Practice Questions

Q84.Let 𝐵 be the centre of the circle 𝑥2 + 𝑦2 - 2𝑥+ 4𝑦+ 1 = 0 . Let the tangents at two points 𝑃 and 𝑄 on the area 𝛥APQ circle intersect at the point 𝐴( 3, 1 ) . Then 8 is equal to . area 𝛥BPQ

Q84.A square ABCD has all its vertices on the curve x2y2 = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is

Q84.If the variable line 3x + 4y = α lies between the two circles (x −1)2 + (y −1)2 = 1 and (x −9)2 + (y −1)2 = 4, without intercepting a chord on either circle, then the sum of all the integral values of α is

Q84.If the minimum area of the triangle formed by a tangent to the ellipse x2 = 1 and the co-ordinate axis is + 4a2 b2 kab, then k is equal to ___________.

Q84.If A = 0 and (I2 + A)(I2 −A)−1 = then 13(a2 + b2) is equal to _____ . θ [ b a ], [tan( 2 ) 0 ]

Q85. n n n ⎧ if 0 ≤k ≤n . If Let denote nCk and = (k ), (k ) [ k ] ⎨ ⎩0, otherwise 9 12 8 13 Ak = ∑9 + ∑8 and A4 −A3 = 190p, then p is equal to _______. i=0( i )[ 12 −k + i ] i=0( i )[ 13 −k + i ]

Q85.Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S →S such that f(m ⋅n) = f(m) ⋅f(n) for every m, n ∈S and m ⋅n ∈S , is equal to _____.

Q85.A function f is defined on [−3, 3] as x , 2 −x2}, −2 ≤x ≤2 f(x) = {min{ [|x|] , 2 < |x| ≤3 where [x] denotes the greatest integer ≤x. The number of points, where f is not differentiable in (−3, 3) is ___ .

Q85.Let 𝑀 be any 3 × 3 matrix with entries from the set 0, 1, 2. The maximum number of such matrices, for which the sum of diagonal elements of 𝑀𝑇𝑀 is seven, is______.

Q85.Consider the following frequency distribution: Class: 0 −6 6 −12 12 −18 18 −24 24 −30 Frequency: a b 12 9 5 If mean = 30922 and median = 14, then the value (a −b)2 is equal to Q86. 0 1 0 Let A = ⎡ 1 0 0 ⎤. Then the number of 3 × 3 matrices B with entries from the set {1, 2, 3, 4, 5} and 0 0 1 ⎣ ⎦ satisfying AB = BA is ________.

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.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 : [−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

Q87.If ∫π0 (sin3 x)e−sin2 xdx =

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 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.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 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 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 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 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. 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.Let y = y(x) be the solution of the differential equation xdy −ydx = √(x2 −y2)dx, x ≥1 , with y(1) = 0. If the area bounded by the line x = 1, x = eπ, y = 0 and y = y(x) is αe2π + β, then the value of 10(α + β) is equal to ___ . −b = 0 be (−3, 5, 2).

Q88.If 𝑎𝑥+ 𝑥- 2𝑑𝑥= 22, 𝑎> 2 and 𝑥 denotes the greatest integer ≤𝑥, then -𝑎𝑥+ 𝑥𝑑𝑥 is equal to ∫-𝑎 ∫𝑎