SurdsAQA A-Level Maths: Subtopic test

1. 1
   Let $p=\sqrt{50}$ and $q=\sqrt{8}$.

   (a)

   Write $p$ in the form $a\sqrt b$, where $b$ is as small as possible.

   [1 mark]

   • A$25\sqrt2$
   • B$5\sqrt2$
   • C$5\sqrt{10}$
   • D$10\sqrt5$

   (b)

   Find $p+q$, giving your answer in the form $a\sqrt2$.

   [1 mark]

   • A$7\sqrt2$
   • B$\sqrt{58}$
   • C$10\sqrt2$
   • D$3\sqrt2$

   (c)

   Express $\dfrac{p}{q}$ as a fraction in its simplest form.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A rectangle has length $(3+\sqrt5)$ cm and width $(3-\sqrt5)$ cm.

   (a)

   Find the area of the rectangle, in cm$^2$.

   [1 mark]

   • A$14$
   • B$6$
   • C$9-\sqrt5$
   • D$4$

   (b)

   Find the exact length of a diagonal of the rectangle, in cm.

   [1 mark]

   • A$28$
   • B$\sqrt{14}$
   • C$2\sqrt7$
   • D$6$

   (c)

   Express length divided by width in the form $a+b\sqrt5$, where $a$ and $b$ are rational.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangle has area $(8+2\sqrt7)$ cm$^2$ and length $(\sqrt7-1)$ cm.

   (a)

   Find the width of the rectangle, giving your answer in the form $\dfrac{a+b\sqrt7}{c}$ where $a$, $b$ and $c$ are integers.

   [3 marks]

   (b)

   Find the exact perimeter of the rectangle, giving your answer in the form $\dfrac{a+b\sqrt7}{c}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two numbers are given by $x=\dfrac{2}{3-\sqrt5}$ and $y=\dfrac{2}{3+\sqrt5}$.

   (a)

   (i) Show that $x=\dfrac{3+\sqrt5}{2}$.
   (ii) Show that $x+y=3$ and $xy=1$.

   [6 marks]

   (b)

   Given that $x+y=3$ and $xy=1$:
   (i) find the value of $x^2+y^2$ without evaluating $x^2$ or $y^2$ separately;
   (ii) hence show that $x-y=\sqrt5$.

   [6 marks]

   Total for question 4: 12 marks

End of questions