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N2019 P1 Q9

N2019 P1 Q9

11 marks
Past Year National Exams
Free
  1. The complex number \(w\) can be expressed as \(\cos \,\theta +\mathrm{i sin}\,\theta .\)
    1. Show that \(w+\frac{1}{w}\) is a real number.[2]
    2. Show that \(\frac{w-1}{w+1}\) can be expressed as \(k\tan \frac{1}{2}\theta ,\) where \(k\) is a complex number to be found.[4]
  2. The complex number \(z\) has modulus \(1\). Find the modulus of the complex number \(\frac{z-3\mathrm{i}}{1+3\mathrm{i}z}.\)[5]

Video Solution:

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Answer:\(w + \frac{1}{w} = 2\cos\theta\) (real); \(k = i\); \(\left|\frac{z-3i}{1+3iz}\right| = 1\)

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