# Mathematics Questions and Answers – Binomial Theorem – General and Middle Terms

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Binomial Theorem – General and Middle Terms”.

1. What is the general term of (x – y)xy?
a) x – yCr (xxy – r . yr)
b) xyCr (xx – y – r . -yr)
c) xyCr (xxy – r . -yr)
d) x – yCr (xx – y – r . yr)

Explanation: The general term of a binomial series is given by nCr an – r br.
Here a = x, b = -y and n = xy
Therefore the general term is given by xyCr (xxy – r . -yr).

2. What is the value of n, if the coefficients of the second term of (x – y)3 is equal to the third term of the expansion (x + y)n?
a) –2
b) 3
c) 4
d) 5

Explanation: Coefficient of the second term of (x – y)3 is 3C1 and the coefficient of the third term of the expansion (x + y)n is nC2.
3C1 = nC2
3 = $$\frac{n!}{2!(n – 2)!}$$
6 = $$\frac{n(n-1)(n-2)!}{(n – 2)!}$$
6 = n2 – n
n2 – n – 6 = 0
n2 – 3n + 2n – 6 = 0
(n – 3) (n + 2) = 0
n = 3, – 2
Since n cannot be negative, n = 3.

3. Which term will be the middle term of (xyz – x)2n?
a) (n + 1)th term
b) (n + 2)th term
c) nth term
d) (n – 1)th term

Explanation: Clearly 2n is an even number and the binomial has 2n + 1 terms.
The middle term for a binomial with even power, is the term equal to (n/2 + 1) where n is number of terms.
In this case, (2n/2 + 1) = n + 1.
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4. What is the middle term of (4 + 2x)6?
a) 11240 x2
b) 10240 x3
c) 12240 x4
d) 10340 x4

Explanation: The middle term will be the 4th term
4th term = 6C3 (4)6 – 3(2x)3
= 20 (64) (8x3)
= 10240 x3

5. What is the middle term of (x2 + x)3?
a) 3x4
b) 6x4
c) 4x4
d) 3x6

Explanation:
Since the power is odd, there will be even number of terms and two middle terms.
r = $$\frac{n + 1}{2}$$ and r = $$\frac{n – 1}{2}$$ Here n = 3
Therefore, r = 2 and r = 1.
When r = 2, 3C2 (x2)3 – 2(x)2 = 3x4
When r = 1, 3C1 (x2)3 – 1(x)1 = 3x5
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6. Which of the following values of n are possible, if the middle term of (x + 3y)n is the fifth term.
a) 6, 7 or 8
b) 7, 8 or 10
c) 7, 8 or 9
d) 8, 9 or 10

Explanation: If n is the number of terms and is even, then the middle term is the $$(\frac{n}{2} + 1)^{th}$$ term.
Else if n is the number of terms and is odd, there are two middle terms which are the $$(\frac{n + 1}{2})^{th}$$ term and the $$(\frac{n + 3}{2})^{th}$$ term.
Case 1: $$\frac{n}{2} + 1$$ = 5 ; n = 8
Case 2: $$\frac{n + 1}{2}$$ = 5 ; n = 9
Case 3: $$\frac{n + 3}{2}$$ = 5 ; n = 7

7. What is the even value of n, if the middle term of (a + b)2n – 3 is 11?
a) 12
b) 10
c) 20
d) 22

Explanation: If 2n – 3 is even, then the middle term is the $$(\frac{2n-3}{2}+1)^{th}$$ term.
Else if 2n – 3 is odd, there are two middle terms which are the $$(\frac{2n-3+1}{2})^{th}$$ term and the $$(\frac{2n-3+3}{2})^{th}$$ term.
Case 1: $$\frac{2n-3}{2}$$ + 1= 11 ; n = 11.5
Case 2: $$\frac{2n-2}{2}$$ = 11 ; n = 12
Case 3: $$\frac{2n}{2}$$ = 11 ; n = 11

8. What is the value of n if the middle term (x + 2y)2n + 1 is the 19th term?
a) 33
b) 34
c) 35
d) 38

Explanation: Clearly (2n + 1) is an odd number. Therefore this is a case of binomial with an odd power.
For a binomial expansion with odd power, there are two middle terms.
Case 1: $$\frac{n+1}{2}$$ = 19
Therefore n = 37
Case 2: $$\frac{n+3}{2}$$ = 19
Therefore n = 35

9. If the general term is 91C2 x89, what is the expansion?
a) (x)91
b) (x – 2)90
c) (x – 1)91
d) (x + 1)90

Explanation: The general term of an expansion is nCr xn – r yr.
Clearly here n is 91 and the first term is x raised to the power 89.
The second term is raised to power 2.
y2 = 1
y = +1 or -1
Therefore the expansion can either be (x + 1)91 or (x – 1)91.

10. What is the middle term of (xyz + 3)80?
a) 80C41 (xyz)41 (3)39
b) 80C40 (xyz)40 (3)40
c) 80C39 (xyz)39 (3)40
d) 80C41 (xyz)41 (3)40

Explanation: Since the power is even, there are odd number of terms.
The middle term is the $$(\frac{n}{2} + 1)^{th}$$ term.
= $$(\frac{80}{2} + 1)^{th}$$ term
= 41st term
The 41st term = 80C40 (xyz)40 (3)40

11. What is the coefficient of the middle term of (z + y)3x, if 3x is considered to be even and the middle term is the 4th term?
a) 7C3
b) 6C2
c) 6C3
d) 7C2

Explanation: Since the middle term is the fourth term
Considering 3x to be even, $$(\frac{3x + 1}{2})^{th}$$ term = 4
x = 7/3
Therefore, the fourth term coefficients are 7C3

12. What is the fourth term of (x – 5y)96?
a) 125 96C3 x93 y3
b) 625 96C3 x93 y4
c) 625 96C4 x92 y4
d) 125 96C4 x92 y4

Explanation: Tr + 1 = nCr xn – r yr
Here first term is 4 and second term is 5y.
n = 96
r = 3
Therefore, Tr + 1 = 96C3 x96 – 3 (5y)3
= 125 96C3 x93 y3

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