DP IB Maths: AA HL

Topic Questions

1.9 Further Complex Numbers

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

Consider the complex number omega equals negative 1 plus 4 straight i .

Show that  omega is a root of the cubic equation

z cubed plus 5 z squared plus 23 z plus 51 equals 0
2b
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4 marks

Find the other two roots of the cubic equation in part (a).

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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

Consider the equation open parentheses z minus 2 close parentheses squared equals straight i comma space straight z element of straight complex numbers.

(i)
Verify that omega subscript 1 equals 2 plus e to the power of straight i straight pi over 4 end exponent is a root of this equation.
(ii)
Find the second root of the equation, expressing your answer in the form omega subscript 2 equals a plus e to the power of straight i theta end exponent where  a element of straight real numbers and  theta greater than 0.
4b
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3 marks

The roots omega subscript 1 and omega subscript 2 are represented by the points A and B respectively on an Argand diagram.

Find AB.

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5
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8 marks

Consider the equation z to the power of 4 plus open parentheses 1 minus 4 straight i close parentheses z squared minus 4 straight i equals 0, 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.

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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

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.

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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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5 marks

Let z subscript 1 equals r subscript 1 open parentheses cos space theta subscript 1 plus straight i space sin space theta subscript 1 close parentheses space a n d space z subscript 2 equals r subscript 2 open parentheses cos theta subscript 2 plus straight i space sin theta subscript 2 close parentheses.

Prove the results

(i)
open vertical bar z subscript 1 over z subscript 2 close vertical bar equals open vertical bar z subscript 1 over z subscript 2 close vertical bar
(ii)
arg straight z subscript 1 over straight z subscript 2 equals arg space z subscript 1 minus arg space z subscript 2
8c
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1 mark

Using the results from part (b), describe fully the geometrical interpretation of dividing z subscript 1 by z subscript 2.

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9a
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3 marks

Sketch omega subscript 1 comma space omega subscript 2 space and space omega subscript 3 on the Argand diagram below.

q9b-1-9-ib-aa-hl-further-complex-numbers-hard-maths-dig

 

9b
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4 marks

omega subscript 1 comma space omega subscript 2 spaceand omega subscript 3 represent the vertices of a triangle.

Find the area of the triangle.

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10a
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5 marks

The complex numbers z subscript 1 equals a comma space z subscript 2 equals 3 minus 2 straight i and z subscript 3 are roots of the cubic equation z cubed plus p z squared plus q z minus 26 equals 0 comma where a comma space p comma space q element of straight real numbers

Find the values of a comma space p and q.

10b
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3 marks

Express z subscript 1 comma space z subscript 2 and z subscript 3 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 or equal than 2 straight pi.

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11
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8 marks

Let  f open parentheses z close parentheses equals z to the power of 4 plus a z cubed plus 6 z squared plus b z plus 65,  where a and b are real constants.

Given that  z equals 3 plus 2 straight i  is a root of the equation f open parentheses z close parentheses equals 0,  show the roots f open parentheses z close parentheses equals 0 on the Argand diagram below.

q11-1-9-ib-aa-hl-further-complex-numbers-hard-maths-dig

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1a
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3 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 spacein the form w equals a plus b straight i.

1b
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4 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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3 marks

Express w 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.

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2
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6 marks

Solve the equation z cubed equals 27 straight i, giving your answers in the form a plus b straight i.

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3a
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4 marks

Let z subscript 1 equals 6   cis 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 cis theta comma find 

(i)
z subscript 1 z subscript 2

(ii)
z subscript 1 over z subscript 2.
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   cis left parenthesis straight pi over 3 right parenthesis and begin mathsize 16px style z subscript 2 equals 2 plus 2 straight i end style.

Express

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

(ii)
z subscript 2 in the form r   cis 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   cis 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 straight e to the power of straight i open parentheses straight pi over 3 close parentheses end exponent and z subscript 2 equals 3   cis open parentheses nπ over 12 close parentheses comma space n element of straight integer numbers to the power of plus. 

Find the value of  z subscript 1 z subscript 2 for  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 number z equals negative 1 plus square root of 3 straight i

Express 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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4 marks

Find the three roots of the equation z cubed equals negative 1 plus square root of 3 straight i, expressing your answers 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.

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8a
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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 b element of straight real numbers.

8b
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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
8c
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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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9a
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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.

9b
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2 marks

Find the modulus and argument of z w.

