A1.★★ Rationalise the denominator and simplify fully: $\dfrac{3}{\sqrt5-\sqrt2}$.   Answer:

A2.★★ Express $\dfrac{\sqrt3+1}{\sqrt3-1}$ in the form $a+b\sqrt3$ where $a,b$ are integers.   Answer:

A3.★★ Simplify fully: $\sqrt{75}-\sqrt{12}+\dfrac{6}{\sqrt3}$.   Answer:

A4.★★ Expand and simplify $\big(2\sqrt5-3\big)^2$.   Answer:

A5.★★ A right-angled triangle has hypotenuse $\sqrt{50}$ cm and one shorter side $\sqrt{18}$ cm. Find the exact length of the third side, in simplest surd form.   Answer:

A6.★★★ (a) Show that $\big(\sqrt7+\sqrt3\big)\big(\sqrt7-\sqrt3\big)=4$.   (b) Hence simplify $\dfrac{1}{\sqrt7+\sqrt3}+\dfrac{1}{\sqrt7-\sqrt3}$.   Answer (b):

A7.★★★ Solve $\sqrt{x}=\sqrt{x-5}+1$. (Remember to check your answer.)   $x=$

B1.★★★ From a point $A$ the angle of elevation to the top of a vertical tower is $30°$. From $B$, which is $40$ m closer along the same straight line, it is $50°$. Find the height of the tower (2 d.p.). (Hint: let height $=h$, distance from $B$ to the foot $=d$; use $\tan50=\tfrac{h}{d}$ and $\tan30=\tfrac{h}{d+40}$.)   Height $=$

B2.★★★ A ship sails $12$ km from $P$ on a bearing of $050°$T, then $16$ km on a bearing of $140°$T to reach $Q$. (a) Show the two legs are at right angles. (b) Find $PQ$. (c) Find the bearing of $Q$ from $P$ (nearest degree).

B3.★★★ A right pyramid has a square base of side $10$ cm and apex $12$ cm above the centre. (a) Find a slant edge (apex to base corner), exact & to 2 d.p. (b) Find the angle a slant edge makes with the base (1 d.p.). (Hint: centre-to-corner $=$ half the base diagonal.)

B4.★★★ A drone hovers $80$ m above a straight road. The angle of depression to a car is $25°$, and to a person beyond the car is $15°$. Find the distance between the car and the person (2 d.p.).   Answer:

B5.★★ An equilateral triangle has side length $2\sqrt3$ cm. Find its exact area (surd form).   Area $=$