Practice Questions

Q64.If n 1⋅3+2⋅5+3⋅7+....upto terms = 95 then the value of n is α is equal to

Q64.The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is (1) 720 (2) 126(5!)2 (3) 7(360)2 (4) 7(720)2

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

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 (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.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.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.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 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.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.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

Q66.If n+1 1 nCn + n1 nCn−1+. . . + 21 nC1 +n C0 = 102310 then n is equal to (1) 9 (2) 8 (3) 7 (4) 6

Q66.Let the coefficients of three consecutive terms in the binomial expansion of (1 + 2x)n be in the ratio 2 : 5 : 8 . Then the coefficient of the term, which is in the middle of these three terms, is

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

Q66.Let he sum of the coefficient of first three terms in the expansion of (x − x23 ) n; x = 0, n ∈N be 376 . Then, the coefficient of x4 is equal to: π +

Q66.For k ∈N, if the sum of the series 1 + k4 + k28 + 13k3 + 19k4 +. . . . . . is 10, then the value of k is is 1024 times 1011th term from

Q66.The sum of the common terms of the following three arithmetic progressions. 3, 7, 11, 15, … … … … , 399 2, 5, 8, 11, . . . . . . . . . 359 and 2, 7, 12, 17, … … , 197 , is equal to _____ .

Q66.Let α be the constant term in the binomial expansion of (√x − x 32 ) , n ≤15. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x−n is λα, then λ is equal to ________.

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

Q66.The compound statement ( ~ ( 𝑃∧𝑄) ) ∨( ( ~𝑃) ∧𝑄) ⇒( ( ~𝑃) ∧( ~𝑄) ) is equivalent to (1) ( ( ~𝑃) ∨𝑄) ∧( ( ~𝑄) ∨𝑃) (2) ( ~𝑄) ∨𝑃 (3) ( ( ~𝑃) ∨𝑄) ∧( ~𝑄) (4) ( ~𝑃) ∨𝑄