Practice Questions

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.A man starts walking from the point 𝑃( - 3, 4 ) , touches the 𝑥-axis at 𝑅, and then turns to reach at the point 𝑄( 0, 2 ) , The man is walking at a constant speed. If the man reaches the point 𝑄 in the minimum time, then 50 ( 𝑃𝑅) 2 + ( 𝑅𝑄) 2 is equal to ______ .

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 ________.

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.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.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.Let I be an identity matrix of order 2 × 2 and P = [25 −1−3 ] P n = 5I −8P is equal to ___ .

Q85.The missing value in the following figure is

Q85.Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10 , and C1(α, β) and C2(γ, δ), C1 ≠C2 are their centres, then |(α + β)(γ + δ)| is equal to = 1.

Q85.Let n be an odd natural number such that the variance of 1, 2, 3, 4, … , n is 14. Then n 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.Consider the function f(x) = sin(x−2)P(x) , P ′′(x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to __________.

Q85. x y z Let A = ⎡y z x ⎤, where x, y and z are real numbers such that x + y + z > 0 and xyz = 2 . If A2 = I3 , z x y ⎣ ⎦ then the value of x3 + y3 + z3 is

Q85.For integers n and r, let (r ) = { 0, otherwise . The maximum value of k for which the sum 10 15 12 13 ∑k + ∑k+1i=0 is maximum, is equal to _________. i=0( i )(k −i ) ( i )(k + 1 −i ) JEE Main 2021 (24 Feb Shift 2) JEE Main Previous Year Paper

Q85.In ΔABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of Δ ABC is 30 cm2 and R and r are respectively the radii of circumcircle and incircle of ΔABC, then the value of 2R + r (in cm) is equal to ______ . and B = be two 2 × 1 matrices with real entries such that A = XB, where

Q85.Let A = {n ∈N ∣n2 ≤n + 10, 000}, B = {3k + 1 ∣k ∈N} and C = {2k ∣k ∈N}, then the sum of all the elements of the set A ∩(B −C) is equal to ________. Q86. ⎡ 1 1 1⎤ If A = 0 1 1 and M = A + A2 + A3 + … + A20, then the sum of all the elements of the matrix M is ⎣ 0 0 1⎦ equal to _______. α + β is equal to α −βe ∫10 √tetdt, then

Q85.The mean of 10 numbers 7 × 8, 10 × 10, 13 × 12, 16 × 14, … is

Q85.The term independent of 𝑥 in the expansion of - , where 𝑥≠0, 1 is equal to 𝑥2 / 3 - 𝑥1 / 3 + 1 𝑥- 𝑥1 / 2

Q85.A tangent line 𝐿 is drawn at the point 2, - 4 on the parabola 𝑦2 = 8𝑥. If the line 𝐿 is also tangent to the circle 𝑥2 + 𝑦2 = 𝑎, then 𝑎 is equal to . 3 = 𝐼- 𝐴3}, where 𝐼

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. 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.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.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.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.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