4.02a Complex numbers: real/imaginary parts, modulus, argument

251 questions

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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 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 2010 June Q8
9 marks Standard +0.8
8 The variable complex number \(z\) is given by $$z = 1 + \cos 2 \theta + i \sin 2 \theta$$ where \(\theta\) takes all values in the interval \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).
  1. Show that the modulus of \(z\) is \(2 \cos \theta\) and the argument of \(z\) is \(\theta\).
  2. Prove that the real part of \(\frac { 1 } { z }\) is constant.
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 2011 June Q7
8 marks Standard +0.3
7
  1. Find the roots of the equation $$z ^ { 2 } + ( 2 \sqrt { } 3 ) z + 4 = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the modulus and argument of each root.
  3. Showing all your working, verify that each root also satisfies the equation $$z ^ { 6 } = - 64$$
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 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 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 2008 November Q10
12 marks Standard +0.8
10 The complex number \(w\) is given by \(w = - \frac { 1 } { 2 } + \mathrm { i } \frac { \sqrt { } 3 } { 2 }\).
  1. Find the modulus and argument of \(w\).
  2. The complex number \(z\) has modulus \(R\) and argument \(\theta\), where \(- \frac { 1 } { 3 } \pi < \theta < \frac { 1 } { 3 } \pi\). State the modulus and argument of \(w z\) and the modulus and argument of \(\frac { z } { w }\).
  3. Hence explain why, in an Argand diagram, the points representing \(z , w z\) and \(\frac { z } { w }\) are the vertices of an equilateral triangle.
  4. In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number \(4 + 2 \mathrm { i }\). Find the complex numbers represented by the other two vertices. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
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 Q6
9 marks Moderate -0.8
6 The complex number \(z\) is given by $$z = ( \sqrt { } 3 ) + \mathrm { i } .$$
  1. Find the modulus and argument of \(z\).
  2. The complex conjugate of \(z\) is denoted by \(z ^ { * }\). Showing your working, express in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real,
    1. \(2 z + z ^ { * }\),
    2. \(\frac { \mathrm { i } z ^ { * } } { z }\).
    3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z\) and \(\mathrm { i } z ^ { * }\) respectively. Prove that angle \(A O B = \frac { 1 } { 6 } \pi\).
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 2011 November Q6
8 marks Standard +0.3
6 The complex number \(w\) is defined by \(w = - 1 + \mathrm { i }\).
  1. Find the modulus and argument of \(w ^ { 2 }\) and \(w ^ { 3 }\), showing your working.
  2. The points in an Argand diagram representing \(w\) and \(w ^ { 2 }\) are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form \(| z - ( a + b \mathrm { i } ) | = k\).
CAIE P3 2012 November Q9
10 marks Standard +0.3
9 The complex number \(1 + ( \sqrt { } 2 ) \mathrm { i }\) is denoted by \(u\). The polynomial \(x ^ { 4 } + x ^ { 2 } + 2 x + 6\) is denoted by \(\mathrm { p } ( x )\).
  1. Showing your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\), and write down a second complex root of the equation.
  2. Find the other two roots of the equation \(\mathrm { p } ( x ) = 0\).
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.
CAIE P3 2015 November Q9
10 marks Standard +0.3
9
  1. It is given that \(( 1 + 3 \mathrm { i } ) w = 2 + 4 \mathrm { i }\). Showing all necessary working, prove that the exact value of \(\left| w ^ { 2 } \right|\) is 2 and find \(\arg \left( w ^ { 2 } \right)\) correct to 3 significant figures.
  2. On a single Argand diagram sketch the loci \(| z | = 5\) and \(| z - 5 | = | z |\). Hence determine the complex numbers represented by points common to both loci, giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).
CAIE P3 2019 November Q7
9 marks Standard +0.3
7
  1. Find the complex number \(z\) satisfying the equation $$z + \frac { \mathrm { i } z } { z ^ { * } } - 2 = 0$$ 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 single Argand diagram sketch the loci given by the equations \(| z - 2 \mathrm { i } | = 2\) and \(\operatorname { Im } z = 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
    2. In the first quadrant the two loci intersect at the point \(P\). Find the exact argument of the complex number represented by \(P\).
CAIE P3 Specimen Q9
10 marks Standard +0.3
9 The complex number \(3 - \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).
  1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
  2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
  4. Find the exact value of the \(x\)-coordinate of \(M\).
  5. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.
CAIE P3 2020 June Q10
12 marks Standard +0.3
10
  1. The complex number \(u\) is defined by \(u = \frac { 3 \mathrm { i } } { a + 2 \mathrm { i } }\), where \(a\) is real.
    1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
    2. Find the exact value of \(a\) for which \(\arg u ^ { * } = \frac { 1 } { 3 } \pi\).
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 2 \mathbf { i } | \leqslant | z - 1 - \mathbf { i } |\) and \(| z - 2 - \mathbf { i } | \leqslant 2\).
    2. Calculate the least value of \(\arg z\) for points in this region.
      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 2022 June Q7
8 marks Standard +0.3
7 The complex number \(u\) is defined by \(u = \frac { \sqrt { 2 } - a \sqrt { 2 } \mathrm { i } } { 1 + 2 \mathrm { i } }\), where \(a\) is a positive integer.
  1. Express \(u\) in terms of \(a\), in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    It is now given that \(a = 3\).
  2. 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 \(\theta\).
  3. Using your answer to part (b), find the two square roots of \(u\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
CAIE P3 2022 June Q5
8 marks Standard +0.3
5 The complex number \(3 - \mathrm { i }\) is denoted by \(u\).
  1. Show, on an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. State the type of quadrilateral formed by the points \(O , A , B\) and \(C\).
  2. Express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  3. By considering the argument of \(\frac { u ^ { * } } { u }\), or otherwise, prove that \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
CAIE P3 2023 June Q5
6 marks Standard +0.3
5 The complex number \(2 + y \mathrm { i }\) is denoted by \(a\), where \(y\) is a real number and \(y < 0\). It is given that \(\mathrm { f } ( a ) = a ^ { 3 } - a ^ { 2 } - 2 a\).
  1. Find a simplified expression for \(\mathrm { f } ( a )\) in terms of \(y\).
  2. Given that \(\operatorname { Re } ( \mathrm { f } ( a ) ) = - 20\), find \(\arg a\).
CAIE P3 2024 March Q3
6 marks Standard +0.3
3 It is given that \(z = - \sqrt { 3 } + \mathrm { i }\).
  1. Express \(z ^ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  2. The complex number \(\omega\) is such that \(z ^ { 2 } \omega\) is real and \(\left| \frac { z ^ { 2 } } { \omega } \right| = 12\). Find the two possible values of \(\omega\), giving your answers in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\) and \(- \pi < \alpha \leqslant \pi\).
CAIE P3 2023 November Q4
5 marks Standard +0.3
4 The complex number \(u\) is defined by \(u = \frac { 3 + 2 \mathrm { i } } { a - 5 \mathrm { i } }\), where \(a\) is real.
  1. Express \(u\) in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are in terms of \(a\).
  2. Given that \(\arg u = \frac { 1 } { 4 } \pi\), find the value of \(a\).