(a) The complex number \(3 + 2 \mathrm { i }\) is denoted by \(w\) and the complex conjugate of \(w\) is denoted by \(w ^ { * }\). Find
the modulus of \(w\),
the argument of \(w ^ { * }\), giving your answer in radians, correct to 2 decimal places.
(b) Find the complex number \(u\) given that \(u + 2 u ^ { * } = 3 + 2 \mathrm { i }\).
(c) Sketch, on an Argand diagram, the locus given by \(| z + 1 | = | z |\). [0pt]
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Find the value of \(k\) such that \(\left( \begin{array} { l } 1 2 1 \end{array} \right)\) and \(\left( \begin{array} { r } - 2 3 k \end{array} \right)\) are perpendicular.
Two lines have equations \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 3 2 7 \end{array} \right) + \lambda \left( \begin{array} { r } 1 - 1 3 \end{array} \right)\) and \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6 5 2 \end{array} \right) + \mu \left( \begin{array} { r } 2 1 - 1 \end{array} \right)\).
Find the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
The vector \(\left( \begin{array} { l } 1 a b \end{array} \right)\) is perpendicular to the lines \(l _ { 1 }\) and \(l _ { 2 }\).
Find the values of \(a\) and \(b\). [0pt]
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