Complex numbers: Practice Questions

Given that the complex number \( z = a + bi \) has a modulus \( |z| = 5 \) and an argument \( \arg(z) = \pi \), find the values of \( a \) and \( b \).

The complex number \(w\) has modulus \(r\) and argument \(\theta\), where \(0 < \theta < \frac{\pi}{2}\). Given that \(w = \frac{2 + i}{a + 2i}\) for some real constant \(a\), and that \(\theta = \frac{\pi}{4}\), find the value of \(r^2\).

The complex number \(z\) satisfies the equation \(z^2 - (4 + 2i)z + (3 + 6i) = 0\). One root of this equation is \(3 + 3i\). Find the other root in the form \(x + iy\).

The complex number \(z\) is defined by \(z = \frac{a + 6i}{1 + 2i}\), where \(a\) is a real constant. Given that \(z\) is a purely imaginary number, find the value of \(a\).

In an Argand diagram, the locus of points representing the complex number \(z\) is defined by the equation \(|z - 2 - 2i| = 2\). A second locus is defined by the equation \(\arg(z - 2) = \frac{\pi}{2}\). Find the complex number represented by the point of intersection of these two loci.

The complex number \( z \) satisfies the equation \( 3z + 2z^* = 15 - 4i \), where \( z^* \) denotes the complex conjugate of \( z \). Find \( z \) in the form \( x + iy \).

Find the Cartesian equation of the locus of points representing complex numbers \( z = x + iy \) that satisfy the equation \( |z - 3i| = 2|z - 3| \), simplifying your answer.

Find the two square roots of the complex number \( 7 - 24i \), expressing your answers in the form \( x + iy \) where \( x \) and \( y \) are real constants.

(a) Find the two square roots of the complex number \( -15 + 8i \), giving your answers in the form \( a + bi \).
(b) Let these square roots be \( w_1 \) and \( w_2 \), where \( \text{Re } w_1 > 0 \). On an Argand diagram, sketch the loci given by:
(i) \( |z - w_1| = |z - w_2| \),
(ii) \( \text{arg}(z - w_1) = -\frac{\pi}{4} \).
(c) Find the complex number represented by the point of intersection of these two loci, giving your answer in the form \( x + iy \).