Divide $12 \sqrt{30}$ by $3 \sqrt{5}$.

Let $x$ be a rational number and $y$ be an irrational number. Is $xy$ necessarily irrational? Justify your answer with an example.

State whether the following statement is true or false:
Every whole number is a rational number.

Find the value of $a$ :
$\frac{3-\sqrt{5}}{3+2 \sqrt{5}}=a \sqrt{5}-\frac{19}{11}$

Rationalise the denominator in each of the following:
$\frac{1}{7-4 \sqrt{3}}$

Classify the following numbers as rational or irrational:
$(i)$ $2-\sqrt{5}$
$(ii)$ $(3+\sqrt{23})-\sqrt{23}$
$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$
$(iv)$ $\frac{1}{\sqrt{2}}$
$(v)$ $2 \pi$