Practice Questions

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.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.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 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.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.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 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

Q63.The coefficient of x70 in x2(1 + x)98 + x3(1 + x)97 + x4(1 + x)96 + … + x54(1 + x)46 is 99Cp −46Cq . Then a possible value of p + q is : (1) 55 (2) 83 (3) 61 (4) 68

Q63.The 20th term from the end of the progression 20, 191 181 173 … , - 1291 is :- 4, 2, 4, 4 (1) -118 (2) -110 (3) -115 (4) -100

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.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.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 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

Q64.Let 2nd, 8th and 44th, terms of a non-constant 𝐴. 𝑃. be respectively the 1st, 2nd and 3rd terms of 𝐺. 𝑃. If the first term of A.P. is 1 then the sum of first 20 terms is equal to- (1) 980 (2) 960 (3) 990 (4) 970

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 two straight lines drawn from the origin O intersect the line 3x + 4y = 12 at the points P and Q such that △OPQ is an isosceles triangle and ∠POQ = 90∘ . If l = OP2 + PQ2 + QO2 , then the greatest integer less than or equal to l is : (1) 42 (2) 46 (3) 44 (4) 48

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 sin x = −35 , where π < x < 3π2 , then 80 (tan2 x −cos x) is equal to (1) 108 (2) 109 (3) 18 (4) 19 JEE Main 2024 (08 Apr Shift 1) JEE Main Previous Year Paper

Q64.Let 𝛼, 𝛽, 𝛾, 𝛿∈𝑍 and let 𝐴𝛼, 𝛽, 𝐵1, 0, 𝐶𝛾, 𝛿 and 𝐷1, 2 be the vertices of a parallelogram 𝐴𝐵𝐶𝐷. If 𝐴𝐵= √10 and the points 𝐴 and 𝐶 lie on the line 3𝑦= 2𝑥+ 1, then 2𝛼+ 𝛽+ 𝛾+ 𝛿 is equal to (1) 10 (2) 5 (3) 12 (4) 8

Q64.Let 𝑚 and 𝑛 be the coefficients of seventh and thirteenth terms respectively in the expansion of 3 + 2 3𝑥 2𝑥 3 1 . Then 𝑛 3 is: 𝑚 (1) 4 (2) 1 9 9 1 9 (3) (4) 4 4

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.A line passing through the point A(9, 0) makes an angle of 30° with the positive direction of x-axis. If this line is rotated about A through an angle of 15° in the clockwise direction, then its equation in the new position is (1) y + x = 9 (2) x + y = 9 √3−2 √3−2 (3) x + y = 9 (4) y + x = 9 √3+2 √3+2

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

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