Practice Questions

Q58.Number of compounds from the following which will not dissolve in cold NaHCO3 and NaOH solutions but will dissolve in hot NaOH solution is _____ . JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper

Q59.All structures given below are of vitamin C. Most stable of them is: (1) (2) (3) (4)

Q60.Compound A, C5H10O5 , given a tetraacetate with AC2 O and oxidation of A with Br2 −H2O gives an acid, C5H10O6 . Reduction of A with HI gives isopentane. The possible structure of A is : JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper (1) (2) (3) (4)

Q61.Let α, β be the roots of the quadratic equation x2 + √6x + 3 = 0. Then α15+β15+α10+β10α23+β23+α14+β14 (1) 81 (2) 9 (3) 72 (4) 729

Q61.The number of real roots of the equation √𝑥2 - 4𝑥+ 3 + √𝑥2 - 9 = √4𝑥2 - 14𝑥+ 6, is: (1) 0 (2) 1 (3) 3 (4) 2

Q62.Let a, b be two real numbers such that ab < 0 . If the complex number 1+aib+i is of unit modulus and a + ib lies on the circle |z −1| = |2z| , then a possible value of 1+[a]4b , where [t] is greatest integer function, is : (1) 0 (2) −1 (3) 1 (4) 21

Q63.The sum to 10 terms of the series 1 2 3 + + + … is :- 1 + 12 + 14 1 + 22 + 24 1 + 32 + 34 59 55 (1) (2) 111 111 (3) 56 (4) 58 111 111

Q64.Let 𝑥1, 𝑥2, … , 𝑥100 be in an arithmetic progression, with 𝑥1 = 2 and their mean equal to 200 . If 𝑦𝑖= 𝑖𝑥𝑖- 𝑖, 1 ≤𝑖≤100, then the mean of 𝑦1, 𝑦2, … , 𝑦100 is (1) 10100 (2) 10101 . 50 (3) 10049 . 50 (4) 10051 . 50

Q64.Let a circle 𝐶1 be obtained on rolling the circle 𝑥2 + 𝑦2 - 4𝑥- 6𝑦+ 11 = 0 upwards 4 units on the tangent T to it at the point 3, 2. Let 𝐶2 be the image of 𝐶1 in 𝑇. Let 𝐴 and 𝐵 be the centers of circles 𝐶1 and 𝐶2 respectively, and 𝑀 and 𝑁 be respectively the feet of perpendiculars drawn from 𝐴 and 𝐵 on the 𝑥-axis. Then the area of the trapezium AMNB is: (1) 22 + √2 (2) 41 + √2 (3) 3 + 2√2 (4) 21 + √2

Q64.Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is _____ .

Q64.Let a, b, c > 1, a3, b3 and c3 be in A. P. and loga b, logc a and logb c be in G. P. If the sum of first 20 terms of an A. P., whose first term is a+4b+c3 and the common difference is a−8b+c10 is −444, then abc is equal to (1) 343 (2) 216 (3) 343 (4) 125 8 8

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.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.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 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

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

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.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

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 S = {θ ∈[0, 2π) : tan(πcosθ) + tan(πsinθ) = 0} , then ∑θ∈S sin2(θ 4 ) is equal to

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.Let the centre of a circle 𝐶 be 𝛼, 𝛽 and its radius 𝑟 < 8. Let 3𝑥+ 4𝑦= 24 and 3𝑥– 4𝑦= 32 be two tangents and 4𝑥+ 3𝑦= 1 be a normal to 𝐶. Then ( 𝛼 - 𝛽+ 𝑟) is equal to (1) 7 (2) 5 (3) 6 (4) 9 𝑒𝑎𝑥- cos(𝑏𝑥) - 𝑐𝑥𝑒-𝑐𝑥 2

Q68.The set of all values of a2 for which the line x + y = 0 bisects two distinct chords drawn from a point P( 1+a2 , 1−a2 ) on the circle 2x2 + 2y2 −(1 + a)x −(1 −a)y = 0 , is equal to : (1) (8, ∞) (2) (0, 4] (3) (4, ∞) (4) (2, 12] JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper

Q68.Let K be the sum of the coefficients of the odd powers of x in the expansion of (1 + x)99 . Let a be the middle 200 1 200C99K 2lm + = n , where m and n are odd numbers, then the ordered term in the expansion of (2 √2 ) . If a pair (l, n) is equal to: (1) (50, 51) (2) (51, 99) (3) (50, 101) (4) (51, 101)

Q68.The points of intersection of the line ax + by = 0 , (a ≠b) and the circle x2 + y2 −2x = 0 are A(α, 0) and B(1, β). The image of the circle with AB as a diameter in the line x + y + 2 = 0 is : (1) x2 + y2 + 5x + 5y + 12 = 0 (2) x2 + y2 + 3x + 5y + 8 = 0 (3) x2 + y2 + 3x + 3y + 4 = 0 (4) x2 + y2 −5x −5y + 12 = 0 y = mx + c, m > 0, of the curves x = 2y2