Math topic中文

Modulus argument conjugate

Modulus, argument and conjugate of a complex number.

Practice questions

  1. 1Solve z4=−4−4√3iz^4 = -4 - 4√3\mathrm{i}, expressing your answers in the form reiθr\mathrm{e}^{\mathrm{i}\theta}, where r>0r > 0 an…
  2. 2已知复数z1=1−3iz_{1}=1-3i，z2=3+iz_{2}=3+i，则()(\quad) A. ∣z1+z2∣=6  |z_{1}+z_{2}|=6\:\: B. z‾1−z2=−2+2i{\overline{z}}_{1}-{z}_{2}=-2+2i C. z1z2=6−8iz_{1}z_{2}=6-8i …
  3. 3Solve the equation iz5=−32i z^5 = -32, giving your roots in the form reiθre^{i\theta}, where r>0r>0 and −π<θ≤π-\pi<\theta\le\pi. Ske…
  4. 4Show that for any complex number z=eiθz = e^{i\theta}, zn+1zn=2cos⁡nθ.z^{n} + \dfrac{1}{z^{n}} = 2\cos n\theta . By taking n=1n = 1, sh…
  5. 5已知复数z=x+yi(x,y∈R)z=x+yi(x,y\in R)，则()(\quad) A. z2⩾0z^{2}\geqslant 0 B. zz的虚部是yiyi C. 若z=1+2iz=1+2i，则x=1x=1，y=2y=2 D. ∣z∣=x2+y2|z|=\sqrt{x^{2}+y^{2}}…
  6. 6Solve z4=−4−43 iz^{4} = -4 - 4\sqrt{3}\,i, expressing your answers in the form reiθre^{i\theta}, where r>0r > 0 and −π<θ≤π-\pi < \theta \le \pi…
  7. 7[Cambridge/H2 Maths 2007] The complex number ww is such that ww∗+2w=3+4iww^* + 2w = 3 + 4i, where w∗w^* is the complex conju…
  8. 8Do not use a calculator in answering this question. Given that x=1+3ix = 1 + 3\mathrm{i} is a root of the equation x3+ax2+18x+b=0,x^3 + ax^2 + 18x + b = 0,…
  9. 9Find, in a simplified form, the cartesian equation of the locus of the point P representing the complex number z where ∣z+3∣=2 ℜ(z)|z+3|=2\,\Re(z)…
  10. 10若z=1+2iz=1+2i，则∣z2∣=()|z^{2}|=(\quad) A. 5+4i5+4i B. −3+4i-3+4i C. 2525 D. 55
  11. 11设复数zz满足(1−i)z=1+i(1-i)z=1+i，则∣z∣=()|z|=() A. 00 B. 11 C. 2 \sqrt {2} D. 22
  12. 12Do not use a calculator in answering this question. Three complex numbers are z1=1+iz_1 = 1 + i, z2=3−iz_2 = \sqrt{3} - i and z3=cos⁡π3+isin⁡π3z_3 = \cos \dfrac{\pi}{3} + i\sin \dfrac{\pi}{3}…
  13. 13设z=1−i1+i+2iz=\dfrac{1-{i}}{1+{i}}+2{i}，则∣z∣=|z|= A. 00 B. 12\dfrac{1}{2} C. 11 D. 2\sqrt{2}
  14. 14The complex number zz satisfies arg⁡(z−1−2i)=θ\arg(z-1-2i)=\theta, where θ\theta is a fixed angle in the interval −π<θ≤π-\pi < \theta \le \pi…
  15. 15Do not use a calculator in answering this question. It is given that complex numbers z1z_1 and z2z_2 are the roots of th…
  16. 16Sketch on a single Argand diagram, the loci defined by −π4<arg⁡(z+1+2i)≤π4-\frac{\pi}{4} < \arg(z + 1 + 2\text{i}) \le \frac{\pi}{4} and…
  17. 17The complex number zz has modulus 2 and argument π8\frac{\pi}{8}. It is also given that w=1+iw = 1 + \mathrm{i}. Given tha…
  18. 18Solve the equation iz5=−32iz^5=-32, giving your roots in the form reiθre^{i\theta}, where r>0r>0 and −π<θ≤π-\pi<\theta\le\pi. Sketch…
  19. 19已知复数 z=3+4iz = 3 + 4i，求 ∣z∣|z|、zˉ\bar{z} 和 z2z^2。
  20. 20The complex number zz satisfies the equation 2 z3=1+22a−1 i,\sqrt{2}\,z^{3} = 1 + \dfrac{2}{2a - 1}\,i , where aa is a real numbe…
