4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)

68 questions

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Edexcel FP1 2009 June Q1
8 marks Moderate -0.8
  1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by
$$z _ { 1 } = 2 - i \quad \text { and } \quad z _ { 2 } = - 8 + 9 i$$
  1. Show \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram. Find, showing your working,
  2. the value of \(\left| z _ { 1 } \right|\),
  3. the value of \(\arg z _ { 1 }\), giving your answer in radians to 2 decimal places,
  4. \(\frac { Z _ { 2 } } { Z _ { 1 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real.
Edexcel FP1 2012 June Q7
11 marks Moderate -0.3
7. $$z = 2 - \mathrm { i } \sqrt { } 3$$
  1. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. Use algebra to express
  2. \(z + z ^ { 2 }\) in the form \(a + b \mathrm { i } \sqrt { } 3\), where \(a\) and \(b\) are integers,
  3. \(\frac { z + 7 } { z - 1 }\) in the form \(c + d \mathrm { i } \sqrt { } 3\), where \(c\) and \(d\) are integers. Given that $$w = \lambda - 3 \mathrm { i }$$ where \(\lambda\) is a real constant, and \(\arg ( 4 - 5 \mathrm { i } + 3 w ) = - \frac { \pi } { 2 }\),
  4. find the value of \(\lambda\).
Edexcel FP2 2003 June Q13
11 marks Standard +0.3
13. Given that \(z = 3 - 3 i\) express, in the form \(a + i b\), where \(a\) and \(b\) are real numbers,
  1. \(z ^ { 2 }\),
    (2)
  2. \(\frac { 1 } { z }\).
    (2)
  3. Find the exact value of each of \(| z | , \left| z ^ { 2 } \right|\) and \(\left| \frac { 1 } { z } \right|\).
    (2) The complex numbers \(z , z ^ { 2 }\) and \(\frac { 1 } { z }\) are represented by the points \(A , B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.
  4. Show the points \(A , B , C\) and \(D\) on an Argand diagram.
  5. Prove that \(\triangle O A B\) is similar to \(\triangle O C D\).
Edexcel FP2 2010 June Q4
10 marks Standard +0.3
4. $$z = - 8 + ( 8 \sqrt { } 3 ) i$$
  1. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
  2. find \(z ^ { 3 }\),
  3. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
Edexcel FP2 2015 June Q2
9 marks Standard +0.3
2. $$z = - 2 + ( 2 \sqrt { 3 } ) \mathrm { i }$$
  1. Find the modulus and the argument of \(z\). Using de Moivre's theorem,
  2. find \(z ^ { 6 }\), simplifying your answer,
  3. find the values of \(w\) such that \(w ^ { 4 } = z ^ { 3 }\), giving your answers in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\).
Edexcel FP2 Q8
11 marks Standard +0.8
8. A complex number \(z\) satisfies the equation $$| z - 5 - 12 i | = 3$$
  1. Describe in geometrical terms with the aid of a sketch, the locus of the point which represents \(z\) in the A rgand diagram. For points on this locus, find
  2. the maximum and minimum values for \(| z |\),
  3. the maximum and minimum values for arg \(z\), giving your answers in radians to 2 decimal places.
Edexcel F3 2021 October Q7
11 marks Challenging +1.8
7. A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1$$ where \(a\) is a positive constant.
The eccentricity of \(H\) is \(e\).
  1. Determine an expression for \(e ^ { 2 }\) in terms of \(a\). The line \(l\) is the directrix of \(H\) for which \(x > 0\) The points \(A\) and \(A ^ { \prime }\) are the points of intersection of \(l\) with the asymptotes of \(H\).
  2. Determine, in terms of \(e\), the length of the line segment \(A A ^ { \prime }\). The point \(F\) is the focus of \(H\) for which \(x < 0\) Given that the area of triangle \(A F A ^ { \prime }\) is \(\frac { 164 } { 3 }\)
  3. show that \(a\) is a solution of the equation $$30 a ^ { 3 } - 164 a ^ { 2 } + 375 a - 4100 = 0$$
  4. Hence, using algebra and making your reasoning clear, show that the only possible value of \(a\) is \(\frac { 20 } { 3 }\)
OCR FP3 2007 June Q1
3 marks Moderate -0.8
1
  1. By writing \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), show that \(z z ^ { * } = | z | ^ { 2 }\).
  2. Given that \(z z ^ { * } = 9\), describe the locus of \(z\).
OCR FP1 2006 June Q6
7 marks Moderate -0.5
6 In an Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Hence find, in the form \(x + \mathrm { i } y\), the complex number representing the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
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 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 FP2 2006 January Q2
18 marks Challenging +1.2
2 In this question, \(\theta\) is a real number with \(0 < \theta < \frac { 1 } { 6 } \pi\), and \(w = \frac { 1 } { 2 } \mathrm { e } ^ { 3 \mathrm { j } \theta }\).
  1. State the modulus and argument of each of the complex numbers $$w , \quad w ^ { * } \quad \text { and } \quad \mathrm { j } w .$$ Illustrate these three complex numbers on an Argand diagram.
