Practice Questions

Q84.Let [t] denote the greatest integer ≤t and {t} denote the fractional part of t . Then integral value of α for α2[x]+{x}+[x]−1 which the left hand limit of the function f(x) = [1 + x] + 2[x]+{x} at x = 0 is equal to α −43 is _____

Q84.Let the coefficients of the middle terms in the expansion of 4 6 + , β > 0 , βx) , (1 −3βx)2 and (1 −β2 x) ( √61 respectively form the first three terms of an A.P. If d is the common difference of this A.P., then 50 −2d is β2 equal to _____ .

Q84.If the coefficients of x and x2 in the expansion of (1 + x)p(1 −x)q, p, q ≤15 , are −3 and −5 respectively, then the coefficient of x3 is equal to ______.

Q84.If x→1(lim sin(3x2−4x+1)−x2+12x3−7x2+ax+b ) = −2, then the value of (a −b) is equal to

Q84.An ellipse E : x2a2 + y2b2 = 1 passes through the vertices of the hyperbola H : x249 −y264 = −1 and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H . Let the product of the eccentricities of E and H be 1 . If l is the length of the latus rectum of the ellipse E , then the 2 value of 113l is equal to _______.

Q84.If sin2(10°) sin(20°) sin(40°) sin(50°) sin(70°) = α− 161 sin(10°), then 16 + α−1 is equal to _____.

Q84.Let a circle C : (x −h)2 + (y −k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to _____.

Q84.If the mean deviation about the mean of the numbers 1, 2, 3, … … , 𝑛, where 𝑛 is odd, is , then 𝑛 is equal 𝑛 to ______.

Q84.The number of solutions of the equation sin x = cos2 x in the interval (0, 10) is ______. 2k . If (I −M 2)N = −2I , then the

Q84.The number of solutions of the equation 2θ −cos2 θ + √2 = 0 in R is equal to ______.

Q84.Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of , in the increasing powers of α , then α is 4√2 + 1 be 4√6 : 1. If the sixth term from the beginning is ( n 1 ) 4√3 4√3 4√3 equal to _______.

Q85.If 1 + (2 + 49C1 + 49C2 + … . +49C49)(50C2 + 50C4 + … . . +50C50) is equal to 2n. m, where m is odd, then n + m is equal to _____ .

Q85.Let the lines y + 2x = √11 + 7√7 and 2y + x = 2√11 + 6√7 be normal to a circle C , then the value of C : (x −h)2 + (y −k)2 = r2 . If the line √11y −3x = 5√773 + 11 is tangent to the circle (5h −8k)2 + 5r2 is equal to ______.

Q85.Let M = 0 −α , where α is a non-zero real number and N = ∑49k=1 M [α 0 ] positive integral value of α is ______.

Q85.The number of functions f , from the set A = {x ∈N : x2 −10x + 9 ≤0} to the set B = {n2 : n ∈N} such that f(x) ≤(x −3)2 + 1 , for every x ∈A , is _______.

Q85.Let 𝐴= 2 -2 and𝐵= -1 2 . Then the number of elements in the set {𝑛, 𝑚: 𝑛, 𝑚∈1, 2, … … . 10 and 1 -1 -1 2 𝑛𝐴𝑛+ 𝑚𝐵𝑚= 𝐼} is _____.

Q85.The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x −y = 3 respectively. If its orthocentre is (2, a), −12 < a < 2 , then p is equal to

Q85.Let H : x2 −y2 = 1, a > 0, b > 0 , be a hyperbola such that the sum of lengths of the transverse and the a2 b2 H is √11 + , then value of a2 + b2 is equal to ______. 2 conjugate axes is 4(2√2 √14). If the eccentricity ) + 2 Q86. 50 tan(3 tan−1( 21 cos−1( √51 ))+4√2 tan( 21 tan−1(2√2)) is equal to ______.

Q85.Let x = sin(2 tan−1 α) and y = sin( 12 tan−1 43 ). If S = {α ∈R : y2 = 1 −x}, then ∑α∈S 16α3 is equal to _______.

Q85.If 40C0 + 41C1 + 42C2 + ⋯+ 60C20 = mn × 60C20 where m & n are co-prime, then m + n is equal to and let L2 be the line passing through the origin and

Q85.The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If 𝜎 is the standard deviation of the data after omitting the two wrong observations from the data, then 38𝜎2 is equal to _______. JEE Main 2022 (26 Jul Shift 2) JEE Main Previous Year Paper

Q85.The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is _______.

Q85.The maximum number of compound propositions, out of p ∨r ∨s, p ∨r ∨~s, p ∨~q ∨s, ~p ∨~r ∨s, ~p ∨~r ∨~s, ~p ∨q ∨~s, q ∨r ∨~s, q ∨~r ∨~s, ~p ∨~q ∨~s that can be made simultaneously true by an assignment of the truth values to p, q, r and s, is equal to

Q85.The number of values of 𝑥 in the interval 4, 4 for which 14 cosec2 𝑥- 2sin2𝑥= 21 - 4cos2𝑥 holds, is ______.

Q85.Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is