Practice Questions

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

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

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.The remainder when (2023)2023 is divided by 35 is

Q68.If 𝑃( ℎ, 𝑘) be point on the parabola 𝑥= 4𝑦2, which is nearest to the point 𝑄( 0, 33 ) , then the distance of 𝑃 from the directrix of the parabola 𝑦2 = 4 ( 𝑥+ 𝑦) is equal to: (1) 2 (2) 4 (3) 8 (4) 6

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.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.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.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.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 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.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 value of 36(4 cos2 9∘−1 )(4 cos2 27∘−1 )(4 cos2 81∘−1 )(4 cos2 243∘−1 ) is (1) 54 (2) 18 (3) 27 (4) 36

Q68.Negation of p ∧(q ∧~(p ∧q)) is (1) (~(p ∧q)) ∨p (2) p ∨q (3) ~(p ∨q) (4) (~(p ∧q)) ∧q

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