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.The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1)(1, 1) is 18 units, is a circle of diameter d. Then d2 is equal to −1), for x ≤0 2 (x

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.If √3(cos2 x) = (√3 −1) ________.

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 n be an odd natural number such that the variance of 1, 2, 3, 4, … , n is 14. Then n is equal to ________.

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.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.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.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.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.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. 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.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.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.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.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 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.The missing value in the following figure is

Q85.Let I be an identity matrix of order 2 × 2 and P = [25 −1−3 ] P n = 5I −8P 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. 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.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 X1, X2, … , X18 be eighteen observations such that ∑18i=1(Xi −α) = 36 and ∑18i=1 (Xi −β)2 = 90 , where α and β are distinct real numbers. If the standard deviation of these observations is 1 , then the value of |α −β| is _______. Q87. ⎡ 1 0 0 ⎤ ⎡1 0 0⎤ If the matrix A = 0 2 0 satisfies the equation A20 + αA19 + βA = 0 4 0 for some real numbers ⎣ 3 0 −1 ⎦ ⎣0 0 1⎦ α and β, then β −α is equal to ______.