4.02k Argand diagrams: geometric interpretation

446 questions

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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 2003 June Q5
8 marks Standard +0.3
5 The complex number 2 i is denoted by \(u\). The complex number with modulus 1 and argument \(\frac { 2 } { 3 } \pi\) is denoted by \(w\).
  1. Find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(w , u w\) and \(\frac { u } { w }\).
  2. Sketch an Argand diagram showing the points \(U , A\) and \(B\) representing the complex numbers \(u\), \(u w\) and \(\frac { u } { w }\) respectively.
  3. Prove that triangle \(U A B\) is equilateral.
CAIE P3 2005 June Q3
7 marks Moderate -0.3
3
  1. Solve the equation \(z ^ { 2 } - 2 \mathrm { i } z - 5 = 0\), giving your answers in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.
  2. Find the modulus and argument of each root.
  3. Sketch an Argand diagram showing the points representing the roots.
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 2008 June Q5
7 marks Standard +0.8
5 The variable complex number \(z\) is given by $$z = 2 \cos \theta + \mathrm { i } ( 1 - 2 \sin \theta ) ,$$ where \(\theta\) takes all values in the interval \(- \pi < \theta \leqslant \pi\).
  1. Show that \(| z - \mathrm { i } | = 2\), for all values of \(\theta\). Hence sketch, in an Argand diagram, the locus of the point representing \(z\).
  2. Prove that the real part of \(\frac { 1 } { z + 2 - \mathrm { i } }\) is constant for \(- \pi < \theta < \pi\).
CAIE P3 2009 June Q7
8 marks Standard +0.3
7
  1. Solve the equation \(z ^ { 2 } + ( 2 \sqrt { } 3 ) \mathrm { i } z - 4 = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Sketch an Argand diagram showing the points representing the roots.
  3. Find the modulus and argument of each root.
  4. Show that the origin and the points representing the roots are the vertices of an equilateral triangle.
CAIE P3 2010 June Q7
9 marks Challenging +1.2
7 The complex number \(2 + 2 \mathrm { i }\) is denoted by \(u\).
  1. Find the modulus and argument of \(u\).
  2. Sketch an Argand diagram showing the points representing the complex numbers 1, i and \(u\). Shade the region whose points represent the complex numbers \(z\) which satisfy both the inequalities \(| z - 1 | \leqslant | z - \mathrm { i } |\) and \(| z - u | \leqslant 1\).
  3. Using your diagram, calculate the value of \(| z |\) for the point in this region for which \(\arg z\) is least.
CAIE P3 2011 June Q8
10 marks Challenging +1.2
8 The complex number \(u\) is defined by \(u = \frac { 6 - 3 \mathrm { i } } { 1 + 2 \mathrm { i } }\).
  1. Showing all your working, find the modulus of \(u\) and show that the argument of \(u\) is \(- \frac { 1 } { 2 } \pi\).
  2. For complex numbers \(z\) satisfying \(\arg ( z - u ) = \frac { 1 } { 4 } \pi\), find the least possible value of \(| z |\).
  3. For complex numbers \(z\) satisfying \(| z - ( 1 + \mathrm { i } ) u | = 1\), find the greatest possible value of \(| z |\).
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 Q4
7 marks Moderate -0.3
4 The complex number \(u\) is defined by \(u = \frac { ( 1 + 2 \mathrm { i } ) ^ { 2 } } { 2 + \mathrm { i } }\).
  1. Without using a calculator and showing your working, express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Sketch an Argand diagram showing the locus of the complex number \(z\) such that \(| z - u | = | u |\).
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 2013 June Q7
9 marks Standard +0.8
7
  1. Without using a calculator, solve the equation $$3 w + 2 \mathrm { i } w ^ { * } = 17 + 8 \mathrm { i }$$ where \(w ^ { * }\) denotes the complex conjugate of \(w\). Give your answer in the form \(a + b \mathrm { i }\).
  2. In an Argand diagram, the loci $$\arg ( z - 2 \mathrm { i } ) = \frac { 1 } { 6 } \pi \quad \text { and } \quad | z - 3 | = | z - 3 \mathrm { i } |$$ intersect at the point \(P\). Express the complex number represented by \(P\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), giving the exact value of \(\theta\) and the value of \(r\) correct to 3 significant figures.
CAIE P3 2013 June Q7
8 marks Standard +0.3
7 The complex number \(z\) is defined by \(z = a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex conjugate of \(z\) is denoted by \(z ^ { * }\).
  1. Show that \(| z | ^ { 2 } = z z ^ { * }\) and that \(( z - k \mathrm { i } ) ^ { * } = z ^ { * } + k \mathrm { i }\), where \(k\) is real. In an Argand diagram a set of points representing complex numbers \(z\) is defined by the equation \(| z - 10 \mathrm { i } | = 2 | z - 4 \mathrm { i } |\).
  2. Show, by squaring both sides, that $$z z ^ { * } - 2 \mathrm { i } z ^ { * } + 2 \mathrm { i } z - 12 = 0$$ Hence show that \(| z - 2 i | = 4\).
  3. Describe the set of points geometrically.
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 2015 June Q8
9 marks Standard +0.3
8 The complex number \(w\) is defined by \(w = \frac { 22 + 4 \mathrm { i } } { ( 2 - \mathrm { i } ) ^ { 2 } }\).
  1. Without using a calculator, show that \(w = 2 + 4 \mathrm { i }\).
  2. It is given that \(p\) is a real number such that \(\frac { 1 } { 4 } \pi \leqslant \arg ( w + p ) \leqslant \frac { 3 } { 4 } \pi\). Find the set of possible values of \(p\).
  3. The complex conjugate of \(w\) is denoted by \(w ^ { * }\). The complex numbers \(w\) and \(w ^ { * }\) are represented in an Argand diagram by the points \(S\) and \(T\) respectively. Find, in the form \(| z - a | = k\), the equation of the circle passing through \(S , T\) and the origin.
CAIE P3 2015 June Q7
9 marks Standard +0.8
7 The complex number \(u\) is given by \(u = - 1 + ( 4 \sqrt { } 3 ) \mathrm { i }\).
  1. Without using a calculator and showing all your working, find the two square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the real numbers \(a\) and \(b\) are exact.
  2. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the relation \(| z - u | = 1\). Determine the greatest value of \(\arg z\) for points on this locus. \(8 \quad\) Let \(f ( x ) = \frac { 5 x ^ { 2 } + x + 6 } { ( 3 - 2 x ) \left( x ^ { 2 } + 4 \right) }\).
CAIE P3 2015 June Q8
9 marks Standard +0.3
8 The complex number 1 - i is denoted by \(u\).
  1. Showing your working and without using a calculator, express $$\frac { \mathrm { i } } { u }$$ in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(| z - u | = | z |\) and \(| z - \mathrm { i } | = 2\).
  3. Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).
CAIE P3 2016 June Q10
11 marks Standard +0.3
10
  1. Showing all your working and without the use of a calculator, find the square roots of the complex number \(7 - ( 6 \sqrt { } 2 ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    1. On an Argand diagram, sketch the loci of points representing complex numbers \(w\) and \(z\) such that \(| w - 1 - 2 \mathrm { i } | = 1\) and \(\arg ( z - 1 ) = \frac { 3 } { 4 } \pi\).
    2. Calculate the least value of \(| w - z |\) for points on these loci.
CAIE P3 2016 June Q10
10 marks Standard +0.3
10
  1. Showing all necessary working, solve the equation \(\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    1. On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation \(| z | = | z - 4 - 3 \mathrm { i } |\).
    2. Find the complex number represented by the point on the locus where \(| z |\) is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.
CAIE P3 2017 June Q7
8 marks Standard +0.3
7 Throughout this question the use of a calculator is not permitted.
The complex numbers \(u\) and \(w\) are defined by \(u = - 1 + 7 \mathrm { i }\) and \(w = 3 + 4 \mathrm { i }\).
  1. Showing all your working, find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - 2 w\) and \(\frac { u } { w }\).
    In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , w\) and \(u - 2 w\) respectively.
  2. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\).
  3. State fully the geometrical relation between the line segments \(O B\) and \(C A\).
CAIE P3 2017 June Q11
10 marks Standard +0.8
11 Throughout this question the use of a calculator is not permitted.
  1. The complex numbers \(z\) and \(w\) satisfy the equations $$z + ( 1 + \mathrm { i } ) w = \mathrm { i } \quad \text { and } \quad ( 1 - \mathrm { i } ) z + \mathrm { i } w = 1$$ Solve the equations for \(z\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. The complex numbers \(u\) and \(v\) are given by \(u = 1 + ( 2 \sqrt { 3 } ) \mathrm { i }\) and \(v = 3 + 2 \mathrm { i }\). In an Argand diagram, \(u\) and \(v\) are represented by the points \(A\) and \(B\). A third point \(C\) lies in the first quadrant and is such that \(B C = 2 A B\) and angle \(A B C = 90 ^ { \circ }\). Find the complex number \(z\) represented by \(C\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
CAIE P3 2019 June Q10
13 marks Standard +0.3
10 Throughout this question the use of a calculator is not permitted.
The complex number \(( \sqrt { } 3 ) + \mathrm { i }\) is denoted by \(u\).
  1. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\). Hence or otherwise state the exact values of the modulus and argument of \(u ^ { 4 }\).
  2. Verify that \(u\) is a root of the equation \(z ^ { 3 } - 8 z + 8 \sqrt { } 3 = 0\) and state the other complex root of this equation.
  3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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 \(\frac { 1 } { 4 } \pi < \arg ( z - u ) < \frac { 1 } { 2 } \pi\).