Practice Questions

Q63.The value of ∑𝑟=22 0 22𝐶𝑟· 23𝐶𝑟 is (1) 45𝐶23 (2) 44𝐶23 (3) 45𝐶24 (4) 44𝐶22

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

Q63.The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is (1) 6 (2) 12 (3) 120 (4) 72

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.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.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

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.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 s1, s2, s3. . . . , s10 respectively be the sum of 12 terms of 10 A. Ps whose first terms are 1, 2, 3, . . . . , 10 and the common differences are 1, 3, 5, . . . , 19 respectively. Then ∑10i=1 si is equal to (1) 7220 (2) 7360 (3) 7260 (4) 7380

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.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 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.The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is : (1) 89 (2) 84 (3) 86 (4) 79

Q63.The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is (1) 21 (2) 20 (3) 22 (4) 18

Q63.The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is (1) 120 (2) 168 (3) 220 (4) 48 13+23+33......upto n terms

Q63.If the coefficient of 𝑥15 in the expansion of 𝑎𝑥3 + 1 is equal to the coefficient of 𝑥-15 in the expansion of 𝑏𝑥 3 1 15 1 𝑎𝑥 3 - , where 𝑎 and 𝑏 are positive real numbers, then for each such ordered pair 𝑎, 𝑏: 𝑏𝑥3 (1) 𝑎= 𝑏 (2) 𝑎𝑏= 1 (3) 𝑎= 3𝑏 (4) 𝑎𝑏= 3

Q63.The number of triplets 𝑥, 𝑦, 𝑧 where 𝑥, 𝑦, 𝑧 are distinct non negative integers satisfying 𝑥+ 𝑦+ 𝑧= 15, is (1) 80 (2) 136 (3) 114 (4) 92

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.If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is 9 (1) 7 (2) 2 (3) 3 (4) 14

Q64.Let 𝑎1, 𝑎2, 𝑎3, … . be a G.P. of increasing positive numbers. Let the sum of its 6th and 8th terms be 2 and the + 𝑎4𝑎4 + 𝑎6 is equal to product of its 3rd and 5th terms be 19. Then 6𝑎2 (1) 3 (2) 3√3 (3) 2 (4) 2√2

Q64.Let a tangent to the curve 𝑦2 = 24𝑥 meet the curve 𝑥𝑦 = 2 at the points 𝐴 and 𝐵. Then the mid- points of such line segments 𝐴𝐵 lie on a parabola with the (1) directrix 4𝑥= 3 (2) directrix 4𝑥= - 3 3 (3) Length of latus rectum (4) Length of latus rectum 2 2 Q65. 1 1 1 1 sin2𝑡 𝑡→01lim sin 2𝑡+ 2 sin 2𝑡+ 3 sin 2𝑡. . . . . . 𝑛 sin 2𝑡 is equal to (1) 𝑛2 + 𝑛 (2) 𝑛 𝑛𝑛+ 1 (3) (4) 𝑛2 2

Q64.If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is (1) 103 (2) 102 (3) 101 (4) 104

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.The value of 1 1 1 1 1 + + + … . + + is 1!50! 3!48! 5!46! 49!2! 51!1! (1) 250 (2) 250 50! 51! (3) 251 (4) 251 51! 50!

Q64.The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is (1) 720 (2) 126(5!)2 (3) 7(360)2 (4) 7(720)2