Chapter 4 . Complex Numbers and Quadratic Equations

Class 11 Maths Chapter 4 . Complex Numbers and Quadratic Equations Exercise 4.1 Solutions :

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

1.       2.       3.    4.    5.    6.    7.    8.     9 .   10. 

Solution :   1. We have,   

     

2. We have, 

[Since,  ]

+0.i        

3. We have, 

4. We have,

  

5. We have,

  

6. We have,

 

7. We have,

 8. We have,

  

9 . We have,   

10. We have, 

Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.
11.                    12.                 13.

Solution:  11. We have,        

The multiplicative inverse of   is

12. we have,              

The multiplicative inverse  of   is

13.We have , 

The multiplicative inverse of  is 

14. Express the following expression in the form of   :      

     

Solution: We have,

 

Class 11 Maths Chapter 5 . Complex Numbers and Quadratic Equations Miscellaneous Exercise Solutions :

1. Evaluate:    .

Solution :  We have, 

2. For any two complex numbers  and  , prove that  .

Solution :  let ,  and  

Here,  , ,  and  

Now ,

 

 

LHS :

 

  RHS  Proved.

3. Reduce  to the standard form .

Solution : We have, 

4. If prove that

Solution : We have, 

 

 

  (Squaring both side)

5. If   , find  .

Solution : Given,

We have,  

6. If  , prove that .

Solution : We have, 

 

7. Let  . Find (i)       (ii) 

Solution : (i) Given,    , 

Since,

We have, 

Now , 

(ii)  Since,

We have, 

Now,    

8. Find the real numbers  and  if  is the conjugate of .

Solution:  The conjugate of  is   .

We have,

 ………. (i)

and  ……… (ii)

From (i),  we get 

Therefore, The value of  and  .

9. Find the modulus of  .

Solution:  let 

10. If  , then show that .

Solution : We have,

  

and 

  

11. If  and  are different complex numbers with  , then find .

Solution : Given ,  

   

We have,  

    [  ]

        [  ]

12. Find the number of non-zero integral solutions of the equation  .

Solution:  We have,

13. If  , then show that

Solution : We have ,

14. If  , then find the least positive integral value of  .

Solution : We have , 

   [Since, ]

Therefore, the value of m is 4 .