Practice Questions

Q86.Let the mean and variance of four numbers 3, 7, x and y (x > y) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x −y is ______.

Q86.Let a, b ∈R, b ≠0 . Defined a function, f(x) = tansin2x−sinπ 2x , for x > 0 {a bx3 If f is continuous at x = 0, then 10 −ab is equal to x = 0 is equal to

Q86.Let X1, X2, … , X18 be eighteen observations such that ∑18i=1(Xi −α) = 36 and ∑18i=1 (Xi −β)2 = 90 , where α and β are distinct real numbers. If the standard deviation of these observations is 1 , then the value of |α −β| is _______. Q87. ⎡ 1 0 0 ⎤ ⎡1 0 0⎤ If the matrix A = 0 2 0 satisfies the equation A20 + αA19 + βA = 0 4 0 for some real numbers ⎣ 3 0 −1 ⎦ ⎣0 0 1⎦ α and β, then β −α is equal to ______.

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.The value of ∫2−2 3x2 −3x −6

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

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.If ∫π0 (sin3 x)e−sin2 xdx =

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

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

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

Q87.The minimum value of 𝛼 for which the equation sin𝑥+ 1 - sin𝑥= 𝛼 has at least one solution in 0, 2 is______.

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 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 ___ .