Practice Questions

Q67.If 2n+1P n−1 : 2n−1P n = 11 : 21 , then n2 + n + 15 is equal to :

Q67.Statement ( 𝑃⇒𝑄) ∧( 𝑅⇒𝑄) is logically equivalent to (1) 𝑃⇒𝑅∨𝑄⇒𝑅 (2) 𝑃∧𝑅⇒𝑄 (3) 𝑃⇒𝑅∧𝑄⇒𝑅 (4) 𝑃∨𝑅⇒𝑄

Q67.Let R be a rectangle given by the lines 𝑥= 0, 𝑥= 2, 𝑦= 0 and 𝑦= 5. Let A𝛼, 0 and B0, 𝛽, 𝛼∈0, 2 and 𝛽∈0, 5, be such that the line segment 𝐴𝐵 divides the area of the rectangle 𝑅 in the ratio 4: 1. Then, the mid- point of 𝐴𝐵 lies on a (1) straight line (2) parabola (3) hyperbola (4) circle

Q67.If the co-efficient of x9 in 11 11 − βx3 1 ) are equal, then (αβ)2 is + βx1 ) and the co-efficient of x−9 in (αx (αx3 equal to : f

Q67.Let 𝐴 be the point 1, 2 and 𝐵 be any point on the curve 𝑥2 + 𝑦2 = 16. If the centre of the locus of the point 𝑃, which divides the line segment 𝐴 𝐵 in the ratio 3: 2 is the point 𝐶𝛼, 𝛽, then the length of the line segment 𝐴𝐶 is (1) 3√5 (2) 4√5 5 5 (3) 2√5 (4) 6√5 5 5

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 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.if the coefficients of three consecutive terms in the expansion of (1 + x)n are the ratio 1 : 5 : 20 then the coefficient of the fourth term is (1) 2436 (2) 5481 (3) 1827 (4) 3654 is α then [α] is

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

Q67.The negation of the expression 𝑞∨( ( ~𝑞) ∧𝑝) is equivalent to (1) ( ~𝑝) ∧( ~𝑞) (2) 𝑝∧( ~𝑞) (3) ( ~𝑝) ∨( ~𝑞) (4) ( ~𝑝) ∨𝑞

Q67.The negation of the statement 𝑝∨𝑞∧𝑞∨~𝑟 is (1) 𝑝∨𝑟∧~𝑞 (2) ~𝑝) ∨𝑟∧~𝑞 (3) ~𝑝∨~𝑞∨~𝑟 (4) ~𝑝∨~𝑞∧~𝑟

Q67.The number of common tangents, to the circles x2 + y2 −18x −15y + 131 = 0 and x2 + y2 −6x −6y −7 = 0 , is (1) 3 (2) 1 (3) 4 (4) 2

Q67.If the coefficients of x7 in (ax2 + 2bx1 ) 11 3bx2 and x−7 in (ax 1 ) (1) 729ab = 32 (2) 32ab = 729 (3) 64ab = 243 (4) 243ab = 64

Q67.The sum, of the coefficients of the first 50 terms in the binomial expansion of (1 −x)100, is equal to (1) 101C50 (2) 99C49 (3) −101C50 (4) −99C49

Q67.The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48 , is (1) 472 (2) 432 (3) 507 (4) 400 JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper

Q67.Let 𝑅 be a relation on 𝑁× 𝑁 defined by 𝑎, 𝑏𝑅𝑐, 𝑑 if and only if 𝑎𝑑𝑏- 𝑐= 𝑏𝑐𝑎- 𝑑. Then 𝑅 is (1) symmetric but neither reflexive nor transitive (2) transitive but neither reflexive nor symmetric (3) reflexive and symmetric but not transitive (4) symmetric and transitive but not reflexive Q68. 1 0 0 Let 𝐴= 0 4 -1 . Then the sum of the diagonal elements of the matrix 𝐴+ 𝐼11 is equal to: 0 12 -3 (1) 6144 (2) 4094 (3) 4097 (4) 2050

Q67.If the term without x in the expansion of 23 + 22 (x x3α ) is 7315 , then |α| is equal to _____ . m 21 . + 5√2(x−2) log2 3) powers of 2(x−2) log2 3 , be

Q68.The equations of the sides AB, BC & CA of a triangle ABC are 2x + y = 0 , x + py = 21a (a ≠0) and x −y = 3 respectively. Let P(2, a) be the centroid of the triangle ABC , then (BC)2 is equal to

Q68.If the point (α, 7√33 ) lies on the curve traced by the mid-points of the line segments of the lines α is equal to x cos θ + y sin θ = 7, θ ∈(0, 2π ) between the co-ordinates axes, then (1) −7 (2) −7√3 (3) 7√3 (4) 7

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

Q68.The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and 𝜎2 respectively. If the variance of all the 30 numbers in the two sets is 13, then 𝜎2 is equal to (1) 10 (2) 11 (3) 9 (4) 12

Q68.If 𝐴 and 𝐵 are two non-zero 𝑛× 𝑛 matrices such that 𝐴2 + 𝐵= 𝐴2𝐵, then (1) 𝐴𝐵= 𝐼 (2) 𝐴2𝐵= 𝐼 (3) 𝐴2 = 𝐼 or 𝐵= 𝐼 (4) 𝐴2𝐵= 𝐵𝐴2

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.Let sets 𝐴 and 𝐵 have 5 elements each. Let the mean of the elements in sets 𝐴 and 𝐵 be 5 and 8 respectively and the variance of the elements in sets 𝐴 and 𝐵 be 12 and 20 respectively. A new set 𝐶 of 10 elements is formed by subtracting 3 from each element of 𝐴 and adding 2 to each element of 𝐵. Then the sum of the mean and variance of the elements of 𝐶 is (1) 40 (2) 32 (3) 38 (4) 36 JEE Main 2023 (11 Apr Shift 1) JEE Main Previous Year Paper

Q68.If lim = 17, then 5𝑎2 + 𝑏2 is equal to 𝑥→0 1 - cos ( 2𝑥) (1) 64 (2) 72 (3) 68 (4) 76