Region shading with multiple inequalities

The region \(R\) on an Argand diagram satisfies both \(|z + 2\text{i}| \leq 3\) and \(-\frac{\pi}{6} \leq \arg(z) \leq \frac{\pi}{2}\)

1. Sketch \(R\) on the Argand diagram below. [3 marks] \includegraphics{figure_10a}
2. Find the maximum value of \(|z|\) in the region \(R\), giving your answer in exact form. [5 marks]

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]

Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{z : |z| \leq 4\sqrt{2}\} \cap \left\{z : \frac{1}{4}\pi \leq \arg z \leq \frac{3}{4}\pi\right\}.$$ \includegraphics{figure_9}

1. Find, in modulus-argument form, the complex number represented by the point P. [2]
2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   lie within this region. [4]

1. Shade on an Argand diagram the set of points $$\left\{z \in \mathbb{C} : |z - 4i| \leqslant 3\right\} \cap \left\{z \in \mathbb{C} : -\frac{\pi}{2} < \arg(z + 3 - 4i) \leqslant \frac{\pi}{4}\right\}$$ [5]

The complex number \(w\) satisfies \(|w - 4i| = 3\).

1. Find the maximum value of \(\arg w\) in the interval \((-\pi, \pi]\). Give your answer in radians correct to 2 decimal places. [2]

You are given the complex number \(w = 2 + 2\sqrt{3}i\).

1. Express \(w\) in modulus-argument form. [3]
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$-\frac{\pi}{2} \leq \arg z \leq \frac{\pi}{3} \text{ and } |z| \leq 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(|z - w|\). [9]

The loci \(C_1\) and \(C_2\) are given by \(|z - (3 + 2i)| = 2\) and \(\arg(z - (3 + 2i)) = \frac{5\pi}{6}\) respectively.

1. Sketch \(C_1\) and \(C_2\) on a single Argand diagram. [4]
2. Find, in surd form, the number represented by the point of intersection of \(C_1\) and \(C_2\). [3]
3. Indicate, by shading, the region of the Argand diagram for which $$|z - (3 + 2i)| \leq 2 \text{ and } \frac{5\pi}{6} \leq \arg(z - (3 + 2i)) \leq \pi.$$ [2]

Draw the region of the Argand diagram for which \(|z - 3 - 4i| \leq 5\) and \(|z| \leq |z - 2|\). [4]

The region \(R\) of an Argand diagram is defined by the inequalities $$0 \leqslant \arg(z + 4\mathrm{i}) \leqslant \frac{1}{4}\pi \quad \text{and} \quad |z| \leqslant 4.$$ Draw a clearly labelled diagram to illustrate \(R\). [4]