Further Complex Numbers (DP IB Maths: AA 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.

Consider the complex number .

Show that   is a root of the cubic equation

Find the other two roots of the cubic equation in part (a).

Consider  where .

Show that .

Consider the equation .

(i)

Verify that  is a root of this equation.

(ii)

Find the second root of the equation, expressing your answer in the form  where  and  .

The roots and  are represented by the points A and B respectively on an Argand diagram.

Find AB.

Consider the equation , where .

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

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 .

Find all the powers .

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

Let .

Write down the value of .

Let .

Prove the results

(i)

(ii)

Using the results from part (b), describe fully the geometrical interpretation of dividing  by .

Sketch on the Argand diagram below.

and  represent the vertices of a triangle.

Find the area of the triangle.

The complex numbers and  are roots of the cubic equation  where . 

Find the values of and .

Express and  in the form , where  and .

Let  ,  where  and  are real constants.

Given that    is a root of the equation ,  show the roots  on the Argand diagram below.

Consider the equation  ,  where . 

Given that one of the distinct roots is , find

(i)

the values of  and 

(ii)

the roots  and , giving your answers in the form , where .

On an Argand diagram  and  are represented by the points A, B and C respectively.

Find the area of the triangle ABC.

Consider the complex numbers  and , where , . 

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

Consider the equation ,  where  and  . 

Given that one root is  and another root is , find the possible values of and .

Consider the complex number .

Use De Moivre’s theorem to find the value of .

Use the principle of mathematical induction to prove, for all , that

Show that the result in part (b) is true for all

Consider the equation .

Given that the product of the roots is 31, find the roots of the equation, expressing your answers in the form , where  and .

Let S be the sum of the roots found in part (a).

Show that and find the value of .

The roots  are represented on an Argand diagram. 

Describe the geometrical shape made by the five roots.

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

By first expressing $1+\sqrt{3}\mathrm{i}$ and $-1+\mathrm{i}$ in the form $r \mathrm{cis} \theta$ where $r>0$ and $-\pi <\theta \le \pi$, show that $\tan \frac{5\pi }{12}=2+\sqrt{3}$.

Consider the complex number .

Find the modulus and argument of z.

Solve  for .

Let .

Show that

(i)

,

(ii)

.

Use the results found in part (a) to find the sum of the infinite series

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)