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9709 P33 - Jun 2014 - Q7
1986
(a) The complex number \(\frac{3 - 5i}{1 + 4i}\) is denoted by \(u\). Showing your working, express \(u\) in the form \(x + iy\), where \(x\) and \(y\) are real.
(b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(|z - 2 - i| \leq 1\) and \(|z - i| \leq |z - 2|\).
(ii) Calculate the maximum value of \(\arg z\) for points lying in the shaded region.
Solution
(a) To express \(u = \frac{3 - 5i}{1 + 4i}\) in the form \(x + iy\), multiply the numerator and denominator by the conjugate of the denominator:
(b)(i) On the Argand diagram, plot the point \(2 + i\) and draw a circle with center \(2 + i\) and radius 1. The inequality \(|z - 2 - i| \leq 1\) represents this circle.
The inequality \(|z - i| \leq |z - 2|\) represents the region on or closer to the line perpendicular bisector of the segment joining \(i\) and \(2\). Shade the region inside the circle and on the correct side of the bisector.
(b)(ii) The maximum value of \(\arg z\) is the angle between the tangents from the origin to the circle. This angle is \(0.927\) radians (or \(53.1^\circ\)).
