Math topic中文

Extension number system

Extension of the number system from real numbers to complex numbers. A complex number is written in cartesian form $z = x + iy$ , where $x, y \in \mathbb{R}$ and $i$ is the imaginary unit.

Practice questions

1Solve $z^4 = -4 - 4√3\mathrm{i}$ , expressing your answers in the form $r\mathrm{e}^{\mathrm{i}\theta}$ , where $r > 0$ an…
2Solve $z^{4} = -4 - 4\sqrt{3}\,i$ , expressing your answers in the form $re^{i\theta}$ , where $r > 0$ and $-\pi < \theta \le \pi$ …
3[Cambridge/H2 Maths 2007] The complex number $w$ is such that $ww^* + 2w = 3 + 4i$ , where $w^*$ is the complex conju…
4Do not use a calculator in answering this question. Given that $x = 1 + 3\mathrm{i}$ is a root of the equation $x^3 + ax^2 + 18x + b = 0,$ …
5Find, in a simplified form, the cartesian equation of the locus of the point P representing the complex number z where $|z+3|=2\,\Re(z)$ …
6Do not use a calculator in answering this question. Three complex numbers are $z_1 = 1 + i$ , $z_2 = \sqrt{3} - i$ and $z_3 = \cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}$ …
7The complex numbers $a$ and $b$ are given by $a = -(1 + \sqrt{3}\text{i})$ and $b = \frac{1}{2}(1 - \text{i})$ . Without…
8Show that $\frac{\text{e}^{\text{i}\theta}}{1 - \text{e}^{\text{i}\theta}}$ can be expressed as $k\left(\text{i}\cot\frac{\theta}{2} - 1\right)$ …
9Do not use a calculator in answering this question. The complex number $z$ has modulus 1 and argument $\theta$ , where $\dfrac{\pi}{2}<\theta<\pi$ …
10Solve the equation $z^{5}+32=0$ , expressing your answers in the form $re^{i\theta}$ , where $r>0$ and $-\pi<\theta\le\pi$ …
11An arithmetic series has first term $a$ and common difference $d$ , where $a > 0$ and $d \ne 0$ . The first, sixth and nin…
12The complex number $z$ is such that $az^2 + bz + a = 0$ where $a$ and $b$ are real constants. It is given that $z = z_0$ …
13Do not use a calculator in answering this question. The complex number $z$ has modulus 1 and argument $\theta$ , wher…
14In a single Argand diagram, sketch the following loci, labelling each locus clearly. $|z| = 5,$ $|z + 8| = |z - 8i|.$ Th…
15On a single Argand diagram sketch the loci given by (a) $|z - 12 - 12 ext{i}| = 12$ , (b) $| ext{i}z - 1 - 12 ext{i}| = 5$ …
16It is known that $w$ satisfies the equation $|2w - 1| \le |w + 1|$ . If $w \in \mathbb{R}$ , find the range of values of $w$ …
17In a single Argand diagram, sketch the following loci, labelling each locus clearly. $|z| = 5$ . $|z + 8| = |z - 8i|$ . Th…
18A complex number $z$ is given by $z = r \mathrm{e}^{\mathrm{i}\frac{\pi}{n}}$ , where $0 < r < 1$ and $n$ is a positive i…