Practice Questions

Q86.Let f : R →R satisfy the equation f(x + y) = f(x) ⋅f(y) for all x, y ∈R and f(x) ≠0 for any x ∈R. If the function f is differentiable at x = 0 and f ′(0) = 3 , then lim h1 (f(h) −1) is equal to ___ . h→0

Q86. x + a −c x + b x + a Let a, b, c, d be in arithmetic progression with common difference λ. If x −1 x + c x + b = 2 , then x −b + d x + d x + c value of λ2 is equal to________.

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.If a rectangle is inscribed in an equilateral triangle of side length 2√2 as shown in the figure, then the square of the largest area of such a rectangle is _____. JEE Main 2021 (25 Jul Shift 2) JEE Main Previous Year Paper

Q86.If the system of equations kx + y + 2z = 1 3x −y −2z = 2 −2x −2y −4z = 3 has infinitely many solutions, then k is equal to ______ .

Q86.If R is the least value of a such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and S is the greatest value of a such that the function f(x) = x2 + ax + 1 is decreasing on [1, 2], then the value of |R −S| is is + x )dx

Q86.The number of elements in the set {𝐴= 𝑎 𝑏 𝑎, 𝑏, 𝑑∈{ - 1, 0, 1} and (𝐼- 𝐴) 0 𝑑: is 2 × 2 identity matrix, is .

Q86.The total number of 3 × 3 matrices A having enteries from the set (0, 1, 2, 3) such that the sum of all the diagonal entries of AAT is 9, is equal to

Q86.Let P(a sec θ, b tan θ) and Q(a sec ϕ, b tan ϕ) where θ + ϕ = π2 , be two points on the hyperbola x2a2 −y2b2 If the ordinate of the point of intersection of normals at P and Q is −k( a2+b22b ), then k is equal to

Q86.The value of the integral ∫π0 |sin 2x|dx is ________.

Q86.Let A = [a1a2 ] [b1b2 ] 1 1 −1 2 X = and k ∈R. If a21 + a22 = 3 (b21 + b22) and (k2 + 1)b22 ≠−2 b1b2 , then the value of k is √3 [1 k ], __________. and g(x) =

Q86.Let f : [0, 3] →R be defined by f(x) = min{x −[x], 1 + [x] −x} where [x] is the greatest integer less than or equal to x. Let P denote the set containing all x ∈[0, 3] where f is discontinuous, and Q denote the set containing all x ∈(0, 3) where f is not differentiable. Then the sum of number of elements in P and Q is equal to _____.

Q86.If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to ___ . dx is

Q86.Let 𝑓( 𝑥) = 𝑥6 + 2𝑥4 + 𝑥3 + 2𝑥+ 3, 𝑥∈R. Then the natural number 𝑛 for which lim 𝑥𝑛𝑓( 1 ) - 𝑓( 𝑥) = 44 is 𝑥→1 𝑥- 1 _____ . 2

Q86.If x→0[lim αxex−β loge(1+x)+γx2e−xx sin2 x ] = 10, α, β, γ ∈R, then the value of α + β + γ is __________. if i < jQ87. ⎧ (−1)j−i 2 if i = j then det(3 Adj (2A−1)) is equal to Let A = {aij} be a 3 × 3 matrix, where aij = ⎨ ⎩ (−1)i+j if i > j ________.

Q86.The maximum value of z in the following equation z = 6xy + y2, where 3x + 4y ≤100 and 4x + 3y ≤75 for x ≥0 and y ≥0 is 2 [[x2] −cos x]dx is ___________.

Q86.Consider the following frequency distribution : class 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 Frequency 𝛼 110 54 30 𝛽 If the sum of all frequencies is 584 and median is 45, then |𝛼- 𝛽| is equal to . JEE Main 2021 (25 Jul Shift 1) JEE Main Previous Year Paper

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 a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point (−5, 0). If the locus of the point P is a circle of radius r, then 4r2 (in the nearest integer) is equal to __________.

Q86.If the system of linear equations 2x + y −z = 3 x −y −z = α 3x + 3y + βz = 3 has infinitely many solutions, then |α + β −αβ| is equal to __________. + αx dydx + βy = 0, then |α −β| is equal to _______.

Q87.If ∫π0 (sin3 x)e−sin2 xdx =

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

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