Class 11 Maths MCQ – Binomial Theorem

This set of Class 11 Maths Chapter 8 Multiple Choice Questions & Answers (MCQs) focuses on “Binomial Theorem”. These MCQs are created based on the latest CBSE syllabus and the NCERT curriculum, offering valuable assistance for exam preparation.

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.

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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