Practice Questions

Q83.If the coefficient of x10 in the binomial expansion of ( 5 14 + x 13 ) is 5kl, where l, k ∈N and l is coprime to 5, then k is equal to ______.

Q83.The number of elements in the set S = θ ∈[−4π, 4π] : 3 cos2 2θ + 6 cos 2θ −10 cos2 θ + 5 = 0 is ______.

Q83.If 2×3×4 1 + 3×4×51 + 4×5×61 + … + 100×101×102 1 = 101k , then 34k is equal to _______.

Q83.Let 𝑎1 = 𝑏1 = 1, 𝑎𝑛= 𝑎𝑛- 1 + 2 and 𝑏𝑛= 𝑎𝑛+ 𝑏𝑛- 1 for every natural number 𝑛≥2. Then ∑𝑛=15 1 𝑎𝑛· 𝑏𝑛 is equal to _____ . 1 15 10Q84. 1 1 - 𝑥 If the maximum value of the term independent of 𝑡 in the expansion of 𝑡2𝑥 5 + , 𝑥≥0, is 𝐾, then 8 K 𝑡 is equal to _____ . JEE Main 2022 (25 Jul Shift 1) JEE Main Previous Year Paper 8 6

Q83.The series of positive multiples of 3 is divided into sets : {3}, {6, 9, 12}, {15, 18, 21, 24, 27}, … Then the sum of the elements in the 11th set is 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.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 [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.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 a1(> 0), a2, a3, a4, a5 are in a G.P. , a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4 , then a2 + a4 + 2a5 is equal to _____. JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper

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

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 _______.

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