Practice Questions

Q87.If y1/4 + y−1/4 = 2x, and (x2 −1) dx2d2y

Q87.Let 𝑓( 𝑥) be a polynomial of degree 3 such that 𝑓𝑘= - for 𝑘= 2, 3, 4, 5 . Then the value of 𝑘 52 - 10 𝑓( 10 ) is equal to _____ .

Q87.The area bounded by the lines y = ||x −1| −2| and y = 2 is _____.

Q87.Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A →A such that f(1) + f(2) = 3 −f(3) is equal to

Q87.Let a curve y = f(x) pass through the point (2, (loge 2)2) and have slope x loge2y x for all positive real values of x. Then the value of f(e) is equal to _____. → → → → is perpendicular to and is perpendicular to + 3 −5 −4

Q87.If the variance of 10 natural numbers 1, 1, 1, … , 1, k is less than 10, then the maximum possible value of k is ___________. 1 ) x is 1

Q87.Let 𝑓𝑥 be a cubic polynomial with 𝑓1 = - 10, 𝑓-1 = 6, and has a local minima at 𝑥= 1, and 𝑓'𝑥 has a local minima at 𝑥= - 1 . Then 𝑓3 is equal to .

Q87.If [⋅] represents the greatest integer function, then the value of ∫ 0√π

Q87.Let f : R →R and g : R →R be defined as f(x) = { |xx +−1|,a, xx <≥00 { (x −1)2x + 1,+ b, xx ≥0< 0 , where a, b are non-negative real numbers. If gof(x) is continuous for all x ∈R, then a + b is equal to ______ .

Q87.If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then d2ydx2 at JEE Main 2021 (26 Aug Shift 1) JEE Main Previous Year Paper

Q87.Let f : (0, 2) →R be defined as f(x) = log2(1 + tan( πx4 )). Then, lim n2 (f( n1 ) + f( n2 ) + … . +f(1)) is equal to ________. n→∞

Q87.If ∫π0 (sin3 x)e−sin2 xdx =

Q87.If f(x) = ∫ dx, (x ≥0), f(0) = 0 and f(1) = K1 , then the value of K is (x2+1+2x7)2

Q87.Let In = ∫e1 x19(log equal to _______.

Q87.Let [t] denote the greatest integer ≤t. Then the value of 8 ⋅∫1−12 ([2x] x > −2, ϕ(0) = 4, then ϕ(2) is

Q87.Let P(x) be a real polynomial of degree 3 which vanishes at x = −3. Let P(x) have local minima at x = 1 , local maxima at x = −1 and ∫1−1 P(x)dx = 18 , then the sum of all the coefficients of the polynomial P(x) is equal to ___ .

Q87.Let 𝑀= 𝐴= 𝑎 𝑏 𝑎, 𝑏, 𝑐, 𝑑∈±3, ± 2, ± 1, 0. Define 𝑓: 𝑀→𝑍, as 𝑓𝐴= det 𝐴, for all 𝐴∈𝑀 where 𝑍 is 𝑐 𝑑: set of all integers. Then the number of 𝐴∈𝑀 such that 𝑓𝐴= 15 is equal to . 0 𝑖 𝑛𝑎 𝑏 𝑎 𝑏

Q88.For real numbers α, β, γ and δ, if (x2−1)+tan−1( x2+1x ) x2+1 γ(x2−1) x2+1 + β + δ + C where C is ∫ x2+1 dx = α loge(tan−1( x )) tan−1( x ) tan−1( x ) (x4+3x2+1) tan−1( x ) an arbitrary constant, then the value of 10(α + βγ + δ) is equal to ______ . → = 8 , then

Q88.If xϕ(x) = ∫x5 (3t2 −2ϕ′(t))dt, JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper

Q88.If the curve, y = y(x) represented by the solution of the differential equation (2xy2 −y)dx + x dy = 0, passes through the intersection of the lines, 2x −3y = 1 and 3x + 2y = 8, then |y(1)| is equal to ___ .

Q88.Let 𝑆= 𝑛∈𝑁, 𝑏, 𝑐, 𝑑∈𝑅, where 𝑖= √-1 . Then the number of 2 - digit 1 0 𝑐 𝑑= 𝑐 𝑑∀𝑎, numbers in the set 𝑆 is

Q88.If →a = αˆi + βˆj + 3ˆk, →b= −βˆi −αˆj −ˆk and →c= ˆi −2ˆj −ˆk such that →a⋅→b= 1 and →b⋅→c= −3, then → 1 × is equal to _______. 3 ((→a b) ⋅→c)

Q88.The number of distinct real roots of the equation 3x4 + 4x3 −12x2 + 4 = 0 is _________. + C, x > 0 where C is the constant of integration, then the +

Q88.Let [t] denote the greatest integer ≤t . The number of points where the function 𝑓(𝑥) = [𝑥]𝑥2 - 1 + sin 𝜋 - [𝑥+ 1], 𝑥∈( - 2, 2) is not continuous is _____ . [𝑥] + 3

Q88.Let a function g : [0, 4] →R be defined as max {t3 −6t2 + 9t −3}, 0 ≤x ≤3 ⎧ g(x) = 0≤t≤x ⎨ ⎩ 4 −x, 3 < x ≤4 then the number of points in the interval (0, 4) where g(x) is NOT differentiable, is _________. JEE Main 2021 (20 Jul Shift 2) JEE Main Previous Year Paper