Chapter 7. Binomial Theorem

Class 11 Maths Chapter 7.  Binomial Theorem Exercise 7.1 Solutions :

Expand each of the expressions in Exercises 1 to 5.

1.      2.     3.        4.     5. 

Solution : 1. We have,

2. We have, 

3. We have,     

4. We have , 

5. We have,

Using binomial theorem, evaluate each of the following:
6.         7.      8.     9.

Solution: 6. We have,           

 

7. We have

              

8. We have,               

9. We have

10. Using Binomial Theorem, indicate which number is larger  or 1000.

Solution:  We have,

 other positive terms

 other positive terms

 other positive terms > 1000 .

Hence,  .
11. Find  . Hence, evaluate  .

Solution :  We have, 

Now ,

 

Similarly ,

Here,   and

 

12. Find  . Hence or otherwise evaluate    .

Solution: We have,

Now ,

[Using ]

and 

Here,   

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

Solution: We have,

  

Now, 

  

Hence,  is divisible by 64 .

14. Prove that  .

Solution:  We have ,  

 

We know that ,

Here,  , then

 

Class 11 Maths Chapter 8.  Binomial Theorem Miscellaneous Exercise Solutions :

1. If  and  are distinct integers, prove that  is a factor of  , whenever  is a positive integer. [Hint write  and expand]

Solution :  We have,

+

+

 is factor of  .

2. Evaluate   .

Solution : let  and

Now, we find   

 

Similarly ,

  Now, 

3. Find the value of    .

Solution : let   and

Now, we find   

 

   

So, 

4. Find an approximation of  using the first three terms of its expansion.

Solution : We have,

  ( negligible other terms)

5. Expand using Binomial Theorem    .

Solution: We have,

Now,

and 

 

6. Find the expansion of   using binomial theorem .

Solution:  We have,