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

OCR FP1 2008 June Q2

7 marks Standard +0.3

2 The complex number \(3 + 4 \mathrm { i }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by
   1. \(| z - a | = | a |\),
   2. \(\arg ( z - 3 ) = \arg a\).

OCR FP1 2013 June Q1

6 marks Moderate -0.5

1 The complex number \(3 + a \mathrm { i }\), where \(a\) is real, is denoted by \(z\). Given that \(\arg z = \frac { 1 } { 6 } \pi\), find the value of \(a\) and hence find \(| z |\) and \(z ^ { * } - 3\).

OCR FP1 Specimen Q3

8 marks Moderate -0.3

3 The complex number \(2 + \mathrm { i }\) is denoted by \(z\), and the complex conjugate of \(z\) is denoted by \(z ^ { * }\).

1. Express \(z ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.
2. Show that \(4 z - z ^ { 2 }\) simplifies to a real number, and verify that this real number is equal to \(z z ^ { * }\).
3. Express \(\frac { z + 1 } { z - 1 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.

OCR FP1 Specimen Q6

10 marks Standard +0.3

6 In an Argand diagram, the variable point \(P\) represents the complex number \(z = x + \mathrm { i } y\), and the fixed point \(A\) represents \(a = 4 - 3 \mathrm { i }\).

1. Sketch an Argand diagram showing the position of \(A\), and find \(| a |\) and \(\arg a\).
2. Given that \(| z - a | = | a |\), sketch the locus of \(P\) on your Argand diagram.
3. Hence write down the non-zero value of \(z\) corresponding to a point on the locus for which
   1. the real part of \(z\) is zero,
   2. \(\quad \arg z = \arg a\).

OCR MEI FP1 2005 January Q8

12 marks Standard +0.3

8 Two complex numbers are given by \(\alpha = 2 - \mathrm { j }\) and \(\beta = - 1 + 2 \mathrm { j }\).

1. Find \(\alpha + \beta , \alpha \beta\) and \(\frac { \alpha } { \beta }\) in the form \(a + b \mathrm { j }\), showing your working.
2. Find the modulus of \(\alpha\), leaving your answer in surd form. Find also the argument of \(\alpha\).
3. Sketch the locus \(| z - \alpha | = 2\) on an Argand diagram.
4. On a separate Argand diagram, sketch the locus \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).

OCR MEI FP1 2006 January Q2

5 marks Easy -1.2

2

1. Given that \(z = a + b \mathrm { j }\), express \(| z |\) and \(z ^ { * }\) in terms of \(a\) and \(b\).
2. Prove that \(z z ^ { * } - | z | ^ { 2 } = 0\).

OCR MEI FP1 2006 January Q8

11 marks Standard +0.3

8 You are given that the complex number \(\alpha = 1 + \mathrm { j }\) satisfies the equation \(z ^ { 3 } + 3 z ^ { 2 } + p z + q = 0\), where \(p\) and \(q\) are real constants.

1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\) in the form \(a + b \mathrm { j }\). Hence show that \(p = - 8\) and \(q = 10\).
2. Find the other two roots of the equation.
3. Represent the three roots on an Argand diagram.

OCR MEI FP1 2007 January Q8

11 marks Moderate -0.3

8 It is given that \(m = - 4 + 2 \mathrm { j }\).

1. Express \(\frac { 1 } { m }\) in the form \(a + b \mathrm { j }\).
2. Express \(m\) in modulus-argument form.
3. Represent the following loci on separate Argand diagrams.
   (A) \(\arg ( z - m ) = \frac { \pi } { 4 }\) (B) \(0 < \arg ( z - m ) < \frac { \pi } { 4 }\)

OCR MEI FP1 2008 January Q2

5 marks Moderate -0.8

2 You are given that \(\alpha = - 3 + 4 \mathrm { j }\).

1. Calculate \(\alpha ^ { 2 }\).
2. Express \(\alpha\) in modulus-argument form.

OCR MEI FP1 2005 June Q2

5 marks Moderate -0.8

2 Find the roots of the quadratic equation \(x ^ { 2 } - 8 x + 17 = 0\) in the form \(a + b \mathrm { j }\).
Express these roots in modulus-argument form.

Edexcel F1 2021 June Q2

7 marks Standard +0.3

2. The complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are given by $$\mathrm { z } _ { 1 } = 2 - \mathrm { i } \quad \mathrm { z } _ { 2 } = p - \mathrm { i } \quad \mathrm { z } _ { 3 } = p + \mathrm { i }$$ where \(p\) is a real number.

1. Find \(\frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } } \right| = 2 \sqrt { 5 }\)
2. find the possible values of \(p\).

CAIE P3 2020 Specimen Q6

8 marks Moderate -0.5

6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.

1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. State the argument of \(\frac { u } { v }\).
   In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u - v\) respectively.
3. State fully the geometrical relationship between \(O C\) and \(B A\).
4. Show that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.

OCR FP3 2015 June Q4

9 marks Standard +0.8

4 In an Argand diagram, the complex numbers \(0 , z\) and \(z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }\) are represented by the points \(O , A\) and \(B\) respectively.

1. Sketch a possible Argand diagram showing the triangle \(O A B\). Show that the triangle is isosceles and state the size of angle \(A O B\). The complex numbers \(1 + \mathrm { i }\) and \(5 + 2 \mathrm { i }\) are represented by the points \(C\) and \(D\) respectively. The complex number \(w\) is represented by the point \(E\), such that \(C D = C E\) and angle \(D C E = \frac { 1 } { 6 } \pi\).
2. Calculate the possible values of \(w\), giving your answers exactly in the form \(a + b \mathrm { i }\).

OCR MEI FP2 2012 June Q2

18 marks Standard +0.8

2

1. 1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
   2. Beginning with an expression for \(\left( z + \frac { 1 } { z } \right) ^ { 4 }\), find the constants \(A , B , C\) in the identity $$\cos ^ { 4 } \theta \equiv A + B \cos 2 \theta + C \cos 4 \theta$$
   3. Use the identity in part (ii) to obtain an expression for \(\cos 4 \theta\) as a polynomial in \(\cos \theta\).
2. 1. Given that \(z = 4 \mathrm { e } ^ { \mathrm { j } \pi / 3 }\) and that \(w ^ { 2 } = z\), write down the possible values of \(w\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\). Show \(z\) and the possible values of \(w\) in an Argand diagram.
   2. Find the least positive integer \(n\) for which \(z ^ { n }\) is real. Show that there is no positive integer \(n\) for which \(z ^ { n }\) is imaginary.
      For each possible value of \(w\), find the value of \(w ^ { 3 }\) in the form \(a + \mathrm { j } b\) where \(a\) and \(b\) are real.

OCR FP1 2010 January Q3

5 marks Moderate -0.3

3 The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }\). Find \(z\), giving your answer in the form \(x + \mathrm { i } y\).

OCR FP1 2012 January Q1

4 marks Easy -1.2

1 The complex number \(a + 5 \mathrm { i }\), where \(a\) is positive, is denoted by \(z\). Given that \(| z | = 13\), find the value of \(a\) and hence find \(\arg z\).

OCR FP1 2009 June Q3

4 marks Easy -1.2

3 The complex numbers \(z\) and \(w\) are given by \(z = 5 - 2 \mathrm { i }\) and \(w = 3 + 7 \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find

1. \(4 z - 3 w\),
2. \(z ^ { * } w\).

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
   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 $$| z - a | \geqslant 3 \sqrt { 2 } \quad \text { and } \quad 0 \leqslant \arg ( z - a ) \leqslant \frac { 1 } { 4 } \pi$$

OCR FP1 2011 June Q5

8 marks Standard +0.3

5 The complex number \(1 + \mathrm { i } \sqrt { 3 }\) is denoted by \(a\).

1. Find \(| a |\) and \(\arg a\).
2. Sketch on a single Argand diagram the loci given by \(| z - a | = | a |\) and \(\arg ( z - a ) = \frac { 1 } { 2 } \pi\).

OCR FP1 2014 June Q2

5 marks Moderate -0.8

2 The complex number \(7 + 3 \mathrm { i }\) is denoted by \(z\). Find

1. \(| z |\) and \(\arg z\),
2. \(\frac { z } { 4 - \mathrm { i } }\), showing clearly how you obtain your answer.

OCR FP1 2015 June Q1

3 marks Moderate -0.8

1 The complex number \(x + \mathrm { i } y\) is denoted by \(z\). Express \(3 z z ^ { * } - | z | ^ { 2 }\) in terms of \(x\) and \(y\).

OCR MEI FP1 2013 January Q2

4 marks Moderate -0.3

2 Given that \(z = a + b \mathrm { j }\), find \(\operatorname { Re } \left( \frac { z } { z ^ { * } } \right)\) and \(\operatorname { Im } \left( \frac { z } { z ^ { * } } \right)\).

OCR MEI FP1 2010 June Q8

10 marks Moderate -0.3

8 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = \sqrt { 3 } + \mathrm { j }\) and \(\beta = 3 \mathrm { j }\).

1. Find the modulus and argument of \(\alpha\) and \(\beta\).
2. Find \(\alpha \beta\) and \(\frac { \beta } { \alpha }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.
3. Plot \(\alpha , \beta , \alpha \beta\) and \(\frac { \beta } { \alpha }\) on a single Argand diagram.

OCR MEI FP1 2011 June Q2

8 marks Moderate -0.8

2 You are given that \(z = 3 - 2 \mathrm { j }\) and \(w = - 4 + \mathrm { j }\).

1. Express \(\frac { z + w } { w }\) in the form \(a + b \mathrm { j }\).
2. Express \(w\) in modulus-argument form.
3. Show \(w\) on an Argand diagram, indicating its modulus and argument.

OCR MEI FP1 2012 June Q2

7 marks Standard +0.3

2 You are given that \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers. \(z _ { 1 } = 3 + 3 \sqrt { 3 } \mathrm { j }\), and \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 3 }\).

1. Find the modulus and argument of \(z _ { 1 }\), giving your answers exactly.
2. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), where \(a\) and \(b\) are to be given exactly.
3. Explain why, when plotted on an Argand diagram, \(z _ { 1 } , z _ { 2 }\) and the origin lie on a straight line.