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

4 The complex number \(z _ { 1 }\) is \(3 - 2 \mathrm { j }\) and the complex number \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 4 }\).

1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), giving \(a\) and \(b\) in exact form.
2. Represent \(z _ { 1 } , z _ { 2 } , z _ { 1 } + z _ { 2 }\) and \(z _ { 1 } - z _ { 2 }\) on a single Argand diagram.

8 You are given the complex number \(w = 2 + 2 \sqrt { 3 } \mathrm { j }\).

1. Express \(w\) in modulus-argument form.
2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).

9 The set \(C\) consists of the set of all complex numbers excluding 1 and - 1 . The operation ⊕ is defined on the elements of \(C\) by \(\mathrm { a } \oplus \mathrm { b } = \frac { \mathrm { a } + \mathrm { b } } { \mathrm { ab } + 1 }\) where \(\mathrm { a } , \mathrm { b } \in \mathrm { C }\).

1. Determine the identity element of \(C\) under ⊕.
2. For each element \(x\) in \(C\) show that it has an inverse element in \(C\).
3. Show that \(\oplus\) is associative on \(C\).
4. Explain why \(( C , \oplus )\) is not a group.
5. Find a subset, \(D\), of \(C\) such that \(( D , \oplus )\) is a group of order 3 . \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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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 $$z - 3 \mathrm { i } z ^ { * } = 16$$

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 $$3 \mathrm { i } z + 2 z ^ { * } = 7 + 8 \mathrm { i }$$

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.

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 $$( 1 - 2 \mathrm { i } ) z - z ^ { * } = 10 ( 2 + \mathrm { i } )$$

   PART	REFERENCE
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3 The complex number \(z\) satisfies the equation \(5 ( z - \mathrm { i } ) = ( - 1 + 2 \mathrm { i } ) z ^ { * }\).
Determine \(z\), giving your answer in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.

7 On an Argand diagram, the point A represents the complex number \(z\) with modulus 2 and argument \(\frac { 1 } { 3 } \pi\). The point B represents \(\frac { 1 } { z }\).

1. Sketch an Argand diagram showing the origin O and the points A and B .
2. The point C is such that OACB is a parallelogram. C represents the complex number \(w\). Determine each of the following.

5 The Argand diagram below shows the points representing 1 and \(z\), where \(| z | = 2\). \includegraphics[max width=\textwidth, alt={}, center]{26cec6f9-78a7-4f0b-969a-13ad02510c25-3_577_595_312_242} Mark the points representing the following complex numbers on the copy of the diagram in the Printed Answer Booklet, labelling them clearly.

• \(\mathrm { Z } ^ { * }\)
• \(\frac { 1 } { z }\)
• \(1 + z\)
• iz

8 In an Argand diagram, the point P representing the complex number \(w\) lies on the locus defined by \(\left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\}\). You are given that \(\operatorname { Re } ( w ) = 1\).

1. Find \(w\). The point P also lies on the locus defined by \(\{ \mathrm { z } : | \mathrm { z } + 3 - 9 \mathrm { i } | = \mathrm { k } \}\), where \(k\) is a constant.
2. Find the complex number represented by the other point of intersection of the loci defined by $$\{ z : | z + 3 - 9 i | = k \} \text { and } \left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\} .$$

2 Fig. 2 shows two complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) represented on an Argand diagram. \begin{figure}[h] \end{figure}

1. On the copy of Fig. 2 in the Printed Answer Booklet, mark points representing each of the following complex numbers.
   In the case where \(z _ { 1 } = 1 + 2 \mathrm { i }\) and \(z _ { 2 } = 3 + \mathrm { i }\), find \(\frac { z _ { 2 } - z _ { 1 } } { z _ { 1 } ^ { * } }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.

