Further Complex Numbers (DP IB Maths: AI HL)

Exam Questions

4 hours28 questions
1a
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2 marks

Consider w equals z subscript 1 over z subscript 2, where z subscript 1 equals 2 plus 2 square root of 3 straight i and z subscript 2 equals 2 plus 2 straight i. 

Express w in the form w equals a plus b straight i.

1b
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2 marks

Write the complex numbers z subscript 1 and z subscript 2 in the form  r e to the power of straight i theta end exponent comma space r greater or equal than 0 comma space minus pi less than theta less than pi.

1c
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2 marks

Express w in the form  r e to the power of i theta end exponent comma space r greater or equal than 0 comma space minus pi less than theta less than pi.

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2a
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4 marks

Consider the equation z to the power of 4 minus 1 equals 15, where z element of straight complex numbers.

Find the four distinct roots of the equation, giving your answers in the form a plus b straight i, where a comma space b element of straight real numbers.

2b
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2 marks

Represent the roots found in part (a) on the Argand diagram below.

q8b_1-9_ib-maths-aa-hl

2c
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2 marks

Find the area of the polygon whose vertices are represented by the four roots on the Argand diagram.

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3a
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4 marks

Let z subscript 1 equals 6   c i s left parenthesis straight pi over 6 right parenthesis and z subscript 2 equals 3 square root of 2 e to the power of straight i open parentheses straight pi over 4 close parentheses end exponent. 

Giving your answers in the form r c i s theta comma find 

(i)
z subscript 1 z subscript 2

(ii)
begin mathsize 16px style z subscript 1 over z subscript 2. end style
3b
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2 marks

Write z subscript 1and z subscript 2 in the form a plus b straight i.

3c
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2 marks

Find z subscript 1 plus z subscript 2 comma giving your answer in the form a plus b straight i.

3d
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2 marks

It is given that z subscript 1 superscript asterisk times and z subscript 2 superscript asterisk times are the complex conjugates of z subscript 1and z subscript 2 respectively. 

Find z subscript 1 superscript asterisk times plus z subscript 2 superscript asterisk times comma giving your answer in the form a plus b straight i.

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4a
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2 marks

Let z subscript 1 equals 2   c i s left parenthesis straight pi over 3 right parenthesis and z subscript 2 equals 2 plus 2 straight i.

Express

(i)
z subscript 1in the form a plus b straight i 

(ii)
z subscript 2 in the form r   c i s theta

4b
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2 marks

Find w subscript 1 equals z subscript 1 plus z subscript 2 comma giving your answer in the form a plus b straight i.

4c
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3 marks

Find w subscript 2 equals z subscript 1 z subscript 2 comma giving your answer in the form r   c i s theta.

4d
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2 marks

Sketch w subscript 1 and w subscript 2 on a single Argand diagram.

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5a
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3 marks

It is given that that z subscript 1 equals 2 e to the power of straight i open parentheses straight pi over 3 close parentheses end exponent and z subscript 2 equals 3   c i s left parenthesis fraction numerator n straight pi over denominator 12 end fraction right parenthesis comma space n element of straight integer numbers to the power of plus. 

Find the value of z subscript 1 z subscript 2for n equals 3. 

5b
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3 marks

Find the least value of n such that z subscript 1 z subscript 2 element of straight real numbers to the power of plus.

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6a
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5 marks

Consider the complex number w equals z subscript 1 over z subscript 2  where z subscript 1 equals 3 minus square root of 3 straight i and z subscript 2 equals 2   c i s open parentheses fraction numerator 2 straight pi over denominator 3 end fraction close parentheses. 

Express w in the form r   c i s theta.

6b
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3 marks

Sketch z subscript 1 comma space z subscript 2  and w on the Argand diagram below. 

q6b_1-9_ib-maths-aa-hl

6c
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2 marks

Find the smallest positive integer value of n such that w to the power of n is a real number. 

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7a
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4 marks

Consider the complex numbers w equals 3 open parentheses cos straight pi over 3 minus isin straight pi over 3 close parentheses and z equals 3 minus square root of 3 straight i

Write w and z in the form r space cis space theta, where r greater than 0 and negative straight pi less than straight theta less or equal than straight pi.

7b
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2 marks

Find the modulus and argument of z w.

7c
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2 marks

Write down the value of z w.

