Practice Questions

Q85.Let L be a common tangent line to the curves 4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31 . Then the square of the slope of the line L is ______.

Q85.Let A = [ ac db ] and B = [ αβ ] ≠[ 00] such that AB = B and a + d =2021, then the value of ad −bc is equal to ______ .

Q85.If the point on the curve y2 = 6x, nearest to the point (3, 32 ) is (α, β), then 2(α + β) is equal to _________.

Q85.If f(x) = sin(cos−1( 1+22x1−22x )) and its first derivative with respect to b are integers, then the minimum value of a2 −b2 is _______.

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.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.The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is 5x8+7x6

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

Q86. lim 𝑛 tan-1 1 is equal to_______. 𝑛→∞tan ∑𝑟= 1 1 + 𝑟+ 𝑟2 JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper 4 1 𝜋

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 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.Let f : [−1, 1] →R be defined as f(x) = ax2 + bx + c for all x ∈[−1, 1], where a, b, c ∈R such that f(−1) = 2, f ′(−1) = 1 and for x ∈(−1, 1) the maximum value of f ′′(x) is 21 . If f(x) ≤α, x ∈[−1, 1], then the least value of α is equal to x )ndx, where n ∈N . If (20)I10 = αI9 + βI8, for natural numbers α and β, then α −β

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.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 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.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) =