Complex numbers：練習問題

Consider the complex number equation \(z^2 - 2z + k = 0\), where \(k\) is a real constant. If one root of this equation is \(z = 1 + 3i\), find the value of \(k\) and the other root.

Find the value of the modulus and the principal argument of the complex number \(z = \frac{(1 + i\tan\theta)^2}{1 - i\tan\theta}\), where \(0 < \theta < \frac{\pi}{2}\).

A complex number \(z\) satisfies the equation \(2z + z^* = 6 - i\), where \(z^*\) denotes the complex conjugate of \(z\). Find the value of \(z\) in the form \(x + iy\).

The locus of points representing the complex number \(z\) in the Argand diagram is given by the equation \(|z - 4| = |z - 2i|\). Find the Cartesian equation of this locus.

The roots of the equation \(z^n = 1\) are represented by the vertices of a regular polygon in the Argand diagram. For \(n=6\), find the sum of the squares of the distances from each vertex to the point representing the complex number \(w = 2\).

Given that the complex number \(z = 2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\), express \(z^3\) in the form \(a + bi\).

Given the complex number \(z = \frac{3+i}{1-2i}\),
(a) Express \(z\) in the form \(x + iy\).
(b) Find the modulus \(|z|\) and the argument \(\arg(z)\) in radians.

Given that \( z = \cos \theta + i \sin \theta \):
(a) Show that \( z^n + z^{-n} = 2 \cos(n\theta) \) and \( z^n - z^{-n} = 2i \sin(n\theta) \).
(b) Use the identity \( (z - z^{-1})^4 = z^4 - 4z^2 + 6 - 4z^{-2} + z^{-4} \) to express \( \sin^4 \theta \) in the form \( A \cos 4\theta + B \cos 2\theta + C \).
(c) Hence, find the exact value of \( \int_{0}^{\pi/2} \sin^4 \theta \, d\theta \).