Multi-part questions with division as one step

Edexcel F1 2017 January Q5

Moderate -0.3

The complex number $z$ is given by

$$z = - 7 + 3 i$$ Find

$| z |$
$\arg z$, giving your answer in radians to 2 decimal places. Given that $\frac { z } { 1 + \mathrm { i } } + w = 3 - 6 \mathrm { i }$
find the complex number $w$, giving your answer in the form $a + b \mathrm { i }$, where $a$ and $b$ are real numbers. You must show all your working.
Show the points representing $z$ and $w$ on a single Argand diagram.

Edexcel F1 2022 January Q2

Moderate -0.8

2. The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by $$z _ { 1 } = 3 + 5 i \text { and } z _ { 2 } = - 2 + 6 i$$

Show $z _ { 1 }$ and $z _ { 2 }$ on a single Argand diagram.
Without using your calculator and showing all stages of your working,
1. determine the value of $\left| z _ { 1 } \right|$
2. express $\frac { z _ { 1 } } { z _ { 2 } }$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are fully simplified fractions.
Hence determine the value of $\arg \frac { Z _ { 1 } } { Z _ { 2 } }$ Give your answer in radians to 2 decimal places.

Edexcel F1 2024 June Q4

Standard +0.3

In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} The complex number $z$ is defined by $$2 = - 3 + 4 i$$

Determine $\left| z ^ { 2 } - 3 \right|$
Express $\frac { 50 } { z ^ { * } }$ in the form $k z$, where $k$ is a positive integer.
Hence find the value of $\arg \left( \frac { 50 } { z ^ { * } } \right)$ Give your answer in radians to 3 significant figures.

Edexcel FP1 2012 January Q1

Moderate -0.5

Given that $z _ { 1 } = 1 - \mathrm { i }$,
1. find $\arg \left( z _ { 1 } \right)$.
Given also that $z _ { 2 } = 3 + 4 \mathrm { i }$, find, in the form $a + \mathrm { i } b , a , b \in \mathbb { R }$,
$z _ { 1 } z _ { 2 }$,
$\frac { z _ { 2 } } { z _ { 1 } }$. In part (b) and part (c) you must show all your working clearly.

Edexcel FP1 2013 January Q2

Moderate -0.3

2. $$z = \frac { 50 } { 3 + 4 \mathrm { i } }$$ Find, in the form $a + \mathrm { i } b$ where $a , b \in \mathbb { R }$,

$z$,
$z ^ { 2 }$. Find
$| z |$,
$\arg z ^ { 2 }$, giving your answer in degrees to 1 decimal place.

Edexcel FP1 Q3

Moderate -0.3

3. $z = 1 + \mathrm { i } \sqrt { 3 }$ Express in the form $a + \mathrm { i } b$, where $a$ and $b$ are real.

$z ^ { 2 } + z$,
$\frac { z } { 3 - z }$,
giving the exact values of $a$ and $b$ in each part.

OCR FP1 2006 January Q1

Moderate -0.8

1

Express $( 1 + 8 i ) ( 2 - i )$ in the form $x + i y$, showing clearly how you obtain your answer.
Hence express $\frac { 1 + 8 i } { 2 + i }$ in the form $x + i y$.

OCR MEI FP1 2005 January Q8

Standard +0.3

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

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

CAIE P3 2020 Specimen Q6

Moderate -0.5

6 Th cm plexm b rs $1 + B$ ad $4 + \quad 2$ are $d \mathbf { b }$ ed $\forall u$ ad $v$ resp ctie ly.

Fid $\frac { \mathrm { u } } { \mathrm { V } }$ irt b fo $\mathrm { m } x + \mathrm { i } y , \mathrm { w } \mathbf { b }$ re $x$ ad $y$ are real.
State th argn en $6 \frac { u } { v }$. In an Arg nd id ag am, with o ign $O$, th $\dot { \mathrm { p } } \mathrm { ns } A , B$ ad $C$ represen th cm p ex m b rs $u , v$ ad $u - v$ resp ctie ly.
State fullyt bg m etrical relatio h申 tween $O C$ ad $B A$.
Sth the tag e $A O B = \frac { 1 } { 4 } \pi$ rad as.

OCR FP1 2011 January Q2

Moderate -0.8

2 The complex numbers $z$ and $w$ are given by $z = 4 + 3 \mathrm { i }$ and $w = 6 - \mathrm { i }$. Giving your answers in the form $x + \mathrm { i } y$ and showing clearly how you obtain them, find

$3 z - 4 w$,
$\frac { z ^ { * } } { w }$.

OCR FP1 2010 June Q4

Moderate -0.3

4 The complex numbers $a$ and $b$ are given by $a = 7 + 6 \mathrm { i }$ and $b = 1 - 3 \mathrm { i }$. Showing clearly how you obtain your answers, find

$| a - 2 b |$ and $\arg ( a - 2 b )$,
$\frac { b } { a }$, giving your answer in the form $x + \mathrm { i } y$.

OCR FP1 2012 June Q1

Moderate -0.8

1 The complex numbers $z$ and $w$ are given by $z = 6 - \mathrm { i }$ and $w = 5 + 4 \mathrm { i }$. Giving your answers in the form $x + \mathrm { i } y$ and showing clearly how you obtain them, find

$z + 3 w$,
$\frac { Z } { W }$.

