Practice Questions

Q65.The number of elements in the set 𝑆= 𝜃∈[0, 2𝜋]: 3cos4𝜃- 5cos2𝜃- 2sin6𝜃+ 2 = 0 is (1) 10 (2) 8 (3) 12 (4) 9

Q66.Let SK = 1+2+...+KK and ∑nj=1 S 2j = An (Bn2 + Cn + D) where A, B, C, D ∈ N and A Has least value then (1) A + C + D is not divisible by D (2) A + B = 5(D −C) (3) A + B + C + D is divisible by 5 (4) A + B is divisible by D

Q66.Consider ellipses 𝐸𝑘: 𝑘𝑥2 + 𝑘2𝑦2 = 1, 𝑘= 1, 2, … , 20. Let 𝐶𝑘 be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse 𝐸𝑘. If 𝑟𝑘 is the radius of the circle 𝐶𝑘, then the value of ∑𝑘=20 1 12 is 𝑟𝑘 (1) 3080 (2) 2870 (3) 3210 (4) 3320

Q66.If ar is the coefficient of x10−r in the Binomial expansion of (1 + x)10 , then ∑10r=1 r3( ar−1 2 (1) 4895 (2) 1210 (3) 5445 (4) 3025

Q67.Let the centre of a circle 𝐶 be 𝛼, 𝛽 and its radius 𝑟 < 8. Let 3𝑥+ 4𝑦= 24 and 3𝑥– 4𝑦= 32 be two tangents and 4𝑥+ 3𝑦= 1 be a normal to 𝐶. Then ( 𝛼 - 𝛽+ 𝑟) is equal to (1) 7 (2) 5 (3) 6 (4) 9 𝑒𝑎𝑥- cos(𝑏𝑥) - 𝑐𝑥𝑒-𝑐𝑥 2

Q67.Let PQ be a focal chord of the parabola y2 = 36x of length 100, making an acute angle with the positive x− axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line PQ? (1) (−6, 45) (2) (6, 29) (3) (3, 33) (4) (−3, 43) y2 + 4 = 1 meet the y−axis at the points A

Q67.Let S = {θ ∈[0, 2π) : tan(πcosθ) + tan(πsinθ) = 0} , then ∑θ∈S sin2(θ 4 ) is equal to

Q67.Let 𝑦= 𝑥+ 2, 4𝑦= 3𝑥+ 6 and 3𝑦= 4𝑥+ 1 be three tangent lines to the circle ( 𝑥- ℎ) 2 + ( 𝑦- 𝑘) 2 = 𝑟2. Then ℎ+ 𝑘 is equal to : (1) 5 (2) 5 ( 1 + √2 ) (3) 6 (4) 5√2

Q68.Let K be the sum of the coefficients of the odd powers of x in the expansion of (1 + x)99 . Let a be the middle 200 1 200C99K 2lm + = n , where m and n are odd numbers, then the ordered term in the expansion of (2 √2 ) . If a pair (l, n) is equal to: (1) (50, 51) (2) (51, 99) (3) (50, 101) (4) (51, 101)

Q68.The points of intersection of the line ax + by = 0 , (a ≠b) and the circle x2 + y2 −2x = 0 are A(α, 0) and B(1, β). The image of the circle with AB as a diameter in the line x + y + 2 = 0 is : (1) x2 + y2 + 5x + 5y + 12 = 0 (2) x2 + y2 + 3x + 5y + 8 = 0 (3) x2 + y2 + 3x + 3y + 4 = 0 (4) x2 + y2 −5x −5y + 12 = 0 y = mx + c, m > 0, of the curves x = 2y2

Q68.Let P(a1, b1) and Q(a2, b2) be two distinct points on a circle with center C(√2, √3). Let and OC be perpendicular to both CP and CQ. If the area of the triangle OCP is √35 , then a21 + a22 + b21 + b22 2 is equal to __________

Q68.The set of all values of a2 for which the line x + y = 0 bisects two distinct chords drawn from a point P( 1+a2 , 1−a2 ) on the circle 2x2 + 2y2 −(1 + a)x −(1 −a)y = 0 , is equal to : (1) (8, ∞) (2) (0, 4] (3) (4, ∞) (4) (2, 12] JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper

Q68.Let the sixth term in the binomial expansion of (√2log2(10−3x) If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of x is _____ .