  21. 21Find the set of values of α\alpha for which the expression 2−isin⁡2α1+2isin⁡2α\dfrac{2 - i\sin 2\alpha}{1 + 2i\sin 2\alpha} is purely re…
  22. 22Do not use a calculator in answering this question. It is given that α=−3+i\alpha = -\sqrt{3} + \mathrm{i} and β=cos⁡π3+isin⁡π3\beta = \cos\frac{\pi}{3} + \mathrm{i}\sin\frac{\pi}{3}…
  23. 23Given that z1=−i2z_1 = -\frac{\text{i}}{2} is a root of the equation 2z3+(i−8)z2+az+13i=02z^3 + (\text{i}-8)z^2 + az + 13\text{i} = 0, find th…
  24. 24设ii为虚数单位，则复数i−2i \dfrac {i-2}{i}的共轭复数是 ___ ．
  25. 25The complex number aa has modulus rr and argument θ\theta, where r∈R+r \in \mathbb{R}^+ and 0<θ<π20 < \theta < \frac{\pi}{2}…
  26. 26The point A represents a fixed complex number a such that −π2<arg⁡a<0-\dfrac{\pi}{2}<\arg a<0. The complex numbers iaia and a+iaa+ia…
  27. 27Do not use a calculator in answering this question. Showing your working, find the complex numbers zz and ww which…
  28. 28The point AA represents the complex number aa with ∣a∣=1|a|=1 and θ=arg⁡(a),  0<θ<π4\theta=\arg(a),\; 0<\theta<\dfrac{\pi}{4}. The point…
  29. 29已知复数z1=2+aiz_{1}=2+ai，z2=2−iz_{2}=2-i，若∣z1∣<∣z2∣|z_{1}| < |z_{2}|，则实数aa的取值范围是 ___ ．
  30. 30A complex number zz varies with tt such that z=2cos⁡t+i(3sin⁡t),where 0≤t<2π.z = 2\cos t + \mathrm{i}(3\sin t), \quad \text{where } 0 \le t < 2\pi.…
  31. 31Solve the equation z5+16=163 iz^5 + 16 = 16\sqrt{3}\, i, giving the roots in the form reiθr e^{i\theta}, where r>0r>0 and −π<θ≤π-\pi < \theta \le \pi…
  32. 32The complex numbers aa and bb are given by a=−(1+3i)a = -(1 + \sqrt{3}\text{i}) and b=12(1−i)b = \frac{1}{2}(1 - \text{i}). Without…
  33. 33On a single Argand diagram, sketch the locus of points representing the complex number zz such that ∣z−4−2i∣≤2and∣z−3∣≤∣z−5∣.|z-4-2i| \le 2\quad\text{and}\quad |z-3| \le |z-5|.…
  34. 34Do not use a graphing calculator in answering this question. The complex number ww is a root to the equation z4+az3+46z2+bz+125=0,z^4 + a z^3 + 46 z^2 + b z + 125 = 0 ,…
  35. 35Given that z1=r1(cos⁡θ1+isin⁡θ1)z_1 = r_1(\cos\theta_1 + \mathrm{i}\sin\theta_1) and z2=r2(cos⁡θ2+isin⁡θ2)z_2 = r_2(\cos\theta_2 + \mathrm{i}\sin\theta_2), pr…
  36. 36若复数zz满足(2−i)z=4+3i(i(2-i)z=4+3i(i为虚数单位))，则∣z∣=|z|=
  37. 37Show that eiθ1−eiθ\frac{\text{e}^{\text{i}\theta}}{1 - \text{e}^{\text{i}\theta}} can be expressed as k(icot⁡θ2−1)k\left(\text{i}\cot\frac{\theta}{2} - 1\right)…
  38. 38The complex number zz satisfies the relations ∣z∣≤2|z|\le 2 and π6≤arg⁡z≤π2\dfrac{\pi}{6}\le \arg z \le \dfrac{\pi}{2}. Illustrate…
  39. 39Given that z=λ−4i1−λiz = \frac{\lambda - 4i}{1 - \lambda i}, λ∈R\lambda \in \mathbb{R} and arg⁡(z)=π\arg(z) = \pi, find the value of zz…
  40. 40Do not use a calculator in answering this question. The complex number zz has modulus 1 and argument θ\theta, where π2<θ<π\dfrac{\pi}{2}<\theta<\pi…
