Region shading with multiple inequalities

8. The point \(P\) represents a complex number \(z\) on an Argand diagram, where $$| z - 6 + 3 i | = 3 | z + 2 - i |$$

1. Show that the locus of \(P\) is a circle, giving the coordinates of the centre and the radius of this circle. The point \(Q\) represents a complex number \(z\) on an Argand diagram, where $$\tan [ \arg ( z + 6 ) ] = \frac { 1 } { 2 }$$
2. On the same Argand diagram, sketch the locus of \(P\) and the locus of \(Q\).
3. On your diagram, shade the region which satisfies both $$| z - 6 + 3 \mathrm { i } | > 3 | z + 2 - \mathrm { i } | \text { and } \tan [ \arg ( z + 6 ) ] > \frac { 1 } { 2 }$$ (2)(Total 14 marks)

8. A complex number \(z\) satisfies the equation $$| z - 5 - 12 i | = 3$$

1. Describe in geometrical terms with the aid of a sketch, the locus of the point which represents \(z\) in the A rgand diagram. For points on this locus, find
2. the maximum and minimum values for \(| z |\),
3. the maximum and minimum values for arg \(z\), giving your answers in radians to 2 decimal places.

4

1. Sketch, on an Argand diagram, the locus given by \(| z - 1 + \mathrm { i } | = \sqrt { 2 }\).
2. Shade on your diagram the region given by \(1 \leqslant | z - 1 + \mathrm { i } | \leqslant \sqrt { 2 }\).

8 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 3 | = 3\) and arg \(( z - 1 ) = \frac { 1 } { 4 } \pi\) respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 | \leqslant 3 \text { and } 0 \leqslant \arg ( z - 1 ) \leqslant \frac { 1 } { 4 } \pi$$

6 \includegraphics[max width=\textwidth, alt={}, center]{2ba2e0bf-d20a-41ab-a77c-86a08e700b40-2_885_803_1425_630} The Argand diagram above shows a half-line \(l\) and a circle \(C\). The circle has centre 3 i and passes through the origin.

1. Write down, in complex number form, the equations of \(l\) and \(C\).
   [0pt]
2. Write down inequalities that define the region shaded in the diagram. [The shaded region includes the boundaries.]

8

1. On a single Argand diagram, sketch the locus of points for which
   (A) \(| z - 3 \mathrm { j } | = 2\),
   (B) \(\quad \arg ( z + 1 ) = \frac { 1 } { 4 } \pi\).
2. Indicate clearly on your Argand diagram the set of points for which $$| z - 3 \mathrm { j } | \leqslant 2 \quad \text { and } \quad \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi .$$
3. (A) By drawing an appropriate line through the origin, indicate on your Argand diagram the point for which \(| z - 3 j | = 2\) and \(\arg z\) has its minimum possible value.
   (B) Calculate the value of \(\arg z\) at this point.

6

1. Sketch on a single Argand diagram the loci given by
   1. \(\quad | z | = | z - 8 |\),
   2. \(\quad \arg ( z + 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
   3. Indicate by shading the region of the Argand diagram for which $$| z | \leqslant | z - 8 | \quad \text { and } \quad 0 \leqslant \arg ( z + 2 i ) \leqslant \frac { 1 } { 4 } \pi$$

6 The complex number \(3 - 3 \mathrm { i }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by
   1. \(| z - a | = 3 \sqrt { 2 }\),
   2. \(\quad \arg ( z - a ) = \frac { 1 } { 4 } \pi\).
   3. Indicate, by shading, the region of the Argand diagram for which $$| z - a | \geqslant 3 \sqrt { 2 } \quad \text { and } \quad 0 \leqslant \arg ( z - a ) \leqslant \frac { 1 } { 4 } \pi$$

7 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(\arg ( z - 2 - 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) and \(| z | = | z - 10 |\) respectively.

1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Indicate, by shading, the region of the Argand diagram for which $$0 \leqslant \arg ( z - 2 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 4 } \pi \text { and } | z | \geqslant | z - 10 | .$$

8

1. Fig. 8 shows an Argand diagram. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{df275813-15de-496f-9742-427a9e03f431-3_892_899_1048_664} \captionsetup{labelformat=empty} \caption{Fig. 8}
   \end{figure}
   1. Write down the equation of the locus represented by the circumference of circle B.
   2. Write down the two inequalities that define the shaded region between, but not including, circles A and B.
2. 1. Draw an Argand diagram to show the region where $$\frac { \pi } { 4 } < \arg ( z - ( 2 + \mathrm { j } ) ) < \frac { 3 \pi } { 4 }$$
   2. Determine whether the point \(43 + 47 \mathrm { j }\) lies within this region.

