Math topic中文

Four operations

The four operations of complex numbers: addition, subtraction, multiplication and division of complex numbers expressed in cartesian form.

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…
2已知复数 $z_{1}=1-3i$ ， $z_{2}=3+i$ ，则 $(\quad)$ A. $|z_{1}+z_{2}|=6\:\:$ B. ${\overline{z}}_{1}-{z}_{2}=-2+2i$ C. $z_{1}z_{2}=6-8i$ …
3Solve the equation $i z^5 = -32$ , giving your roots in the form $re^{i\theta}$ , where $r>0$ and $-\pi<\theta\le\pi$ . Ske…
4Given that $z = 2 - \text{i}$ is a root of the equation $4z^4 - 12z^3 + 17z^2 + pz + q = 0$ , where $p$ and $q$ are real,…
5Show that for any complex number $z = e^{i\theta}$ , $z^{n} + \dfrac{1}{z^{n}} = 2\cos n\theta .$ By taking $n = 1$ , sh…
6已知复数 $z=x+yi(x,y\in R)$ ，则 $(\quad)$ A. $z^{2}\geqslant 0$ B. $z$ 的虚部是 $yi$ C. 若 $z=1+2i$ ，则 $x=1$ ， $y=2$ D. $|z|=\sqrt{x^{2}+y^{2}}$ …
7已知 $a∈R$ ，若 $\dfrac {1+ai}{2+i}$ 为实数，则 $a=$ ___ ．
8Solve $z^{4} = -4 - 4\sqrt{3}\,i$ , expressing your answers in the form $re^{i\theta}$ , where $r > 0$ and $-\pi < \theta \le \pi$ …
9One of the roots of the equation $2z^4 - 14z^3 + 33z^2 - 26z + p = 0\text{, where } p \text{ is a constant}$ is $3 + \mathrm{i}$ …
10已知 $A(1\:,\:2)$ ， $B(a\:,1)$ ， $C(2\:,\:3)$ ， $D(-1,b)($ 其中 $a$ ， $b\in R)$ 是复平面上的四点，且向量 $\overset{\rightarrow}{{AB}}{,}{\overset{\rightarrow}{{CD}}{,}}$ …
11已知 $i$ 为虚数单位，则 $\dfrac{i}{1+i}$ 的虚部等于 $()$ A. $\dfrac{1}{2}i$ B. $\dfrac{1}{2}$ C. $\dfrac{1}{4}$ D. $-\dfrac{1}{4}i$ …
12The complex number $z$ satisfies the equation $\mathrm{f}(z) = 0$ , where $\mathrm{f}(z) = (1 - a\mathrm{i})z^2 - 2\mathrm{i}z - 10 - 20\mathrm{i}$ …
13Do not use a calculator in answering this question. Given that $x = 1 + 3\mathrm{i}$ is a root of the equation $x^3 + ax^2 + 18x + b = 0,$ …
14One of the roots of the equation $2z^{4} - 14z^{3} + 33z^{2} - 26z + p = 0\text{, where } p \text{ is a constant}$ is…
15Find, in a simplified form, the cartesian equation of the locus of the point P representing the complex number z where $|z+3|=2\,\Re(z)$ …
16若 $z=1+2i$ ，则 $|z^{2}|=(\quad)$ A. $5+4i$ B. $-3+4i$ C. $25$ D. $5$
17设复数 $z$ 满足 $(1-i)z=1+i$ ，则 $|z|=()$ A. $0$ B. $1$ C. $\sqrt {2}$ D. $2$
18已知 ${i}$ 为虚数单位，复数 $z={{\left( a-{i} \right)}^{2}}$ ， $a\in \mathrm{R}$ ，若复数 $z$ 是纯虚数，则 $\left| z \right|=$ A. $1$ B.…
19在复平面内，复数 $z= \dfrac {1}{3+i}$ 对应的点位于 $()$ A. 第一象限 B. 第二象限 C. 第三象限 D. 第四象限
