Simplify $$\frac{5 - 2 \sqrt{3}}{\sqrt{3} - 1}$$ giving your answer in the form $p + q \sqrt{3}$, where $p$ and $q$ are rational numbers. - Edexcel - A-Level Maths Pure - Question 6 - 2011 - Paper 2

Expanding the numerator:

$(5 - 2 \sqrt{3})(\sqrt{3} + 1) = 5\sqrt{3} + 5 - 2 \cdot 3 - 2 \sqrt{3} = 5\sqrt{3} + 5 - 6 - 2\sqrt{3} = 3\sqrt{3} - 1$

The denominator simplifies as follows:

$(\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2$

Now, we combine the results:

$\frac{3\sqrt{3} - 1}{2} = \frac{-1}{2} + \frac{3}{2}\sqrt{3}$

Thus, in the form $p + q\sqrt{3}$, we have:

$p = -\frac{1}{2}, \: q = \frac{3}{2}$.