IGCSE Math B 1.D Surds Topic Practice

Without using a calculator, find the value of $\frac{\sqrt{12}+\sqrt{300}}{\sqrt{27}}$
Show your working clearly.

Without using a calculator and showing your working clearly, find the value of the integer a so that
$$\sqrt{180}-\sqrt{27}-\sqrt{20}+\sqrt{147}=a(\sqrt5+\sqrt3).$$

Given that $\mathbf a=\binom{7}{-3}$, find the exact value of $|\mathbf a|$.

Here is a right-angled triangle.

Given that $\tan \theta=\sqrt{8}$
express $3(\sin \theta+\cos \theta)$ in the form $m+\sqrt{n}$ where m and n are integers.
Show your working clearly.

Diagram NOT accurately drawn

Without using a calculator and showing all your working, express
$$3\sqrt{180}-2\sqrt{245}$$
in the form $\sqrt{a}$, where a is an integer.

Express
$$\frac{\sqrt{35}+3\sqrt5-2\sqrt7-6}{\sqrt5-2}$$
in the form $m+\sqrt n$, where m and n are integers.
Show your working clearly.

Diagram NOT accurately drawn.

Figure 3 shows triangle ABC.
$$AB=4\sqrt3\text{ cm},\quad BC=3\sqrt5\text{ cm},\quad AC=x\text{ cm},\quad \angle BAC=15^\circ.$$
Given that the exact value of $\cos15^\circ=\frac{\sqrt6+\sqrt2}{4}$.

Show that $(3+2\sqrt3)^2=21+12\sqrt3$.