9c
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2 marks

Write down the value of z w.

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10a
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2 marks

Let z equals 12 plus 16 straight i, where a comma b element of straight real numbers.

Verify that 4 plus 2 straight i and negative 4 minus 2 straight i are the second roots of z.

10b
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4 marks

Hence, or otherwise, find two distinct roots of the equation w squared plus 4 w plus left parenthesis 1 minus 4 straight i right parenthesis equals 0, where w element of straight complex numbers. Give your answer in the form a plus b straight i, where a comma b element of straight real numbers.

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11a
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1 mark

The complex numbers omega subscript 1 equals 3 and omega subscript 2 equals 2 minus 2 straight i are roots of the cubic equation omega cubed plus p omega squared plus q omega plus r equals 0 comma where space p comma space q comma space r element of straight real numbers.

Write down the third root, w subscript 3, of the equation.

11b
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4 marks

Find the values of space p comma space qand r.

11c
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4 marks

Express w subscript 1 comma w subscript 2 and w subscript 3 in the form r   c i s theta.

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1a
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6 marks

Consider the equation  p z cubed plus q z squared plus 8 p cubed z plus 5 q equals 0,  where z element of straight complex numbers comma space p element of straight real numbers

Given that one of the distinct roots is z subscript 1 equals 5 over 2, find

(i)
the values of p and q
(ii)
the roots z subscript 1 and z subscript 2, giving your answers in the form a plus b straight i, where a comma b space element of straight real numbers.
1b
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2 marks

On an Argand diagram z subscript 1 comma space z subscript 2 and z subscript 3 are represented by the points A, B and C respectively.

Find the area of the triangle ABC.

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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 iIm 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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3
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8 marks

Consider the equation z to the power of 4 minus 5 a z cubed plus 25 a z squared minus 20 a b z plus 24 a b equals 0,  where a comma space b element of straight integer numbers and  z element of straight complex numbers

Given that one root is a plus a straight i and another root is b plus b straight i, find the possible values of a spaceand b.

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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.

Use De Moivre’s theorem to find the value of z cubed.

4b
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5 marks

Use the principle of mathematical induction to prove, for all n element of straight natural numbers, that

open parentheses cos space theta plus straight i space sin space theta close parentheses to the power of n equals cos space n theta plus straight i space sin space n theta
4c
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2 marks

Show that the result in part (b) is true for all n element of straight integer numbers.

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5a
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5 marks

Consider the equation open parentheses z plus a close parentheses to the power of 5 plus 1 equals 0 comma space z element of straight complex numbers.

Given that the product of the roots is 31, find the roots of the equation, expressing your answers in the form omega subscript n equals b plus e to the power of straight i theta end exponent, where b element of straight real numbers and theta greater than 0.

5b
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3 marks

Let S be the sum of the roots found in part (a).

Show that Im open parentheses straight S close parentheses equals 0 and find the value of Re open parentheses straight S close parentheses .

5c
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1 mark

The roots omega subscript 1 comma space omega subscript 2 comma... comma space omega subscript 5 are represented on an Argand diagram. 

Describe the geometrical shape made by the five roots.

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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 fraction numerator 5 pi over denominator 12 end fraction equals 2 plus square root of 3.

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8a
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5 marks

Consider the complex number z equals 1 plus cos space 2 theta plus straight i space sin space 2 theta.

Find the modulus and argument of z.

8b
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2 marks

Solve z equals 0 for negative pi less than theta less or equal than pi.

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9a
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4 marks

Let z equals e to the power of straight i theta end exponent.

Show that

(i)
z plus z squared plus z cubed plus midline horizontal ellipsis plus z to the power of n equals cos space theta plus cos space 2 theta plus midline horizontal ellipsis plus cos space n theta plus straight i open parentheses sin space theta plus sin space 2 theta plus midline horizontal ellipsis plus sin space n theta close parentheses,
(ii)
open parentheses 2 minus z close parentheses open parentheses 2 minus z to the power of asterisk times close parentheses equals 5 minus 4 space cos space theta.
9b
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4 marks

Use the results found in part (a) to find the sum of the infinite series

fraction numerator sin space theta over denominator 2 end fraction plus fraction numerator sin space 2 theta over denominator 2 squared end fraction plus fraction numerator sin space 3 theta over denominator 2 cubed end fraction plus fraction numerator sin space 4 theta over denominator 2 to the power of 4 end fraction plus midline horizontal ellipsis

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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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