Practice Questions

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 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 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.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 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.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.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.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.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.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.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 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

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

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

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.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.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.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.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.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 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.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 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.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