Practice Questions

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 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.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.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 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 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}}

Q88.Let f : R →R satisfy f(x + y) = 2xf(y) + 4y(f(x), ∀x, y ∈R. If f(2) = 3 , then 14 ⋅ff ′(4)′(2) (2−x2) dx √2

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.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 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.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 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.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.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 ______.

Q89.Let p and p + 2 be prime numbers and let p! (p + 1)! (p + 2)! Δ = (p + 1)! (p + 2)! (p + 3)! (p + 2)! (p + 3)! (p + 4)! Then the sum of the maximum values of α and β , such that pα and (p + 2)β divide Δ , is _______.

Q89.Let y = y(x) be the solution of the differential equation −1 < x < 1 (1 −x2)dy = (xy + (x3 + 2)√1 −x2)dx, 1 and y(0) = 0. If ∫ 2 √1 −x2y(x)dx = k then k−1 is equal to −12

Q89.Let f be a differentiable function satisfying f(x) = 2 ∫√30 f( λ2x3 )dλ, √3 passes through the point (α, 6), then α is equal to _______. → → → →

Q89.Let y = y(x) be the solution curve of the differential equation = 0, 0 < x < √π2 sin(2x2) loge(tan x2)dy + (4xy −4√2x sin(x2 −π4 ))dx , which passes through the point (√π6 , 1). Then y(√π3 ) is equal to _______. y−2

Q89.The largest value of 𝑎, for which the perpendicular distance of the plane containing the lines →𝑟= ^𝑖+ ^𝑗+ 𝜆 ^𝑖+ 𝑎 ^𝑗- ^𝑘and →𝑟= ^𝑖+ ^𝑗+ 𝜇- ^𝑖+ ^𝑗- 𝑎𝑘 from the point 2, 1, 4 is √3, is ______.

Q89.Let →𝑏= ^𝑖+ ^𝑗+ 𝜆 ^𝑘, 𝜆∈ℝ. If →𝑎 is a vector such that →𝑎× →𝑏= 13 ^𝑖- ^𝑗- 4 ^𝑘 and →𝑎· →𝑏+ 21 = 0, then →𝑏- →𝑎· ^𝑘- ^𝑗+ →𝑏+ →𝑎· ^𝑖- ^𝑘 is equal to 1 1

Q89.Let S = {1, 2, 3, 4} . Then the number of elements in the set { f : S × S →S : f is onto and f(a, b) = f(b, a) ≥a∀(a, b) ∈S × S } is

Q90.If the probability that a randomly chosen 6 -digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96p is equal to _____. JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper