Question 1

Consider the complex numbers $z{_1}=2cis150^∘$ and $z{_2}=-1+i $.
Calculate $\frac{z{_1}}{z{_2}}$  giving your answer in modulus-argument.

Accepted Answer

$\sqrt2 cis15^∘$

Answer

$\sqrt2 cis135^∘$

Answer

$\sqrt3 cis15^∘$

Answer

$2 cis15^∘$

Question 2

Given $w = 2 + 2i$  and $z=cos(\frac{5π}{6})+isin(\frac{5π}{6})$, find in its simplest form $w^4 z^6$.

Accepted Answer

$w^4 z^6=64$

Answer

$w^4 z^6=-64$

Answer

$w^4 z^6=-32$

Answer

$w^4 z^6=32$

Question 3

Find the roots of $z^{24}=1$ which satisfy the condition $0<arg(z)<\fracπ2$, expressing your answers in the form $re^{iθ}$, where $r, θ∈R^+$.

Accepted Answer

$z=e \frac{πi}{12}$ or $e \frac{2πi}{12 }$or $e \frac{3πi}{12}$ or $e\frac {4πi}{12}$ or $e\frac {5πi}{12}$

Answer

$z=e \frac{πi}{12}$ or $e \frac{2πi}{12 }$or $e \frac{3πi}{12}$

Answer

$z=e \frac{πi}{12}$ or $e \frac{3πi}{12}$ or $e\frac {5πi}{12}$ or $e\frac {7πi}{12}$

Answer

$z=e \frac{πi}{6}$ or $e \frac{2πi}{6 }$or $e \frac{3πi}{6}$ or $e\frac {4πi}{6}$ or $e\frac {5πi}{6}$

Question 4

Consider $w=2(cos\fracπ3+i sin\fracπ/3)$.
Express $w^2$ and $w^3$ in modulus-argument form.

Accepted Answer

$w^2=4cis(\frac{2π}{3})$;  $w^3=8cis(π)$

Answer

$w^2=2cis(\frac{2π}{3})$;  $w^3=8cis(π)$

Answer

$w^2=4cis(\frac{2π}{3})$;  $w^3=4cis(π)$

Answer

$w^2=4cis(\frac{π}{3})$;  $w^3=8cis(π)$

Question 5

Consider the complex numbers $z{_1}=1+\sqrt{3} i$, $z{_2}=1+i$ and $w=\frac{z{{_1}}}{z{_2}}$ .
Write down the modulus of w.

Accepted Answer

$|w|=\sqrt2$

Answer

$|w|=3$

Answer

$|w|=2$

Answer

$|w|=\sqrt3$

Question 6

Consider the complex numbers $z{_1}=1+\sqrt{3} i$, $z{_2}=1+i$ and $w=\frac{z{{_1}}}{z{_2}}$ .  By expressing $z{_1}$ and $z{_2}$ in modulus-argument form write down the argument of $w$.

Accepted Answer

$argw=\frac{π}{12}$

Answer

$argw=\frac{\sqrt2π}{12}$

Answer

$argw=\frac{π}{4}$

Answer

$argw=\frac{π}{3}$

Question 7

Solve $2sin(x+60^∘)=cos(x+30^∘)$, $0^∘≤x≤180^∘$.

Accepted Answer

$x=150^∘$

Answer

$x=330^∘$

Answer

$x=210^∘$

Answer

$x=30^∘$

Question 8

Let $z=1-cos2θ-isin2θ$, $z∈C, 0≤θ≤π$.
Find the modulus and argument of $z$ in terms of $θ$. Express each answer in its simplest form.

Accepted Answer

$arg(z)=θ-\fracπ2$

Answer

$arg(z)=θ+\fracπ2$

Answer

$arg(z)=θ-\fracπ3$

Answer

$arg(z)=θ+\fracπ3$

Question 9

Use de Moivre’s theorem to find the value of $(cos(\fracπ3)+isin(\fracπ3))^3$

Accepted Answer

$-1$

Answer

$2$

Answer

$-2$

Answer

$1$

Question 10

Let $z=cosθ+isinθ$.
Find an expression in terms of $θ$ for $(z)^n+(z*)^n$, $n∈Z^+$ where $z*$ is the complex conjugate of $z$.

Accepted Answer

$2cos(nθ)$

Answer

$cos(nθ)$

Answer

$2sin(nθ)$

Answer

$2cos(nθ)+2sin(nθ)$

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