Practice Questions

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 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

Q64.The negation of the statement ~p ∧(p ∨q) is: (1) ~p ∨q (2) ~p ∧q (3) p ∨~q (4) p ∧~q

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, 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.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 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 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 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

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.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.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 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 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.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.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 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

Q65.Let an ellipse 𝐸: 𝑥2 + 𝑦2 = 1, 𝑎2 > 𝑏2, passes through 3 1 and has eccentricity 1 If a circle, centered at √ 2, √3. 𝑎2 𝑏2 2 focus 𝐹( 𝛼, 0 ) , 𝛼> 0, of 𝐸 and radius √3, intersects 𝐸 at two points 𝑃 and 𝑄, then 𝑃𝑄2 is equal to : (1) 8 (2) 4 3 3 16 (3) (4) 3 3

Q65.All possible values of θ ∈[0, 2π] for which sin 2θ + tan 2θ > 0 lie in : (1) (0, π2 ) ∪(π, 3π2 ) (2) (0, π2 ) ∪( π2 , 3π4 ) ∪(π, 7π6 ) (3) (0, π4 ) ∪( π2 , 3π4 ) ∪(π, 5π4 ) ∪( 3π2 , 7π4 ) (4) (0, π4 ) ∪( π2 , 3π4 ) ∪( 3π2 , 11π6 )

Q65.The sum of solutions of the equation 1+sin x = |tan 2x|, x ∈(−π2 , π2 ) −{−π4 , π4 } is: (1) 10 π (2) −7π30 (3) −π15 (4) −11π30

Q65.Two tangents are drawn from a point P to the circle x2 + y2 −2x −4y + 4 = 0, such that the angle between these tangents is tan−1( 125 ), where tan−1( 125 ) ∈(0, π). If the centre of the circle is denoted by C and these tangents touch the circle at points A and B, then the ratio of the areas of ΔPAB and ΔCAB is : (1) 11 : 4 (2) 9 : 4 (3) 3 : 1 (4) 2 : 1

Q65.The point P(a, b) undergoes the following three transformations successively: (a) reflection about the line y = x. (b) translation through 2 units along the positive direction of x− axis. (c) rotation through angle π4 about the origin in the anti-clockwise direction. , 2a + b is equal to: 7 ), then the value of If the co-ordinates of the final position of the point P are (−1√2 √2 (1) 13 (2) 9 (3) 5 (4) 7

Q65.If x→∞(√x2 (1) (1, −12 ) (2) (−1, 21 ) (3) (−1, −12 ) (4) (1, 21 )

Q65.If nP r = nP r+1 and nCr = nCr−1, then the value of r is equal to: (1) 1 (2) 4 (3) 2 (4) 3

Q65.Let A(1, 4) and B(1, −5) be two points. Let P be a point on the circle ((x −1))2 + (y −1)2 = 1 , such that (PA)2 + (PB)2 have maximum value, then the points, P, A and B lie on (1) a hyperbola (2) a straight line (3) an ellipse (4) a parabola xf(a)−af(x) equals: