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

Sort by: Default | Easiest first | Hardest first
CAIE P3 2002 June Q9
11 marks Standard +0.3
9 The complex number \(1 + i \sqrt { } 3\) is denoted by \(u\).
  1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Hence, or otherwise, find the modulus and argument of \(u ^ { 2 }\) and \(u ^ { 3 }\).
  2. Show that \(u\) is a root of the equation \(z ^ { 2 } - 2 z + 4 = 0\), and state the other root of this equation.
  3. Sketch an Argand diagram showing the points representing the complex numbers \(i\) and \(u\). Shade the region whose points represent every complex number \(z\) satisfying both the inequalities $$| z - \mathrm { i } | \leqslant 1 \quad \text { and } \quad \arg z \geqslant \arg u .$$
CAIE P3 2007 June Q8
10 marks Standard +0.3
8 The complex number \(\frac { 2 } { - 1 + \mathrm { i } }\) is denoted by \(u\).
  1. Find the modulus and argument of \(u\) and \(u ^ { 2 }\).
  2. Sketch an Argand diagram showing the points representing the complex numbers \(u\) and \(u ^ { 2 }\). Shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z | < 2\) and \(\left| z - u ^ { 2 } \right| < | z - u |\).
CAIE P3 2010 June Q8
9 marks Standard +0.3
8
  1. The equation \(2 x ^ { 3 } - x ^ { 2 } + 2 x + 12 = 0\) has one real root and two complex roots. Showing your working, verify that \(1 + i \sqrt { } 3\) is one of the complex roots. State the other complex root.
  2. On a sketch of an Argand diagram, show the point representing the complex number \(1 + \mathrm { i } \sqrt { } 3\). On the same diagram, shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 - i \sqrt { } 3 | \leqslant 1\) and \(\arg z \leqslant \frac { 1 } { 3 } \pi\).
    1. Express \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in partial fractions.
    2. Hence obtain the expansion of \(\frac { 4 + 5 x - x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2011 June Q7
9 marks Standard +0.3
7
  1. The complex number \(u\) is defined by \(u = \frac { 5 } { a + 2 \mathrm { i } }\), where the constant \(a\) is real.
    1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Find the value of \(a\) for which \(\arg \left( u ^ { * } \right) = \frac { 3 } { 4 } \pi\), where \(u ^ { * }\) denotes the complex conjugate of \(u\).
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z | < 2\) and \(| z | < | z - 2 - 2 \mathrm { i } |\).
CAIE P3 2012 June Q10
11 marks Standard +0.3
10
  1. The complex numbers \(u\) and \(w\) satisfy the equations $$u - w = 4 \mathrm { i } \quad \text { and } \quad u w = 5$$ Solve the equations for \(u\) and \(w\), giving all answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 2 + 2 \mathrm { i } | \leqslant 2 , \arg z \leqslant - \frac { 1 } { 4 } \pi\) and \(\operatorname { Re } z \geqslant 1\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
    2. Calculate the greatest possible value of \(\operatorname { Re } z\) for points lying in the shaded region.
CAIE P3 2014 June Q7
9 marks Standard +0.3
7
  1. The complex number \(\frac { 3 - 5 \mathrm { i } } { 1 + 4 \mathrm { i } }\) is denoted by \(u\). Showing your working, express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 2 - \mathrm { i } | \leqslant 1\) and \(| z - \mathrm { i } | \leqslant | z - 2 |\).
    2. Calculate the maximum value of \(\arg z\) for points lying in the shaded region.
CAIE P3 2019 June Q8
9 marks Standard +0.3
8 Throughout this question the use of a calculator is not permitted.
The complex number \(u\) is defined by $$u = \frac { 4 \mathrm { i } } { 1 - ( \sqrt { } 3 ) \mathrm { i } }$$
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
  2. Find the exact modulus and argument of \(u\).
  3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | < 2\) and \(| z - u | < | z |\).
CAIE P3 2016 March Q10
11 marks Standard +0.3
10
  1. Find the complex number \(z\) satisfying the equation \(z ^ { * } + 1 = 2 \mathrm { i } z\), where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z + 1 - 3 \mathrm { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
    2. Determine the difference between the greatest and least values of \(\arg z\) for points lying in this region.
CAIE P3 2019 March Q7
10 marks Standard +0.3
7
  1. Showing all working and without using a calculator, solve the equation $$( 1 + \mathrm { i } ) z ^ { 2 } - ( 4 + 3 \mathrm { i } ) z + 5 + \mathrm { i } = 0$$ Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. The complex number \(u\) is given by $$u = - 1 - \mathrm { i }$$ On a sketch of an Argand diagram show the point representing \(u\). Shade the region whose points represent complex numbers satisfying the inequalities \(| z | < | z - 2 \mathrm { i } |\) and \(\frac { 1 } { 4 } \pi < \arg ( z - u ) < \frac { 1 } { 2 } \pi\).
