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 – y}C_{r} (x^{xy – r} . y^{r})

b) ^{xy}C_{r} (x^{x – y – r} . -y^{r})

c) ^{xy}C_{r} (x^{xy – r} . -y^{r})

d) ^{x – y}C_{r} (x^{x – y – r} . y^{r})

View Answer

Explanation: The general term of a binomial series is given by

^{n}C

_{r}a

^{n – r}b

^{r}.

Here a = x, b = -y and n = xy

Therefore the general term is given by

^{xy}C

_{r}(x

^{xy – r}. -y

^{r}).

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

Explanation: Coefficient of the second term of (x – y)

^{3}is

^{3}C

_{1}and the coefficient of the third term of the expansion (x + y)

^{n}is

^{n}C

_{2}.

^{3}C

_{1}=

^{n}C

_{2}

3 = \(\frac{n!}{2!(n – 2)!}\)

6 = \(\frac{n(n-1)(n-2)!}{(n – 2)!}\)

6 = n

^{2}– n

n

^{2}– n – 6 = 0

n

^{2}– 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) n^{th} term

d) (n – 1)^{th} term

View Answer

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 x^{2}

b) 10240 x^{3}

c) 12240 x^{4}

d) 10340 x^{4}

View Answer

Explanation: The middle term will be the 4

^{th}term

4

^{th}term =

^{6}C

_{3}(4)

^{6 – 3}(2x)

^{3}

= 20 (64) (8x

^{3})

= 10240 x

^{3}

5. What is the middle term of (x^{2} + x)^{3}?

a) 3x^{4}

b) 6x^{4}

c) 4x^{4}

d) 3x^{6}

View Answer

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,

^{3}C

_{2}(x

^{2})

^{3 – 2}(x)

^{2}= 3x

^{4}

When r = 1,

^{3}C

_{1}(x

^{2})

^{3 – 1}(x)

^{1}= 3x

^{5}

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

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

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 19^{th} term?

a) 33

b) 34

c) 35

d) 38

View Answer

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 ^{91}C_{2} x^{89}, what is the expansion?

a) (x)^{91}

b) (x – 2)^{90}

c) (x – 1)^{91}

d) (x + 1)^{90}

View Answer

Explanation: The general term of an expansion is

^{n}C

_{r}x

^{n – r}y

^{r}.

Clearly here n is 91 and the first term is x raised to the power 89.

The second term is raised to power 2.

y

^{2}= 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) ^{80}C_{41} (xyz)^{41} (3)^{39}

b) ^{80}C_{40} (xyz)^{40} (3)^{40}

c) ^{80}C_{39} (xyz)^{39} (3)^{40}

d) ^{80}C_{41} (xyz)^{41} (3)^{40}

View Answer

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

= 41

^{st}term

The 41

^{st}term =

^{80}C

_{40}(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 4^{th} term?

a) ^{7}C_{3}

b) ^{6}C_{2}

c) ^{6}C_{3}

d) ^{7}C_{2}

View Answer

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

^{7}C

_{3}

12. What is the fourth term of (x – 5y)^{96}?

a) 125 ^{96}C_{3} x^{93} y^{3}

b) 625 ^{96}C_{3} x^{93} y^{4}

c) 625 ^{96}C_{4} x^{92} y^{4}

d) 125 ^{96}C_{4} x^{92} y^{4}

View Answer

Explanation: T

_{r + 1}=

^{n}C

_{r}x

^{n – r}y

^{r}

Here first term is 4 and second term is 5y.

n = 96

r = 3

Therefore, T

_{r + 1}=

^{96}C

_{3}x

^{96 – 3}(5y)

^{3}

= 125

^{96}C

_{3}x

^{93}y

^{3}

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