Further Complex Numbers (DP IB Maths: AI HL)

Consider the equation where .

Find the value of  for which one of the two distinct roots is .

Find the range of values of for which the equation has two distinct, real roots.

Let , where .

Find when

(i)

(ii)

On an Argand diagram the point  can be transformed to the point  by two transformations. Describe the two transformations and the order in which they are applied.

Hence, or otherwise, find the value of  when .

Consider  where .

Show that .

It is given that and

Giving your answers in the form , use technology to find the values of

(i)

(ii)

, for .

Find the least value of such that .

Let

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

Let  and .

i)

Write  in the form .

ii)

Hence, find   in the form ,  giving the exact value of  ,where  and .

Consider the complex numbers  and  .

Express

(i)

in the form 

(ii)

in the form , where  and .

Find the exact value of .

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

Without drawing an Argand diagram, describe the geometrical relationship between  and .

Use technology to find all the powers

Find the area of the shape made by the powers  when plotted on an Argand diagram.

Give your answer as an exact value.

Let .

Write down the value of .

Let and 

Prove the results

(i)

(ii)

 

Using technology, or otherwise, show that

The current, , in an AC circuit can be modelled by the equation  where  is the frequency and  is the phase shift. 

Two AC voltage sources of the same frequency generate currents  and .

Write down the maximum value and phase shift of the two currents  and  when they are each connected to the circuit alone.

The two AC voltage sources are connected to the circuit at the same time and the total current can be expressed as .

Write down the maximum value and phase shift of .

The height of a wave in metres, relative to a particular boat, can be modelled by the function , where  is the time in seconds. Observers on the boat are tracking a jumping dolphin. The height of the dolphin’s jumps can be modelled by the function .

Find an expression for the height the dolphin can reach, at time  seconds, when the height of the dolphin’s jump is affected by the height of the waves. Give your answer in the form 

Use technology to find the time when the dolphin first reaches its maximum height and write down the maximum height the dolphin reaches.

Find the time interval in the first two seconds when the height of dolphin will be above the wave

Let .

(i)

Find the values of and , giving your answers in the form  , where  and θ  is given as an exact value.

(ii)

Plot and   on an Argand diagram.

Find the exact value of   such that successive integer powers of the complex number lie on a unit circle.

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)