  2. Show that \(( 1 + w ) \left( 1 + w ^ { * } \right) = \frac { 5 } { 4 } + \cos 3 \theta\). Infinite series \(C\) and \(S\) are defined by $$\begin{aligned} & C = \cos 2 \theta - \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 8 \theta - \frac { 1 } { 8 } \cos 11 \theta + \ldots \\ & S = \sin 2 \theta - \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 8 \theta - \frac { 1 } { 8 } \sin 11 \theta + \ldots \end{aligned}$$
  3. Show that \(C = \frac { 4 \cos 2 \theta + 2 \cos \theta } { 5 + 4 \cos 3 \theta }\), and find a similar expression for \(S\).
OCR MEI FP2 2007 January Q2
18 marks Challenging +1.8
2
  1. You are given the complex numbers \(w = 3 \mathrm { e } ^ { - \frac { 1 } { 12 } \pi \mathrm { j } }\) and \(z = 1 - \sqrt { 3 } \mathrm { j }\).
    1. Find the modulus and argument of each of the complex numbers \(w , z\) and \(\frac { w } { z }\).
    2. Hence write \(\frac { w } { z }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
  2. In this part of the question, \(n\) is a positive integer and \(\theta\) is a real number with \(0 < \theta < \frac { \pi } { n }\).
    1. Express \(\mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { j } \theta } + \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta }\) in simplified trigonometric form, and hence, or otherwise, show that $$1 + \mathrm { e } ^ { \mathrm { j } \theta } = 2 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta } \cos \frac { 1 } { 2 } \theta$$ Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \binom { n } { 3 } \cos 3 \theta + \ldots + \binom { n } { n } \cos n \theta \\ & S = \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \binom { n } { 3 } \sin 3 \theta + \ldots + \binom { n } { n } \sin n \theta \end{aligned}$$
    2. Find \(C\) and \(S\), and show that \(\frac { S } { C } = \tan \frac { 1 } { 2 } n \theta\).
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 \(p\).
OCR FP3 2009 January Q2
5 marks Standard +0.3
2
  1. Express \(\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
  2. Hence find the smallest positive value of \(n\) for which \(\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }\) is real and positive.
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 2015 June Q2
5 marks Moderate -0.8
2 Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 13 = 0\).
Find the modulus and argument of each root.
OCR MEI FP1 2016 June Q2
6 marks Standard +0.3
2 The complex number \(z _ { 1 }\) is \(2 - 5 \mathrm { j }\) and the complex number \(z _ { 2 }\) is \(( a - 1 ) + ( 2 - b ) \mathrm { j }\), where \(a\) and \(b\) are real.
  1. Express \(\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } }\) in the form \(x + y \mathrm { j }\), giving \(x\) and \(y\) in exact form. You must show clearly how you obtain your
    answer.
  2. Given that \(\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } } = z _ { 2 }\), find the exact values of \(a\) and \(b\).
AQA FP1 2012 January Q3
8 marks Easy -1.2
3
  1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
    1. \(x ^ { 2 } + 9 = 0\);
    2. \(( x + 2 ) ^ { 2 } + 9 = 0\).
    1. Expand \(( 1 + x ) ^ { 3 }\).
    2. Express \(( 1 + 2 \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
    3. Given that \(z = 1 + 2 \mathrm { i }\), find the value of $$z ^ { * } - z ^ { 3 }$$
AQA FP1 2013 January Q2
9 marks Moderate -0.3
2
  1. Solve the equation \(w ^ { 2 } + 6 w + 34 = 0\), giving your answers in the form \(p + q \mathrm { i }\), where \(p\) and \(q\) are integers.
  2. It is given that \(z = \mathrm { i } ( 1 + \mathrm { i } ) ( 2 + \mathrm { i } )\).
    1. Express \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are integers.
    2. Find integers \(m\) and \(n\) such that \(z + m z ^ { * } = n \mathrm { i }\).
AQA FP1 2007 June Q3
6 marks Moderate -0.5
3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$z - 3 \mathbf { i } z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
  2. Find the complex number \(z\) such that $$z - 3 \mathrm { i } z ^ { * } = 16$$
AQA FP1 2008 June Q2
6 marks Moderate -0.5
2 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$3 \mathrm { i } z + 2 z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
  2. Find the complex number \(z\) such that $$3 \mathrm { i } z + 2 z ^ { * } = 7 + 8 \mathrm { i }$$
AQA FP1 2009 June Q3
7 marks Moderate -0.5
3 The complex number \(z\) is defined by $$z = x + 2 \mathrm { i }$$ where \(x\) is real.
  1. Find, in terms of \(x\), the real and imaginary parts of:
    1. \(z ^ { 2 }\);
    2. \(z ^ { 2 } + 2 z ^ { * }\).
  2. Show that there is exactly one value of \(x\) for which \(z ^ { 2 } + 2 z ^ { * }\) is real.
AQA FP1 2010 June Q2
6 marks Moderate -0.3
2 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$( 1 - 2 i ) z - z ^ { * }$$
  2. Hence find the complex number \(z\) such that $$( 1 - 2 \mathrm { i } ) z - z ^ { * } = 10 ( 2 + \mathrm { i } )$$
    PARTREFERENCE
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