$$\begin{aligned}
& l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 
10 \\
3
\end{array} \right) + \lambda \left( \begin{array} { c } 
2 \\
- 2 \\
1
\end{array} \right) \\
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 
5 \\
2 \\
4
\end{array} \right) + \mu \left( \begin{array} { c } 
3 \\
1 \\
- 2
\end{array} \right)
\end{aligned}$$

$P$ is the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the position vector of $P$.
\item Find, correct to 1 decimal place, the acute angle between $l _ { 1 }$ and $l _ { 2 }$.\\
$Q$ is a point on $l1$ which is 12 metres away from $P \cdot R$ is the point on $l2$ such that $QR$ is perpendicular to $l1$.
\item Determine the length $QR$.\\[0pt]
\\
5. (a) Express $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in three partial fractions.\\
(b)\\
Hence find the first three terms in the expansion of $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$.\\
(c) State the set of values for which the expansion in part (b) is valid.\\[0pt]
\\
6.

The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)$.\\
(a) Find the value of a such that $\mathbf { A B } = \mathbf { B A }$.\\
(b) Prove by counter example that matrix multiplication for $2 \times 2$ matrices is not commutative.\\
(c) A triangle of area 4 square units is transformed by the matrix $\mathbf { B }$. Find the area of the image of the triangle following this transformation.
\item Find the equations of the invariant lines of the form $y = m x$ for the transformation represented by matrix $\mathbf { B }$.\\[0pt]
\\
7. (a) In this question you must show detailed reasoning.

Find the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.\\
(b) The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z | = | z - 2 i |$ and $| z - 2 | = \sqrt { 5 }$ respectively.\\
i. Sketch on a single Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$, showing any intercepts with the imaginary axis.\\
ii. Indicate, by shading on your Argand diagram, the region $\{ z : | z | \leqslant | z - 2 \mathrm { i } | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \}$.\\
(c) i. Show that both of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfy $| z - 2 | < \sqrt { 5 }$.\\
ii. State, with a reason, which root of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfies $| z | < | z - 2 i |$.\\
(d) On the same Argand diagram as part (b), indicate the positions of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.\\[0pt]
\\[0pt]
\\[0pt]
\\[0pt]

\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM 2024 Q10}}