Practice Questions

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

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

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.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.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.The statement ~𝑝∨~𝑝∧𝑞 is equivalent to (1) ~𝑝∧𝑞 (2) 𝑝∧𝑞∧~𝑝 (3) ~𝑝∧𝑞∧𝑞 (4) ~𝑝∨𝑞 JEE Main 2023 (10 Apr Shift 2) JEE Main Previous Year Paper

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.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.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.The statement ( 𝑝∧( ~𝑞) ) ∨( ( ~𝑝) ∧ 𝑞) ∨( ( ~𝑝) ∧ ( ~𝑞) ) is equivalent to _____ (1) ~𝑝∨𝑞 (2) ~𝑝∨~𝑞 (3) 𝑝∨~𝑞 (4) 𝑝∨𝑞 Q70. 1 2 3 Let for 𝐴= 𝛼3 1 , 𝐴= 2. If |2 adj ( 2 adj ( 2𝐴) ) | = 32𝑛, then 3𝑛+ 𝛼 is equal to 1 1 2 (1) 9 (2) 11 (3) 12 (4) 10

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.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

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 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.For the system of linear equations 2𝑥- 𝑦+ 3𝑧= 5 3𝑥+ 2𝑦- 𝑧= 7 4𝑥+ 5𝑦+ 𝛼𝑧= 𝛽, which of the following is NOT correct? (1) The system has infinitely many solutions for (2) The system has infinitely many solutions for 𝛼= – 5 and 𝛽= 9 𝛼= - 6 and 𝛽= 9 (3) The system in inconsistent for 𝛼= – 5 and (4) The system has a unique solution for 𝛼≠– 5 𝛽= 8 and 𝛽= 8

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.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.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.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

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.If the tangents at the points P and Q on the circle x2 + y2 −2x + y = 5 meet at the point R( 94 , 2), then the area of the triangle PQR is (1) 5 (2) 13 4 8 (3) 5 (4) 13 8 4