1

1. The complex number \(\mathrm { a } + \mathrm { ib }\) is denoted by \(z\).
   1. Write down \(z ^ { * }\).
   2. Find \(\operatorname { Re } ( \mathrm { iz } )\).
2. The complex number \(w\) is given by \(w = \frac { 5 + \mathrm { i } \sqrt { 3 } } { 2 - \mathrm { i } \sqrt { 3 } }\).
   1. In this question you must show detailed reasoning. Express \(w\) in the form \(\mathrm { x } + \mathrm { iy }\).
   2. Convert \(w\) to modulus-argument form.

2 Two complex numbers are given by \(u = - 1 + \mathrm { i }\) and \(v = - 2 - \mathrm { i }\).

1. 1. Find \(\mathrm { u } - \mathrm { v }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
   2. In this question you must show detailed reasoning. Find \(\frac { \mathrm { u } } { \mathrm { v } }\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
2. Express \(u\) in exact modulus-argument form.

6 The complex number \(z\) satisfies the equation \(z ^ { 2 } - 4 \mathrm { i } z ^ { * } + 11 = 0\).
Given that \(\operatorname { Re } ( z ) > 0\), find \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Section B (108 marks)
Answer all the questions.

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively in Argand diagrams and $$w = z ^ { 2 } - 1$$

1. Show that \(v = 2 x y\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 3 x\). Find the equation of the locus of \(Q\).

1. The complex number \(z\) is given by \(z = 3 + \lambda \mathrm { i }\), where \(\lambda\) is a positive constant. The complex conjugate of \(z\) is denoted by \(\bar { z }\).

Given that \(z ^ { 2 } + \bar { z } ^ { 2 } = 2\), find the value of \(\lambda\).

9. The complex numbers \(z\) and \(w\) are represented by the points \(P ( x , y )\) and \(Q ( u , v )\) respectively, in Argand diagrams, and \(w = 1 - z ^ { 2 }\).

1. Find expressions for \(u\) and \(v\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 4 x\). Find the equation of the locus of \(Q\).
3. Find the perpendicular distance of the point corresponding to \(z = 2 + 5 \mathrm { i }\) in the \(( u , v )\)-plane, from the locus of \(Q\).

1. The complex numbers \(z , v\) and \(w\) are related by the equation

$$z = \frac { v } { w }$$ Given that \(v = - 16 + 11 \mathrm { i }\) and \(w = 5 + 2 \mathrm { i }\), find \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).

1. The complex numbers \(z\) and \(w\) are represented, respectively, by the points \(P ( x , y )\) and \(Q ( u , v )\) in Argand diagrams and

$$w = \frac { z } { 1 - z }$$

1. Show that \(v = \frac { y } { ( 1 - x ) ^ { 2 } + y ^ { 2 } }\) and obtain an expression for \(u\) in terms of \(x\) and \(y\).
2. The point \(P\) moves along the line \(y = 1 - x\). Find and simplify the equation of the locus of \(Q\).

1. Given that

$$\begin{aligned} z _ { 1 } & = 2 + 3 \\ \left| z _ { 1 } z _ { 2 } \right| & = 39 \sqrt { 2 } \\ \arg \left( z _ { 1 } z _ { 2 } \right) & = \frac { \pi } { 4 } \end{aligned}$$ where \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers,

1. write \(z _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) Give the exact value of \(r\) and give the value of \(\theta\) in radians to 4 significant figures.
2. Find \(z _ { 2 }\) giving your answer in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are integers.

1. The infinite series C and S are defined by

$$\begin{aligned} & \mathrm { C } = \cos \theta + \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 9 \theta + \frac { 1 } { 8 } \cos 13 \theta + \ldots \\ & \mathrm { S } = \sin \theta + \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 9 \theta + \frac { 1 } { 8 } \sin 13 \theta + \ldots \end{aligned}$$ Given that the series C and S are both convergent,

1. show that $$\mathrm { C } + \mathrm { iS } = \frac { 2 \mathrm { e } ^ { \mathrm { i } \theta } } { 2 - \mathrm { e } ^ { 4 \mathrm { i } \theta } }$$
2. Hence show that $$\mathrm { S } = \frac { 4 \sin \theta + 2 \sin 3 \theta } { 5 - 4 \cos 4 \theta }$$