NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem Ex 8.1

NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem Ex 8.1

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Question 1:

Expand the expression (1– 2x)⁵

Answer:

By using Binomial Theorem, the expression (1– 2x)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 (2x – 3)⁶

Answer:

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

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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.