Practice Questions

Q66.Let SK = 1+2+...+KK and ∑nj=1 S 2j = An (Bn2 + Cn + D) where A, B, C, D ∈ N and A Has least value then (1) A + C + D is not divisible by D (2) A + B = 5(D −C) (3) A + B + C + D is divisible by 5 (4) A + B is divisible by D

Q66.Let x = 13 9 13) and (7√2 9) . If (8√3 (1) [x] + [y] is even (2) [x] is odd but [y] is even (3) [x] is even but [y] is odd (4) [x] and [y] are both odd Q67. 50th root of a number x is 12 and 50th root of another number y is 18 . Then the remainder obtained on dividing (x + y) by 25 is ________. O be the origin

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

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.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.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.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 ar is the coefficient of x10−r in the Binomial expansion of (1 + x)10 , then ∑10r=1 r3( ar−1 2 (1) 4895 (2) 1210 (3) 5445 (4) 3025

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.Let the ellipse 𝐸: 𝑥2 + 9𝑦2 = 9 intersect the positive 𝑥- and 𝑦-axes at the points 𝐴 and 𝐵 respectively. Let the major axis of 𝐸 be a diameter of the circle 𝐶. Let the line passing through 𝐴 and 𝐵 meet the circle 𝐶 at the 𝑚 point 𝑃. If the area of the triangle with vertices 𝐴, 𝑃 and the origin 𝑂 is 𝑛, where 𝑚 and 𝑛 are coprime, then 𝑚- 𝑛 is equal to (1) 16 (2) 15 (3) 17 (4) 18

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.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.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.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.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.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.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.If the orthocentre of the triangle, whose vertices are 1, 2, 2, 3 and 3, 1 is 𝛼, 𝛽, then the quadratic equation whose roots are 𝛼+ 4𝛽 and 4𝛼+ 𝛽, is (1) 𝑥2 - 19𝑥+ 90 = 0 (2) 𝑥2 - 18𝑥+ 80 = 0 (3) 𝑥2 - 22𝑥+ 120 = 0 (4) 𝑥2 - 20𝑥+ 99 = 0

Q66.If (α, β) is the orthocenter of the triangle ABC with vertices A(3, –7), B(–1, 2) and C(4, 5), then 9α −6β + 60 is equal to (1) 25 (2) 35 (3) 30 (4) 40

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.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

Q67.Let 𝑦= 𝑥+ 2, 4𝑦= 3𝑥+ 6 and 3𝑦= 4𝑥+ 1 be three tangent lines to the circle ( 𝑥- ℎ) 2 + ( 𝑦- 𝑘) 2 = 𝑟2. Then ℎ+ 𝑘 is equal to : (1) 5 (2) 5 ( 1 + √2 ) (3) 6 (4) 5√2

Q67.Let PQ be a focal chord of the parabola y2 = 36x of length 100, making an acute angle with the positive x− axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line PQ? (1) (−6, 45) (2) (6, 29) (3) (3, 33) (4) (−3, 43) y2 + 4 = 1 meet the y−axis at the points A

Q67.If the 1011th term from the end in the binomial expansion of ( 4x5 − 2x5 ) 2022 the beginning, then 32|x| is equal to (1) 15 (2) 10 (3) 12 (4) 8

Q67.The constant term in the expansion of 5 + x71 + 3x2) is _____ . (2x