Practice Questions

Q63.If 𝑒cos2𝑥+ cos4𝑥+ cos6𝑥+ . . . . ∞log𝑒2 satisfies the equation 𝑡2 - 9𝑡+ 8 = 0, then the value of 2sin𝑥 where sin𝑥+ √3cos𝑥, 0 < 𝑥< 𝜋2, is equal to (1) 3 (2) 1 2 2 (3) √3 (4) 2√3

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

Q64.Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point (−4, 1) and having their centres on the circumference of the circle x2 + y2 + 2x + 4y −4 = 0. If r1 = a + b√2, then r2 a + b is equal to: (1) 3 (2) 11 (3) 5 (4) 7

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 < 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.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.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.If 0 < x, y < π and cos x + cos y −cos(x + y) = 23 , then sin x + cos y is equal to: (1) 1 (2) √3 2 2 (3) 1−√3 (4) 1+√3 2 2 JEE Main 2021 (25 Feb Shift 2) JEE Main Previous Year Paper

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 [x] denote greatest integer less than or equal to x . If for n ∈N, (1 −x + x3) n = ∑3nj=0 ajxj , then [ 3n2 ] [ 3n−12 ] ∑ j=0 a2j + 4 ∑ j=0 a2j+1 is equal to : (1) 2 (2) 2n−1 (3) 1 (4) n

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.Let the circle S : 36x2 + 36y2 −108x + 120y + C = 0 be such that it neither intersects nor touches the co- ordinate axes. If the point of intersection of the lines, x −2y = 4 and 2x −y = 5 lies inside the circle S, then: (1) 25 9 < C < 133 (2) 100 < C < 165 (3) 81 < C < 156 (4) 100 < C < 156 = 1, a > b. Let E2 be another ellipse such that it touches the end points of major axis of E1

Q64.If two tangents drawn from a point P to the parabola y2 = 16(x −3) are at right angles, then the locus of point P is: (1) x + 4 = 0 (2) x + 2 = 0 (3) x + 3 = 0 (4) x + 1 = 0 = b, then the ordered pair (a, b) is: lim −x + 1 −ax)

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

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.If the fourth term in the expansion of (x + xlog2 x) 7 is 4480, then the value of x where x ∈N is equal to: (1) 2 (2) 4 (3) 3 (4) 1

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.The negation of the statement ~p ∧(p ∨q) is: (1) ~p ∨q (2) ~p ∧q (3) p ∨~q (4) p ∧~q

Q64.If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x2 + y2 = 1 is a circle of radius r, then r is equal to (1) 1 (2) 1 4 (3) 1 (4) 1 3 2

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.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.A circle C touches the line x = 2y at the point (2, 1) and intersects the circle C1 : x2 + y2 + 2y −5 = 0 at two points P and Q such that PQ is a diameter of C1 . Then the diameter of C is : (1) 4√15 (2) √285 (3) 15 (4) 7√5 = 1 having eccentricity √52 . If the tangent and

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.Let a parabola 𝑃 be such that its vertex and focus lie on the positive 𝑥-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from 𝑂( 0, 0 ) to the parabola 𝑃 which meet 𝑃 at 𝑆 and 𝑅, then the area (in sq. units) of Δ𝑆𝑂𝑅 is equal to : (1) 16√2 (2) 16 (3) 32 (4) 8√2