Q: Passage: Let u be a positive integer and f be a real number lying between 0 and 1. Further, ( 2 +1 )^ 10 =u+f and ( 2 -1 )^ 10 =v. Question: What is the multiplicative inverse of ( 2 +1 )^ 20 ?

The multiplicative inverse of ( 2 +1 )^ 20 is 1 ( 2 +1 )^ 20 . Rationalizing the base gives 1 2 +1 = 2 -1 ( 2 +1 ) ( 2 -1 ) = 2 -1. Thus, 1 ( 2 +1 )^ 20 = ( 2 -1 )^ 20 . Given that v = ( 2 -1 )^ 10 . Squaring both sides yields v^2 = ( 2 -1 )^ 20 . Therefore, the multiplicative inverse of ( 2 +1 )^ 20 is v^2. Answer: v^2

Q: If the sum of binomial coefficients in the expansion of (x + y)^n is 256, then the greatest binomial coefficient occurs in which one of the following terms?

The sum of binomial coefficients in the expansion of (x + y)^n is 2^n. 2^n = 256 2^n = 2^8 n = 8 In the expansion of (x + y)^8, the number of terms is n + 1 = 9. The greatest binomial coefficient occurs at the middle term. Since n = 8 is even, the middle term is the ( n 2 + 1)-th term. Middle term = 8 2 + 1 = 5 The greatest binomial coefficient occurs in the fifth term. Answer: Fifth

Q: If k < ( 2 +1)^3 < k+2, where k is a natural number, then what is the value of k?

Let x = ( 2 +1)^3. Let x = I + f, where I is the integer part and 0 Let f' = ( 2 -1)^3. Since 0 Now, evaluate x - f': x - f' = ( 2 +1)^3 - ( 2 -1)^3 x - f' = (2 2 + 6 + 3 2 + 1) - (2 2 - 6 + 3 2 - 1) x - f' = (7 + 5 2 ) - (5 2 - 7) = 14 Since x = I + f, we get I + f - f' = 14, which implies f - f' = 14 - I. Since 14 - I is an integer and -1 Thus, f = f' and I = 14. This gives x = 14 + f, meaning 14 We are given the inequality k Checking the given options: For k = 11, the interval is (11, 13), wh

Q: What is the remainder when 5^ 99 is divided by 13?

To find the remainder when 5^ 99 is divided by 13, we evaluate the powers of 5 modulo 13. 5^1 5 13 5^2 = 25 -1 13 Squaring both sides gives: 5^4 (-1)^2 1 13 We can express 5^ 99 in terms of 5^4 as follows: 5^ 99 = 5^ 96 5^3 = (5^4)^ 24 125 Substituting 5^4 1 13 : 5^ 99 (1)^ 24 125 13 5^ 99 125 13 Dividing 125 by 13 yields 125 = 13 9 + 8. Thus, 125 8 13 . The remainder is 8. Answer: 8

Question 1

A set S contains (2n+1) elements. If the number of subsets of S which contain at most n elements is 1024, then what is the value of n ?

Accepted Answer

The number of elements in the set S is 2n+1. The number of subsets of S containing at most n elements is given by the sum: _ k=0 ^ n ^ 2n+1 C_ k We know the sum of all binomial coefficients is: _ k=0 ^ 2n+1 ^ 2n+1 C_ k = 2^ 2n+1 Using the symmetry property of binomial coefficients, ^ n C_ r = ^ n C_ n-r , we have: _ k=0 ^ n ^ 2n+1 C_ k = _ k=n+1 ^ 2n+1 ^ 2n+1 C_ k Therefore, the sum of the first half of the coefficients is exactly half of the total sum: _ k=0 ^ n ^ 2n+1 C_ k = 2^ 2n+1 2 = 2^ 2n

Question 2

Passage: Let u be a positive integer and f be a real number lying between 0 and 1. Further, ( 2 +1 )^ 10 =u+f and ( 2 -1 )^ 10 =v. Question: Consider the following statements : I. (u+v+f) is an intege

Accepted Answer

Given ( 2 +1)^ 10 = u+f and ( 2 -1)^ 10 = v. Adding both equations, we get: u+f+v = ( 2 +1)^ 10 + ( 2 -1)^ 10 Using the binomial expansion (x+y)^n + (x-y)^n = 2(^ n C_ 0 x^n + ^ n C_ 2 x^ n-2 y^2 + ), we have: u+f+v = 2(^ 10 C_ 0 ( 2 )^ 10 + ^ 10 C_ 2 ( 2 )^8 + + ^ 10 C_ 10 ) Since all powers of 2 are even, the right-hand side is an even integer. Let this integer be I. Therefore, u+f+v = I, which means (u+v+f) is an integer. Thus, Statement I is correct. Since u is given as a positive integer an

Question 3

Passage: Let u be a positive integer and f be a real number lying between 0 and 1. Further, ( 2 +1 )^ 10 =u+f and ( 2 -1 )^ 10 =v. Question: What is the value of u ?

Accepted Answer

Given ( 2 +1)^ 10 = u + f and ( 2 -1)^ 10 = v. Adding the two equations: u + f + v = ( 2 +1)^ 10 + ( 2 -1)^ 10 Using the binomial expansion (x+y)^n + (x-y)^n = 2 [^ n C_ 0 x^n + ^ n C_ 2 x^ n-2 y^2 + ]: u + f + v = 2 [^ 10 C_ 0 ( 2 )^ 10 + ^ 10 C_ 2 ( 2 )^8 + ^ 10 C_ 4 ( 2 )^6 + ^ 10 C_ 6 ( 2 )^4 + ^ 10 C_ 8 ( 2 )^2 + ^ 10 C_ 10 ] u + f + v = 2 [1(32) + 45(16) + 210(8) + 210(4) + 45(2) + 1(1)] u + f + v = 2 [32 + 720 + 1680 + 840 + 90 + 1] u + f + v = 2 [3363] = 6726 Since 0 We are given 0 Since

Question 4

What is the greatest value of r satisfying the inequality ^ 15 C_ r+1 >2 ^ 15 C_r ?

Accepted Answer

Given inequality is ^ 15 C_ r+1 > 2 ^ 15 C_r Using the property ^ n C_ r+1 = n-r r+1 ^ n C_r, we get: 15-r r+1 ^ 15 C_r > 2 ^ 15 C_r Since ^ 15 C_r > 0, we can divide both sides by ^ 15 C_r: 15-r r+1 > 2 15 - r > 2(r + 1) 15 - r > 2r + 2 3r r r Since r must be an integer, the greatest possible value of r is 4. Answer: 4

Question 5

Passage: Let u be a positive integer and f be a real number lying between 0 and 1. Further, ( 2 +1 )^ 10 =u+f and ( 2 -1 )^ 10 =v. Question: What is the value of uv ?

Accepted Answer

Given ( 2 +1 )^ 10 = u+f and ( 2 -1 )^ 10 = v. Multiplying these two equations, we get: (u+f)v = ( 2 +1 )^ 10 ( 2 -1 )^ 10 (u+f)v = (( 2 )^2 - 1^2 )^ 10 = 1^ 10 = 1 uv + fv = 1 uv = 1 - fv Since f is the fractional part, 0 Also, v = ( 2 -1 )^ 10 , and since 0 Because both f and v lie strictly between 0 and 1, their product fv also lies strictly between 0 and 1, i.e., 0 Therefore, uv = 1 - fv must satisfy: 0 Answer: 0<uv<1

Question 6

Passage: Let u be a positive integer and f be a real number lying between 0 and 1. Further, ( 2 +1 )^ 10 =u+f and ( 2 -1 )^ 10 =v. Question: What is the value of (v+f) ?

Accepted Answer

Given ( 2 +1)^ 10 = u+f and ( 2 -1)^ 10 = v. Adding both equations: u+f+v = ( 2 +1)^ 10 + ( 2 -1)^ 10 Expanding using the binomial theorem: u+f+v = 2 [^ 10 C_0 ( 2 )^ 10 + ^ 10 C_2 ( 2 )^8 + + ^ 10 C_ 10 ] The right hand side is an even integer. Let it be 2k. u+f+v = 2k f+v = 2k - u Since 2k and u are integers, f+v must be an integer. We are given 0 Also, 0 Adding the inequalities for f and v: 0 Since f+v is an integer strictly between 0 and 2, the only possible value is 1. Therefore, v+f = 1. A

Question 7

Passage: Let u be a positive integer and f be a real number lying between 0 and 1. Further, ( 2 +1 )^ 10 =u+f and ( 2 -1 )^ 10 =v. Question: What is the multiplicative inverse of ( 2 +1 )^ 20 ?

Accepted Answer

The multiplicative inverse of ( 2 +1 )^ 20 is 1 ( 2 +1 )^ 20 . Rationalizing the base gives 1 2 +1 = 2 -1 ( 2 +1 ) ( 2 -1 ) = 2 -1. Thus, 1 ( 2 +1 )^ 20 = ( 2 -1 )^ 20 . Given that v = ( 2 -1 )^ 10 . Squaring both sides yields v^2 = ( 2 -1 )^ 20 . Therefore, the multiplicative inverse of ( 2 +1 )^ 20 is v^2. Answer: v^2

Question 8

If the sum of binomial coefficients in the expansion of (x + y)^n is 256, then the greatest binomial coefficient occurs in which one of the following terms?

Accepted Answer

The sum of binomial coefficients in the expansion of (x + y)^n is 2^n. 2^n = 256 2^n = 2^8 n = 8 In the expansion of (x + y)^8, the number of terms is n + 1 = 9. The greatest binomial coefficient occurs at the middle term. Since n = 8 is even, the middle term is the ( n 2 + 1)-th term. Middle term = 8 2 + 1 = 5 The greatest binomial coefficient occurs in the fifth term. Answer: Fifth

Question 9

If k < ( 2 +1)^3 < k+2, where k is a natural number, then what is the value of k?

Accepted Answer

Let x = ( 2 +1)^3. Let x = I + f, where I is the integer part and 0 Let f' = ( 2 -1)^3. Since 0 Now, evaluate x - f': x - f' = ( 2 +1)^3 - ( 2 -1)^3 x - f' = (2 2 + 6 + 3 2 + 1) - (2 2 - 6 + 3 2 - 1) x - f' = (7 + 5 2 ) - (5 2 - 7) = 14 Since x = I + f, we get I + f - f' = 14, which implies f - f' = 14 - I. Since 14 - I is an integer and -1 Thus, f = f' and I = 14. This gives x = 14 + f, meaning 14 We are given the inequality k Checking the given options: For k = 11, the interval is (11, 13), wh

Question 10

What is the remainder when 5^ 99 is divided by 13?

Accepted Answer

To find the remainder when 5^ 99 is divided by 13, we evaluate the powers of 5 modulo 13. 5^1 5 13 5^2 = 25 -1 13 Squaring both sides gives: 5^4 (-1)^2 1 13 We can express 5^ 99 in terms of 5^4 as follows: 5^ 99 = 5^ 96 5^3 = (5^4)^ 24 125 Substituting 5^4 1 13 : 5^ 99 (1)^ 24 125 13 5^ 99 125 13 Dividing 125 by 13 yields 125 = 13 9 + 8. Thus, 125 8 13 . The remainder is 8. Answer: 8

Question 11

What is the number of rational terms in the expansion of (3^ 1 2 + 5^ 1 4 )^ 12 ?

Accepted Answer

The general term in the expansion of (3^ 1 2 + 5^ 1 4 )^ 12 is given by: T_ r+1 = ^ 12 C_ r (3^ 1 2 )^ 12-r (5^ 1 4 )^ r T_ r+1 = ^ 12 C_ r 3^ 12-r 2 5^ r 4 For the term to be rational, the exponents of 3 and 5 must be integers. Therefore, 12-r 2 and r 4 must be integers. This requires r to be a multiple of 4. Since 0 r 12, the possible values of r are 0, 4, 8, 12. There are 4 such values of r, which means there are 4 rational terms in the expansion. Answer: 4

Question 12

Which one of the following is the greatest coefficient in the expansion of (1 + x)^ 100 ?

Accepted Answer

The general term in the expansion of (1 + x)^ n is given by T_ r+1 = ^ n C_ r x^ r , where the coefficient of x^ r is ^ n C_ r . For an even integer n, the greatest binomial coefficient is the middle term, which is ^ n C_ n/2 . Given n = 100, which is even, the greatest coefficient is ^ 100 C_ 100/2 = ^ 100 C_ 50 . This corresponds to the coefficient of x^ 50 . Answer: The coefficient of x^ 50

Binomial Theorem — Previous Year Questions & Solutions