8

1. Sketch on an Argand diagram the locus, \(C\), of points for which \(| z - 4 | = 3\).
2. By drawing appropriate lines through the origin, indicate on your Argand diagram the point A on the locus \(C\) where \(\arg z\) has its maximum value. Indicate also the point B on the locus \(C\) where \(\arg z\) has its minimum value.
3. Given that \(\arg z = \alpha\) at A and \(\arg z = \beta\) at B , indicate on your Argand diagram the set of points for which \(\beta \leqslant \arg z \leqslant \alpha\) and \(| z - 4 | \geqslant 3\).
4. Calculate the value of \(\alpha\) and the value of \(\beta\).

8

1. Indicate on an Argand diagram the set of points \(z\) for which \(| z - ( - 8 + 15 \mathrm { j } ) | < 10\).
2. Using the diagram, show that \(7 < | z | < 27\).
3. Mark on your Argand diagram the point, \(P\), at which \(| z - ( - 8 + 15 \mathrm { j } ) | = 10\) and \(\arg z\) takes its maximum value. Find the modulus and argument of \(z\) at \(P\).

8 Fig. 8 shows an Argand diagram. \begin{figure}[h] \end{figure}

1. Write down the equation of the locus represented by the perimeter of the circle in the Argand diagram.
2. Write down the equation of the locus represented by the half-line \(\ell\) in the Argand diagram.
3. Express the complex number represented by the point P in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
4. Use inequalities to describe the set of points that fall within the shaded region (excluding its boundaries) in the Argand diagram.

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

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

4 Indicate, on a single Argand diagram

1. the set of points for which \(\arg ( z - ( - 1 - \mathrm { j } ) ) = \frac { \pi } { 4 }\),
2. the set of points for which \(| z - ( 1 + 2 j ) | = 2\),
3. the set of points for which \(| z - ( 1 + 2 j ) | \geqslant 2\) and \(0 \leqslant \arg ( z - ( - 1 - j ) ) \leqslant \frac { \pi } { 4 }\).

5 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z + 3 - 4 \mathrm { j } | = 5\) and arg \(( z + 3 - 6 \mathrm { j } ) = \frac { 1 } { 2 } \pi\) respectively.

1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
2. Write down the complex number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Indicate, by shading on your sketch, the region satisfying $$| z + 3 - 4 \mathrm { j } | \geqslant 5 \text { and } \frac { 1 } { 2 } \pi \leqslant \arg ( z + 3 - 6 \mathrm { j } ) \leqslant \frac { 3 } { 4 } \pi .$$

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

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

3 In this question you must show detailed reasoning.

1. Find the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\). The loci \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\) are given by \(| z | = | z - 2 \mathrm { i } |\) and \(| z - 2 | = \sqrt { 5 }\) respectively.
2. 1. Sketch on a single Argand diagram the loci \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\), showing any intercepts with the imaginary axis.
   2. Indicate, by shading on your Argand diagram, the region $$\{ z : | z | \leqslant | z - 2 i | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \} .$$
3. 1. Show that both of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfy \(| z - 2 | < \sqrt { 5 }\).
   2. State, with a reason, which root of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\) satisfies \(| z | < | z - 2 i |\).
4. On the same Argand diagram as part (b), indicate the positions of the roots of the equation \(2 z ^ { 2 } - 2 z + 5 = 0\).

2 The locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \left\{ z : 0 \leqslant \arg ( z + i ) \leqslant \frac { 1 } { 4 } \pi \right\}\).

1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
2. Determine whether the complex number \(1.2 + 0.8\) is is \(C _ { 1 }\). The locus \(C _ { 2 }\) is the set of complex numbers represented by the interior of the circle with radius 2 and centre 3 . The locus \(C _ { 2 }\) is illustrated on the Argand diagram below. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-2_698_920_1009_239}
3. Use set notation to define \(C _ { 2 }\).
4. Determine whether the complex number \(1.2 + 0.8\) is in \(C _ { 2 }\).

4 Draw the region in an Argand diagram for which \(| z | \leq 2\) and \(| z | > | z - 3 i |\).

1

1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 6 \mathrm { i } | = 3$$
2. It is given that \(z\) satisfies the equation \(| z - 6 \mathrm { i } | = 3\).
   1. Write down the greatest possible value of \(| z |\).
   2. Find the greatest possible value of \(\arg z\), giving your answer in the form \(p \pi\), where \(- 1 < p \leqslant 1\).

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 \(| w | ^ { 2 }\)