Region shading with inequalities

9 Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{ z : | z | \leqslant 4 \sqrt { 2 } \} \cap \left\{ z : \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi \right\} .$$ \begin{figure}[h] \end{figure}

1. Find, in modulus-argument form, the complex number represented by the point P .
2. Find, in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q .
3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
   • \(3 + 5 \mathrm { i }\)
   • \(5.5 ( \cos 0.8 + \mathrm { i } \sin 0.8 )\) lie within this region.

5 An Argand diagram is shown below. The circle has centre at the point representing \(1 + 3 i\), and the half line intersects the circle at the origin and at the point representing \(4 + 4 \mathrm { i }\). \includegraphics[max width=\textwidth, alt={}, center]{c4484913-14bf-4bf4-a290-0301586333ce-3_748_917_351_242} State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries.

6 On separate Argand diagrams, sketch the set of points represented by each of the following.

1. \(| z - 1 - 2 i | \leqslant 4\).
2. \(\quad \arg ( z + \mathrm { i } ) = \frac { 1 } { 3 } \pi\).

1. (a) Shade on an Argand diagram the set of points

$$\{ z \in \mathbb { C } : | z - 1 - \mathrm { i } | \leqslant 3 \} \cap \quad z \in \mathbb { C } : \frac { \pi } { 4 } \leqslant \arg ( z - 2 ) \leqslant \frac { 3 \pi } { 4 }$$ The complex number \(w\) satisfies $$| w - 1 - \mathrm { i } | = 3 \text { and } \arg ( w - 2 ) = \frac { \pi } { 4 }$$ (b) Find, in simplest form, the exact value of \(| w | ^ { 2 }\)

1. A curve \(C\) in the complex plane is described by the equation

$$| z - 1 - 8 i | = 3 | z - 1 |$$

1. Show that \(C\) is a circle, and find its centre and radius.
2. Using the answer to part (a), determine whether \(z = 3 - 3 \mathrm { i }\) satisfies the inequality $$| z - 1 - 8 i | \geqslant 3 | z - 1 |$$
3. Shade, on an Argand diagram, the set of points that satisfies both $$| z - 1 - 8 i | \geqslant 3 | z - 1 | \quad \text { and } \quad 0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 4 }$$

1. Sketch on an Argand diagram the region defined by

$$z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg ( z + 2 ) < \frac { \pi } { 4 } \cap \{ z \in \mathbb { C } : - 1 < \operatorname { Re } ( z ) \leqslant 1 \}$$ On your sketch

• shade the part of the diagram that is included in the region
• use solid lines to show the parts of the boundary that are included in the region
• use dashed lines to show the parts of the boundary that are not included in the region

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$z _ { 1 } = - 4 + 4 i$$

1. Express \(\mathrm { z } _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r \in \mathbb { R } , r > 0\) and \(0 \leqslant \theta < 2 \pi\) $$z _ { 2 } = 3 \left( \cos \frac { 17 \pi } { 12 } + i \sin \frac { 17 \pi } { 12 } \right)$$
2. Determine in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers,
   1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
   2. \(\left( z _ { 2 } \right) ^ { 4 }\)
3. Show on a single Argand diagram
   1. the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(\frac { z _ { 1 } } { z _ { 2 } }\)
   2. the region defined by \(\left\{ z \in \mathbb { C } : \left| z - z _ { 1 } \right| < \left| z - z _ { 2 } \right| \right\}\)

1. The point \(P\) in the complex plane represents a complex number \(z\) such that

$$| z + 9 | = 4 | z - 12 i |$$ Given that, as \(z\) varies, the locus of \(P\) is a circle,

1. determine the centre and radius of this circle.
2. Shade on an Argand diagram the region defined by the set $$\{ z \in \mathbb { C } : | z + 9 | < 4 | z - 12 i | \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg \left( z - \frac { 3 + 44 i } { 5 } \right) < \frac { \pi } { 4 } \right\}$$

2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 1 | = 5\) and \(\arg ( z + 4 + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) respectively.

1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Indicate by shading on your Argand diagram the following set of points. $$\{ z : | z - 1 | \leqslant 5 \} \cap \left\{ z : 0 \leqslant \arg ( z + 4 + 4 i ) \leqslant \frac { 1 } { 4 } \pi \right\}$$

8

1. The complex number \(w\) is such that $$\arg ( w + 2 \mathrm { i } ) = \tan ^ { - 1 } \frac { 1 } { 2 }$$ It is given that \(w = x + \mathrm { i } y\), where \(x\) and \(y\) are real and \(x > 0\) Find an equation for \(y\) in terms of \(x\) 8
2. The complex number \(z\) satisfies both $$- \frac { \pi } { 2 } \leq \arg ( z + 2 \mathrm { i } ) \leq \tan ^ { - 1 } \frac { 1 } { 2 } \quad \text { and } \quad | z - 2 + 3 \mathrm { i } | \leq 2$$ The region \(R\) is the locus of \(z\) Sketch the region \(R\) on the Argand diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-10_1015_1020_1683_511} 8
3. \(\quad z _ { 1 }\) is the point in \(R\) at which \(| z |\) is minimum. 8
4. 1. Calculate the exact value of \(\left| z _ { 1 } \right|\) 8
5. (ii) Express \(z _ { 1 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.

4

1. A locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \{ \mathrm { z } : | \mathrm { z } + \mathrm { i } | \leqslant \mid \mathrm { z } - 2 \}\).
   1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
   2. Find the cartesian equation of the boundary line of the region representing \(C _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\).
2. A locus \(C _ { 2 }\) is defined by \(C _ { 2 } = \{ \mathrm { z } : | \mathrm { z } + 1 | \leqslant 3 \} \cap \{ \mathrm { z } : | \mathrm { z } - 2 \mathrm { i } | \geqslant 2 \}\). Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 2 }\).

1 Indicate by shading on an Argand diagram the region $$\{ z : | z | \leqslant | z - 4 | \} \cap \{ z : | z - 3 - 2 i | \leqslant 2 \} .$$