Practice Questions

Q87.Two tangent lines l1 and l2 are drawn from the point (2, 0) to the parabola 2y2 = −x. If the lines l1 and l2 are also tangent to the circle (x −5)2 + y2 = r, then 17r2 is equal to y2

Q87.The sum of all the elements of the set {α ∈{1, 2, … . . 100} : HCF(α, 24) = 1} is a, b ∈{1, 2, 3, … and let Tn = {A ∈S : An(n+1) = I} . Then the number of 100}}

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}

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.If 𝑡 denotes the greatest integer ≤𝑡, then number of points, at which the function 𝑓𝑥= 42𝑥+ 3 + 1 9𝑥+ - 12𝑥+ 20 is not differentiable in the open interval -20, 20, is ______. 2

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 ____.

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

Q88.The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______.

Q88.The value of the integral dx is equal to ______. π4 48 ∫π0 ( 3πx22 −x3) 1+cos2sin x x

Q88.Let A = {1, a1, a2 … … a18, 77} be a set of integers with 1 < a1 < a2 < … . . < a18 < 77. Let the set A + A = {x + y : x, y ∈A} contain exactly 39 elements. Then, the value of a1 + a2 + … . . +a18 is equal to ______.

Q88.Let S = {(−10 ab ); 100 elements in n=1Tn∩ is _____.

Q88.Let f(x) = min{[x −1], [x −2], … , [x −10]} where [t] denotes the greatest integer ≤t. Then ∫100 f(x)dx + ∫100 (f(x))2dx + ∫100 |f(x)|dx is equal _______. to x > 0 and f(1) = √3 . If y = f(x)

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.Let f be a twice differentiable function on R. If f ′(0) = 4 and f(x) + ∫x0 (x −t)f ′(t)dt = (e2x + e−2x) cos 2x + a2 x, then (2a + 1)5a2 is equal to _______. n ∈N . Then the sum of all the elements of the set

Q88.Let 𝑓: 0, 1 →𝑅 be a twice differentiable function in 0, 1 such that 𝑓0 = 3 and 𝑓1 = 5. If the line 𝑦= 2𝑥+ 3 intersects the graph of 𝑓 at only two distinct points in 0, 1, then the least number of points 𝑥∈0, 1, at which 𝑓''𝑥= 0, is √3 15𝑥3

Q88.Let 𝑓𝑥= 4𝑥2 - 8𝑥+ 5, if 8𝑥2 - 6𝑥+ 1 ≥0 , where 𝛼 denotes the greatest integer less than or equal to 𝛼. 4𝑥2 - 8𝑥+ 5, if 8𝑥2 - 6𝑥+ 1 < 0 Then the number of points in 𝑅 where 𝑓 is not differentiable is _____ . 1 𝑛+ 1𝑘- 1

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 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.If the sum of all the roots of the equation e2x −11ex −45e−x + 812 = 0 is loge P , then P is equal to _____.

Q88.Suppose 𝑦= 𝑦𝑥 be the solution curve to the differential equation 𝑑𝑦 𝑦= 2 - 𝑒-𝑥 such that lim is finite. 𝑑𝑥- 𝑥→∞𝑦𝑥 If 𝑎 and 𝑏 are respectively the 𝑥- and 𝑦- intercept of the tangent to the curve at 𝑥= 0, then the value of 𝑎- 4𝑏 is equal to _______.

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.Let y = y(x) be the solution of the differential equation dx 2 2 cos4 x−cos 2x with y( π4 ) = π232 . If y( π3 ) = π218 e−tan−1(α) , then the value of 3α2 is equal to ______.

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

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