[Legacy] Indices and surds

Find the perimeter of the trapezium, giving your answer in its simplest form.

Find the area of the trapezium, giving your answer in the form $p\sqrt{7}+q$, where $p$ and $q$ are rational numbers.

Find $\cot \angle DBC$, giving your answer in the form $r\sqrt{7}+s$, where $r$ and $s$ are simplified rational numbers.

You are given that $\cos 120° = -\frac{1}{2}$, $\sin 120° = \frac{\sqrt{3}}{2}$ and $\tan 120° = -\sqrt{3}$.

In the triangle $ABC$, $AB = 5\sqrt{3} - 6$, $BC = 5\sqrt{3} + 6$ and angle $ABC = 120°$. Find $AC$, giving your answer in the form $a\sqrt{b}$ where $a$ and $b$ are integers greater than 1.

You are given that $\cos 30° = \frac{\sqrt{3}}{2}$, $\sin 30° = \frac{1}{2}$ and $\tan 30° = \frac{1}{\sqrt{3}}$.

In the triangle $PQR$, $PQ = 3 + 2\sqrt{5}$ and angle $PQR = 30°$. Given that the area of this triangle is $\frac{2 + 5\sqrt{5}}{4}$, find $QR$, giving your answer in the form $c + d\sqrt{5}$, where $c$ and $d$ are integers.

Find the exact coordinates of the points of intersection of the curve $y = x^2 + 2\sqrt{5}x - 20$ and the line $y = 3\sqrt{5}x + 10$.

It is given that $\tan\theta = \frac{\sqrt{3}-1}{2+\sqrt{3}}$, for $0 < \theta < \frac{\pi}{2}$. Find $\operatorname{cosec}^2\theta$ in the form $a+b\sqrt{3}$, where $a$ and $b$ are constants.

Show that the coordinates of $A$ can be written in the form $(p+q\sqrt{3}, r+s\sqrt{3})$, where $p$, $q$, $r$ and $s$ are integers.

Find the $x$-coordinate of the stationary point on the curve, giving your answer in the form $a+b\sqrt{3}$, where $a$ and $b$ are rational numbers.