Practice Questions

Q63.If the sum of an infinite GP, a, ar, ar2, ar3, … is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, … is: (1) 25 (2) 9 2 2 (3) 1 (4) 5 2 2

Q63.The number of solutions of sin7 x + cos7 x = 1, x ∈[0, 4π] is equal to (1) 11 (2) 7 (3) 5 (4) 9

Q63. cosec 18° is a root of the equation: (1) x2 −2x −4 = 0 (2) 4x2 + 2x −1 = 0 (3) x2 + 2x −4 = 0 (4) x2 −2x + 4 = 0

Q63.The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is: (1) 26664 (2) 122664 (3) 122234 (4) 22264

Q63.If for x, y ∈R, x > 0, y = log10 x + log10 x1/3 + log10 x1/9 + … upto ∞ terms and 2+4+6+…+2y3+6+9+…+3y = log104 x , then the ordered pair (x, y) is equal to (1) (106, 6) (2) (106, 9) (3) (102, 3) (4) (104, 6)

Q63.If 15 sin4 α + 10 cos4 α = 6, for some α ∈R, then the value of 27 sec6 α + 8 cosec6 α is equal to : (1) 350 (2) 500 (3) 400 (4) 250

Q63.Team ′A′ consists of 7 boys and n girls and Team ′B′ has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to: (1) 5 (2) 2 (3) 4 (4) 6

Q63.In an increasing geometric series, the sum of the second and the sixth term is 252 and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to: (1) 35 (2) 32 (3) 26 (4) 30 1 10 1 (1−x) 10 where x ∈(0, 1) is: 5 + t )

Q63.Let A(a, 0), B(b, 2b + 1) and C(0, b), b ≠0, |b| ≠1 , be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is: (1) −2b (2) 2b2 b+1 b+1 (3) −2b2 (4) 2b b+1 b+1

Q63.The value of 2 sin( 8π ) sin( 2π8 ) sin( 3π8 ) sin( 5π8 ) sin( 6π8 ) sin( 7π8 ) is : (1) 1 (2) 1 4√2 8 (3) 1 (4) 1 8√2 4 JEE Main 2021 (26 Aug Shift 2) JEE Main Previous Year Paper

Q64.If 0 < x < 1, then 23 x2 + 53 x3 + 74 x4 + … , is equal to (1) x( 1−xx+1 ) + loge(1 −x) (2) x( 1−x1+x ) + loge(1 −x) (3) 1−x1+x + loge(1 −x) (4) 1−x1+x + loge(1 −x) Q65. ∑20k=0 (20Ck) 2 is equal to (1) 40C21 (2) 41C20 (3) 40C20 (4) 40C19

Q64.The lowest integer which is greater than + is (1 10100 ) (1) 3 (2) 4 (3) 2 (4) 1

Q64.If 20Cr is the co-efficient of xr in the expansion of (1 + x)20 , then the value of ∑20r=0 r2(20Cr) is equal to: (1) 420 × 218 (2) 380 × 218 (3) 380 × 219 (4) 420 × 219 cos x

Q64.If 0 < θ, ϕ < π2 , x = ∑∞n=0 cos2n θ, y = ∑∞n=0 sin2n ϕ and z = ∑∞n=0 cos2n θ ⋅sin2n ϕ then : (1) xy −z = (x + y)z (2) xy + yz + zx = z (3) xy + z = (x + y) z (4) xyz = 4

Q64.If p and q are the lengths of the perpendiculars from the origin on the lines, x cosec α −y sec α = k cot 2α and x sin α + y cos α = k sin 2α respectively, then k2 is equal to : (1) 2p2 + q2 (2) p2 + 2q2 (3) 4q2 + p2 (4) 4p2 + q2

Q64.Let the lengths of intercepts on x -axis and y -axis made by the circle x2 + y2 + ax + 2ay + c = 0, (a < 0) be 2√2 and 2√5 , respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to : (1) √11 (2) √7 (3) √6 (4) √10

Q64.Let 𝑎1, 𝑎2, … , 𝑎21 be an 𝐴. 𝑃. such that ∑𝑛= 1 9 𝑎𝑛𝑎𝑛+ 1 is equal to : (1) 57 (2) 48 (3) 36 (4) 72 𝜋 𝜋

Q64.The maximum value of the term independent of t in the expansion of (tx (1) 10! (2) 10! √3(5!)2 3(5!)2 (3) 2.10! (4) 2.10! 3√3(5!)2 3(5!)2

Q64.A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the 1 coordinate axes is 4. Three stones 𝐴, 𝐵 and 𝐶 are placed at the points 1, 1, 2, 2 and 4, 4 respectively. Then which of these stones is / are on the path of the man? (1) 𝐶 only (2) All the three (3) 𝐵 only (4) 𝐴 only

Q64.If sin θ + cos θ = 21 , then 16(sin(2θ) + cos(4θ) + sin(6θ)) is equal to: (1) 23 (2) −27 (3) −23 (4) 27

Q64.The coefficient of x256 in the expansion of (1 −x)101(x2 + x + 1)100 is: (1) 100C16 (2) 100C15 (3) −100C16 (4) −100C15

Q64.Let 𝑎1, 𝑎2, 𝑎3, … be an A.P. If 𝑎1 + 𝑎2 + … + 𝑎10 100 , 𝑝≠10, then 𝑎11 is equal to : 𝑎1 + 𝑎2 + … + 𝑎𝑝= 𝑝2 𝑎10 19 100 (1) (2) 21 121 (3) 21 (4) 121 19 100

Q64.The number of solutions of the equation x + 2 tan x = π2 in the interval [0, 2π] is (1) 3 (2) 4 (3) 2 (4) 5

Q64.If α, β are natural numbers such that 100α −199β = (100)(100) + (99)(101) + (98)(102) + … . . +(1)(199), then the slope of the line passing through (α, β) and origin is: (1) 540 (2) 550 (3) 530 (4) 510 Q65. 1 + 1 + 1 + … + 1 is equal to 32−1 52−1 72−1 (201)2−1 (1) 101 (2) 25 404 101 (3) 101 (4) 99 408 400 JEE Main 2021 (18 Mar Shift 1) JEE Main Previous Year Paper

Q64.Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of ΔABC , then (R + r) is equal to : (1) 9 (2) 7√2 √2 (3) 2√2 (4) 3√2