Mathematics Extension 2

In which quadrant is the complex number $(-3+3i)^3$ located on the Argand plane?

(A) The first quadrant

(B) The second quadrant

(C) The third quadrant

(D) The fourth quadrant

Let $\alpha$ be a real number and suppose that $z$ is a complex number such that

You may assume that

for all positive integer $n$. Let

Show that

Given that $z^n=a+bi$ and $|z|=1$, where $a$ and $b$ are real, which of the following is equal to $z^{-n}$?

A. $a+bi$

B. $a-bi$

C. $-a+bi$

D. $-a-bi$

It is known that a particular complex number $z$ is NOT a real number. Which of the following could be true for this number $z$?

A. $\bar z=iz$

B. $\bar z=|z^2|$

C. $\operatorname{Re}(iz)=\operatorname{Im}(z)$

D. $\operatorname{Arg}(z^3)=\operatorname{Arg}(z)$

Given that $z$ is a complex number such that $\operatorname{Re}(z)\ne0$, which of the following is equivalent to
$\frac{4z\bar z}{(z+\bar z)^2}?$

A. $1+\left(\frac{\operatorname{Im}(z)}{\operatorname{Re}(z)}\right)^2$ B. C. D.

Find $\frac{-1+i\sqrt3}{1+i}$ in Cartesian form.

Express $\dfrac{3-i}{2+i}$ in the form $x+iy$, where $x$ and $y$ are real numbers.

By letting $(x+iy)^2=7+24i$, where $x,y$ are real, find $\sqrt{7+24i}$ in Cartesian form.

If $z=-\sqrt3+i$ and $w=1+i$, write $\frac zw$ in the form $x+iy$, where $x$ and $y$ are real numbers.

The complex numbers $z=5+i$ and $w=2-4i$ are given. Find $\frac{\bar z}{w}$, giving your answer in Cartesian form.

Find the two square roots of $-i$, giving the answers in the form $x+iy$, where $x$ and $y$ are real numbers.

Consider the complex numbers $w=-1+4i$ and $z=2-i$. Evaluate $w\bar{z}$.

Solve $z^2+3z+(3-i)=0$, giving your answer(s) in the form $a+bi$, where $a$ and $b$ are real.