Practice Questions

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

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 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.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

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 1011th term from the end in the binomial expansion of ( 4x5 − 2x5 ) 2022 the beginning, then 32|x| is equal to (1) 15 (2) 10 (3) 12 (4) 8

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

Q67.The relation 𝑅= 𝑎, 𝑏: 𝑔𝑐𝑑𝑎, 𝑏= 1, 2𝑎≠𝑏, 𝑎, 𝑏∈ℤ is: (1) transitive but not reflexive (2) symmetric but not transitive (3) reflexive but not symmetric (4) neither symmetric nor transitive

Q67.The constant term in the expansion of 5 + x71 + 3x2) is _____ . (2x

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.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.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.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.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.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.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

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.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.If 𝐴 is a 3 × 3 matrix and 𝐴= 2, then 3 adj 3𝐴𝐴2 is equal to (1) 312 · 611 (2) 312 · 610 (3) 310 · 611 (4) 311 · 610

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 [t] denote the greatest integer ≤t. if the constant term in the expansion of (3x2 − 2x51 ) 7 equal to _____ JEE Main 2023 (08 Apr Shift 1) JEE Main Previous Year Paper

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.The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is (1) 1072 (2) 1792 (3) 1216 (4) 1456 JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper

Q68.Let a circle of radius 4 be concentric to the ellipse 15𝑥2 + 19𝑦2 = 285. Then the common tangents are inclined to the minor axis of the ellipse at the angle (1) π (2) π 3 4 π π (3) (4) 6 12