1. 1
   A translation of a shape is described by the vector $\mathbf{a} = \binom{4}{-3}$ and a second translation is described by the vector $\mathbf{b} = \binom{-2}{5}$.

   (a)

   Find $\mathbf{a} + \mathbf{b}$ as a column vector.

   [1 mark]

   • A$\binom{2}{2}$
   • B$\binom{6}{-8}$
   • C$\binom{2}{-2}$
   • D$\binom{-2}{2}$

   (b)

   Find $2\mathbf{a}$ as a column vector.

   [1 mark]

   • A$\binom{2}{-1.5}$
   • B$\binom{8}{6}$
   • C$\binom{4}{-3}$
   • D$\binom{8}{-6}$

   (c)

   Find the magnitude of vector $\mathbf{a}$.

   [1 mark]

   • A25
   • B5
   • C7
   • D1

   Total for question 1: 3 marks

2. 2
   In triangle OAB, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. M is the midpoint of AB.

   (a)

   Find $\overrightarrow{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

   [2 marks]

   (b)

   Find $\overrightarrow{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   OPQR is a parallelogram with $\overrightarrow{OP} = \mathbf{p}$ and $\overrightarrow{OR} = \mathbf{r}$. X is the point on PQ such that $\overrightarrow{PX} = \frac{1}{3}\overrightarrow{PQ}$.

   (a)

   Find $\overrightarrow{OQ}$ in terms of $\mathbf{p}$ and $\mathbf{r}$.

   [3 marks]

   (b)

   Show that $\overrightarrow{OX} = \mathbf{p} + \frac{1}{3}\mathbf{r}$.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   In quadrilateral OABC, $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OC} = \mathbf{c}$, and $\overrightarrow{OB} = 2\mathbf{a} + \mathbf{c}$. E is the midpoint of OC, and F is the point on AB such that $\overrightarrow{AF} = \frac{1}{2}\overrightarrow{AB}$.

   (a)

   Find $\overrightarrow{AB}$ in terms of $\mathbf{a}$ and $\mathbf{c}$.

   [4 marks]

   (b)

   Find $\overrightarrow{OF}$ in terms of $\mathbf{a}$ and $\mathbf{c}$.

   [4 marks]

   (c)

   Find the position vector of the midpoint of EF in terms of $\mathbf{a}$ and $\mathbf{c}$, and hence determine whether this midpoint lies on the line OB.

   [5 marks]

   Total for question 4: 13 marks

End of questions