1. 1
   The line $L$ has vector equation $\mathbf{r} = (\mathbf{i} + 2\mathbf{k}) + t(2\mathbf{i} + \mathbf{j} - \mathbf{k})$, where $t\in\mathbb{R}$. The plane $\Pi$ has equation $x + 2y + z = 9$.

   (a)

   Find the value of $t$ at the point where $L$ meets $\Pi$.

   [1 mark]

   • A$3$
   • B$-1$
   • C$2$
   • D$-2$

   (b)

   Find the coordinates of the point where $L$ meets $\Pi$.

   [1 mark]

   • A$(4, 2, -2)$
   • B$(5, 2, 0)$
   • C$(7, 3, -1)$
   • D$(-1, -1, 3)$

   (c)

   Find the acute angle between $L$ and $\Pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The planes $\Pi_1$ and $\Pi_2$ have equations $x + 2y + 2z = 5$ and $2x - y + 2z = 1$ respectively. They meet in the line $M$.

   (a)

   Find the acute angle between $\Pi_1$ and $\Pi_2$, correct to one decimal place.

   [1 mark]

   • A$26.4^\circ$
   • B$116.4^\circ$
   • C$87.2^\circ$
   • D$63.6^\circ$

   (b)

   Which vector is a direction vector for $M$?

   [1 mark]

   • A$6\mathbf{i} + 2\mathbf{j} - 5\mathbf{k}$
   • B$6\mathbf{i} - 2\mathbf{j} - 5\mathbf{k}$
   • C$3\mathbf{i} + \mathbf{j} + 4\mathbf{k}$
   • D$2\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}$

   (c)

   Find a vector equation of $M$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three planes have equations $\Pi_1: x + y + z = 6$, $\Pi_2: 2x - y + z = 3$ and $\Pi_3: x + 4y + 2z = k$, where $k\in\mathbb{R}$.

   (a)

   Find the value of $k$ for which the three planes meet in a line.

   [3 marks]

   (b)

   (i) For this value of $k$, find a vector equation of the line in which the planes meet.
   (ii) Describe the geometrical arrangement of the three planes when $k \neq 15$. Justify your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a laboratory, a laser is mounted at the point S$(3, -1, 8)$ and fires a narrow beam in the direction of the vector $-2\mathbf{i} + 3\mathbf{j} + \mathbf{k}$. A flat mirror lies in the plane $x - 2y + 2z = 3$. Distances are in metres.

   (a)

   (i) Find the coordinates of the point P where the beam meets the mirror.
   (ii) Find the acute angle between the beam and the mirror.

   [6 marks]

   (b)

   The laser is turned so that its beam from S meets the mirror at right angles. Find the coordinates of the point F where the beam now meets the mirror, and hence find the shortest distance from S to the mirror.

   [6 marks]

   Total for question 4: 12 marks

End of questions