Modulus and argument with equations

The complex number \(w\) is given by $$w = 10 - 5\text{i}$$

1. Find \(|w|\). [1]
2. Find \(\arg w\), giving your answer in radians to 2 decimal places. [2]

The complex numbers \(z\) and \(w\) satisfy the equation $$(2 + \text{i})(z + 3\text{i}) = w$$

1. Use algebra to find \(z\), giving your answer in the form \(a + b\text{i}\), where \(a\) and \(b\) are real numbers. [4]

Given that $$\arg(\lambda + 9\text{i} + w) = \frac{\pi}{4}$$ where \(\lambda\) is a real constant,

1. find the value of \(\lambda\). [2]

Given that \(\frac{z + 2i}{z - \lambda i} = i\), where \(\lambda\) is a positive, real constant,

1. show that \(z = \left( \frac{\lambda}{2} + 1 \right) + i \left( \frac{\lambda}{2} - 1 \right)\). [5]

Given also that \(\arg z = \arctan \frac{1}{3}\), calculate

1. the value of \(\lambda\), [3]
2. the value of \(|z|^2\). [2]