Practice Questions

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.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.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.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.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.The number of elements in the set 𝑆= 𝜃∈[0, 2𝜋]: 3cos4𝜃- 5cos2𝜃- 2sin6𝜃+ 2 = 0 is (1) 10 (2) 8 (3) 12 (4) 9

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

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

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.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.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

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

Q66.Consider ellipses 𝐸𝑘: 𝑘𝑥2 + 𝑘2𝑦2 = 1, 𝑘= 1, 2, … , 20. Let 𝐶𝑘 be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse 𝐸𝑘. If 𝑟𝑘 is the radius of the circle 𝐶𝑘, then the value of ∑𝑘=20 1 12 is 𝑟𝑘 (1) 3080 (2) 2870 (3) 3210 (4) 3320

Q66.For the two positive numbers a, b, if a, b and 181 are in a geometric progression, while a1 , 10 and 1b are in an arithmetic progression, then, 16a + 12b is equal to _____ . Q67. ∑6k=0 51−kC3 is equal to (1) 51C4 −45C4 (2) 51C3 −45C3 (3) 52C4 −45C4 (4) 52C3 −45C3