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9709 P33 - Nov 2022 - Q5
1923
(a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z + 2| \leq 2\) and \(\text{Im} \, z \geq 1\).
(b) Find the greatest value of \(\arg z\) for points in the shaded region.
Solution
(a) The inequality \(|z + 2| \leq 2\) represents a circle on the Argand diagram with center at \(-2\) and radius 2. The inequality \(\text{Im} \, z \geq 1\) represents the region above the line \(y = 1\). The shaded region is the intersection of the circle and the half-plane above \(y = 1\).
(b) To find the greatest value of \(\arg z\), consider the point on the circle \(|z + 2| = 2\) that is tangent to the line \(y = 1\). This point is where the argument is maximized. The argument \(\arg z\) is given by \(\arctan\left(\frac{y}{x}\right)\). The correct point is identified, and the calculation yields \(\frac{11}{12}\pi\), which is approximately 2.88 radians or 165°.