20Do not use a calculator in answering this question. Three complex numbers are $z_1 = 1 + i$ , $z_2 = \sqrt{3} - i$ and $z_3 = \cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}$ …
21设 $z=\dfrac{1-{i}}{1+{i}}+2{i}$ ，则 $|z|=$ A. $0$ B. $\dfrac{1}{2}$ C. $1$ D. $\sqrt{2}$
22The equation $z^{3}-2z+k=0$ , where $k$ is a real constant, has a root $z = 1 + ai$ , where $a$ is a positive real constan…
23Do not use a calculator in answering this question. It is given that complex numbers $z_1$ and $z_2$ are the roots of th…
24Sketch on a single Argand diagram, the loci defined by $-\frac{\pi}{4} < \arg(z + 1 + 2\text{i}) \le \frac{\pi}{4}$ and…
25设复数 $z=1+ \sqrt {2}i$ ，则 $z^{2}-2z$ 等于 $()$ A. $3 i$ B. $-3 i$ C. $3$ D. $-3$
26The complex number $z$ has modulus 2 and argument $\frac{\pi}{8}$ . It is also given that $w = 1 + \mathrm{i}$ . Given tha…
27Solve the equation $iz^5=-32$ , giving your roots in the form $re^{i\theta}$ , where $r>0$ and $-\pi<\theta\le\pi$ . Sketch…
28已知复数 $z = 3 + 4i$ ，求 $|z|$ 、 $\bar{z}$ 和 $z^2$ 。
29设a，b，c∈R，则复数（a+bi）（c+di）为实数的充要条件是（） A. ad-bc=0 B. ac-bd=0 C. ac+bd=0 D. ad+bc=0
30The complex number $z$ satisfies the equation $\sqrt{2}\,z^{3} = 1 + \dfrac{2}{2a - 1}\,i ,$ where $a$ is a real numbe…
31Find the set of values of $\alpha$ for which the expression $\dfrac{2 - i\sin 2\alpha}{1 + 2i\sin 2\alpha}$ is purely re…
32Do not use a calculator in answering this question. It is given that $\alpha = -\sqrt{3} + \mathrm{i}$ and $\beta = \cos\frac{\pi}{3} + \mathrm{i}\sin\frac{\pi}{3}$ …
33Given that $z_1 = -\frac{\text{i}}{2}$ is a root of the equation $2z^3 + (\text{i}-8)z^2 + az + 13\text{i} = 0$ , find th…
34设 $i$ 为虚数单位，则复数 $\dfrac {i-2}{i}$ 的共轭复数是 ___ ．
35Do not use a calculator in answering this question. Showing your working, find the complex numbers $z$ and $w$ which…
36已知 $z$ 是复数，若 $z+2i$ 为实数 $(i$ 为虚数单位 $)$ ，且 $z-4$ 为纯虚数． (1)求复数 $z$ ； (2)若复数 $(z+mi)^{2}$ 在复平面上对应的点在第四象限，求实数 $m$ 的取值范围．
37已知 $\overset{-}{z}$ 是复数 $z$ 的共轭复数，若 $z+2\overset{-}{z}$ 在复平面上的对应点位于第一象限，则 $z$ 的对应点位于 $(\quad)$ A. 第一象限 B. 第二象限 C. 第三象限…
38A complex number $z$ varies with $t$ such that $z = 2\cos t + \mathrm{i}(3\sin t), \quad \text{where } 0 \le t < 2\pi.$ …
39设复数 $\dfrac {i-2}{1+i}=a+bi(a,b∈R)$ 则 $a+b=()$ A. $1$ B. $2$ C. $-1$ D. $-2$
40若复数 $\dfrac {a+i}{b-i}=2-i$ 其中 $a$ ， $b$ 是实数，则复数 $a+bi$ 在复平面内所对应的点位于 $()$ A. 第一象限 B. 第二象限 C. 第三象限 D. 第四象限
