Practice Questions

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

Q87.Let f and g be twice differentiable even functions on (−2, 2) such that f( 41 ) = 0, f( 21 ) = 0, f(1) = 1 and g( 34 ) = 0, g(1) = 2 Then, the minimum number of solutions of f(x)g′′(x) + f ′(x)g′′(x) = 0 in (−2, 2) is equal to _____.

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

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 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.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.Let Max Min Max , = α1 + α2 loge( 158 ), then { 9−x25−x } 5−x } { 9−x25−x x}dx = β. If ∫2α−1β−83 0⩽x⩽2 = α and 0⩽x⩽2{ α1 + α2 is equal to ______

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.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 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 𝐴 be a 3 × 3 matrix having entries from the set -1, 0, 1. The number of all such matrices 𝐴 having sum of all the entries equal to 5, is _____ Q88. 1 𝑥25 Let 𝑓: 𝑅→𝑅 be a function defined by 𝑓𝑥= 21 - 2 + 𝑥25 50. If the function 𝑔𝑥= 𝑓𝑓𝑓𝑥+ 𝑓𝑓𝑥, then the 2 greatest integer less than or equal to 𝑔1 is ______.

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

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