Q68.Let the tangent and normal at the point (3√3, 1) on the ellipse x236 and B respectively. Let the circle C be drawn taking AB as a diameter and the line x = 2√5 intersect C at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point (α, β), then α2 −β2 is equal to (1) 61 (2) 60 (3) 304 (4) 314 5 5

Q68.Let f(θ) = 3(sin4( 3π2 −θ) + sin4(3π + θ)) −2(1 −sin2 2θ) and S = {θ ∈[0, π] ′(θ) = −√32 }. If 4β = ∑θ∈S θ then f(β) is equal to (1) 11 (2) 5 8 4 (3) 9 (4) 3 8 2

Q69.If the line l1 : 3y −2x = 3 is the angular bisector of the lines l2 : x −y + 1 = 0 and l3 : αx + βy + 17 = 0 , then α2 + β2 −α −β is equal to ............

Q69.If m and n respectively are the numbers of positive and negative value of θ in the interval [−π, π] that satisfy the equation cos 2θ cos 2θ = cos 3θ cos 9θ2 , then mn is equal to _____ .

Q69.Let A(0, 1), B(1, 1) and C(1, 0) be the mid-points of the sides of a triangle with incentre at the point D. If the α and β are rational numbers, then focus of the parabola y2 = 4ax passing through D is (α + β√2, 0), where α is equal to β2 (1) 8 (2) 12 (3) 6 (4) 29 JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper

Q69.Let C(α, β) be the circumcentre of the triangle formed by the lines 4x + 3y = 69 , 4y −3x = 17 , and x + 7y = 61 . Then (α −β)2 + α + β is equal to (1) 18 (2) 17 (3) 15 (4) 16

Q69.For a triangle 𝐴𝐵𝐶, the value of cos2𝐴+ cos2𝐵+ cos2𝐶 is least. If its inradius is 3 and incentre is 𝑀, then which of the following is NOT correct? (1) Perimeter of ∆𝐴𝐵𝐶 is 18√3 (2) sin2𝐴+ sin2𝐵+ sin2𝐶= sin𝐴+ sin𝐵+ sin𝐶 (3) →MA · →MB = - 18 (4) area of ∆𝐴𝐵𝐶 is 27√3 2

Q69.For the system of linear equations 𝑥+ 𝑦+ 𝑧= 6 𝛼𝑥+ 𝛽𝑦+ 7𝑧= 3 𝑥+ 2𝑦+ 3𝑧= 14 which of the following is NOT true ? (1) If 𝛼= 𝛽= 7, then the system has no solution (2) If 𝛼= 𝛽 and 𝛼≠7 then the system has a unique solution. (3) There is a unique point ( 𝛼, 𝛽) on the line (4) For every point ( 𝛼, 𝛽) ≠( 7, 7 ) on the line 𝑥+ 2𝑦+ 18 = 0 for which the system has x - 2y + 7 = 0, the system has infinitely many infinitely many solutions solutions.

Q69.Let m1 and m2 be the slopes of the tangents drawn from the point P(4, 1) to the hyperbola H : 25y2 −x216 = 1 If Q is the point from which the tangents drawn to H have slopes |m1| and |m2| and they make positive (PQ)2 intercepts α and β on the x− axis, then αβ is equal to _______.

Q70.Let the determinant of a square matrix A of order m be m −n , where m and n satisfy 4m + n = 22 and 17m + 4n = 93 . If det(n adj(adj(mA))) = 3a5b6c , then a + b + c is equal to (1) 84 (2) 96 (3) 101 (4) 109

Q70.The line x = 8 is the directrix of the ellipse E : x2 + y2 = 1 with the corresponding focus (2, 0). If the a2 b2 x -axis at tangent to E at the point P in the first quadrant passes through the point (0, 4√3) and intersects the Q, then (3PQ)2 is equal to _____ .

Q70.The vertices of a hyperbola H are (±6, 0) and its eccentricity is √52 . Let N be the normal to H at a point in the first quadrant and parallel to the line √2x + y = 2√2 . If d is the length of the line segment of N between H and the y -axis then d2 is equal to _____ .