Argand Diagrams (CIE A Level Maths: Pure 3)

Exam Questions

4 hours36 questions
1a
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2 marks

z subscript 1 equals 3 plus 4 i  and  z subscript 2 equals 5 minus 3 i.

Work out the values of z subscript 1 plus z subscript 2  and  z subscript 1 minus z subscript 2.

1b
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3 marks

An Argand diagram is a way to represent complex numbers as points or vectors in two dimensional space.  An Argand diagram is based around a standard set of x comma y Cartesian coordinate axes, with the real axis replacing the x-axis and the imaginary axis replacing the y-axis.

A complex number z given in a plus b i form, where a and b are real numbers, may be represented on an Argand diagram as a point with coordinates left parenthesis a comma b right parenthesis.

Show the complex numbers z subscript 1 comma space z subscript 2 comma space z subscript 1 plus z subscript 2 and z subscript 1 minus z subscript 2 as points on an Argand diagram.

1c
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3 marks

A complex number z given in a plus b i form, where a and b are real numbers, may also be represented on an Argand diagram by a position vector connecting the origin to the point with coordinates left parenthesis a comma b right parenthesis.

On an Argand diagram, show the complex numbers z subscript 1 comma space z subscript 2 comma space z subscript 1 plus z subscript 2and z subscript 1 minus z subscript 2 in position vector form.

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2
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4 marks

The modulus of a complex number z is the distance of the point z from 0 (i.e. from the origin) in an Argand diagram.

For a complex number z given in a plus b i form, where a and b are real numbers, the modulus of z may be calculated by using the formula

 vertical line z vertical line equals square root of a squared plus b squared end root

Calculate the modulus of each of the following, giving your answers as exact values:

space left parenthesis straight i right parenthesis space space 3 minus 4 i space space space space space space space left parenthesis ii right parenthesis space space 1 plus i space space space space space space space left parenthesis iii right parenthesis space space minus 7 i space space space space space space space space space space space space space space left parenthesis iv right parenthesis space space minus 24 over 25 plus 7 over 25 space straight i
left parenthesis straight v right parenthesis space space 2 space space space space space space space space space space space space space space left parenthesis vi right parenthesis space space 23 i space space space space space space space left parenthesis vii right parenthesis space space minus 5 minus 7 i space space space space space space space left parenthesis viii right parenthesis space space minus 15

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3a
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2 marks

The argument of a complex number z is the angle in an Argand diagram measured (in radians!) from the positive real axis to the position vector of z. The positive direction is defined to be anticlockwise from the positive real axis (so a negative argument means a clockwise measurement). 

The following diagram summarises the method for finding the principal argument of a complex number, i.e. the argument that lies in the interval  negative pi less than a r g space z less or equal than pi:

q3a-8-2-easy-cie-a-level-maths-pure

Write down the principal arguments of

left parenthesis straight i right parenthesis space space 5 straight i space space space space space space space left parenthesis ii right parenthesis space space 3 space space space space space space space left parenthesis iii right parenthesis space space minus 7 space space space space space space space left parenthesis iv right parenthesis space space minus 4 straight i

3b
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4 marks

Find the principal arguments of the following complex numbers, giving your answers as exact values:

left parenthesis straight i right parenthesis space space 2 plus 2 straight i space space space space space space space space space left parenthesis ii right parenthesis space space square root of 3 minus straight i end root space space space space space space space space space left parenthesis iii right parenthesis space space minus 4 plus 4 square root of 3 straight i space space space space space space space left parenthesis iv right parenthesis space space minus 7 over 2 minus 7 over 2 space straight i

3c
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4 marks

Find the principal arguments of the following complex numbers, giving your answers correct to 3 significant figures:

left parenthesis straight i right parenthesis space space 3 plus 4 straight i space space space space space space space left parenthesis ii right parenthesis space space minus 8 minus 15 straight i space space space space space space space left parenthesis iii right parenthesis space space 12 minus 5 space straight i space space space space space space space left parenthesis iv right parenthesis space space minus 7 over 25 plus 24 over 25 space straight i

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4a
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1 mark

The complex number z is such that  z equals left parenthesis 7 minus p right parenthesis plus left parenthesis 1 plus 7 p right parenthesis i,  where  p  is a real number constant.

Show that  vertical line z vertical line equals square root of 50 plus 50 p squared end root.

4b
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2 marks

Given that vertical line z vertical line equals 10,  find the two possible values of p.

