Practice Questions

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.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 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.The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x)n+2 , which are in the ratio 1 : 3 : 5 , is equal to (1) 92 (2) 63 (3) 41 (4) 25

Q68.Among the statements : (S1) : 20232022 −19992022 is divisible by 8 . (S2) : 13(13)n −11n −13 is divisible by 144 for infinitely many n ∈N (1) Only (S2) is correct (2) Only (S1) is correct (3) Both (S1) and (S2) are correct (4) Both (S1) and (S2) are incorrect

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 remainder when (2023)2023 is divided by 35 is

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 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.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 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

Q69.The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is (1) 11 (2) 13 (3) 12 (4) 14

Q69.The value of tan 9 o −tan 27 o −tan 63 o + tan 81 o is _____.

Q69.The parabolas : ax2 + 2bx + cy = 0 and d2 + 2ex + fy = 0 intersect on the line y = 1. If a, b, c, d, e, f are positive real numbers and a, b, c are in G. P., then (1) d, e, f are in A.P. (2) ad , eb , fc are in G.P. (3) a d , eb , fc are in A.P. (4) d, e, f are in G.P.

Q69.Let S be the set of all a ∈N such that the area of the triangle formed by the tangent at the point P(b, c), b, c ∈N , on the parabola y2 = 2ax and the lines x = b, y = 0 is 16 unit2 , then ∑a∈S a is equal to _____ .

Q69.The distance of the point (6, −2√2) from the common tangent and x = 1 + y2 is (1) 1 (2) 5 3 (3) 14 (4) 5√3 3

Q69.If the x-intercept of a focal chord of the parabola y2 = 8x + 4y + 4 is 3 , then the length of this chord is equal to _____ .

Q69.The locus of the middle points of the chords of the circle C1 : (x −4)2 + (y −5)2 = 4 which subtend an angle θi at the centre of the circle Ci , is a circle of radius ri . If θ1 = π3 , θ3 = 2π3 and r12 = r22 + r32 , then θ2 is equal to π 3π (1) (2) 4 4 (3) π (4) π 6 2

Q69.An organization awarded 48 medals in event '𝐴', 25 in event '𝐵' and 18 in event '𝐶'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events? (1) 15 (2) 21 (3) 10 (4) 9

Q69.From the top 𝐴 of a vertical wall 𝐴𝐵 of height 30 m, the angles of depression of the top 𝑃 and bottom 𝑄 of a vertical tower 𝑃𝑄 are 15∘ and 60∘ respectively, 𝐵 and 𝑄 are on the same horizontal level. If 𝐶 is a point on 𝐴𝐵 such that 𝐶𝐵= 𝑃𝑄, then the area (in m2) of the quadrilateral 𝐵𝐶𝑃𝑄 is equal to JEE Main 2023 (06 Apr Shift 1) JEE Main Previous Year Paper (1) 300 ( √3 - 1 ) (2) 300 ( √3 + 1 ) (3) 600 ( √3 - 1 ) (4) 200 ( √3 - 1 )

Q69.A light ray emits from the origin making an angle 30° with the positive x -axis. After getting reflected by the line x + y = 1 , if this ray intersects x-axis at Q, then the abscissa of Q is (1) 2 (2) 2 (√3−1) 3+√3 (3) 2 (4) √3 3−√3 2(√3+1) JEE Main 2023 (29 Jan Shift 1) JEE Main Previous Year Paper

Q69.The set of all values of λ for which the equation cos2 2x −2 sin4 x −2 cos2 x = λ (1) [−2, −1] (2) [−2, −32 ] (3) [−1, −12 ] (4) [−32 , −1]

Q69.Let 𝑁 denote the number that turns up when a fair die is rolled. If the probability that the system of equations 𝑥+ 𝑦+ 𝑧= 12𝑥+ 𝑁𝑦+ 2𝑧= 23𝑥+ 3𝑦+ 𝑁𝑧= 3 has unique solution is 𝑘 then the sum of value of 𝑘 and all possible values of 𝑁 is 6, JEE Main 2023 (24 Jan Shift 1) JEE Main Previous Year Paper (1) 18 (2) 19 (3) 20 (4) 21

Q69.Among the statements: 𝑆1: 𝑝∨𝑞⇒𝑟⇔𝑝⇒𝑟 𝑆2: 𝑝∨𝑞⇒𝑟⇔𝑝⇒𝑟∨𝑞⇒𝑟 (1) Only ( 𝑆1 ) is a tautology (2) Neither ( 𝑆1 ) nor ( 𝑆2 ) is a tautology (3) Only ( 𝑆2 ) is a tautology (4) Both ( 𝑆1 ) and ( 𝑆2 ) are tautologies

Q69.In a triangle ABC , if cos A + 2 cos B + cos C = 2 and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then cos A −cos C is equal to (1) 9 (2) 10 7 7 (3) 5 (4) 3 7 7