1. 1
   The line $L$ has vector equation $\mathbf{r}=\begin{pmatrix}1\\ -2\\ 3\end{pmatrix}+\lambda\begin{pmatrix}2\\ 1\\ -2\end{pmatrix}$, $\lambda\in\mathbb{R}$.

   (a)

   Which of the following points lies on $L$?

   [1 mark]

   • A$(3,-1,5)$
   • B$(2,1,-2)$
   • C$(-1,-1,5)$
   • D$(5,0,-1)$

   (b)

   Which of the following is a Cartesian equation of $L$?

   [1 mark]

   • A$\frac{x-1}{2}=\frac{y+2}{1}=\frac{z-3}{-2}$
   • B$\frac{x+1}{2}=\frac{y-2}{1}=\frac{z+3}{-2}$
   • C$\frac{x-2}{1}=\frac{y-1}{-2}=\frac{z+2}{3}$
   • D$\frac{x-1}{2}=\frac{y+2}{1}=\frac{z-3}{2}$

   (c)

   The line $M$ has equation $\mathbf{r}=\mu\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}$. Find the acute angle between $L$ and $M$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In two dimensions, the line $\ell$ passes through the points $P(3,-1)$ and $Q(7,5)$.

   (a)

   Which of the following is a vector equation of $\ell$?

   [1 mark]

   • A$\mathbf{r}=\begin{pmatrix}3\\ -1\end{pmatrix}+\lambda\begin{pmatrix}7\\ 5\end{pmatrix}$
   • B$\mathbf{r}=\begin{pmatrix}4\\ 6\end{pmatrix}+\lambda\begin{pmatrix}3\\ -1\end{pmatrix}$
   • C$\mathbf{r}=\begin{pmatrix}3\\ -1\end{pmatrix}+\lambda\begin{pmatrix}4\\ 6\end{pmatrix}$
   • D$\mathbf{r}=\begin{pmatrix}7\\ 5\end{pmatrix}+\lambda\begin{pmatrix}-4\\ 6\end{pmatrix}$

   (b)

   Which of the following is an equation of $\ell$ in the form $y=mx+c$?

   [1 mark]

   • A$y=\frac23x-3$
   • B$y=\frac32x-\frac{11}{2}$
   • C$y=\frac32x+\frac72$
   • D$y=-\frac32x+\frac72$

   (c)

   Find the coordinates of the point where $\ell$ crosses the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ship moves with constant velocity. Its position $t$ hours after noon is given by $\mathbf{r}=\begin{pmatrix}2\\ 5\end{pmatrix}+t\begin{pmatrix}3\\ -4\end{pmatrix}$, where distances are in kilometres, $\mathbf{i}$ points east and $\mathbf{j}$ points north. A lighthouse stands at the point $H(12,0)$.

   (a)

   Find the speed of the ship and its position at 14:30.

   [3 marks]

   (b)

   Find the time at which the ship is closest to the lighthouse, and the distance between them at that time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Engineers are planning straight tunnels underground. Coordinates are in metres, with the $z$-axis vertically upwards and the $xy$-plane horizontal. The first tunnel starts at the origin $A(0,0,0)$ and passes through the point $B(6,3,-2)$. A second tunnel follows the line $\mathbf{r}=\begin{pmatrix}10\\ 0\\ -3\end{pmatrix}+\mu\begin{pmatrix}2\\ -1\\ 2\end{pmatrix}$.

   (a)

   (i) Write down a vector equation for the line of the first tunnel.
   (ii) Write the equation of this line in parametric form and in Cartesian form.
   (iii) Find the coordinates of the point on the first tunnel that is 5 m below the level of $A$.

   [6 marks]

   (b)

   A calculator may be used in this part.
   (i) Find the acute angle between the two tunnels.
   (ii) Find the angle at which the first tunnel descends below the horizontal.

   [6 marks]

   Total for question 4: 12 marks

End of questions