[Legacy] Indices and surds

DO NOT USE A CALCULATOR IN THIS QUESTION.

Write $(5 - \sqrt{3})(\sqrt{6} + \sqrt{2})^{-2}$ in the form $a + b\sqrt{3}$, where $a$ and $b$ are constants.

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Write $\frac{16+11\sqrt{10}}{2+\sqrt{10}} + 1$ in the form $p + q\sqrt{10}$, where $p$ and $q$ are integers.

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A cylinder has base radius $(2 + \sqrt{3})\,\text{m}$ and volume $\pi(16 + 9\sqrt{3})\,\text{m}^3$. Find the exact value of its height, giving your answer in its simplest form.

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A curve has equation $y = \frac{6 + \sqrt{x}}{3 + \sqrt{x}}$ where $x \geq 0$. Find the exact value of $y$ when $x = 6$. Give your answer in the form $a + b\sqrt{c}$, where $a$, $b$ and $c$ are integers.

Find the exact solution of the equation

$$\frac{p^{\frac{3}{2}} + p^{\frac{1}{2}}}{p^{-\frac{1}{2}}} = 4$$

Solve the following simultaneous equations. Give your answers in the form $a + b\sqrt{3}$, where $a$ and $b$ are rational.

$$\begin{aligned} x + y &= 3 \\ 2x - \sqrt{3}y &= 5 \end{aligned}$$

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The point $(1 - \sqrt{5}, p)$ lies on the curve

$$y = \frac{10 + 2\sqrt{5}}{x^2}$$

Find the exact value of $p$, simplifying your answer.

Find the value of $x$ such that

$$\frac{4^{x+1}}{2^{x-1}} = 32^{\frac{x}{3}} \times 8^{\frac{1}{3}}$$

Solve the following simultaneous equations.

$$\begin{aligned} 3^x \times 9^{y-1} &= 243 \\ 8 \times 2^{y - \frac{1}{2}} &= \frac{2^{2x+1}}{4\sqrt{2}} \end{aligned}$$