NCERT Solutions for Class 11 Maths Chapter 8

Binomial Theorem Class 11

Chapter 8 Binomial Theorem Exercise 8.1, 8.2, miscellaneous Solutions

 Exercise 8.1 : Solutions of Questions on Page Number : 166                                                                                                               

Q1 :

Expand the expression (1- 2x)5

Answer :

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

Q2 :

Expand the expression

Answer :

By using Binomial Theorem, the expression                       can be expanded as

Q3 :

Expand the expression (2x - 3)6

Answer :

Q4 :

Expand the expression

Answer :

By using Binomial Theorem, the expression                        can be expanded as

Q5 :

Expand

Answer :

By using Binomial Theorem, the expression                     can be expanded as

Q6 :

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

Q7 :

Using Binomial Theorem, evaluate (102)5

Answer :

102can 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

Q8 :

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

Q9 :

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

Q10 :

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

Answer :

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

Q11 :

Find (a + b)4– (a– b)4. Hence, evaluate                                                       .

Answer :

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

Q12 :

Find (x+ 1)6+ (x – 1)6. Hence or otherwise evaluate                                             .

Answer :

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

By putting                 , we obtain

Q13 :

Show that                             is divisible by 64, whenever nis 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

Q14 :

Prove that                               .

Answer :

By Binomial Theorem,

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

 Exercise 8.2 : Solutions of Questions on Page Number : 171                                                                                                               

Q1 :

Find the coefficient of x5in (x + 3)8

Answer :

It is known that (r + 1)th  term, (Tr+1), in the binomial expansion of (a + b)n is given by                                     . Assuming that x5occurs in the (r+ 1)thterm of the expansion (x+ 3)8, we obtain

Comparing the indices of xin x5and in Tr+1, we obtain

r= 3

Q2 :

Find the coefficient of a5b7in (a - 2b)12

Answer :

It is known that (r + 1)th  term, (Tr+1), in the binomial expansion of (a + b)n is given by                                     . Assuming that a5b7occurs in the (r+ 1)thterm of the expansion (a– 2b)12, we obtain

Comparing the indices of aand b in a5b7 and in Tr+1, we obtain

r= 7

Thus, the coefficient

Q3 :

Write the general term in the expansion of (x2- y)6

Answer :

It is known that the general term Tr+1 {which is the (r + 1)th  term} in the binomial expansion of (a + b)n is given by          .

Thus, the general term in the expansion of (x2– y6) is

Q4 :

Write the general term in the expansion of (x2- yx)12, x ≠0

Answer :

It is known that the general term Tr+1 {which is the (r + 1)th  term} in the binomial expansion of (a + b)n is given by          .

Thus, the general term in the expansion of(x2– yx)12is

Q5 :

Find the 4thterm in the expansion of (x- 2y)12 .

Answer :

It is known that (r + 1)th  term, (Tr+1), in the binomial expansion of (a + b)n is given by                                     . Thus, the 4thterm in the expansion of (x– 2y)12is

Q6 :

Find the 13thterm in the expansion of                                               .

Answer :

It is known that (r + 1)th  term, (Tr+1), in the binomial expansion of (a + b)n is given by                                     .

Thus, 13thterm in the expansion of                                is

Q7 :

Find the middle terms in the expansions of

Answer :

It is known that in the expansion of (a+ b)n, if n is odd, then there are two middle terms, namely,                      term

and                            term.

Q8 :

Find the middle terms in the expansions of

Answer :

It is known that in the expansion (a+ b)n, if n is even, then the middle term is                      term.

Therefore, the middle term in the expansion of                          is                                   term

Q9 :

In the expansion of (1 + a)m + n, prove that coefficients of amand anare equal.

Answer :

It is known that (r + 1)th  term, (Tr+1), in the binomial expansion of (a + b)n is given by                                     . Assuming that amoccurs in the (r+ 1)thterm of the expansion (1 + a)m+ n, we obtain

Comparing the indices of ain amand in Tr + 1, we obtain

r= m

Therefore, the coefficient of amis

Assuming that anoccurs in the (k+ 1)thterm of the expansion (1 + a)m+n, we obtain

Comparing the indices of ain anand in Tk+ 1, we obtain

k= n

Therefore, the coefficient of anis

Q10 :

The coefficients of the (r- 1)th, rthand (r + 1)thterms in the expansion of (x+ 1)nare in the ratio 1:3:5. Find nand r.

