Complex Numbers (DP IB Maths: AA HL)

Consider the quadratic equation . 

The roots of the equation are    and   where  

Find the value of and .

Sketch and  on the Argand diagram below, be sure to include an appropriate scale.

Consider the complex numbers  and .

Find

(i)

(ii)

(iii)

(iv)

Consider the complex numbers  and  . 

Find the modulus and argument of .

Consider the complex numbers  and .   

Work out the following:

(i)

(ii)

(iii)

For part (iii) give your answer in the form ,  where  and  are real numbers.

Write down the complex conjugate of and describe the geometrical relationship between  and .

Find all possible real values for  and  such that

(i)

(ii)

(iii)

For a general complex number ,  where ,  show that

(i)

(ii)

For the complex numbers and ,  where  , show that

Consider the complex numbers  and .

Find

(i)

(ii)

(iii)

(iv)

Consider the complex numbers  and  where . 

Find the possible values of and .

Using the answers gained in part (a), write down values for and  that will satisfy the equation

Consider the complex numbers . 

Represent the complex numbers  and on an Argand diagram.

The points  and are represented by the points  and  on the Argand diagram respectively.

Find the angle

Consider the complex numbers  and  , where .

Find the possible values of and .

Find the modulus of .

Let .

Given that , express in the form , where .

Find , giving your answer in the form , where

Consider the complex numbers and .

Find

(i)

(ii)

The complex numbers  and  are represented by the points  and respectively on an Argand diagram with origin . 

Determine whether the angle made by with the positive horizontal axis is greater than or less than the angle made by  with the positive horizontal axis. Give a reason for your answer.

Consider the complex number .

Write down, in terms of ,

(i)

(ii)

In the case where , find the modulus and argument of .

Consider the complex numbers   and .

Express  in the form , where .

Find

(i)

(ii)

(iii)

, giving your answer as an exact value.

Consider a general complex number ,  where   ,  and  . 

Show that

(i)

(ii)

(iii)

Consider the equation , where , .

Find an expression in terms of and  for .

Find in terms of  given that  is purely real.

Consider the complex numbers  and . 

Find the modulus of   giving your answer as an exact value.

The argument of is given as , where .  Find the value of .

Consider the complex numbers  

Express  in the form , where .

In the case where  is purely imaginary, represent  and  on an Argand diagram.

Consider the complex numbers and   where

Find the values of   and .

Find the modulus of , giving your answer as an exact value.

Find the argument of , giving your answer in the range   .

Consider the complex numbers and . 

Find the values of and such that  and .

Consider the complex numbers ,  and , where 
and .

Find the values of and  in terms of . 

Given that , find the possible values of .

Given additionally that  radians correct to 2 decimal places, determine the exact value of  .