CAIE P3 2009 November Q7
10 marks Standard +0.3
7 The complex number \(- 2 + \mathrm { i }\) is denoted by \(u\).
  1. Given that \(u\) is a root of the equation \(x ^ { 3 } - 11 x - k = 0\), where \(k\) is real, find the value of \(k\).
  2. Write down the other complex root of this equation.
  3. Find the modulus and argument of \(u\).
  4. Sketch an Argand diagram showing the point representing \(u\). Shade the region whose points represent the complex numbers \(z\) satisfying both the inequalities $$| z | < | z - 2 | \quad \text { and } \quad 0 < \arg ( z - u ) < \frac { 1 } { 4 } \pi$$
CAIE P3 2010 November Q3
6 marks Moderate -0.3
3 The complex number \(w\) is defined by \(w = 2 + \mathrm { i }\).
  1. Showing your working, express \(w ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real. Find the modulus of \(w ^ { 2 }\).
  2. Shade on an Argand diagram the region whose points represent the complex numbers \(z\) which satisfy $$\left| z - w ^ { 2 } \right| \leqslant \left| w ^ { 2 } \right|$$
CAIE P3 2012 November Q10
11 marks Standard +0.3
10
  1. Without using a calculator, solve the equation \(\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }\).
    1. Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where $$| z - 4 - 4 i | \leqslant 2$$
    2. For the complex numbers represented by points in the region \(R\), it is given that $$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$ Find the values of \(p , q , \alpha\) and \(\beta\), giving your answers correct to 3 significant figures. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
      University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P3 2017 November Q7
8 marks Standard +0.3
7 Throughout this question the use of a calculator is not permitted.
The complex number \(1 - ( \sqrt { } 3 ) \mathrm { i }\) is denoted by \(u\).
  1. Find the modulus and argument of \(u\).
  2. Show that \(u ^ { 3 } + 8 = 0\).
  3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Re } z \geqslant 2\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
    [0pt] [4] \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 8 x ^ { 2 } + 9 x + 8 } { ( 1 - x ) ( 2 x + 3 ) ^ { 2 } }\).
CAIE P3 2021 June Q2
4 marks Moderate -0.3
2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 1 - i | \leqslant 1\) and \(\arg ( z - 1 ) \leqslant \frac { 3 } { 4 } \pi\).
CAIE P3 2023 June Q3
4 marks Standard +0.3
3 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3\) and \(| z | \geqslant | z - 4 \mathrm { i } |\).
CAIE P3 2024 June Q6
7 marks Standard +0.3
6
  1. On an Argand diagram shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 4 - 3 i | \leqslant 2\) and \(\arg ( z - 2 - i ) \geqslant \frac { 1 } { 3 } \pi\).
  2. Calculate the greatest value of \(\arg z\) for points in this region.
CAIE P3 2022 March Q2
4 marks Moderate -0.5
2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 2 - 3 \mathrm { i } | \leqslant 2\) and \(\arg z \leqslant \frac { 3 } { 4 } \pi\).
CAIE P3 2023 March Q2
5 marks Standard +0.8
2
  1. On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(- \frac { 1 } { 3 } \pi \leqslant \arg ( z - 1 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 3 } \pi\) and \(\operatorname { Re } z \leqslant 3\).
  2. Calculate the least value of \(\arg z\) for points in the region from (a). Give your answer in radians correct to 3 decimal places.
CAIE P3 2024 March Q5
6 marks Standard +0.8
5
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 2 i | \leqslant 3\) and \(| z | \geqslant | 10 - z |\).
  2. Find the greatest value of \(\arg z\) for points in this region.
CAIE P3 2020 November Q2
4 marks Moderate -0.5
2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | \geqslant 2\) and \(| z - 1 + \mathrm { i } | \leqslant 1\).
CAIE P3 2021 November Q5
7 marks Standard +0.3
5
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - 2 \mathbf { i } | \leqslant 1\) and \(\operatorname { Im } z \geqslant 2\).
  2. Find the greatest value of \(\arg z\) for points in the shaded region, giving your answer in degrees.
CAIE P3 2022 November Q2
4 marks Moderate -0.8
2 On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | \leqslant 3 , \operatorname { Re } z \geqslant - 2\) and \(\frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \pi\).
CAIE P3 2022 November Q5
6 marks Standard +0.3
5
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 2 | \leqslant 2\) and \(\operatorname { Im } z \geqslant 1\).
  2. Find the greatest value of \(\arg z\) for points in the shaded region.
CAIE P3 2023 November Q2
4 marks Standard +0.3
2 On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathrm { i } | \leqslant | z + 2 - \mathrm { i } |\) and \(0 \leqslant \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi\).
CAIE P3 2023 November Q4
6 marks Standard +0.3
4
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 3 \mathrm { i } | \leqslant 2\) and \(\operatorname { Re } z \leqslant 3\).
  2. Find the greatest value of \(\arg z\) for points in this region.