Complex arithmetic operations - Complex Numbers Argand & Loci

Edexcel F1 2021 January Q6

11 marks Moderate -0.3

6. The complex number $z$ is defined by $$z = - \lambda + 3 i \quad \text { where } \lambda \text { is a positive real constant }$$ Given that the modulus of $z$ is 5

write down the value of $\lambda$
determine the argument of $z$, giving your answer in radians to one decimal place. In part (c) you must show detailed reasoning.
Solutions relying on calculator technology are not acceptable.
Express in the form $a + \mathrm { i } b$ where $a$ and $b$ are real,
1. $\frac { z + 3 i } { 2 - 4 i }$
2. $\mathrm { Z } ^ { 2 }$
Show on a single Argand diagram the points $A$, $B$, $C$ and $D$ that represent the complex numbers $$z , z ^ { * } , \frac { z + 3 i } { 2 - 4 i } \text { and } z ^ { 2 }$$

Edexcel F1 2015 June Q7

11 marks Moderate -0.3

7. $$z = - 3 k - 2 k \mathrm { i } , \text { where } k \text { is a real, positive constant. }$$

Find the modulus and the argument of $z$, giving the argument in radians to 2 decimal places and giving the modulus as an exact answer in terms of $k$.
Express in the form $a + \mathrm { i } b$, where $a$ and $b$ are real and are given in terms of $k$ where necessary,
1. $\frac { 4 } { z + 3 k }$
2. $z ^ { 2 }$
Given that $k = 1$, plot the points $A , B , C$ and $D$ representing $z , z ^ { * } , \frac { 4 } { z + 3 k }$ and $z ^ { 2 }$ respectively on a single Argand diagram.

Edexcel F1 2017 June Q9

10 marks Moderate -0.3

9. $$z = \frac { 1 } { 5 } - \frac { 2 } { 5 } \mathrm { i }$$

Find the modulus and the argument of $z$, giving the modulus as an exact answer and giving the argument in radians to 2 decimal places. Given that $$\mathrm { zw } = \lambda \mathrm { i }$$ where $\lambda$ is a real constant,
find $w$ in the form $a + \mathrm { i } b$, where $a$ and $b$ are real. Give your answer in terms of $\lambda$.
Given that $\lambda = \frac { 1 } { 10 }$
1. find $\frac { 4 } { 3 } ( z + w )$,
2. plot the points $A , B , C$ and $D$, representing $z , z w , w$ and $\frac { 4 } { 3 } ( z + w )$ respectively, on a single Argand diagram.

Edexcel F1 2022 June Q1

6 marks Moderate -0.8

1. $$z _ { 1 } = 3 + 3 i \quad z _ { 2 } = p + q i \quad p , q \in \mathbb { R }$$ Given that $\left| z _ { 1 } z _ { 2 } \right| = 15 \sqrt { 2 }$

determine $\left| z _ { 2 } \right|$ Given also that $p = - 4$
determine the possible values of $q$
Show $z _ { 1 }$ and the possible positions for $z _ { 2 }$ on the same Argand diagram.

Edexcel F1 Specimen Q1

7 marks Moderate -0.8

The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by

$$z _ { 1 } = 2 + 8 \mathrm { i } \quad \text { and } \quad z _ { 2 } = 1 - \mathrm { i }$$ Find, showing your working,

$\frac { z _ { 1 } } { z _ { 2 } }$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are real,
the value of $\left| \frac { z _ { 1 } } { z _ { 2 } } \right|$,
the value of $\arg \frac { z _ { 1 } } { z _ { 2 } }$, giving your answer in radians to 2 decimal places.

Edexcel FP1 2010 January Q1

7 marks Moderate -0.8

The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by

$$z _ { 1 } = 2 + 8 i \quad \text { and } \quad z _ { 2 } = 1 - i$$ Find, showing your working,

$\frac { Z _ { 1 } } { Z _ { 2 } }$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are real,
the value of $\left| \frac { z _ { 1 } } { z _ { 2 } } \right|$,
the value of $\arg \frac { Z _ { 1 } } { Z _ { 2 } }$, giving your answer in radians to 2 decimal places.

Edexcel FP1 2009 June Q1

8 marks Moderate -0.8

The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by

$$z _ { 1 } = 2 - i \quad \text { and } \quad z _ { 2 } = - 8 + 9 i$$

Show $z _ { 1 }$ and $z _ { 2 }$ on a single Argand diagram. Find, showing your working,
the value of $\left| z _ { 1 } \right|$,
the value of $\arg z _ { 1 }$, giving your answer in radians to 2 decimal places,
$\frac { Z _ { 2 } } { Z _ { 1 } }$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are real.

