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Nov 2012 p33 q10
1995
(a) Without using a calculator, solve the equation \(iw^2 = (2 - 2i)^2\).
(b) (i) Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where \(|z - 4 - 4i| \leq 2\).
(ii) For the complex numbers represented by points in the region \(R\), it is given that \(p \leq |z| \leq q\) and \(\alpha \leq \arg z \leq \beta\). Find the values of \(p, q, \alpha\) and \(\beta\), giving your answers correct to 3 significant figures.
Solution
(a) Expand \(iw^2 = (2 - 2i)^2\) to get \(iw^2 = 4 - 8i + 4i^2 = 4 - 8i - 4 = -8i\). Thus, \(w^2 = -8\). Solving for \(w\), we have \(w = \pm i\sqrt{8}\).
(b) (i) The region \(R\) is a circle centered at \((4, 4)\) with radius 2. Sketch this circle on an Argand diagram.
(ii) The modulus \(|z|\) ranges from the distance from the origin to the nearest point on the circle to the distance to the farthest point. The center is at \((4, 4)\), so the distance from the origin to the center is \(\sqrt{4^2 + 4^2} = \sqrt{32} = 4\sqrt{2}\). The nearest point is \(4\sqrt{2} - 2\) and the farthest is \(4\sqrt{2} + 2\). Thus, \(p = 3.66\) and \(q = 7.66\).
The angles \(\alpha\) and \(\beta\) are found by considering the tangents from the origin to the circle. The angle \(\theta\) is given by \(\sin^{-1}\left(\frac{1}{4\sqrt{2}}\right)\). Thus, \(\alpha = \frac{1}{4}\pi - \sin^{-1}\left(\frac{1}{4\sqrt{2}}\right) \approx 0.424\) and \(\beta = \frac{1}{4}\pi + \sin^{-1}\left(\frac{1}{4\sqrt{2}}\right) \approx 1.15\).
