1. 1
   The plane $\Pi$ passes through the point A$(1, 2, -1)$ and has normal vector $\mathbf{n} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$.

   (a)

   Find a Cartesian equation of $\Pi$.

   [1 mark]

   • A$2x - y + 3z = 3$
   • B$2x - y + 3z = 0$
   • C$x + 2y - z = -3$
   • D$2x - y + 3z = -3$

   (b)

   Which of the following points lies on $\Pi$?

   [1 mark]

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

   (c)

   The point B$(3, k, 2)$ lies on $\Pi$. Find the value of $k$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The points P$(1, 0, 2)$, Q$(3, 1, 0)$ and R$(0, 2, 1)$ lie in the plane $\Pi$.

   (a)

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

   [1 mark]

   • A$\mathbf{r} = \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) + \mu(-\mathbf{i} + 2\mathbf{j} - \mathbf{k})$
   • B$\mathbf{r} = \mathbf{i} + 2\mathbf{k} + \lambda(3\mathbf{i} + \mathbf{j}) + \mu(2\mathbf{j} + \mathbf{k})$
   • C$\mathbf{r} = \mathbf{i} + 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) + \mu(-\mathbf{i} + 2\mathbf{j} - \mathbf{k})$
   • D$\mathbf{r} = \mathbf{i} + 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) + \mu(-2\mathbf{i} - \mathbf{j} + 2\mathbf{k})$

   (b)

   Find $\overrightarrow{PQ}\times\overrightarrow{PR}$, a normal vector to $\Pi$.

   [1 mark]

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

   (c)

   Hence find a Cartesian equation of $\Pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A flat solar panel on a roof is in the shape of a parallelogram ABDC. Relative to an origin on horizontal ground, three of its corners are A$(2, 0, 1)$, B$(4, 3, 1)$ and C$(0, 1, 4)$, where the $z$-axis is vertical and distances are in metres.

   (a)

   Show that the Cartesian equation of the plane containing the panel is $9x - 6y + 8z = 26$.

   [3 marks]

   (b)

   (i) Find the coordinates of D.
   (ii) The panel is held up by a vertical pole whose base is at the point $(3, 2, 0)$ on the ground and whose top touches the panel. Find the length of the pole.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The plane $\Pi_1$ has vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + \mu\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$, where $\lambda, \mu\in\mathbb{R}$. The plane $\Pi_2$ has Cartesian equation $4x + py + qz = 7$, where $p, q\in\mathbb{R}$, and is parallel to $\Pi_1$.

   (a)

   (i) Find a Cartesian equation of $\Pi_1$.
   (ii) Show that the point E$(3, 3, 2)$ lies on $\Pi_1$, and find the values of $\lambda$ and $\mu$ that give E.

   [6 marks]

   (b)

   (i) Write down the value of $p$ and the value of $q$.
   (ii) Show that $\Pi_1$ and $\Pi_2$ have no points in common.
   (iii) Find a vector equation of $\Pi_2$ in the form $\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions