1. 1
   Simplify: $p=\sqrt{20}$; $q=\sqrt{200}-\sqrt{32}$; $r=\sqrt{45}$.

   (a)

   Simplify $p$.

   [1 mark]

   • A$2\sqrt5$
   • B$4\sqrt5$
   • C$5\sqrt2$
   • D$10\sqrt2$

   (b)

   Simplify $q$.

   [1 mark]

   • A$4\sqrt2$
   • B$6\sqrt2$
   • C$8\sqrt2$
   • D$2\sqrt2$

   (c)

   Simplify $r$.

   [1 mark]

   • A15
   • B$9\sqrt5$
   • C$5\sqrt3$
   • D$3\sqrt5$

   Total for question 1: 3 marks

2. 2
   A student is simplifying fractions involving surds: $\frac{10}{\sqrt5}$ and $\frac{6}{2-\sqrt2}$.

   (a)

   Rationalise the denominator of $\frac{10}{\sqrt5}$, simplifying your answer.

   [2 marks]

   (b)

   Rationalise the denominator of $\frac{6}{2-\sqrt2}$, simplifying your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangle has length $(3+\sqrt2)$ cm and width $(4-\sqrt2)$ cm.

   (a)

   Find the perimeter of the rectangle, giving your answer in its simplest form.

   [3 marks]

   (b)

   Find the area of the rectangle. Expand the brackets and simplify, giving your answer in the form $a+b\sqrt2$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A right-angled triangle has two shorter sides of length $\sqrt{18}$ cm and $\sqrt{32}$ cm.

   (a)

   Simplify $\sqrt{18}$ and $\sqrt{32}$, each in simplest surd form, then find the sum of the two shorter sides in the form $a\sqrt2$.

   [4 marks]

   (b)

   Use Pythagoras' theorem to find the length of the hypotenuse, giving your answer in its simplest surd form.

   [4 marks]

   (c)

   A second, similar triangle has a scale factor of 3 applied to the first triangle. Find the length of its hypotenuse in simplest surd form, then rationalise the denominator of $\frac{10}{\text{hypotenuse of second triangle}}$, giving your answer in simplest surd form.

   [5 marks]

   Total for question 4: 13 marks

End of questions