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8
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5 marks

Write 5 cos open parentheses 2 t plus 3 close parentheses plus 4 space cos open parentheses 2 t plus 5 close parentheses in the form A cos open parentheses 2 t plus B close parentheses where A greater than 0 comma space minus straight pi less than B less than straight pi.

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1a
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4 marks

Consider the equation z squared plus p z minus 2 p minus 1 equals 0 comma where z element of straight complex numbers comma space p element of straight real numbers.

Find the value of p for which one of the two distinct roots is z subscript 1 equals 2 plus square root of 3 straight i.

1b
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4 marks

Find the range of values of p for which the equation has two distinct, real roots.

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2a
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4 marks

Let w equals 2 minus iz, where w comma space z element of straight complex numbers.

Find w when

(i)
z equals 2 e to the power of straight pi over 2 i end exponent

(ii)
z equals square root of 2 e to the power of straight pi over 4 straight i end exponent
2b
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4 marks

On an Argand diagram the point z can be transformed to the point w by two transformations. Describe the two transformations and the order in which they are applied.

2c
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2 marks

Hence, or otherwise, find the value of z when w equals 1 plus straight i.

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3
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5 marks

Consider z equals cis space theta where z element of straight complex numbers comma space z not equal to 1.

Show that Re open parentheses fraction numerator 1 plus straight z over denominator 1 minus straight z end fraction close parentheses equals 0.

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4a
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5 marks

It is given that z subscript 1 equals 2 space cis space open parentheses straight pi over 4 close parentheses and z subscript 2 equals square root of 2 space cis open parentheses nπ over 12 close parentheses comma space straight n element of straight integer numbers to the power of plus.

Giving your answers in the form r e to the power of i theta end exponent comma space r greater or equal than 0 comma space minus pi less than theta less or equal than pi, use technology to find the values of

(i)
z subscript 1 cubed

(ii)
open parentheses z subscript 1 z subscript 2 close parentheses cubed, for n equals 2.

 

4b
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3 marks

Find the least value of n such that z subscript 1 z subscript 2 element of space straight real numbers to the power of plus.

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5a
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2 marks

Let z equals 1 plus straight i.

Express z in the form z equals a e to the power of straight i b end exponent, where a comma b element of straight real numbers , giving the exact values of a and b.

5b
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6 marks

Let w subscript 1 equals e to the power of i x end exponent and w subscript 2 equals z w subscript 1.

i)
Write w subscript 1 plus w subscript 2 in the form e to the power of straight i x end exponent open parentheses c plus i d close parentheses.
ii)
Hence, find R e open parentheses w subscript 1 plus w subscript 2 close parentheses  in the form A space cos open parentheses x plus a close parentheses,  giving the exact value of A ,where A greater than 0 and 0 less than alpha less than straight pi over 2.

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6a
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3 marks

Consider the complex numbers w subscript 1 equals z subscript 1 over z subscript 2 comma space z subscript 1 equals fraction numerator square root of 2 e to the power of negative straight pi over 3 straight i end exponent over denominator 3 end fraction and  z subscript 2 equals 2 minus 2 square root of 3 straight i.

Express

(i)
z subscript 1 in the form a plus b straight i
(ii)
z subscript 2 in the form r space cis space theta, where r greater than 0 and negative straight pi less than theta less than straight pi.
6b
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2 marks

Find the exact value of w subscript 1.

6c
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2 marks

Find w subscript 2 equals z subscript 1 z subscript 2, giving your answer in the form r space cis space theta, where r greater than 0 and negative pi less than theta less than pi.

6d
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1 mark

Without drawing an Argand diagram, describe the geometrical relationship between z subscript 1 and z subscript 2.

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7a
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5 marks

z equals fraction numerator square root of 3 over denominator 2 end fraction straight i minus 1 half

Use technology to find all the powers z to the power of n.

7b
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3 marks

Find the area of the shape made by the powers z to the power of n when plotted on an Argand diagram.

Give your answer as an exact value.

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8a
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2 marks

Let z equals cos space theta plus straight i space sin space theta.

Write down the value of z z to the power of asterisk times.