Answer :

It is known that (k + 1)th  term, (Tk+1), in the binomial expansion of (a + b)n is given by                                       .

Therefore, (r – 1)thterm in the expansion of (x+ 1)nis

rth term in the expansion of (x+ 1)nis

(r+ 1)thterm in the expansion of (x+ 1)nis

Therefore, the coefficients of the (r– 1)th, rth, and (r + 1)thterms in the expansion of (x+

1)nare                                                 respectively. Since these coefficients are in the ratio 1:3:5, we obtain

Multiplying (1) by 3 and subtracting it from (2), we obtain 4r – 12 = 0

⇒ r= 3

Putting the value of rin (1), we obtain

n– 12 + 5 = 0

⇒ n= 7

Q11 :

Prove that the coefficient of xnin the expansion of (1 + x)2nis twice the coefficient of xnin the expansion of (1

+ x)2n-1 .

Answer :

It is known that (r + 1)th  term, (Tr+1), in the binomial expansion of (a + b)n is given by                                     . Assuming that xnoccurs in the (r+ 1)thterm of the expansion of (1 + x)2n, we obtain

Comparing the indices of xin xnand in Tr+ 1, we obtain

r= n

Therefore, the coefficient of xnin the expansion of (1 + x)2nis

Assuming that xnoccurs in the (k+1)thterm of the expansion (1 + x)2n – 1, we obtain

Comparing the indices of xin xnand Tk+ 1, we obtain

k= n

Therefore, the coefficient of xnin the expansion of (1 + x)2n –1is

From (1) and (2), it is observed that

Therefore, the coefficient of xnin the expansion of (1 + x)2nis twice the coefficient of xnin the expansion of (1 + x)2n–1. Hence, proved.

Q12 :

Find a positive value of mfor which the coefficient of x2in the expansion (1 + x)mis 6.

Answer :

It is known that (r + 1)th  term, (Tr+1), in the binomial expansion of (a + b)n is given by                                     . Assuming that x2occurs in the (r + 1)thterm of the expansion (1 +x)m, we obtain

Comparing the indices of xin x2and in Tr+ 1, we obtain

r= 2

Therefore, the coefficient of x2is           .

It is given that the coefficient of x2in the expansion (1 + x)mis 6.

Thus, the positive value of m, for which the coefficient of x2in the expansion (1 + x)mis 6, is 4.

 Exercise Miscellaneous : Solutions of Questions on Page Number : 175                                                                                            

Q1 :

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

Answer :

It is known that (r + 1)th  term, (Tr+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

Q2 :

Find aif the coefficients of x2and x3in the expansion of (3 + ax)9are equal.

Answer :

It is known that (r + 1)th  term, (Tr+1), in the binomial expansion of (a + b)n is given by                                     . Assuming that x2occurs in the (r+ 1)thterm in the expansion of (3 + ax)9, we obtain

Comparing the indices of xin x2and in Tr+ 1, we obtain

r= 2

Thus, the coefficient of x2is

Assuming that x3occurs in the (k+ 1)thterm in the expansion of (3 + ax)9, we obtain

Comparing the indices of xin x3and in Tk+ 1, we obtain

k = 3

Thus, the coefficient of x3is

It is given that the coefficients of x2and x3are the same.

Q3 :

Find the coefficient of x5in the product (1 + 2x)6(1 - x)7using binomial theorem.

Answer :

Using Binomial Theorem, the expressions, (1 + 2x)6and (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 x5are

Q4 :

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

Q5 :

Evaluate                                                       .

Answer :

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

Q6 :

Find the value of                                                                     .

Answer :

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

Q7 :

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

Answer :

0.99 = 1 – 0.01

Q8 :

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                                              .

Thus, the value of n is 10.

Q9 :

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

Q10 :

Find the expansion of                                          using binomial theorem.

Answer :

Using Binomial Theorem, the given expression                                           can be expanded as

From (1) and (2), we obtain