Practice Questions

Q62.The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + . . . is (1) 3520 (2) 3450 (3) 3250 (4) 3420

Q62.If 𝑎𝑛= 4𝑛2 - 16𝑛+ 15, then 𝑎1 + 𝑎2 + … . + 𝑎25 is equal to: (1) 51 (2) 49 144 138 50 52 (3) (4) 141 147 1 15

Q62.If for z = α + iβ, |z + 2| = z + 4(1 + i), then α + β and αβ are the roots of the equation (1) x2 + 3x −4 = 0 (2) x2 + 7x + 12 = 0 (3) x2 + x −12 = 0 (4) x2 + 2x −3 = 0

Q62.Let z1 = 2 + 3i and z2 = 3 + 4i . The set S = {z ∈C : |z −z1|2 −|z −z2|2 = |z1 −z2|2} represents a (1) straight line with sum of its intercepts on the (2) hyperbola with the length of the transverse axis 7 coordinate axes equals 14 (3) straight line with the sum of its intercepts on the (4) hyperbola with eccentricity 2 coordinate axes equals −18

Q62.Let C be the circle in the complex plane with centre z0 = 12 (1 + 3i) and radius r = 1. Let z1 = 1 + i and the complex number z2 be outside circle C such that |z1 −z0||z2 −z0| = 1 . If z0, z1 and z2 are collinear, then the smaller value of |z2|2 is equal to (1) 5 (2) 7 2 2 (3) 13 (4) 3 2 2

Q62.Let z be a complex number such that z−2iz+i = 2, z ≠−i. Then z lies on the circle of radius 2 and centre (1) (2, 0) (2) (0, 2) (3) (0, 0) (4) (0, −2)

Q62.Let A = {θ ∈(0, 2π) : 1+2i1−i sinsinθθ is purely imaginary} Then the sum of the elements is in A is (1) 4π (2) 3π (3) π (4) 2π

Q62.Let 𝑤1 be the point obtained by the rotation of 𝑧1 = 5 + 4𝑖 about the origin through a right angle in the anticlockwise direction, and 𝑤2 be the point obtained by the rotation of 𝑧2 = 3 + 5𝑖 about the origin through a right angle in the clockwise direction. Then the principal argument 𝑤1 - 𝑤2 is equal to (1) 𝜋- tan-18 (2) -𝜋+ tan-133 9 5 (3) -𝜋+ tan-18 (4) 𝜋- tan-133 9 5

Q62.Let the first term a and the common ratio 𝑟 of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to (1) 241 (2) 231 (3) 210 (4) 220 1 13 1 13

Q62.For three positive integers 𝑝, 𝑞, 𝑟, 𝑥𝑝𝑞2 = 𝑦𝑞𝑟= 𝑧𝑝2𝑟 and 𝑟= 𝑝𝑞+ 1 such that 1 3, 3log𝑦𝑥, 3 log𝑧𝑦, 7log𝑥𝑧 are in A.P. with common difference 2. The 𝑟- 𝑝- 𝑞 is equal to (1) 2 (2) 6 (3) 12 (4) -6

Q62.If the center and radius of the circle = 2 are respectively 𝛼, 𝛽 and 𝛾, then 3𝛼+ 𝛽+ 𝛾 is equal to 𝑧- 3 (1) 11 (2) 9 (3) 10 (4) 12

Q63.Let x and y be distinct integers where 1 ≤x ≤25 and 1 ≤y ≤25. Then, the number of ways of choosing x and y, such that x + y is divisible by 5 , is _____ .

Q63.All words, with or without meaning, are made using all the letters of the word 𝑀𝑂𝑁𝐷𝐴𝑌. These words are written as in a dictionary with serial numbers. The serial number of the word 𝑀𝑂𝑁𝐷𝐴𝑌 is JEE Main 2023 (13 Apr Shift 2) JEE Main Previous Year Paper (1) 327 (2) 328 (3) 324 (4) 326

Q63.The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is (1) 16800 (2) 33600 (3) 18000 (4) 14800

Q63.The number of five-digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to (1) 132 (2) 120 (3) 72 (4) 96

Q63.Let 𝑎1, 𝑎2, 𝑎3, . . . . , 𝑎𝑛 be n positive consecutive terms of an arithmetic progression. If 𝑑> 0 is its common difference, then lim 𝑑 1 + 1 + … + 1 is 𝑛→∞√ 𝑛 √𝑎1 + √𝑎2 √𝑎2 + √𝑎3 √𝑎𝑛- 1 + √𝑎𝑛 (1) 1 (2) √𝑑 √𝑑 (3) 1 (4) 2 𝑛

Q63.If all the six digit numbers x1x2x3x4x5x6 with 0 < x1 < x2 < x3 < x4 < x5 < x6 are arranged in the increasing order, then the sum of the digits in the 72th number is _______.

Q63.The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is

Q63.If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is (6!)k then k is equal to (1) 2835 (2) 5670 (3) 1890 (4) 945

Q63.All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is (1) 576 (2) 578 (3) 580 (4) 582

Q63.If 𝑆𝑛= 4 + 11 + 21 + 34 + 50 + … to 𝑛 terms, then 60𝑆29 - 𝑆9 is equal to (1) 223 (2) 226 (3) 220 (4) 227

Q63.Number of integral solutions to the equation x + y + z = 21 , where x ≥1, y ≥3, z ≥4 , is equal to _____ .

Q63.Let a1, a2, a3, … …. be an A.P. If a7 = 3, the product (a1a4) is minimum and the sum of its first n terms is zero then n! −4an(n+2) is equal to (1) 381 (2) 9 4 (3) 33 (4) 24 4

Q63.If the coefficient of 𝑥7 in 𝑎𝑥- and the coefficient of 𝑥-5 in 𝑎𝑥+ are equal, then 𝑎4𝑏4 is equal to: 𝑏𝑥2 𝑏𝑥2 (1) 11 (2) 44 (3) 22 (4) 33. 𝜋 2𝜋 4𝜋 8𝜋 16𝜋 Q64.96 cos cos cos cos cos is equal to 33 33 33 33 33 (1) 3 (2) 1 (3) 4 (4) 2

Q63.Let S = {z ∈C −{i, 2i} z2−3iz−2 ∈R}. JEE Main 2023 (11 Apr Shift 2) JEE Main Previous Year Paper