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.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 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 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 (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.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 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.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.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

Q66.A straight line cuts off the intercepts $\mathrm{OA}=\mathrm{a}$ and $\mathrm{OB}=\mathrm{b}$ on the positive directions of $\mathrm{x}$-axis and $\mathrm{y}-$ axis respectively. If the perpendicular from origin $\mathrm{O}$ to this line makes an angle of $\frac{\pi}{6}$ with positive direction of $y$-axis and the JEE Main 2023 (30 Jan Shift 1) JEE Main Previous Year Paper area of $\triangle \mathrm{OAB}$ is $\frac{98}{3} \sqrt{3}$, then $\mathrm{a}^2-\mathrm{b}^2$ is equal to: 392 (1) (2) 196 3 (3) 196 (4) 98 3

Q66.Consider: S1: 𝑝⇒𝑞∨𝑝∧~𝑞 is a tautology. JEE Main 2023 (31 Jan Shift 1) JEE Main Previous Year Paper S2: ~p ⇒~q ∧~p ∨q is a contradiction. Then (1) only S2 is correct (2) both S1 and S2 are correct (3) both S1 and S2 are wrong (4) only S1 is correct

Q66.The straight lines 𝑙1 and 𝑙2 pass through the origin and trisect the line segment of the line 𝐿: 9𝑥+ 5𝑦= 45 between the axes. If 𝑚1 and 𝑚2 are the slopes of the lines 𝑙1 and 𝑙2, then the point of intersection of the line 𝑦= ( 𝑚1 + 𝑚2 ) 𝑥 with 𝐿 lies on (1) 𝑦– 2𝑥= 5 (2) 6𝑥+ 𝑦= 10 (3) 𝑦– 𝑥= 5 (4) 6𝑥– 𝑦= 15

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.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 −

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

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 {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.Let ( 𝛼, 𝛽) be the centroid of the triangle formed by the lines 15𝑥- 𝑦= 82, 6𝑥- 5𝑦= - 4 and 9𝑥+ 4𝑦= 17 . Then 𝛼+ 2𝛽 and 2𝛼- 𝛽 are the roots of the equation (1) 𝑥2 - 7𝑥+ 12 = 0 (2) 𝑥2 - 14𝑥+ 48 = 0 (3) 𝑥2 - 13𝑥+ 42 = 0 (4) 𝑥2 - 10𝑥+ 25 = 0

Q66.The absolute difference of the coefficients of x10 and x7 in the expansion of (2x2 + 2x1 ) 11 is equal to (1) 133 −13 (2) 113 −11 (3) 103 −10 (4) 123 −12 Q67. 25190 −19190 −8190 + 2190 is divisible by (1) neither 14 nor 34 (2) 14 but not by 34 (3) 34 but not by 14 (4) both 14 and 34

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 +