8b
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3 marks

Let z subscript 1 equals r subscript 1 e to the power of straight i theta end exponent and z subscript 2 equals r subscript 2 to the power of straight i open parentheses theta plus straight pi over 2 close parentheses end exponent

Prove the results

(i)
Re open parentheses z subscript 1 plus z subscript 2 close parentheses equals r subscript 1 cos space theta minus r subscript 2 space sin space theta

(ii)
Im open parentheses z subscript 1 plus z subscript 2 close parentheses equals r subscript 1 sin space theta plus r subscript 2 space cos space theta 

 

  

8c
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4 marks

Using technology, or otherwise, show that

Re left parenthesis 2 e to the power of straight i 5 x end exponent plus 6 e to the power of straight i open parentheses 5 x plus 1 close parentheses end exponent equals 7.28 space cos open parentheses 0.77 plus 5 x close parentheses

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9a
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2 marks

The current, I, in an AC circuit can be modelled by the equation I equals a space cos open parentheses b t minus c close parentheses where b is the frequency and c is the phase shift. 

Two AC voltage sources of the same frequency generate currents I subscript A equals 12 space cos open parentheses b t close parentheses and I subscript B equals 15 space cos open parentheses b t minus straight pi over 4 close parentheses.

Write down the maximum value and phase shift of the two currents I subscript A and I subscript B when they are each connected to the circuit alone.

9b
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5 marks

The two AC voltage sources are connected to the circuit at the same time and the total current can be expressed as I subscript A plus I subscript B.

Write down the maximum value and phase shift of I subscript A plus I subscript B.

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10a
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4 marks

The height of a wave in metres, relative to a particular boat, can be modelled by the function h open parentheses t close parentheses equals 0.5 space sin open parentheses 2 t close parentheses, where t is the time in seconds. Observers on the boat are tracking a jumping dolphin. The height of the dolphin’s jumps can be modelled by the function j open parentheses t close parentheses equals 2 space sin open parentheses 2 t minus 0.5 close parentheses.

Find an expression for the height the dolphin can reach, at time t seconds, when the height of the dolphin’s jump is affected by the height of the waves. Give your answer in the form f open parentheses t close parentheses equals A space sin open parentheses b t minus c close parentheses

10b
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3 marks

Use technology to find the time when the dolphin first reaches its maximum height and write down the maximum height the dolphin reaches.

10c
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2 marks

Find the time interval in the first two seconds when the height of dolphin will be above the wave

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1a
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5 marks

Let z equals 2 space cis space fraction numerator 3 straight pi over denominator 4 end fraction.

(i)
Find the values of z squared comma space z cubed and z to the power of 4, giving your answers in the form  a e to the power of straight i theta end exponent, where a element of straight real numbers to the power of plus and θ  is given as an exact value.

(ii)
Plot z comma space z squared comma space z to the power of 3 space end exponentand  z to the power of 4 on an Argand diagram.
1b
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3 marks

Find the exact value of a  such that successive integer powers of the complex number w equals fraction numerator z over denominator a plus straight i end fraction lie on a unit circle.

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2
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4 marks

Consider the complex numbers z and w, where z equals square root of 3 minus straight i comma space Im open parentheses straight z squared over straight w close parentheses equals 0 comma space open vertical bar straight z squared over straight w close vertical bar equals 1 half open vertical bar straight z close vertical bar. 

Use geometrical reasoning to find the two possibilities for w, giving your answers in exponential form.

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3a
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4 marks

Consider the complex numbers z subscript 1 equals 2 e to the power of straight pi over 3 straight i end exponent comma space z subscript 2 equals 3 minus z subscript 1 and z subscript 3 equals z subscript 1 over z subscript 2.

Write z subscript 1 comma space z subscript 2 and  z subscript 3 in the form a plus b straight i where a comma space b element of straight real numbers, giving exact values of a and b.

3b
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5 marks

A scale model of a triangular patio is planned by representing the vertices of the triangle on an Argand diagram as the points z subscript 1 comma space z subscript 2 spaceand z subscript 3. Find the area of the triangle in the model.

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4a
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3 marks

Consider the complex number z equals 1 plus square root of 3 straight i.

(i)
Plot the position of z on an Argand diagram.

(ii)
Express z in the form z equals a e to the power of straight i b end exponent, where a comma b element of straight real numbers, giving the exact value of aand the exact value of b.
4b
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2 marks

Use technology to find the value of square root of z cubed end root . Give your answer in the form z equals a e to the power of straight i b end exponent, where a comma b space element of straight real numbers, giving the exact value of a and the exact value of b.

4c
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2 marks

Find the smallest positive integer k such that z to the power of k is

(i)
a positive real number,

(ii)
a negative real number.
4d
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1 mark

Explain why there is no possible integer value of k such that z to the power of k is purely imaginary.

