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)
View Answer

Answer: c
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).
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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
View Answer

Answer: b
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
View Answer

Answer: a
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
View Answer

Answer: b
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
View Answer

Answer: a
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
View Answer

Answer: c
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
View Answer

Answer: a
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
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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
View Answer

Answer: c
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
View Answer

Answer: c
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.
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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
View Answer

Answer: b
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
View Answer

Answer: a
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
View Answer

Answer: a
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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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter