Practice Questions

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

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 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.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 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.Let the sixth term in the binomial expansion of (√2log2(10−3x) If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of x is _____ .

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.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.Let P(a1, b1) and Q(a2, b2) be two distinct points on a circle with center C(√2, √3). Let and OC be perpendicular to both CP and CQ. If the area of the triangle OCP is √35 , then a21 + a22 + b21 + b22 2 is equal to __________

Q68.If lim = 17, then 5𝑎2 + 𝑏2 is equal to 𝑥→0 1 - cos ( 2𝑥) (1) 64 (2) 72 (3) 68 (4) 76

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 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.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

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.If the line l1 : 3y −2x = 3 is the angular bisector of the lines l2 : x −y + 1 = 0 and l3 : αx + βy + 17 = 0 , then α2 + β2 −α −β is equal to ............

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.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 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 m and n respectively are the numbers of positive and negative value of θ in the interval [−π, π] that satisfy the equation cos 2θ cos 2θ = cos 3θ cos 9θ2 , then mn is equal to _____ .

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 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.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.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