Math topic中文

Equality

Equality of complex numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

Practice questions

  1. 1Given that z=2−iz = 2 - \text{i} is a root of the equation 4z4−12z3+17z2+pz+q=04z^4 - 12z^3 + 17z^2 + pz + q = 0, where pp and qq are real,…
  2. 2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}}…
  3. 3[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…
  4. 4One of the roots of the equation 2z4−14z3+33z2−26z+p=0, where p is a constant2z^4 - 14z^3 + 33z^2 - 26z + p = 0\text{, where } p \text{ is a constant} is 3+i3 + \mathrm{i}…
  5. 5The complex number zz satisfies the equation f(z)=0\mathrm{f}(z) = 0, where f(z)=(1−ai)z2−2iz−10−20i\mathrm{f}(z) = (1 - a\mathrm{i})z^2 - 2\mathrm{i}z - 10 - 20\mathrm{i}…
  6. 6Do 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,…
  7. 7One of the roots of the equation 2z4−14z3+33z2−26z+p=0, where p is a constant2z^{4} - 14z^{3} + 33z^{2} - 26z + p = 0\text{, where } p \text{ is a constant} is…
  8. 8Find, 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)…
  9. 9Do 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}…
  10. 10Find 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…
  11. 11Given 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…
  12. 12Do not use a calculator in answering this question. Showing your working, find the complex numbers zz and ww which…
  13. 13设复数i−21+i=a+bi(a,b∈R) \dfrac {i-2}{1+i}=a+bi(a,b∈R) 则a+b=()a+b=() A. 11 B. 22 C. −1-1 D. −2-2
  14. 14若复数a+ib−i=2−i \dfrac {a+i}{b-i}=2-i其中aa，bb是实数，则复数a+bia+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
  15. 15Solve 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…
  16. 16The 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…
  17. 17Given 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…
  18. 18Do 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…
  19. 19Solve 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…
  20. 20Let 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}…
  21. 21One of the roots of the equation 2z4−14z3+33z2−26z+p=0,2z^{4} - 14z^{3} + 33z^{2} - 26z + p = 0, where pp is a constant, is 3+i3 + i. Base…
  22. 22The cubic equation x3+ax2+bx+c=0x^3 + ax^2 + bx + c = 0, where aa, bb and cc are constants, has roots 3+i3 + \mathrm{i} and 22.…
  23. 23Do 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…
  24. 24The 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…
  25. 25Do not use a calculator in answering this question. The complex number zz has modulus 1 and argument θ\theta, wher…
  26. 26Find the complex number z in the form x+iy, where x, y \in \mathbb{R} such that 3izz∗−1+2i=−i\dfrac{3i z}{z^{*}-1+2i} = -i. [Note:…
  27. 27Two of the roots of the equation z3+az2+bz+c=0z^3 + az^2 + bz + c = 0 are 3ei(−23π)3\text{e}^{\text{i}\left(-\frac{2}{3}\pi\right)} and −2-2…
  28. 28Given 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. Do not u…
  29. 29[Cambridge/9709/2009] The complex number −2+i-2 + i is denoted by uu. Given that uu is a root of the equation x3−11x−k=0x^3 - 11x - k = 0…
  30. 30The 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. Gi…
  31. 31Do not use a calculator in answering this question. Showing your working, find the complex numbers zz and ww which sat…
  32. 32One of the roots of the equation z4−z3−9z2+sz+t=0z^4 - z^3 - 9z^2 + sz + t = 0, where ss and tt are real, is 1+2i1 + 2\text{i}. Find t…
  33. 33设复数zz满足3z−−2z=2+5i(i3\overset{-}{z}-2z=2+5i(i为虚数单位))，则∣z∣=()|z|=(\quad) A. 5\sqrt{5} B. 22 C. 3\sqrt{3} D. 11