1. 1
   In a maths quiz, students are given a set of surd simplification questions to answer without a calculator.

   (a)

   Simplify $\sqrt{48}$.

   [1 mark]

   • A$4\sqrt{3}$
   • B$2\sqrt{12}$
   • C$4\sqrt{6}$
   • D$16\sqrt{3}$

   (b)

   Simplify fully $3\sqrt{2} \times 2\sqrt{8}$.

   [1 mark]

   • A$24\sqrt{2}$
   • B$6\sqrt{16}$
   • C$24$
   • D$5\sqrt{10}$

   (c)

   Simplify $\sqrt{72} \div \sqrt{8}$.

   [1 mark]

   • A$6$
   • B$9$
   • C$\sqrt{9}$
   • D$3$

   Total for question 1: 3 marks

2. 2
   A landscape gardener is designing a rectangular flower bed with a length of $\sqrt{12}$ metres and a width of $\sqrt{3}$ metres.

   (a)

   Calculate the exact area of the flower bed. Give your answer as an integer.

   [2 marks]

   (b)

   The gardener wants to edge the entire perimeter of the flower bed. Simplify $\sqrt{12}$ first, then calculate the exact perimeter of the flower bed, giving your answer in the form $a\sqrt{3}$ metres.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student revising for a non-calculator paper is investigating expressions of the form $(\sqrt{a} + \sqrt{b})^2$.

   (a)

   Show that $(\sqrt{5} + \sqrt{3})^2 = 8 + 2\sqrt{15}$.

   [3 marks]

   (b)

   Show that $(2\sqrt{3} - \sqrt{2})^2$ can be written in the form $p + q\sqrt{6}$, stating the values of $p$ and $q$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A construction engineer is calculating exact lengths for a triangular metal brace, where all measurements involve surds and must be left in exact form before a final decimal check.

   (a)

   Rationalise the denominator of $\dfrac{5}{\sqrt{10}}$, giving your answer in its simplest form.

   [4 marks]

   (b)

   Prove that $\dfrac{1}{3+\sqrt{5}}$ can be written as $\dfrac{3-\sqrt{5}}{4}$, by rationalising the denominator.

   [4 marks]

   (c)

   The length of the brace, in metres, is given by $\dfrac{2+\sqrt{3}}{2-\sqrt{3}}$. By rationalising the denominator, express this length in the form $p + q\sqrt{3}$, where $p$ and $q$ are integers, and hence calculate the length correct to 2 decimal places.

   [5 marks]

   Total for question 4: 13 marks

End of questions