1.6 Further Complex Numbers

Consider the equation , where .

Find the four distinct roots of the equation, giving your answers in the form , where

Represent the roots found in part (a) on the Argand diagram below.

Find the area of the polygon whose vertices are represented by the four roots on the Argand diagram.

Let  and  

Giving your answers in the form find 

(i)

(ii)

Write and in the form 

Find  giving your answer in the form 

It is given that and  are the complex conjugates of and respectively. 

Find giving your answer in the form 

Let  and .

Express

(i)

in the form  

(ii)

in the form

Find giving your answer in the form .

Find  giving your answer in the form 

Sketch and  on a single Argand diagram.

It is given that that  and  

Find the value of for  

Find the least value of  such that 

Consider the complex number   where and  

Express in the form

Sketch  and on the Argand diagram below. 

Find the smallest positive integer value of  such that  is a real number. 

Consider the complex numbers  and . 

Write  and  in the form , where  and .

Find the modulus and argument of .

Write down the value of .

Write  in the form  where .

Consider the complex numbers  and , where  

Use geometrical reasoning to find the two possibilities for w, giving your answers in exponential form.

Consider the complex numbers and .

Write and  in the form where , giving exact values of  and .

A scale model of a triangular patio is planned by representing the vertices of the triangle on an Argand diagram as the points and . Find the area of the triangle in the model.

Consider the complex number .

(i)

Plot the position of  on an Argand diagram.

(ii)

Express in the form , where , giving the exact value of and the exact value of .

Use technology to find the value of . Give your answer in the form , where , giving the exact value of and the exact value of .

Find the smallest positive integer such that is

Explain why there is no possible integer value of such that  is purely imaginary.

Given the points 1 and  on an Argand diagram, where    is a complex number, explain how to find each of the following points by geometrical construction.  In each case provide a sketch to illustrate your answer.

.

Consider the equations and , where .  Find  giving your answer in the form , where  and .

By first expressing  and  in the form where  and , show that .

Two power sources are connected to a single electrical circuit. At time seconds, the voltage, , provided by the first power source is modelled by , and the voltage, , from the second power source can be modelled by the function .

The total voltage in the circuit, ,  is given by .

Find an expression for in the form , where 

Write down the maximum voltage and the phase shift provided by the second power source, giving your answers correct to 2 decimal places.

A popular fast-food chain is considering opening a new restaurant in an up and coming part of New York.  They have looked at competition in the area and predict that the costs , in thousands of USD, to run the new restaurant for the first 100 days after opening could be modelled by the function

where  is the number of days since opening.

The CEO of the company will only give permission for the new restaurant to open if the model predicts that the revenue will be greater than the costs by the 80^th day after opening.

The revenue , in thousands of USD, for the first 100 days is predicted to follow the model

(ii)

For this value of , show that the profits of the hotel can be modelled by the function , giving the values of and  correct to four significant figures.

According to the model found in part (a) (ii), find

The primary square root of a complex number  is defined as , where  and .  If  then the value for  is chosen such that .  Note that the other square root of will then be given by .

Show that

Given that , derive a formula for  in terms of and , and explain why  in this case will always have the same sign (positive, negative, or zero) as .

Hence show that in general

with the choice of the positive or negative value being dependent on the properties of .

Explain what must be true of  for each of the following to be true:

(i)

(ii)

(iii)