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 \(l _ { 2 }\).

1. Find the position vector of \(P\).
2. Find, correct to 1 decimal place, the acute angle between \(/ _ { 1 }\) and \(/ _ { 2 }\).
   \(Q\) is a point on \(/ 1\) which is 12 metres away from \(P \cdot R\) is the point on \(/ 2\) such that \(Q R\) is perpendicular to \(/ 1\).
3. Determine the length \(Q R\).
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   5. (a) Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
4. 
   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 \(x\).
5. State the set of values for which the expansion in part (b) is valid.
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   6. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(= \left( \begin{array} { l l } 1 & a
   3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2
   3 & 3 \end{array} \right)\).
6. Find the value of a such that \(\mathbf { A B } = \mathbf { B A }\).
7. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
8. 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.
9. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).
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   7. (a) In this question you must show detailed reasoning. Find the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
10. 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 \(\{ z : | z | \leqslant | z - 2 \mathrm { i } | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \}\).
11. 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 \(| z | < | z - 2 i |\).
12. On the same Argand diagram as part (b), indicate the positions of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).
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