Practice Questions

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 _______.

Q84.Let 𝑥1, 𝑥2, 𝑥3, … . . , 𝑥20 be in geometric progression with 𝑥1 = 3 and the common ration 12. A new data is constructed replacing each 𝑥𝑖 by 𝑥𝑖- 𝑖2. If 𝑥 is the mean of new data, then the greatest integer less than or equal to 𝑥 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 x→1(lim sin(3x2−4x+1)−x2+12x3−7x2+ax+b ) = −2, then the value of (a −b) is equal to

Q84.If the sum of solutions of the system of equations 2sin2𝜃- cos2𝜃= 0 and 2cos2𝜃+ 3sin𝜃= 0 in the interval 0, 2𝜋 is 𝑘𝜋, then 𝑘 is equal 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

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

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.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 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.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 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.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

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 _______.

Q86.The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecking it was found that, in the observations, 20 was misread as 5 . Then, the correct variance is equal to _____.

Q86.Let A = {n ∈ N : H. C. F. (n, 45) = 1} and let B = {2k : k ∈{1, 2, … , 100}} . Then the sum of all the elements of A ∩B is _____.

Q86.Let the hyperbola H : x2 −y2 = 1 and the ellipse E : 3x2 + 4y2 = 12 be such that the length of latus rectum a2 of H is equal to the length of latus rectum of E . If eH and eE are the eccentricities of H and E respectively, then the value of 12(e2H + e2E) is equal to _____.

Q86.Let the equation of two diameters of a circle 𝑥2 + 𝑦2 - 2𝑥+ 2𝑓𝑦+ 1 = 0 be 2𝑝𝑥- 𝑦= 1 and 2𝑥+ 𝑝𝑦= 4𝑝. Then the slope 𝑚∈0, ∞ of the tangent to the hyperbola 3𝑥2 - 𝑦2 = 3 passing through the centre of the circle is equal to _____. Q87. 2 -1 -1 √3i - 1 Let 𝐴= 1 0 -1 and 𝐵= 𝐴- 𝐼. If 𝜔= , then the number of elements in the set 2 1 -1 0 𝑛∈1, 2, … , 100: 𝐴𝑛+ 𝜔𝐵𝑛= 𝐴+ 𝐵 is equal to _____ .

Q86.Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x2 −2x, and g(f(x)) = 4x2 + 6x + 1, then the value of f(2) + g(2) is ______.