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5a
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3 marks

Given the points 1 and z on an Argand diagram, where z not equal to 0   is a complex number, explain how to find each of the following points by geometrical construction.  In each case provide a sketch to illustrate your answer.

z squared.

5b
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3 marks

open parentheses 2 minus straight i close parentheses z.

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6
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8 marks

Consider the equations u to the power of asterisk times plus 2 v equals 2 straight i and straight i u plus v to the power of asterisk times equals 3, where u comma space v element of straight complex numbers.  Find u over v giving your answer in the form r e to the power of straight i theta end exponent, where r greater than 0 and 0 less than theta less than 2 pi.

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7
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8 marks

By first expressing 1 plus square root of 3 straight i and negative 1 plus straight i in the form r space cis space theta where r greater than 0 and negative pi less than theta less or equal than pi, show that tan open parentheses fraction numerator 5 pi over denominator 12 end fraction close parentheses equals 2 plus square root of 3.

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8a
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4 marks

Two power sources are connected to a single electrical circuit. At time t seconds, the voltage, V subscript 1 , provided by the first power source is modelled by V subscript 1 equals Im open parentheses 3 straight e to the power of 12 ti end exponent close parentheses, and the voltage, V subscript 2, from the second power source can be modelled by the function V subscript 2.

The total voltage in the circuit, V subscript T equals V subscript 1 plus V subscript 2,  is given by V subscript T equals 10 space sin open parentheses 12 t plus 20 close parentheses.

Find an expression for V subscript 2 in the form A sin open parentheses B t plus C close parentheses, where A comma space B comma space C element of straight real numbers.

8b
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2 marks

Write down the maximum voltage and the phase shift provided by the second power source, giving your answers correct to 2 decimal places.

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9a
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6 marks

A popular fast-food chain is considering opening a new restaurant in an up and coming part of New York.  They have looked at competition in the area and predict that the costs C, in thousands of USD, to run the new restaurant for the first 100 days after opening could be modelled by the function

C open parentheses d close parentheses equals 30 space sin open parentheses 0.01 d minus 4.5 close parentheses plus 4 comma space space space 0 less than d less or equal than 100 comma 

where d is the number of days since opening.

The CEO of the company will only give permission for the new restaurant to open if the model predicts that the revenue will be greater than the costs by the 80th day after opening.

The revenue R, in thousands of USD, for the first 100 days is predicted to follow the model

R open parentheses d close parentheses equals a space sin open parentheses 0.01 d plus 0.1 close parentheses plus 4.5 comma space space space space space space space 0 less than d less or equal than 100 comma space space space space a element of straight natural numbers.

(i)
Find the minimum value of  that will ensure the restaurant is making a profit by day 80.

(ii)
For this value of a, show that the profits of the hotel can be modelled by the function P open parentheses d close parentheses equals A sin open parentheses 0.01 d plus b close parentheses plus c, giving the values of A comma space band c correct to four significant figures.
9b
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4 marks

According to the model found in part (a) (ii), find

(i)
the profit the restaurant is predicted to make on day 100, to the nearest thousand USD,

(ii)
the first day for which the loss is less than $20 000 USD.

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10a
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5 marks

The primary square root of a complex number z is defined as square root of z equals x plus straight i y, where x comma space y element of straight real numbers and x greater or equal than 0.  If x equals 0 then the value for y is chosen such that y greater or equal than 0.  Note that the other square root of z will then be given by negative square root of z equals negative x minus straight i y.

Show that

x equals square root of fraction numerator Re open parentheses z close parentheses plus square root of open parentheses Re open parentheses z close parentheses close parentheses squared plus open parentheses Im open parentheses z close parentheses close parentheses squared end root over denominator 2 end fraction end root
10b
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2 marks

Given that x greater than 0, derive a formula for y in terms of xand Im open parentheses z close parentheses, and explain why y in this case will always have the same sign (positive, negative, or zero) as Im open parentheses z close parentheses.

10c
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2 marks

Hence show that in general

y equals plus-or-minus square root of fraction numerator negative Re open parentheses z close parentheses plus square root of open parentheses Re open parentheses z close parentheses close parentheses squared plus open parentheses Im open parentheses z close parentheses close parentheses squared end root over denominator 2 end fraction end root


with the choice of the positive or negative value being dependent on the properties of z.

10d
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3 marks

Explain what must be true of z for each of the following to be true:

(i)
x equals 0 comma space y not equal to 0

(ii)
x not equal to 0 comma space y equals 0

(iii)
x equals 0 comma space y equals 0

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