Practice Questions

Q63.For x ⩾0, the least value of K, for which 41+x + 41−x, K2 , 16x + 16−x are three consecutive terms of an A.P., is equal to : (1) 8 (2) 4 (3) 10 (4) 16

Q63.If the set R = {(a, b) : a + 5b = 42, a, b ∈N} has m elements and ∑mn=1 (1 −in!) = x + iy, where i = √−1 , then the value of m + x + y is (1) 12 (2) 4 (3) 8 (4) 5

Q63.Suppose 28 - 𝑝, 𝑝, 70 - 𝛼, 𝛼 are the coefficient of four consecutive terms in the expansion of ( 1 + 𝑥) 𝑛. Then the value of 2𝛼- 3𝑝 equals (1) 7 (2) 10 (3) 4 (4) 6 𝜋

Q63.The sum of the series + + + . ... up to 10 terms is 1 −3 ⋅12 + 14 1 −3 ⋅22 + 24 1 −3 ⋅32 + 34 (1) 45 (2) - 45 109 109 55 55 (3) (4) - 109 109

Q63.Let three real numbers a, b, c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a > 10 and the arithmetic mean of a, b and c is 8, then the cube of the geometric mean of a, b and c is (1) 128 (2) 316 (3) 120 (4) 312

Q63.Suppose θϵ [0, π4 ] is a solution of 4 cos θ −3 sin θ = 1. Then cos θ is equal to : (1) 4 (2) 6+√6 (3√6+2) (3√6+2) (3) 4 (4) 6−√6 (3√6−2) (3√6−2)

Q63.Let A = {n ∈[100, 700] ∩N : n is neither a multiple of 3 nor a multiple of 4 }. Then the number of elements in A is (1) 290 (2) 280 (3) 300 (4) 310

Q63.If loge a, loge b, loge c are in an A. P. and loge a −loge 2b, loge 2b −loge 3c, loge 3c −loge a are also in an A. P., then a : b : c is equal to (1) 9 : 6 : 4 (2) 16 : 4 : 1 (3) 25 : 10 : 4 (4) 6 : 3 : 2

Q63.Let 𝑆𝑛 denote the sum of the first n terms of an arithmetic progression. If 𝑆10 = 390 and the ratio of the tenth and the fifth terms is 15 : 7, then 𝑆15 −𝑆5 is equal to: (1) 800 (2) 890 (3) 790 (4) 690 1 18 1 1

Q63.If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to (1) 7 (2) 4 (3) 5 (4) 6

Q63.If 𝑛 is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then 𝑛 is equal to: (1) 47 (2) 53 (3) 51 (4) 43

Q63.In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 70 and the 3 product of the third and fifth terms is 49 . Then the sum of the 4th , 6th and 8th terms is equal to : (1) 96 (2) 91 (3) 84 (4) 78

Q63.Let a, ar, ar2 , be an infinite G.P. If ∑∞n=0 arn = 57 and ∑∞n=0 a3r3n = 9747, then a + 18r is equal to (1) 46 (2) 38 (3) 31 (4) 27 is

Q63.If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at 315th position in this arrangement is : (1) NRAGUP (2) NRAPUG (3) NRAPGU (4) NRAGPU

Q63.There are 5 points P1, P2, P3, P4, P5 on the side AB, excluding A and B, of a triangle ABC . Similarly there are 6 points P6, P7, … , P11 on the side BC and 7 points P12, P13, … , P18 on the side CA of the triangle. The number of triangles, that can be formed using the points P1, P2, … , P18 as vertices, is : (1) 776 (2) 796 (3) 751 (4) 771

Q64.The sum of the coefficient of x2/3 and x−2/5 in the binomial expansion of (x2/3 + 12 x−2/5) 9 (1) 21/4 (2) 63/16 (3) 19/4 (4) 69/16

Q64.Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then : (1) P2 = 6√3Q (2) P2 = 36√3Q (3) P = 36√3Q2 (4) P2 = 72√3Q

Q64.If the term independent of x in the expansion of (√ax2 + 2x31 )10 is 105 , then a2 is equal to : (1) 2 (2) 4 (3) 6 (4) 9 JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper cos 36∘+5 sin 18∘

Q64.Let |cos θ cos(60 −θ) cos(60 + θ)| ≤18 , θϵ[0, 2π]. Then, the sum of all θϵ[0, 2π], where cos 3θ attains its maximum value, is : (1) 15π (2) 18π (3) 6π (4) 9π

Q64.If the constant term in the expansion of 12 + , x ≠0, is α × 28 × 5√3, then 25α is equal to : ( 5√3x 2x ) 3√5 (1) 724 (2) 742 (3) 639 (4) 693

Q64. n−1Cr = (k2 −8)nCr+1 if and only if : (1) 2√2 < k ≤3 (2) 2√3 < k ≤3√2 (3) 2√3 < k < 3√3 (4) 2√2 < k < 2√3 JEE Main 2024 (27 Jan Shift 1) JEE Main Previous Year Paper

Q64.Let the first three terms 2, p and q , with q ≠2, of a G.P. be respectively the 7th , 8th and 13th terms of an A.P. If the 5th term of the G.P. is the nth term of the A.P., then n is equal to: (1) 163 (2) 151 (3) 177 (4) 169

Q64.If the coefficients of x4, x5 and x6 in the expansion of (1 + x)n are in the arithmetic progression, then the maximum value of n is: (1) 7 (2) 21 (3) 28 (4) 14

Q64.If each term of a geometric progression a1, a2, a3, … with a1 = 18 and a2 ≠a1 , is the arithmetic mean of the next two terms and Sn = a1 + a2 + … + an , then S20 −S18 is equal to (1) 215 (2) −218 (3) 218 (4) −215

Q64.If α, −π2 < α < π2 is the solution of 4 cos θ + 5 sin θ = 1, then the value of tan α is (1) 10−√10 (2) 10−√10 6 12 (3) √10−10 (4) √10−10 12 6