Practice Questions

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 minimum value of 𝛼 for which the equation sin𝑥+ 1 - sin𝑥= 𝛼 has at least one solution in 0, 2 is______.

Q87.Let F : [3, 5] →R be a twice differentiable function on (3, 5) such that F(x) = e−x ∫x3 (3t2 + 2t + 4F ′(t))dt. If F ′(4) = αeβ−224 , then α + β is equal to _____. (eβ−4)2

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.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.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.An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 JEE Main 2021 (27 Aug Shift 2) JEE Main Previous Year Paper candidates is 15. If μ is the average marks of girls and σ2 is the variance of marks of 50 candidates, then μ + σ2 is equal to Q88. ∫2ex+3e−x4ex+7e−x dx = 141 (ux + v loge(4ex + 7e−x)) + C , where C is a constant of integration, then u + v is equal to

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

Q87.Let A be a 3 × 3 real matrix. If det (2 Adj (2 Adj (Adj (2A)))) = 241, then the value of det (A2) equals ______.

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 T be the tangent to the ellipse E : x2 + 4y2 = 5 at the point P(1, 1). If the area of the region bounded by |α + β + γ| is equal the tangent T , ellipse E , lines x = 1 and x = √5 is α√5 + β + γ cos−1( √51 ), then to______. →

Q87.The number of points, at which the function f(x) = |2x + 1| −3|x + 2| + x2 + x −2 , x ∈R is not differentiable, is

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

Q87.The value of ∫2−2 3x2 −3x −6

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

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.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

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. if |x| ≤2 2 ) . Let f : R →R be a function defined as f(x) = { 3(1 −|x|0 if |x| > 2 Let g : R →R be given by g(x) = f(x + 2) −f(x −2). If n and m denote the number of points in R where g is not continuous and not differentiable, respectively, then n + m is equal to ________.

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.If 𝑎𝑥+ 𝑥- 2𝑑𝑥= 22, 𝑎> 2 and 𝑥 denotes the greatest integer ≤𝑥, then -𝑎𝑥+ 𝑥𝑑𝑥 is equal to ∫-𝑎 ∫𝑎

Q88.Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through a −2√2 b = 3 , then (a2 + b2 + ab) is equal to _____. (3, −3) and (4, −2√2), given that

Q88.If a + α = 1, b + β = 2 and af(x) + αf( x1 ) = bx + xβ , x ≠0, then the value of the expression f(x)+f(x+ x ___________.