1. 1
   A design student is calculating exact lengths of diagonals in geometric figures and needs to simplify and manipulate surd expressions.

   (a)

   Simplify $\sqrt{50}$.

   [1 mark]

   • A$5\sqrt{2}$
   • B$2\sqrt{5}$
   • C$10\sqrt{5}$
   • D$25\sqrt{2}$

   (b)

   Simplify $\sqrt{8} + \sqrt{18}$.

   [1 mark]

   • A$26$
   • B$\sqrt{26}$
   • C$5\sqrt{2}$
   • D$2\sqrt{13}$

   (c)

   Evaluate $625^{-\frac{1}{2}}$.

   [1 mark]

   • A$\frac{1}{625}$
   • B$25$
   • C$-25$
   • D$\frac{1}{25}$

   Total for question 1: 3 marks

2. 2
   A design student is working out exact dimensions and areas of a rectangle whose sides are given as surds.

   (a)

   A rectangle has length $\sqrt{48}$ cm and width $\sqrt{12}$ cm. Simplify each surd, giving your answers in the form $a\sqrt{b}$.

   [2 marks]

   (b)

   Using your simplified surds from part (a), calculate the exact area of the rectangle, giving your answer as an integer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student is rationalising denominators involving surds as part of a coursework exercise on exact calculations.

   (a)

   Rationalise the denominator of $\frac{2}{\sqrt{8}}$, giving your answer in the form $a\sqrt{b}$ where $b$ is as small as possible.

   [3 marks]

   (b)

   Rationalise the denominator of $\frac{1}{2 - \sqrt{3}}$, giving your answer in the form $a + b\sqrt{3}$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A student is applying surd simplification to a geometry problem involving a right-angled triangle, calculating an exact hypotenuse and comparing it with rationalised expressions.

   (a)

   A right-angled triangle has two shorter sides of length $\sqrt{20}$ cm and $\sqrt{45}$ cm. Simplify both surds, then use Pythagoras' theorem to find the exact length of the hypotenuse in the form $a\sqrt{b}$.

   [4 marks]

   (b)

   Show that $\left(\frac{1}{25}\right)^{\frac{3}{2}} = \frac{1}{125}$.

   [4 marks]

   (c)

   A rectangle has area $\sqrt{65}$ (from part (a)) multiplied by $\sqrt{5}$ square cm, and one side of length $\sqrt{5}$ cm. Find the exact length of the other side, giving your answer in simplified surd form, and rationalise any denominator that appears in your working.

   [5 marks]

   Total for question 4: 13 marks

End of questions