Practice Questions

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

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

Q64.If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of 4√2 + 4 1 is √3 √6: 1, then the third term from the beginning is: (1) 30√2 (2) 30√3 (3) 60√2 (4) 60√3

Q64.Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1 . Then the serial number of 35337 is

Q64.Let A1 and A2 be two arithmetic means and G1, G2 and G3 be three geometric means of two distinct positive numbers. Then G41 + G42 + G43 + G21G23 is equal to (1) (A1 + A2)2G1G3 (2) 2(A1 + A2)G1G3 (3) (A1 + A2)G21G23 (4) 2(A1 + A2)G21G23

Q64.If n 1⋅3+2⋅5+3⋅7+....upto terms = 95 then the value of n is α is equal to

Q64.The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is _____ .

Q64.Let the number ( 22 2022 + ( 2022 22 leave the remainder α when divided by 3 and β when divided by 7 ) ) . Then (α2 + β2 ) is equal to (1) 20 (2) 13 (3) 5 (4) 10

Q64.The total number of 4 -digit numbers whose greatest common divisor with 54 is 2 , is

Q64.The sum 12 −2. 32 + 3. 52 −4. 72 + 5. 92 −… . . +15. 292 is _____ . , is

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.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 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.Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed? = p1 p2 p3 . . . pm , where

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

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.The number of 4 -letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is _____.

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.The coefficient of 𝑥301 in 1 + 𝑥500 + 𝑥1 + 𝑥499 + 𝑥21 + 𝑥498 + … . . + 𝑥500 is: (1) 501𝐶302 (2) 500𝐶301 (3) 500𝐶300 (4) 501𝐶200 1 1 1

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.Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is _____ .

Q64.The sum to 20 terms of the series 2 ⋅22 −32 + 2 ⋅42 −52 + 2 ⋅62−. . . . . . . . . . . . is equal to __________.

Q64.Let an be nth term of the series 5 + 8 + 14 + 23 + 35 + 50+. . . . . . .and Sn = ∑nk=1 ak . Then S30 −a40 is equal to (1) 11310 (2) 11260 (3) 11290 (4) 11280

Q65.The 8th common term of the series S1 = 3 + 7 + 11 + 15 + 19 + … S2 = 1 + 6 + 11 + 16 + 21 + … . is + y = + [t] denotes the greatest integer ≤t, then