Given that $1$ and $4+\text{i}$ are roots of the equation, write down the third root.

Verify that the mean of the two complex roots is $4$.

Explain why the equation will have at least one real root for all values of $q$.

Express $z_1$ in terms of $z_2$.

Use the result from part (c)(ii) to show that $a^2-3b=0$.

Sketch the graph of $y=x^4-6x^3-2x^2+58x-51$, stating clearly the coordinates of any maximum and minimum points and intersections with axes.

Find the values of $x$ that satisfy the equation $\left|p\right|=\left|q\right|$.

One of the roots $w_1$ satisfies the condition $\text{Re}\left(w_1\right)=0$.

Given that $w_1=\frac{z}{z-\text{i}}$, express $z$ in the form $a+b\text{i}$, where $a$, $b\in \mathbb{Q}$.

Sketch the graph of $f$. State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of $a$.

the girls do not sit on either end and do not sit together.

the median of $X$.

the girls do not sit together.

Show that $(\omega -3{\omega }^2)({\omega }^2-3\omega )=13$.

Show that $abc=8$.

Find the roots of the equation $w^3=8\text{i}$, $w\in \mathbb{C}$. Give your answers in Cartesian form.

$a+\text{i}b$ where $a$, $b\in \mathbb{R}$.

By writing $x^4-9x^3+24x^2+22x-12$ as a product of one linear and one cubic factor, prove that the equation has at least one complex root.

$\text{E}\left(X\right)$.

Given that $\text{Re}\left(z_1{z_2}^{\ast }\right)=0$, show that ${\text{Z}}_1{\text{OZ}}_2$ is a right-angled triangle.

Hence show that ${z_1}^2+{z_2}^2=z_1{z}_2$.

Consider the equation $z^2+az+12=0$, where $z\in \mathrm{ℂ}$ and $a\in \mathrm{ℝ}$.

Given that $0<\alpha -\theta <\pi$, deduce that only one equilateral triangle ${\text{Z}}_1{\text{OZ}}_2$ can be formed from the point $\text{O}$ and the roots of this equation.

Find, in terms of $a$, the probability that $X$ lies between 1 and 3.

Find the term independent of $x$ in the expansion of $\frac{1}{x^3}{\left(\frac{1}{3x^2}-\frac{x}{2}\right)}^9$.

$r{\text{e}}^{\text{i}\theta }$ where $r$, $\theta \in \mathbb{R}$, $r>0$.

Consider the expansion of ${\left(2+x\right)}^n$, where $n⩾3$ and $n\in \mathbb{Z}$.

The coefficient of $x^3$ is four times the coefficient of $x^2$. Find the value of $n$.

By expressing $z_1$ and $z_2$ in modulus-argument form write down the modulus of $w$;

Use the identity from part (b) to show that the quadratic equation $x^2-6x+1=0$ has roots ${\text{cot}}^2\frac{\mathrm{\pi }}{8}$ and ${\text{cot}}^2\frac{3\mathrm{\pi }}{8}$.

There are more males than females in the group.

Determine the roots of the equation $(z+2\text{i}{)}^3=216\text{i}$, $z\in \mathbb{C}$, giving the answers in the form $z=a\sqrt{3}+b\text{i}$ where $a,b\in \mathbb{Z}$.

the girls do not sit on either end.

$a$.

Use de Moivre’s theorem and the result from part (a) to show that $\cot 4\theta =\frac{{\cot }^4 \theta -6 {\cot }^2 \theta +1}{4 {\cot }^3 \theta -4 \cot \theta }$.

Two of the teachers, Gary and Gerwyn, refuse to go out for a meal together.

Plot the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ on an Argand diagram.

By using de Moivre’s theorem, show that $\alpha =\frac{1}{1-{\mathrm{e}}^{\mathrm{i}{\displaystyle \frac{\mathrm{\pi }}{6}}}}$ is a root of this equation.

Show that $z_1{z_2}^{\ast }=r_1{r}_2{\text{e}}^{\text{i}\left(\alpha -\theta \right)}$ where ${z_2}^{\ast }$ is the complex conjugate of $z_2$.

State what the result in part (f)(ii) tells us when considering this equation $x^4-9x^3+24x^2+22x-12=0$.

By varying the value of $q$ in the equation $x^3-7x^2+qx+1=0$, determine the smallest positive integer value of $q$ required to show that Noah is incorrect.

Using the result from part (c), show that when $q=17$, this equation has at least one complex root.

Use your result from part (f)(ii) to show that the equation $x^4-2x^3+3x^2-4x+5=0$ has at least one complex root.

Determine the value of $\text{Re}\left(\alpha \right)$.

Find the modulus and argument of $z$ in terms of $\theta$. Express each answer in its simplest form.

By expressing $z_1$ and $z_2$ in modulus-argument form write down the argument of $w$.

Show that $\sin {105}^{\circ }+\cos {105}^{\circ }=\frac{1}{\sqrt{2}}$.

Consider the equation $z^4+a{z}^3+b{z}^2+cz+d=0$, where $a$, $b$, $c$, $d\in \mathbb{R}$ and $z\in \mathbb{C}$.

Two of the roots of the equation are log₂6 and $i\sqrt{3}$ and the sum of all the roots is 3 + log₂3.

Show that 6$a$ + $d$ + 12 = 0.

Solve the inequality .

Let $P\left(z\right)=a{z}^3-37z^2+66z-10$, where $z\in \mathbb{C}$ and $a\in \mathbb{Z}$.

One of the roots of $P\left(z\right)=0$ is $3+\text{i}$. Find the value of $a$.

Determine the value of

(i)     $1+\omega +{\omega }^2$;

(ii)     $1+\omega \text{*}+(\omega \text{*}{)}^2$.

Find the smallest positive integer value of $n$, such that $w^n$ is a real number.

Hence find the cube roots of $z$ in modulus-argument form.

Solve the equation, giving the solutions in the form $a+\text{i}b$, where $a\text{, }b\in \mathbb{R}$.

Solve $2\sin (x+{60}^{\circ })=\cos (x+{30}^{\circ }),0^{\circ }⩽x⩽{180}^{\circ }$.

Verify that ${\omega }_1=1+{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{6}}$ is a root of this equation.

Find ${\omega }_2$ and ${\omega }_3$, expressing these in the form $a+{\mathrm{e}}^{\mathrm{i}\theta }$, where $a\in \mathrm{ℝ}$ and $\theta >0$.

Find $\mathrm{AC}$.