OCR FP1 2014 June Q2

Moderate -0.8

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

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

OCR MEI FP1 2009 January Q9

Standard +0.3

9 Two complex numbers, $\alpha$ and $\beta$, are given by $\alpha = 1 + \mathrm { j }$ and $\beta = 2 - \mathrm { j }$.

Express $\alpha + \beta , \alpha \alpha ^ { * }$ and $\frac { \alpha + \beta } { \alpha }$ in the form $a + b \mathrm { j }$.
Find a quadratic equation with roots $\alpha$ and $\alpha ^ { * }$.
$\alpha$ and $\beta$ are roots of a quartic equation with real coefficients. Write down the two other roots and find this quartic equation in the form $z ^ { 4 } + A z ^ { 3 } + B z ^ { 2 } + C z + D = 0$.

OCR MEI FP1 2010 June Q8

Moderate -0.3

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

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

OCR MEI FP1 2016 June Q2

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.

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.
Given that $\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } } = z _ { 2 }$, find the exact values of $a$ and $b$.

OCR Further Pure Core AS 2024 June Q2

Moderate -0.3

2 In this question you must show detailed reasoning.

Express $\frac { 8 + \mathrm { i } } { 2 - \mathrm { i } }$ in the form $\mathrm { a } + \mathrm { bi }$ where $a$ and $b$ are real.
Solve the equation $4 x ^ { 2 } - 8 x + 5 = 0$. Give your answer(s) in the form $\mathrm { c } + \mathrm { di }$ where $c$ and $d$ are real.

AQA FP1 2013 January Q2

Moderate -0.3

2

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.
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 $z + m z ^ { * } = n \mathrm { i }$.

OCR MEI Further Pure Core 2024 June Q2

Moderate -0.8

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

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.
Express $u$ in exact modulus-argument form.

WJEC Further Unit 1 2022 June Q1

Standard +0.3

The complex numbers $z , w$ are given by $z = 3 - 4 \mathrm { i } , w = 2 - \mathrm { i }$.
1. (i) Find the modulus and argument of $z w$.
  (ii) Express $z w$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$.
2. The complex numbers $v , w , z$ satisfy the equation $\frac { 1 } { v } = \frac { 1 } { w } - \frac { 1 } { z }$. Find $v$ in the form $a + \mathrm { i } b$, where $a , b$ are real.
3. The complex conjugate of $v$ is denoted by $\bar { v }$.
Show that $v \bar { v } = k$, where $k$ is a real number whose value is to be determined.

CAIE P3 2016 November Q7

Standard +0.3

Find the modulus and argument of $z$.
Express each of the following in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact:
(a) $z + 2 z ^ { * }$;
(b) $\frac { z ^ { * } } { \mathrm { i } z }$.
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 $A O B$ is equal to $\frac { 1 } { 6 } \pi$.

SPS SPS FM Pure 2022 June Q6

Standard +0.3

6. The complex number $w$ is given by $$w = 10 - 5 \mathrm { i }$$

Find $| w |$.
Find arg $w$, giving your answer in radians to 2 decimal places. The complex numbers $z$ and $w$ satisfy the equation $$( 2 + \mathrm { i } ) ( z + 3 \mathrm { i } ) = w$$
Use algebra to find $z$, giving your answer in the form $a + b \mathrm { i }$, where $a$ and $b$ are real numbers. Given that $$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$ where $\lambda$ is a real constant,
find the value of $\lambda$.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2023 September Q7

Moderate -0.3

7. The complex number $2 - \mathrm { i }$ is denoted by $z$.

Find $| z |$ and arg $z$.
Given that $a z + b z ^ { * } = 4 - 8 \mathrm { i }$, find the values of the real constants $a$ and $b$.
[0pt] [BLANK PAGE]

Edexcel FP1 Q1

Standard +0.3

Given that $z = 22 + 4 \mathrm { i }$ and $\frac { z } { w } = 6 - 8 \mathrm { i }$, find
1. $w$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are real,
2. the argument of $z$, in radians to 2 decimal places.
3. (a) Prove that $\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r - 1 ) = \frac { 1 } { 6 } n ( n - 1 ) ( 2 n + 5 )$.
4. Deduce that $n ( n - 1 ) ( 2 n + 5 )$ is divisible by 6 for all $n > 1$.
  [0pt] [P4 January 2002 Qn 3]
$$\mathrm { f } ( x ) = x ^ { 3 } + x - 3$$ The equation $\mathrm { f } ( x ) = 0$ has a root, $\alpha$, between 1 and 2 .
By considering $\mathrm { f } ^ { \prime } ( x )$, show that $\alpha$ is the only real root of the equation $\mathrm { f } ( x ) = 0$.
Taking 1.2 as your first approximation to $\alpha$, apply the Newton-Raphson procedure once to $\mathrm { f } ( x )$ to obtain a second approximation to $\alpha$. Give your answer to 3 significant figures.
Prove that your answer to part (b) gives the value of $\alpha$ correct to 3 significant figures.

Edexcel FP1 Q19

Standard +0.3

Given that $z = 1 + \sqrt { } 3 \mathrm { i }$ and that $\frac { w } { z } = 2 + 2 \mathrm { i }$, find
1. $w$ in the form $a + \mathrm { i } b$, where $a , b \in \mathbb { R }$,
2. the argument of $w$,
3. the exact value for the modulus of $w$.
On an Argand diagram, the point $A$ represents $z$ and the point $B$ represents $w$.
Draw the Argand diagram, showing the points $A$ and $B$.
Find the distance $A B$, giving your answer as a simplified surd.