## EXERCISE 8.1

#### Page No 166:

#### Question 1:

Expand the expression (1– 2*x*)^{5}

#### Answer:

By using Binomial Theorem, the expression (1– 2*x*)^{5 }can be expanded as

#### Question 2:

Expand the expression

#### Answer:

By using Binomial Theorem, the expression ^{ }can be expanded as

#### Question 3:

Expand the expression (2*x* – 3)^{6}

#### Answer:

By using Binomial Theorem, the expression (2*x* – 3)^{6 }can be expanded as

#### Page No 167:

#### Question 4:

Expand the expression

#### Answer:

By using Binomial Theorem, the expression can be expanded as

#### Question 5:

Expand

#### Answer:

By using Binomial Theorem, the expression ^{ }can be expanded as

#### Question 6:

Using Binomial Theorem, evaluate (96)^{3}

#### Answer:

96 can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, binomial theorem can be applied.

It can be written that, 96 = 100 – 4

#### Question 7:

Using Binomial Theorem, evaluate (102)^{5}

#### Answer:

102 can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied.

It can be written that, 102 = 100 + 2

#### Question 8:

Using Binomial Theorem, evaluate (101)^{4}

#### Answer:

101 can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied.

It can be written that, 101 = 100 + 1

#### Question 9:

Using Binomial Theorem, evaluate (99)^{5}

#### Answer:

99 can be written as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied.

It can be written that, 99 = 100 – 1

#### Question 10:

Using Binomial Theorem, indicate which number is larger (1.1)^{10000} or 1000.

#### Answer:

By splitting 1.1 and then applying Binomial Theorem, the first few terms of (1.1)^{10000} can be obtained as

#### Question 11:

Find (*a* + *b*)^{4} – (*a* – *b*)^{4}. Hence, evaluate.

#### Answer:

Using Binomial Theorem, the expressions, (*a* + *b*)^{4} and (*a* – *b*)^{4}, can be expanded as

#### Question 12:

Find (*x* + 1)^{6} + (*x* – 1)^{6}. Hence or otherwise evaluate.

#### Answer:

Using Binomial Theorem, the expressions, (*x* + 1)^{6} and (*x* – 1)^{6}, can be expanded as

By putting, we obtain

#### Question 13:

Show that is divisible by 64, whenever *n* is a positive integer.

#### Answer:

In order to show that is divisible by 64, it has to be proved that,

, where *k* is some natural number

By Binomial Theorem,

For *a* = 8 and *m* = *n* + 1, we obtain

Thus,

is divisible by 64, whenever *n* is a positive integer.

#### Question 14:

Prove that.

#### Answer:

By Binomial Theorem,

By putting *b* = 3 and *a* = 1 in the above equation, we obtain

Hence, proved.