41Solve the equation $z^5 + 16 = 16\sqrt{3}\, i$ , giving the roots in the form $r e^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ …
42The complex numbers $a$ and $b$ are given by $a = -(1 + \sqrt{3}\text{i})$ and $b = \frac{1}{2}(1 - \text{i})$ . Without…
43Do not use a graphing calculator in answering this question. The complex number $w$ is a root to the equation $z^4 + a z^3 + 46 z^2 + b z + 125 = 0 ,$ …
44Given that $z_1 = r_1(\cos\theta_1 + \mathrm{i}\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + \mathrm{i}\sin\theta_2)$ , pr…
45设复数 $z$ 的共轭复数为 $\overline{z}$ ，且满足 $2z+\overline{z}=3+i$ ， $\omega=\sin\theta-i\cos\theta$ ，复数 $z-\omega$ 对应复平面的向量 $\overrightarrow{OM}.(1)$ …
46若复数 $z$ 满足 $(2-i)z=4+3i(i$ 为虚数单位 $)$ ，则 $|z|=$
47Show that $\frac{\text{e}^{\text{i}\theta}}{1 - \text{e}^{\text{i}\theta}}$ can be expressed as $k\left(\text{i}\cot\frac{\theta}{2} - 1\right)$ …
48Given that $z = \frac{\lambda - 4i}{1 - \lambda i}$ , $\lambda \in \mathbb{R}$ and $\arg(z) = \pi$ , find the value of $z$ …
49Do not use a calculator in answering this question. The complex number $z$ has modulus 1 and argument $\theta$ , where $\dfrac{\pi}{2}<\theta<\pi$ …
50Solve the equation $z^{5}+32=0$ , expressing your answers in the form $re^{i\theta}$ , where $r>0$ and $-\pi<\theta\le\pi$ …
51若 $i$ 为虚数单位，则 $(3-6i)(1+9i)=(\quad)$ A. $51+33i$ B. $51-33i$ C. $57+21i$ D. $57-21i$
52Let $z$ be the complex number $-1+i\sqrt{3}$ . Find $\arg(z),$ the real number $a$ such that $\arg\big(z(z+a)\big)=\frac{5\pi}{6}$ …
53已知 $i$ 是虚数单位，则 $(\dfrac{1-i}{1+i})^{3}=(\quad)$ A. $1$ B. $i$ C. $-i$ D. $-1$
54 $i$ 为虚数单位， $\dfrac{5}{i-2}=(\quad)$ A. $i\text{+}2$ B. $i\text{-}2$ C. $\text{-}2\text{-}i$ D. $2\text{-}i$
55An arithmetic series has first term $a$ and common difference $d$ , where $a > 0$ and $d \ne 0$ . The first, sixth and nin…
56下列四个选项中，正确的为 $(\quad)$ A. 复平面内实轴上的点都表示实数，虚轴上的点都表示纯虚数 B. 已知复数 $z=x+yi(x,y\in R)$ ，且 $|z-2|=\sqrt{3}$ ，则 $\dfrac{y}{x}$ 的…
57The points P and Q are represented by the complex numbers p and q respectively, where $p=\sqrt{2}+\sqrt{2}i\,,\ \arg(q)=\dfrac{2}{3}\pi\text{ and }|q|=2.$ …
58 $(1+i)(2-i)=()$ A. $-3-i$ B. $-3+i$ C. $3-i$ D. $3+i$
59已知 $i$ 是虚数单位，复数 $z=1+2i$ ，则 $i \overline {z}=()$ A. $2-i$ B. $2+i$ C. $-2-i$ D. $-2+i$
60已知复数 $z_{1}=2+ai(a∈R)$ ， $z_{2}=1-2i$ ，若 $\dfrac {z_{1}}{z_{2}}$ 为纯虚数，则 $|z_{1}|=()$ A. $\sqrt {2}$ B. $\sqrt {3}$ …