Practice Questions

Q83.Let a circle C of radius 5 lie below the x-axis. The line L1 = 4x + 3y + 2 passes through the centre P of the circle C and intersects the line L2 : 3x −4y −11 = 0 at Q . The line L2 touches C at the point Q . Then the distance of P from the line 5x −12y + 51 = 0 is

Q83.If two tangents drawn from a point (α, β) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10α + 5)2 + (16β2 + 50)2 equals ______

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.A common tangent T to the curves C1 : x24 + y29 = 1 and C2 : x242 − 143y2 = 1 quadrant. If T touches C1 at (x1, y1) and C2 at (x2, y2), then |2x1 + x2| is equal to _______. JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper Q84. ⎡ α β γ ⎤ Consider a matrix A = α2 β2 γ 2 , where α, β, γ are three distinct natural numbers. ⎣β + γ γ + α α + β⎦ If det(adj(adj(adj(adjA))) = 232 × 316 , then the number of such 3 - tuples (α, β, γ) is _______. (α−β)16(β−γ)16(γ−α)16

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

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

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.Let A = ∑10i=1 ∑10j=1 min{i, j} and B = ∑10i=1 ∑10j=1 max{i, j}. Then A + B is equal to _____.

Q83.Different A.P.'s are constructed with the first term 100, the last term 199, And integral common differences. The sum of the common differences of all such, A.P's having at least 3 terms and at most 33 terms is.

Q83.Let A = {1, 2, 3, 4, 5, 6, 7} . Define B ={ T ⊆A : either 1 ∉T or 2 ∈T } and C ={ T ⊆A : T the sum of all the elements of T is a prime number.} Then the number of elements in the set B ∪C is _______. Q84. 1 a a 1 48 2160 Let A = ⎡0 1 b ⎤ , a, b ∈R. If for some n ∈N, An = ⎡0 1 96 ⎤ then n + a + b is equal to _______. 0 0 1 0 0 1 ⎣ ⎦ ⎣ ⎦

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

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

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 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.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.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.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

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 𝐶𝑟 denote the binomial coefficient of 𝑥𝑟 in the expansion of 1 + 𝑥10. If for 𝛼, 𝛽∈𝑅, 𝛼× 211 𝐶1 𝐶2 𝐶1 + 3 · 2𝐶2 + 5 · 3𝐶3 + … upto 10 terms = (𝐶0 + 2 + 3 + … upto 10 terms) then the value of 2𝛽- 1 𝛼+ 𝛽 is equal to _____. 𝜋 7𝜋

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.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.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 _______.