Polar and Euler form, De Moivre's theorem and complex roots

Let $z=1+\sqrt3\,i$.

(a) Find the modulus and argument of $z$. [2]

(b) Use de Moivre's theorem to find $z^5$ in the form $a+bi$. [3]

(c) Find the smallest positive integer $n$ for which $z^n$ is real. [2]

Let $z=\cos\theta+i\sin\theta$.

(a) Use de Moivre's theorem to show that $z^n+z^{-n}=2\cos n\theta$. [3]

(b) By considering $\left(z+\dfrac1z\right)^4$, show that $\cos^4\theta=\dfrac{\cos4\theta+4\cos2\theta+3}{8}$. [4]

The equation $z^3-5z^2+az+b=0$, where $a,b\in\mathbb{R}$, has $1+2i$ as one root.

(a) Write down another non-real root of the equation. [1]

(b) Find the third root. [2]

(c) Find the values of $a$ and $b$. [3]

The complex numbers $z_1$ and $z_2$ are given by $z_1=6\,\mathrm{e}^{\frac{5\pi i}{6}}$ and $z_2=2\,\mathrm{e}^{\frac{\pi i}{3}}$.

(a) Find $z_1z_2$, giving your answer in the form $r\mathrm{e}^{i\theta}$, where $r>0$ and $-\pi<\theta\le\pi$. [3]

(b) Find $\dfrac{z_1}{z_2}$, giving your answer in the form $a+bi$, where $a,b\in\mathbb{R}$. [2]

(a) Solve the equation $z^4=-8+8\sqrt3\,i$, giving your answers in the form $a+bi$, where $a,b\in\mathbb{R}$. [5]

The roots are represented by the points $P_1,P_2,P_3,P_4$ on an Argand diagram.

(b) Find the area of the quadrilateral $P_1P_2P_3P_4$. [2]

The diagram shows a square $OABC$ on an Argand diagram, where $O$ is the origin and the vertices are labelled anticlockwise. The point $A$ represents the complex number $w=3+2i$.

(a) Describe the geometric effect on a point of multiplying its complex number by $i$. [1]

(b) Hence find the complex number represented by
(i) $C$; [1]
(ii) $B$. [2]

A point $D$ is to be placed so that triangle $OAD$ is equilateral, with $D$ on the same side of $OA$ as $C$.

(c) Find the complex number represented by $D$, giving your answer in the form $a+bi$, where $a,b$ are exact. [3]

(a) Use De Moivre's theorem to show that $\cos3\theta=4\cos^3\theta-3\cos\theta$. [4]

(b) Hence, by using the substitution $x=\cos\theta$, show that the roots of the equation $8x^3-6x-1=0$ are $x=\cos\dfrac{\pi}{9}$, $x=\cos\dfrac{5\pi}{9}$ and $x=\cos\dfrac{7\pi}{9}$. [3]

(c) Hence find the exact value of $\cos\dfrac{\pi}{9}\cos\dfrac{5\pi}{9}\cos\dfrac{7\pi}{9}$. [2]

Let $\omega=\mathrm{e}^{\frac{2\pi i}{5}}$.

(a) Show that $1+\omega+\omega^2+\omega^3+\omega^4=0$. [2]

(b) Hence show that $\cos\dfrac{2\pi}{5}+\cos\dfrac{4\pi}{5}=-\dfrac12$. [3]

(c) Hence find the exact value of $\cos\dfrac{2\pi}{5}$. [3]

Let $z=-2+2i$.

(a) Write $z$ in the form $r\mathrm{e}^{i\theta}$, where $r>0$ and $-\pi<\theta\le\pi$. [2]

(b) Find the smallest positive integer $n$ such that $z^n$ is a positive real number, and state the value of $z^n$ for this $n$. [3]

(c) Find all three values of $z^{\frac23}$, giving your answers in the form $a+bi$. [3]

The polynomial $p(z)=z^4-6z^3+22z^2-30z+13$ has real coefficients. One root of $p(z)=0$ is $2+3i$.

(a) Write down a second root of $p(z)=0$, giving a reason. [1]

(b) Find a real quadratic factor of $p(z)$. [2]

(c) Hence find the remaining roots of $p(z)=0$. [2]

(d) Express the root $2+3i$ in the form $r\mathrm{e}^{i\theta}$, giving $r$ and $\theta$ correct to three significant figures. [2]