Math topic中文

Geometrical effects

Geometrical effects of the following operations on the Argand diagram:

  • conjugation
  • negation
  • addition
  • subtraction
  • multiplication by ii

Practice questions

  1. 1Solve 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…
  2. 2Solve 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…
  3. 3The 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…
  4. 4The 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}…
  5. 5The 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…
  6. 6The 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…
  7. 7A 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.…
  8. 8The 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…
  9. 9Given 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…
  10. 10Do 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…
  11. 11Solve 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…
  12. 12The 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.…
  13. 13The point A on the Argand diagram below represents the complex number w1w_1 with modulus rr and argument θ\theta. It i…
  14. 14A 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…
  15. 15Do not use a calculator in answering this question. The complex number zz has modulus 1 and argument θ\theta, wher…
  16. 16The point AA on the Argand diagram below represents the complex number w1w_1 with modulus rr and argument θ\theta. It…
  17. 17A complex number z varies with t 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 taking x=Re(z)x = \mathrm{Re}(z)…