Edexcel FP1 2010 June Q1

7 marks Moderate -0.5

1. $$z = 2 - 3 \mathrm { i }$$

Show that $z ^ { 2 } = - 5 - 12 \mathrm { i }$. Find, showing your working,
the value of $\left| z ^ { 2 } \right|$,
the value of $\arg \left( z ^ { 2 } \right)$, giving your answer in radians to 2 decimal places.
Show $z$ and $z ^ { 2 }$ on a single Argand diagram.

Edexcel FP1 2015 June Q4

8 marks Moderate -0.8

4. $$z _ { 1 } = 3 \mathrm { i } \text { and } z _ { 2 } = \frac { 6 } { 1 + \mathrm { i } \sqrt { 3 } }$$

Express $z _ { 2 }$ in the form $a + \mathrm { i } b$, where $a$ and $b$ are real numbers.
Find the modulus and the argument of $z _ { 2 }$, giving the argument in radians in terms of $\pi$.
Show the three points representing $z _ { 1 } , z _ { 2 }$ and $\left( z _ { 1 } + z _ { 2 } \right)$ respectively, on a single Argand diagram.

Edexcel FP1 Specimen Q6

10 marks Moderate -0.8

6. Given that $z = - 3 + 4 \mathrm { i }$,

find the modulus of $z$,
the argument of $z$ in radians to 2 decimal places. Given also that $w = \frac { - 14 + 2 \mathrm { i } } { z }$,
use algebra to find $w$, giving your answers in the form $a + \mathrm { i } b$, where $a$ and $b$ are real. The complex numbers $z$ and $w$ are represented by points $A$ and $B$ on an Argand diagram.
Show the points $A$ and $B$ on an Argand diagram.

OCR Further Pure Core AS 2018 June Q3

9 marks Moderate -0.3

3 In this question you must show detailed reasoning.
The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by $z _ { 1 } = 2 - 3 i$ and $z _ { 2 } = a + 4 i$ where $a$ is a real number.

Express $z _ { 1 }$ in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
Find $z _ { 1 } z _ { 2 }$ in terms of $a$, writing your answer in the form $c + \mathrm { id }$.
The real and imaginary parts of a complex number on an Argand diagram are $x$ and $y$ respectively. Given that the point representing $z _ { 1 } z _ { 2 }$ lies on the line $y = x$, find the value of $a$.
Given instead that $z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }$ find the value of $a$.

Edexcel FP1 Q19

11 marks Moderate -0.3

Given that $z = 1 + \sqrt{3}i$ and that $\frac{w}{z} = 2 + 2i$, find

$w$ in the form $a + ib$, where $a, b \in \mathbb{R}$, [3]
the argument of $w$, [2]
the exact value for the modulus of $w$. [2]

On an Argand diagram, the point $A$ represents $z$ and the point $B$ represents $w$.

Draw the Argand diagram, showing the points $A$ and $B$. [2]
Find the distance $AB$, giving your answer as a simplified surd. [2]

OCR Further Pure Core 2 2018 March Q2

5 marks Moderate -0.8

The complex number $2 + i$ is denoted by $z$.

Show that $z^2 = 3 + 4i$. [2]
Plot the following on the Argand diagram in the Printed Answer Booklet.
[1]
State the relationship between $|z^2|$ and $|z|$. [1]
State the relationship between $\arg(z^2)$ and $\arg(z)$. [1]

OCR Further Pure Core 2 2018 December Q1

6 marks Easy -1.2

The Argand diagram below shows the two points which represent two complex numbers, $z_1$ and $z_2$. \includegraphics{figure_1} On the copy of the diagram in the Printed Answer Booklet
- draw an appropriate shape to illustrate the geometrical effect of adding $z_1$ and $z_2$,
- indicate with a cross ($\times$) the location of the point representing the complex number $z_1 + z_2$.
[2]
You are given that $\arg z_3 = \frac{1}{4}\pi$ and $\arg z_4 = \frac{3}{8}\pi$. In each part, sketch and label the points representing the numbers $z_3$, $z_4$ and $z_3z_4$ on the diagram in the Printed Answer Booklet. You should join each point to the origin with a straight line.
1. $|z_3| = 1.5$ and $|z_4| = 1.2$ [2]
2. $|z_3| = 0.7$ and $|z_4| = 0.5$ [2]

Pre-U Pre-U 9794/1 2011 June Q10

9 marks Moderate -0.3

The complex number $z$ is such that $|z| = 2$ and $\arg z = -\frac{3}{4}\pi$. Find the exact value of the real part of $z$ and of the imaginary part of $z$. [2]
The complex numbers $u$ and $v$ are such that $$u = 1 + ia \quad \text{and} \quad v = b - i,$$ where $a$ and $b$ are real and $a < b$. Given that $uv = 7 + 9i$, find the values of $a$ and $b$. [7]