Practice Questions

Q82.The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____.

Q83.The greatest integer less than or equal to the sum of first 100 terms of the sequence 1 5 19 65 … is equal to 3, 9, 27, 81, ______

Q83.If one of the diameters of the circle x2 + y2 −2√2x −6√2y + 14 = 0 is a chord of the circle 2 (x −2√2) 2 = r2 , then the value of r2 is equal to +(y −2√2)

Q83.Let the coefficients of x−1 and x−3 in the expansion of (2x 5 − x 51 ) , x > 0, be mand n respectively. If r is a positive integer such mn2 = 15Cr.2r , then the value of r is equal to ______. JEE Main 2022 (29 Jun Shift 2) JEE Main Previous Year Paper

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.If 2×3×4 1 + 3×4×51 + 4×5×61 + … + 100×101×102 1 = 101k , then 34k 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.Let the eccentricity of the hyperbola 𝑥2 - 𝑦2 = 1 be 5 If the equation of the normal at the point 8 12 on the 𝑎2 𝑏2 4. √5, 5 hyperbola is 8√5𝑥+ 𝛽𝑦= 𝜆, then 𝜆- 𝛽 is equal to _____. 5𝑛+ 1

Q83.If 6 + 10 + 20 + 40 + … . . + 102403 = 2n ⋅m, where m is odd, then m. n is equal to _____ . 312 311 310 39

Q83.The total number of 3 -digit numbers, whose greatest common divisor with 36 is 2 , is ______.

Q83.For 𝑝, 𝑞∈𝑅, consider the real valued function 𝑓𝑥= 𝑥- 𝑝2 - 𝑞, 𝑥∈𝑅 and 𝑞> 0. Let 𝑎1, 𝑎2, 𝑎3 and 𝑎4 be in an arithmetic progression with mean 𝑝 and positive common difference. If 𝑓𝑎𝑖= 500 for all 𝑖= 1, 2, 3, 4, then the absolute difference between the roots of 𝑓𝑥= 0 is

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 _______.

Q83.The number of positive integers k such that the constant term in the binomial expansion of 12 (2x3 + xk3 ) , x ≠0 is 28 ⋅l, where l is an odd integer, is ______.

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.If ∑𝑘=10 1 𝐾210𝐶𝐾 2 = 22000 𝐿, then 𝐿 is equal to _____.

Q83.The remainder on dividing 1 + 3 + 32 + 33 + … + 32021 by 50 is _____.

Q83.If the length of the latus rectum of the ellipse x2 + 4y2 + 2x + 8y −λ = 0 is 4 , and l is the length of its major axis, then λ + l is equal to _____. . Let the major

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.The number of solutions of the equation 2θ −cos2 θ + √2 = 0 in 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.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 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.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 mean deviation about the mean of the numbers 1, 2, 3, … … , 𝑛, where 𝑛 is odd, is , then 𝑛 is equal 𝑛 to ______.