  41. 41Solve the equation z5+32=0z^{5}+32=0, expressing your answers in the form reiθre^{i\theta}, where r>0r>0 and −π<θ≤π-\pi<\theta\le\pi…
  42. 42The complex number zz satisfies ∣z−4−3i∣≤2andπ<arg⁡[(z−4)2]<2π|z-4-3\text{i}| \le 2 \quad \text{and} \quad \pi < \arg\left[(z-4)^2\right] < 2\pi…
  43. 43Let zz be the complex number −1+i3-1+i\sqrt{3}. Find arg⁡(z),\arg(z), the real number aa such that arg⁡(z(z+a))=5π6\arg\big(z(z+a)\big)=\frac{5\pi}{6}…
  44. 44An arithmetic series has first term aa and common difference dd, where a>0a > 0 and d≠0d \ne 0. The first, sixth and nin…
  45. 45The points P and Q are represented by the complex numbers p and q respectively, where p=2+2i , arg⁡(q)=23π and ∣q∣=2.p=\sqrt{2}+\sqrt{2}i\,,\ \arg(q)=\dfrac{2}{3}\pi\text{ and }|q|=2.…
  46. 46已知复数z1=2+ai(a∈R)z_{1}=2+ai(a∈R)，z2=1−2iz_{2}=1-2i，若z1z2 \dfrac {z_{1}}{z_{2}}为纯虚数，则∣z1∣=()|z_{1}|=() A. 2 \sqrt {2} B. 3 \sqrt {3}…
  47. 47Let z=1+3iz = 1 + \sqrt{3}\mathrm{i}, arg⁡(w)=π4\arg(w) = \frac{\pi}{4} and ∣zw∣=22|zw| = 2\sqrt{2}. Find ww in the form x+iyx + \mathrm{i}y…
  48. 48The point A on the Argand diagram below represents the complex number w1w_1 with modulus rr and argument θ\theta. It i…
  49. 49Solve the following simultaneous equations for complex numbers zz and ww. 2iz∗−w=1z+w∗=4\begin{aligned} 2\mathrm{i}z^* - w &= 1 \\ z + w^* &= 4 \end{aligned}…
  50. 50The complex number zz is such that az2+bz+a=0az^2 + bz + a = 0 where aa and bb are real constants. It is given that z=z0z = z_0…
  51. 51The three distinct roots of the equation x3−1=0x^3 - 1 = 0 are denoted by 1,ω1, \omega and ω2\omega^2. Without first finding…
  52. 52Do not use a calculator in answering this question. The complex numbers zz and ww satisfy the following equations. w+z∗=−2+4iw+z^{*}=-2+4i…
  53. 53A complex number zz varies with tt such that z=2cos⁡t+i(3sin⁡t),where 0≤t<2π.z = 2\cos t + i(3\sin t),\quad \text{where } 0 \le t < 2\pi. By takin…
  54. 54Do not use a calculator in answering this question. One of the roots of the equation 3x3+px2+qx+3=03x^3 + px^2 + qx + 3 = 0, whe…
  55. 55Solve the equation z4=3−iz^4 = \sqrt{3} - i, giving the roots in the form reiθre^{i\theta} where r>0r > 0 and −π<θ≤π-\pi < \theta \le \pi…
  56. 56The complex numbers zz and ww are such that z=1+ia,w=−b−iz = 1 + i a, \quad w = -b - i where aa and bb are real and positive. G…
  57. 57Do not use a calculator in answering this question. The complex number zz has modulus 1 and argument θ\theta, wher…
  58. 58[Cambridge/9794/Specimen Paper for 2010] Determine the roots pp and qq of the equation x2−6x+13=0x^2 - 6x + 13 = 0. If pp…
  59. 59The polar form of a complex number zz is given by z=reiθz = r\text{e}^{\text{i}\theta}, where r>0r > 0 and 0<θ≤π0 < \theta \le \pi…
  60. 60若复数满足z−(1+i)=2iz-(1+i)=2i，则zz的模是()(\quad) A. 2\sqrt{2} B. 22 C. 10\sqrt{10} D. 1010