Practice Questions

Q65.Let < an > be a sequence such that a1 + a2+. . . +an = (n+1)(n+2)n2+3n . If 28 ∑10k=1 ak1 p1, p2, . . . pm are the first m prime numbers, then m is equal to JEE Main 2023 (12 Apr Shift 1) JEE Main Previous Year Paper (1) 5 (2) 8 (3) 6 (4) 7

Q65.If the coefficients of 𝑥 and 𝑥2 in ( 1 + 𝑥) 𝑝( 1 - 𝑥) 𝑞 are 4 and -5 respectively, then 2𝑝+ 3𝑞 is equal to (1) 60 (2) 69 (3) 66 (4) 63 𝜋 1 then

Q65.If gcd(m, n) = 1 and 12 −22 + 32 −42+. . . . +(2021)2 −(2022)2 + (2023)2 = 1012m2n then m2 −n2 is equal to (1) 240 (2) 200 (3) 220 (4) 180

Q65.Fractional part of the number 42022 is equal to 15 (1) 8 (2) 4 15 15 (3) 14 (4) 1 15 15 n 6

Q65.A line segment 𝐴𝐵 of length 𝜆 moves such that the points 𝐴 and 𝐵 remain on the periphery of a circle of radius 𝜆. Then the locus of the point, that divides the line segment 𝐴𝐵 in the ratio 2: 3, is a circle of radius (1) 3 (2) 2 5𝜆 3𝜆 (3) √19 𝜆 (4) √19 𝜆 5 7 JEE Main 2023 (10 Apr Shift 1) JEE Main Previous Year Paper

Q65.Let a, b, c and d be positive real numbers such that a + b + c + d = 11 . If the maximum value of a5b3c2d is 3750β, then the value of β is (1) 90 (2) 110 (3) 55 (4) 108

Q65.Let a1, a2, a3, … . be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24 , then a1a9 + a2a4a9 + a5 + a7 is equal to

Q65.The coefficient of 𝑥5 in the expansion of 2𝑥3 - 1 5 is 3𝑥2 (1) 80 (2) 9 9 (3) 8 (4) 26 3

Q65.The combined equation of the two lines 𝑎𝑥+ 𝑏𝑦+ 𝑐= 0 and 𝑎'𝑥+ 𝑏'𝑦+ 𝑐' = 0 can be written as 𝑎𝑥+ 𝑏𝑦+ 𝑐𝑎'𝑥+ 𝑏'𝑦+ 𝑐' = 0. The equation of the angle bisectors of the lines represented by the equation 2𝑥2 + 𝑥𝑦- 3𝑦2 = 0 is (1) 3𝑥2 + 5𝑥𝑦+ 2𝑦2 = 0 (2) 𝑥2 - 𝑦2 + 10𝑥𝑦= 0 (3) 3𝑥2 + 𝑥𝑦- 2𝑦2 = 0 (4) 𝑥2 - 𝑦2 - 10𝑥𝑦= 0

Q65.If 2𝑛𝐶3: 𝑛𝐶3 = 10: 1, then the ratio 𝑛2 + 3𝑛: 𝑛2 - 3𝑛+ 4 is (1) 35: 16 (2) 27: 11 (3) 65: 37 (4) 2: 1

Q65.The 8th common term of the series S1 = 3 + 7 + 11 + 15 + 19 + … S2 = 1 + 6 + 11 + 16 + 21 + … . is + y = + [t] denotes the greatest integer ≤t, then

Q65.The coefficient of x−6 , in the expansion of ( 4x5 + 2x25 ) 9 5 9 x 2 4 is −84 and the coefficient of x−3l is 2αβ where 2 − xl

Q65.If tan15° + + + tan195° = 2a, then the value of 𝑎+ is : tan75° tan105° 𝑎 (1) 4 (2) 4 - 2√3 (3) 2 (4) 5 - 3 2√3

Q65.Let 0 < z < y < x be three real numbers such that x1 , 1y , 1z are in an arithmetic progression and x, √2y, z are in a geometric progression. If xy + yz + zx = 3 xyz, then 3(x + y + z)2 is equal to √2

Q65.If (30C1)2 + 2(30C2)2 + 3(30C3)2. . . . . . . . . . 30(30C30)2 = (30!)2α60! , then (1) 30 (2) 60 (3) 15 (4) 10

Q65.Let f(x) = 2xn + λ, λ ∈R, n ∈N, and f(4) = 133 , f(5) = 255 . Then the sum of all the positive integer divisors of (f(3) −f(2)) is (1) 61 (2) 60 (3) 58 (4) 59

Q65.If the maximum distance of normal to the ellipse 𝑥2 + 𝑦2 = 1, 𝑏< 2, from the origin is 1 , then the eccentricity 4 𝑏2 of the ellipse is: (1) 1 (2) √3 √2 2 (3) 1 (4) √3 2 4

Q65.Let (a + bx + cx2)10 = ∑20i=10 pixi, a, b, c ∈N. If p1 = 20 and p2 = 210, then 2(a + b + c) is equal to (1) 6 (2) 15 (3) 12 (4) 8 JEE Main 2023 (15 Apr Shift 1) JEE Main Previous Year Paper

Q65.The sum ∑∞n=1 2n2+3n+4(2n)! is equal to : (1) 11e 2 + 2e7 (2) 13e4 + 4e5 −4 (3) 11e 2 + 2e7 −4 (4) 13e4 + 4e5

Q65.Let A1, A2, A3 be the three A.P. with the same common difference d and having their first terms as A, A + 1, A + 2, respectively. Let a, b, c be the 7th , 9th , 17th terms of A1, A2, A3 , respectively such that a 7 1 2b 17 1 + 70 = 0 . If a = 29, then the sum of first 20 terms of an AP whose first term is c −a −b and c 17 1 common difference is d , is equal to _____ . 12 JEE Main 2023 (25 Jan Shift 1) JEE Main Previous Year Paper ar ) is equal to

Q65.Let a1 = b1 = 1 and an = an−1 + (n −1), bn = bn−1 + an−1, ∀n ≥2. If S = ∑10n=1( 2nbn ) and T = ∑8n=1 2n−1n then 27(2S −T) is equal to

Q65.The largest natural number n such that 3n divides 66! is _______

Q66.If the constant term in the binomial expansion of ( ) β < 0 is an odd number, then |αl −β| is equal to _____ .

Q66.Let {ak} and {bk}, k ∈N , be two G.P.s with common ratio r1 and r2 respectively such that a1 = b1 = 4 and r1 < r2 . Let ck = ak + bk, k ∈N . If c2 = 5 and c3 = 134 then ∑∞k=1 ck −(12a6 + 8 b4) is equal to

Q66.If (20)19 + 2(21)(20)18 + 3(21)2(20)17+. . . +20(21)19 = k(20)19 , then k is equal to _____. 11 are equal, then −