Region shading with inequalities

A question is this type if and only if it requires shading a region on an Argand diagram defined by one or more inequalities involving modulus and/or argument conditions.

62 questions · Standard +0.2

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CAIE P3 2023 November Q2
5 marks Moderate -0.3
2 On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 1 + 2 i | \leqslant | z |\) and \(| z - 2 | \leqslant 1\).
Edexcel FP2 2006 June Q8
14 marks Standard +0.8
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)
OCR FP1 2007 January Q4
6 marks Moderate -0.3
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 \(1 \leqslant | z - 1 + \mathrm { i } | \leqslant \sqrt { 2 }\).
OCR FP1 2007 June Q8
8 marks Standard +0.3
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 $$| z - 3 | \leqslant 3 \text { and } 0 \leqslant \arg ( z - 1 ) \leqslant \frac { 1 } { 4 } \pi$$
OCR FP1 2013 June Q6
7 marks Standard +0.3
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.]
OCR MEI FP1 2008 January Q8
12 marks Standard +0.3
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.
OCR MEI FP1 2006 June Q4
8 marks Moderate -0.8
4 Indicate, on separate Argand diagrams,
  1. the set of points \(z\) for which \(| z - ( 3 - \mathrm { j } ) | \leqslant 3\),
  2. the set of points \(z\) for which \(1 < | z - ( 3 - \mathrm { j } ) | \leqslant 3\),
  3. the set of points \(z\) for which \(\arg ( z - ( 3 - \mathrm { j } ) ) = \frac { 1 } { 4 } \pi\).
Edexcel FP2 2008 June Q10
12 marks Standard +0.8
10. 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 $$| z + 3 | = | z - i \sqrt { } 3 |$$
  2. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.(5)
  3. On your diagram shade the region which satisfies $$| z - 3 | \geq 2 | z | \text { and } | z + 3 | \geq | z - i \sqrt { } 3 |$$
OCR FP1 2011 January Q6
8 marks Standard +0.3
6
  1. Sketch on a single Argand diagram the loci given by
    (a) \(\quad | z | = | z - 8 |\),
    (b) \(\quad \arg ( z + 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
  2. Indicate by shading the region of the Argand diagram for which $$| z | \leqslant | z - 8 | \quad \text { and } \quad 0 \leqslant \arg ( z + 2 i ) \leqslant \frac { 1 } { 4 } \pi$$
OCR FP1 2013 January Q7
7 marks Standard +0.3
7
  1. Sketch on a single Argand diagram the loci given by
    (a) \(| z | = 2\),
    (b) \(\quad \arg ( z - 3 - \mathrm { i } ) = \pi\).
  2. Indicate, by shading, the region of the Argand diagram for which $$| z | \leqslant 2 \text { and } 0 \leqslant \arg ( z - 3 - i ) \leqslant \pi .$$
OCR FP1 2009 June Q6
11 marks Standard +0.3
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
    (a) \(| z - a | = 3 \sqrt { 2 }\),
    (b) \(\quad \arg ( z - a ) = \frac { 1 } { 4 } \pi\).
  3. Indicate, by shading, the region of the Argand diagram for which $$| z - a | \geqslant 3 \sqrt { 2 } \quad \text { and } \quad 0 \leqslant \arg ( z - a ) \leqslant \frac { 1 } { 4 } \pi$$
OCR FP1 2010 June Q6
6 marks Moderate -0.3
6
  1. Sketch on a single Argand diagram the loci given by
    (a) \(| z - 3 + 4 \mathrm { i } | = 5\),
    (b) \(| z | = | z - 6 |\).
  2. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 + 4 i | \leqslant 5 \quad \text { and } \quad | z | \geqslant | z - 6 | .$$
OCR FP1 2012 June Q7
10 marks Standard +0.3
7 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 3 - 4 \mathrm { i } | = 4\) and \(| z | = | z - 8 \mathrm { i } |\) respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Hence find the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 - 4 i | \leqslant 4 \text { and } | z | \geqslant | z - 8 i | .$$
OCR FP1 2014 June Q7
7 marks Standard +0.3
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 $$0 \leqslant \arg ( z - 2 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 4 } \pi \text { and } | z | \geqslant | z - 10 | .$$
OCR FP1 2015 June Q5
8 marks Standard +0.3
5 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z + 2 | = 2\) and \(\arg ( z + 2 ) = \frac { 5 } { 6 } \pi\) respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Find the complex number represented by the intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Indicate, by shading, the region of the Argand diagram for which $$| z + 2 | \leqslant 2 \text { and } \frac { 5 } { 6 } \pi \leqslant \arg ( z + 2 ) \leqslant \pi .$$
OCR MEI FP1 2010 January Q8
12 marks Standard +0.3
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.
    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.
OCR MEI FP1 2011 January Q4
6 marks Moderate -0.8
4 Represent on an Argand diagram the region defined by \(2 < | z - ( 3 + 2 \mathrm { j } ) | \leqslant 3\).
OCR MEI FP1 2009 June Q8
12 marks Standard +0.3
8 Fig. 8 shows an Argand diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-3_421_586_1105_778} \captionsetup{labelformat=empty} \caption{Fig. 8}
\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.
OCR MEI FP1 2015 June Q4
6 marks Standard +0.3
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 \(0 \leqslant \arg ( z - ( - 1 - j ) ) \leqslant \frac { \pi } { 4 }\).
OCR MEI FP1 2016 June Q5
8 marks Standard +0.3
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 $$| z + 3 - 4 \mathrm { j } | \geqslant 5 \text { and } \frac { 1 } { 2 } \pi \leqslant \arg ( z + 3 - 6 \mathrm { j } ) \leqslant \frac { 3 } { 4 } \pi .$$
OCR Further Pure Core AS Specimen Q4
4 marks Standard +0.3
4 Draw the region of the Argand diagram for which \(| z - 3 - 4 i | \leq 5\) and \(| z | \leq | z - 2 |\).
OCR Further Pure Core 1 2019 June Q2
3 marks Standard +0.3
2 Indicate by shading on an Argand diagram the region \(\{ z : | z | \leqslant | z - 4 | \} \cap \{ z : | z - 3 - 2 i | \leqslant 2 \}\).
OCR Further Pure Core 1 2022 June Q3
11 marks Standard +0.3
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.
    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 $$\{ z : | z | \leqslant | z - 2 i | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \} .$$
    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 \(2 z ^ { 2 } - 2 z + 5 = 0\).
OCR Further Pure Core 1 2024 June Q2
10 marks Standard +0.3
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 \(C _ { 2 }\).
OCR Further Pure Core 1 Specimen Q4
3 marks Standard +0.3
4 Draw the region in an Argand diagram for which \(| z | \leq 2\) and \(| z | > | z - 3 i |\).