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.If √3(cos2 x) = (√3 −1) ________.

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.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.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.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 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 _______.

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

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.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

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

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.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.

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.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.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.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 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.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 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. 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