Practice Questions

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

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 mirror image of a circle c1 : x2 + y2 −2x −6y + α = 0 in line y = x + 1 be c2 : 5x2 + 5y2 + 10gx +10fy + 38 = 0. If r is the radius of circle c2 , then α + 6r2 is equal to ______

Q87.Let 𝑓𝑥= 𝑥- 1𝑥2 - 2𝑥- 3 + 𝑥- 3, 𝑥∈ℝ. If 𝑚 and 𝑀 are respectively the number of points of local minimum and local maximum of 𝑓 in the interval 0, 4, then 𝑚+ 𝑀 is equal to _____.

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

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.For k ∈R, let the solutions of the equation cos(sin−1(x cot(tan−1(cos(sin−1 x))))) = k, 0 < |x| < 1 be α √2 and β, where the inverse trigonometric functions take only principal values. If the solutions of the equation 1 and α , then b is equal to ______. x2 −bx −5 = 0 are 1 + β α2 β2 k2

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.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.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 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 ______.

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 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.Let A = (1−i+ i 10 ) {n ∈{1, 2, … . , 100} : An = A} 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.Let 𝐴= 1 -1 and 𝐵= 𝛽1 , 𝛼, 𝛽∈𝑅. Let 𝛼1 be the value of 𝛼 which satisfies 𝐴+ 𝐵2 = 𝐴2 + 2 2 and 2 𝛼 1 0 2 2 𝛼2 be the value of 𝛼 which satisfies 𝐴+ 𝐵2 = 𝐵2. Then 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.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 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.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 the function f(x) = 2x2 −loge x, x > 0, be decreasing in (0, a) and increasing in (a, 4). A tangent to the parabola y2 = 4ax at a point P on it passes through the point (8a, 8a −1) but does not pass through the point (−1a , 0). If the equation of the normal at P is αx + βy = 1 , then α + β is equal to n ∈N is equal to _______.

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.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.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 area of the region {(x,