Wednesday 21 September 2016

U6FM Hwk

Total: 27 marks. Show all working out. Those not showing mathematical rigour will be penalised.

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  1. Express $\cos 6\theta$ in terms of $\cos^n \theta$

    [3 marks]

  2. Express $\cos^6 \theta$ in terms of $\cos (n\theta)$

    [4 marks]

  3. Solve, giving your answers in the form $re^{i\theta}$ with $ -\pi < \theta \leq \pi$ , \[ z^6=1-\sqrt3 i \] Represent your solutions on an Argand diagram

    [4 marks]

  4. Represent on an Argand diagram the locus of $z$ s.t.
    1. $\arg{(z-3-2i)} = -\frac{5\pi}{6}$

      [3 marks]

    2. $|z+1|=|z-2i|$ also give the cartesian equation

      [3 marks]

    3. $|z+1|=|4z-8i|$ also give the cartesian equation

      [4 marks]

    4. Give the cartesian equation and sketch the locus \[ \arg (\frac{z+3}{z-3}) = 3\pi /4 \]

      [6 marks]

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