 6, 7 and 𝐵= 3, 6, 7, 9. Then the number of elements in the set 𝐶⊆𝐴: 𝐶∩𝐵≠𝜙 is ______

Q86.Let the abscissae of the two points 𝑃 and 𝑄 be the roots of 2𝑥2 - 𝑟𝑥+ 𝑝= 0 and the ordinates of 𝑃 and 𝑄 be the roots of 𝑥2 - 𝑠𝑥- 𝑞= 0. If the equation of the circle described on 𝑃𝑄 as diameter is 2𝑥2 + 𝑦2 - 11𝑥- 14𝑦- 22 = 0, then 2𝑟+ 𝑠- 2𝑞+ 𝑝 is equal to ______.

Q86.If f(θ) = sin θ + ∫ −π2 2 (sin θ + t cos θ) ⋅f(t)dt, then ∫ 0 2 f(θ)dθ is 9−x2

Q86.For the curve C : (x2 + y2 −3) + (x2 −y2 −1) 5 = 0 , the value of 3y′ −y3y′′ , at the point (α, α), α > 0 , on C , is equal to ________.

Q86.Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S →S as f(n) = { 2n2n,−11 ifif nn == 1,6, 2,7, 3,8, 4,9, 510 + 1 , if n is odd Let g : S ≥S be a function such that fog(n) = , then {nn −1 , if n is even g(10)(g(1) + g(2) + g(3) + g(4) + g(5)) is equal to

Q86.The number of distinct real roots of the equation x5(x3 −x2 −x + 1) + x(3x3 −4x2 −2x + 4) −1 = 0 is

Q87.Let f(x) = max{|x + 1|, |x + 2|, … , |x + 5|} . Then ∫0−6 f(x)dx is equal to ______.

Q87.Let the mean and the variance of 20 observations x1, x2, … x20 be 15 and 9, respectively. For α ∈R, if the mean of (x1 + α)2, (x2 + α)2, … , (x20 + α)2 is 178, then the square of the maximum value of α is equal to JEE Main 2022 (29 Jul Shift 1) JEE Main Previous Year Paper ______.

Q87.Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve 4x3 −3xy2 + 6x2 −5xy −8y2 + 9x + 14 = 0 at the point (−2, 3) be A . Then 8A is equal to _______.

Q87.Let R1 and R2 be relations on the set {1, 2, … , 50} such that R1 ={ (p, pn) : p is a prime and n ≥0 is an integer} and R2 ={ (p, pn) : p is a prime and n = 0 or 1 }. Then, the number of elements in R1 −R2 is ____.

Q87.Let f : R →R be a function defined f(x) = e2x+e2e2x . Then f( 1001 ) + f( 1002 ) + f( 1003 ) + … + f( 10099 ) is equal to ______.

Q87.A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is tan−1 34 . Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is _______.

Q87.Let A = (1−i+ i 10 ) {n ∈{1, 2, … . , 100} : An = A} is

Q87.The number of matrices 𝐴= 𝑎 𝑏 where 𝑎, 𝑏, 𝑐, d ∈-1, 0, 1, 2, 3, … … , 10, such that 𝐴= 𝐴-1, is ______. 𝑐 𝑑,

Q87.Let c, k ∈R. If f(x) = (c + 1)x2 + (1 −c2)x + 2k and f(x + y) = f(x) + f(y) −xy, for all x, y ∈R, then the value of |2(f(1) + f(2) + f(3) + … … + f(20))| is equal to ______. √2y π dy + = xetan−1(√2 cot 2x), 0 < x <

Q87.If y(x) = (xx)x, x > 0 then d2x + 20 at x = 1 is equal to dy2 2 2 + y 3 ≤1, x + y ≥0, y y) : x 3 is A , then 256Aπ is ≥0}

Q88.For real numbers a, b(a > b > 0), let x2 y2 = 30π Area {(x, y) : x2 + y2 ≤a2 and a2 + b2 ≥1} and x2 y2 = 18π Area {(x, y) : x2 + y2 ≥b2 and a2 + b2 ≤1} Then the value of (a −b)2 is equal to _____.

Q88.If n(2n + 1) ∫10 (1 −xn)2ndx = 1177 ∫10 (1 −xn)2n+1dx, then JEE Main 2022 (26 Jul Shift 1) JEE Main Previous Year Paper

Q88.If the tangent to the curve 𝑦= 𝑥3 - 𝑥2 + 𝑥 at the point 𝑎, 𝑏 is also tangent to the curve 𝑦= 5𝑥2 + 2x - 25 at the point 2, - 1, then 2𝑎+ 9𝑏 is equal to ______. 2 2 2 2 2

Q88.Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1 and R2 . If max|R1, R2| = R2 , then R1R2 is equal to ______

Q88.If the system of linear equations 2x −3y = γ + 5 αx + 5y = β + 1 , where α, β, γ ∈R has infinitely many solutions, then the value of |9α + 3β + 5γ| is equal to

Q88.Let M and N be the number of points on the curve y5 −9xy + 2x = 0 , where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals _______.

Q88.The value of 𝑏> 3 for which 12 𝑏 1 49 is equal to _____. ∫3 𝑥2 - 1𝑥2 - 4𝑑𝑥= log𝑒 40,

Q88.Let the tangents at the points P and Q on the ellipse x2 S is 2 + 4 = 1 meet at the point R(√2, 2√2 −2). If the focus of the ellipse on its negative major axis, then SP 2 + SQ2 is equal to π dx is equal to

Q88.If the area of the region {(x,