1. 1
   Three lines are given by $L_1:\ \mathbf{r}=\begin{pmatrix}1\\ 2\\ 0\end{pmatrix}+\lambda\begin{pmatrix}2\\ -1\\ 3\end{pmatrix}$, $L_2:\ \mathbf{r}=\begin{pmatrix}3\\ 1\\ 3\end{pmatrix}+\mu\begin{pmatrix}-4\\ 2\\ -6\end{pmatrix}$ and $L_3:\ \mathbf{r}=\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}+t\begin{pmatrix}4\\ -2\\ 6\end{pmatrix}$.

   (a)

   Which statement about $L_1$ and $L_2$ is correct?

   [1 mark]

   • AThey are coincident.
   • BThey are parallel and distinct.
   • CThey intersect at exactly one point.
   • DThey are skew.

   (b)

   Which statement about $L_1$ and $L_3$ is correct?

   [1 mark]

   • AThey are coincident.
   • BThey intersect at exactly one point.
   • CThey are parallel and distinct.
   • DThey are skew.

   (c)

   The point $(k,-1,9)$ lies on $L_1$. Find the value of $k$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In two dimensions, three lines are given by $\ell_1:\ \mathbf{r}=\begin{pmatrix}1\\ 3\end{pmatrix}+s\begin{pmatrix}2\\ -1\end{pmatrix}$, $\ell_2:\ \mathbf{r}=\begin{pmatrix}2\\ -2\end{pmatrix}+t\begin{pmatrix}1\\ 1\end{pmatrix}$ and $\ell_3:\ \mathbf{r}=\begin{pmatrix}3\\ 2\end{pmatrix}+u\begin{pmatrix}-4\\ 2\end{pmatrix}$.

   (a)

   Find the point of intersection of $\ell_1$ and $\ell_2$.

   [1 mark]

   • A$(3,2)$
   • B$(2,-2)$
   • C$(7,0)$
   • D$(5,1)$

   (b)

   Which statement about $\ell_1$ and $\ell_3$ is correct?

   [1 mark]

   • AThey are parallel and distinct.
   • BThey are coincident.
   • CThey intersect at exactly one point.
   • DThey are skew.

   (c)

   The line $\ell_4:\ \mathbf{r}=\begin{pmatrix}0\\ k\end{pmatrix}+v\begin{pmatrix}2\\ -1\end{pmatrix}$ is coincident with $\ell_1$. Find the value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three lines are given by $L_1:\ \mathbf{r}=\begin{pmatrix}1\\ 0\\ 2\end{pmatrix}+\lambda\begin{pmatrix}1\\ 2\\ -1\end{pmatrix}$, $L_2:\ \mathbf{r}=\begin{pmatrix}2\\ 5\\ 0\end{pmatrix}+\mu\begin{pmatrix}1\\ 1\\ 1\end{pmatrix}$ and $L_3:\ \mathbf{r}=\begin{pmatrix}2\\ 5\\ 0\end{pmatrix}+\nu\begin{pmatrix}1\\ p\\ 3\end{pmatrix}$, where $p\in\mathbb{R}$.

   (a)

   Given that $L_1$ and $L_3$ intersect, find the value of $p$.

   [3 marks]

   (b)

   Show that $L_1$ and $L_2$ are skew.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two aircraft, $A$ and $B$, fly in straight lines at constant velocity. Their positions $t$ minutes after 09:00 are $\mathbf{r}_A=\begin{pmatrix}1\\ 2\\ 4\end{pmatrix}+t\begin{pmatrix}2\\ 1\\ 0\end{pmatrix}$ and $\mathbf{r}_B=\begin{pmatrix}-1\\ 15\\ 0\end{pmatrix}+t\begin{pmatrix}3\\ -2\\ 1\end{pmatrix}$, where distances are in kilometres and the third component is the height above the ground.

   (a)

   Show that the flight paths of the two aircraft intersect, and find the coordinates of the point of intersection.

   [6 marks]

   (b)

   (i) Determine whether the aircraft collide.
   (ii) Find the distance between the aircraft at the moment aircraft $B$ passes through the point where the paths cross.

   [6 marks]

   Total for question 4: 12 marks

End of questions