NCERT solution class 11 chapter 8 Binomial Theorem exercise 8.3 mathematics

EXERCISE 8.3

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Page No 175:

Question 1:

Find a, b and n in the expansion of (a + b)ⁿ if the first three terms of the expansion are 729, 7290 and 30375, respectively.

Answer:

It is known that (r + 1)^th term, (T_r+1), in the binomial expansion of (a + b)ⁿ is given by .
The first three terms of the expansion are given as 729, 7290, and 30375 respectively.
Therefore, we obtain

Dividing (2) by (1), we obtain

Dividing (3) by (2), we obtain

From (4) and (5), we obtain

Substituting n = 6 in equation (1), we obtain
a⁶ = 729

From (5), we obtain

Thus, a = 3, b = 5, and n = 6.

Question 2:

Find a if the coefficients of x² and x³ in the expansion of (3 + ax)⁹ are equal.

Answer:

It is known that (r + 1)^th term, (T_r+1), in the binomial expansion of (a + b)ⁿ is given by .
Assuming that x² occurs in the (r + 1)^th term in the expansion of (3 + ax)⁹, we obtain

Comparing the indices of x in x² and in T_{r + 1}, we obtain
r = 2
Thus, the coefficient of x² is

Assuming that x³ occurs in the (k + 1)^th term in the expansion of (3 + ax)⁹, we obtain

Comparing the indices of x in x³ and in T_{k+ 1}, we obtain
k = 3
Thus, the coefficient of x³ is

It is given that the coefficients of x² and x³ are the same.

Thus, the required value of a is.

Question 3:

Find the coefficient of x⁵ in the product (1 + 2x)⁶ (1 – x)⁷ using binomial theorem.

Answer:

Using Binomial Theorem, the expressions, (1 + 2x)⁶ and (1 – x)⁷, can be expanded as


The complete multiplication of the two brackets is not required to be carried out. Only those terms, which involve x⁵, are required.
The terms containing x⁵ are

Thus, the coefficient of x⁵ in the given product is 171.

Question 4:

If a and b are distinct integers, prove that a – b is a factor of aⁿ – bⁿ, whenever n is a positive integer.
[Hint: write aⁿ = (a – b + b)ⁿ and expand]

Answer:

In order to prove that (a – b) is a factor of (aⁿ – bⁿ), it has to be proved that
aⁿ – bⁿ = k (a – b), where k is some natural number
It can be written that, a = a – b + b

This shows that (a – b) is a factor of (aⁿ – bⁿ), where n is a positive integer.

Question 5:

Evaluate.

Answer:

Firstly, the expression (a + b)⁶ – (a – b)⁶ is simplified by using Binomial Theorem.
This can be done as

Question 6:

Find the value of.

Answer:

Firstly, the expression (x + y)⁴ + (x – y)⁴ is simplified by using Binomial Theorem.
This can be done as

Question 7:

Find an approximation of (0.99)⁵ using the first three terms of its expansion.

Answer:

0.99 = 1 – 0.01

Thus, the value of (0.99)⁵ is approximately 0.951.

Question 8:

Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of 

Answer:

In the expansion, ,
Fifth term from the beginning 
Fifth term from the end 
Therefore, it is evident that in the expansion of, the fifth term from the beginning is and the fifth term from the end is.

It is given that the ratio of the fifth term from the beginning to the fifth term from the end is. Therefore, from (1) and (2), we obtain

Thus, the value of n is 10.

Page No 176:

Question 9:

Expand using Binomial Theorem.

Answer:

Using Binomial Theorem, the given expression  can be expanded as

Again by using Binomial Theorem, we obtain

From (1), (2), and (3), we obtain

Question 10:

Find the expansion of using binomial theorem.

Answer:

Using Binomial Theorem, the given expression  can be expanded as

Again by using Binomial Theorem, we obtain

From (1) and (2), we obtain

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