6. The complex number \(w\) is given by
$$w = 10 - 5 \mathrm { i }$$
- Find \(| w |\).
- Find arg \(w\), giving your answer in radians to 2 decimal places.
The complex numbers \(z\) and \(w\) satisfy the equation
$$( 2 + \mathrm { i } ) ( z + 3 \mathrm { i } ) = w$$
- Use algebra to find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
Given that
$$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$
where \(\lambda\) is a real constant,
- find the value of \(\lambda\).
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