Practice Questions

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.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.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 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.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 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.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 −

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.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 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 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.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.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.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 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.Let 𝑆= 𝑥∈- 𝜋 (𝛽- 14 ) 2 is equal to 2, 2: 91 - tan2𝑥+ 9tan2𝑥= 10 and 𝛽= ∑𝑥∈𝑆tan2 3,𝑥 6 (1) 16 (2) 8 (3) 64 (4) 32

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.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.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.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 _____ .

Q67.If the co-efficient of x9 in 11 11 − βx3 1 ) are equal, then (αβ)2 is + βx1 ) and the co-efficient of x−9 in (αx (αx3 equal to : f

Q67.The sum, of the coefficients of the first 50 terms in the binomial expansion of (1 −x)100, is equal to (1) 101C50 (2) 99C49 (3) −101C50 (4) −99C49

Q67.Let 𝐴 be the point 1, 2 and 𝐵 be any point on the curve 𝑥2 + 𝑦2 = 16. If the centre of the locus of the point 𝑃, which divides the line segment 𝐴 𝐵 in the ratio 3: 2 is the point 𝐶𝛼, 𝛽, then the length of the line segment 𝐴𝐶 is (1) 3√5 (2) 4√5 5 5 (3) 2√5 (4) 6√5 5 5

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