Region shading with multiple inequalities

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

$$\{ z \in \mathbb { C } : | z - 4 i | \leqslant 3 \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 2 } < \arg ( z + 3 - 4 i ) \leqslant \frac { \pi } { 4 } \right\}$$ The complex number \(w\) satisfies $$| w - 4 \mathrm { i } | = 3$$ (b) Find the maximum value of \(\arg w\) in the interval \(( - \pi , \pi ]\). Give your answer in radians correct to 2 decimal places.

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 $$| z - 1 - 8 i | \geqslant 3 | z - 1 | \quad \text { and } \quad 0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 4 }$$

5. \begin{figure}[h] \end{figure} Figure 1 shows an Argand diagram.
The set of points, \(A\), that lies within the shaded region, including its boundaries, is defined by $$A = \{ z : p \leqslant \arg ( z ) \leqslant q \} \cap \{ z : | z | \leqslant r \}$$ where \(p , q\) and \(r\) are positive constants.

1. Write down the values of \(p , q\) and \(r\). Given that \(w = - 2 \sqrt { 3 } + 2 \mathrm { i }\) and \(\mathrm { z } \in A\),
2. find the maximum value of \(| w - z | ^ { 2 }\) giving your answer in an exact simplified form.

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. $$\{ z : | z - 1 | \leqslant 5 \} \cap \left\{ z : 0 \leqslant \arg ( z + 4 + 4 i ) \leqslant \frac { 1 } { 4 } \pi \right\}$$

5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$

1. Sketch, on an Argand diagram, the locus of \(z\).
2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 \mathrm { i } ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).

2

1. Indicate on an Argand diagram the region for which \(| z - 4 \mathrm { i } | \leqslant 2\).
2. The complex number \(z\) satisfies \(| z - 4 \mathrm { i } | \leqslant 2\). Find the range of possible values of \(\arg z\).

5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$

1. Sketch, on an Argand diagram, the locus of \(z\).
2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 i ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).

8 The complex number \(z\) satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where \(a\) is real.
Show that \(a\) must lie in the interval \([ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]\), where \(s\) and \(t\) are prime numbers.
[0pt] [6 marks]

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 $$\{ z : | z | \leqslant | z - 4 | \} \cap \{ z : | z - 3 - 2 i | \leqslant 2 \} .$$

2

1. On a single Argand diagram, sketch and clearly label each of the following loci:
   1. \(| z | = 4\),
   2. \(\quad \arg ( z + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
   3. On the same Argand diagram, shade the region \(R\) defined by the inequalities $$| z | \leqslant 4 \quad \text { and } \quad 0 \leqslant \arg ( z + 4 i ) \leqslant \frac { 1 } { 4 } \pi$$

Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm{i}\) is denoted by \(u\).

1. It is given that \(u\) is a root of the equation \(x^3 + ax^2 - 3x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\). [4]
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - u| < 1\) and \(|z| < |z + \mathrm{i}|\). [4]

The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 3| = 2|z|.$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, and give the coordinates of the centre and the radius of the circle.(5)

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z + 3| = |z - i\sqrt{3}|.$$

1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.(5)
2. On your diagram shade the region which satisfies $$|z - 3| \geq 2|z| \text{ and } |z + 3| \geq |z - i\sqrt{3}|.$$ (2)

Indicate, on separate Argand diagrams,

1. the set of points \(z\) for which \(|z-(3-\mathrm{j})| \leqslant 3\), [3]
2. the set of points \(z\) for which \(1 < |z-(3-\mathrm{j})| \leqslant 3\), [2]
3. the set of points \(z\) for which \(\arg(z-(3-\mathrm{j})) = \frac{1}{4}\pi\). [3]