4c
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2 marks

For each of the values of p found in part (b), find the principal argument of the corresponding version of  z.  Give your answers correct to 3 significant figures.

4d
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1 mark

Given additionally that  z lies in the fourth quadrant of the Argand diagram, find the precise value of  p.

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5a
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3 marks

It is possible to express a complex number z in modulus-argument form,  i.e. in the form

straight z equals straight r left parenthesis cos space straight theta plus straight i space sin space straight theta space right parenthesis

where r equals vertical line z vertical line and  theta equals arg space z.

Express the complex number negative 5 plus 12 i  in modulus-argument form.  The value for theta should be in radians correct to 3 significant figures, and should be given in the interval negative pi less than theta less or equal than pi.

5b
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2 marks

A complex number z has modulus 14 and argument negative pi over 6.  Using the fact that

r left parenthesis cos space theta plus i space sin space theta space right parenthesis equals left parenthesis r space cos space theta space right parenthesis plus left parenthesis r space sin space theta space right parenthesis i

express z in the form  a plus b i,  where a and  b are real numbers given as exact values.

5c
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3 marks

By considering the definition of modulus-argument form above, explain why the following numbers are not in modulus-argument form:

(i)
z subscript 1 equals negative 4 open parentheses cos pi over 4 plus straight i space sin pi over 4 close parentheses
(ii)
z subscript 2 equals 5 open parentheses cos pi over 3 minus straight i space sin pi over 3 close parentheses
(iii)
z subscript 3 equals 7 open parentheses cos pi over 6 plus straight i space sin open parentheses negative pi over 6 close parentheses close parentheses

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6
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4 marks

When multiplying and dividing two complex numbers z subscript 1 and  z subscript 2,  the moduli and arguments of the numbers are connected by the following sets of relationships:

vertical line z subscript 1 z subscript 2 vertical line equals vertical line z subscript 1 vertical line vertical line z subscript 2 vertical line space space space space space and space space space space space arg left parenthesis z subscript 1 z subscript 2 right parenthesis equals arg left parenthesis z subscript 1 right parenthesis plus arg left parenthesis z subscript 2 right parenthesis

open vertical bar z subscript 1 over z subscript 2 close vertical bar equals fraction numerator vertical line z subscript 1 vertical line over denominator vertical line z subscript 2 vertical line end fraction space space space space space space space and space space space space space space arg space open parentheses z subscript 1 over z subscript 2 close parentheses equals arg space left parenthesis z subscript 1 right parenthesis minus arg space left parenthesis z subscript 2 right parenthesis

When the numbers are given in modulus-argument form as z subscript 1 equals r subscript 1 left parenthesis cos theta subscript 1 plus space straight i space sin theta subscript 1 right parenthesis and  z subscript 2 equals r subscript 2 left parenthesis cos theta subscript 2 plus straight i space sin theta subscript 2 right parenthesis, where r subscript 1 equals vertical line z subscript 1 vertical line comma space r subscript 2 equals vertical line z subscript 2 vertical line comma space theta subscript 1 equals arg space straight z subscript 1 space space space space space space space space and space space space space space space space space space straight theta subscript 2 equals arg space straight z subscript 2,  these relationships mean that

z subscript 1 z subscript 2 equals r subscript 1 r subscript 2 left parenthesis cos left parenthesis theta subscript 1 plus theta subscript 2 right parenthesis plus i space sin left parenthesis theta subscript 1 plus theta subscript 2 right parenthesis right parenthesis

z subscript 1 over z subscript 2 equals r subscript 1 over r subscript 2 left parenthesis cos left parenthesis theta subscript 1 minus theta subscript 2 right parenthesis plus straight i space sin left parenthesis theta subscript 1 minus theta subscript 2 right parenthesis right parenthesis

Use those results to work out the following, giving your answers in modulus-argument form:

(i)
5 open parentheses cos pi over 3 plus straight i space sin pi over 3 close parentheses cross times 4 open parentheses cos pi over 2 plus straight i space sin space pi over 2 close parentheses

(ii)
12 open parentheses cos pi over 2 plus straight i space sin straight pi over 2 close parentheses divided by 3 open parentheses cos space fraction numerator 3 pi over denominator 4 end fraction plus straight i space sin space fraction numerator 3 pi over denominator 4 end fraction close parentheses

(iii)
square root of 3 space open parentheses cos pi over 6 plus straight i space sin pi over 6 close parentheses cross times square root of 6 open parentheses cos open parentheses negative fraction numerator 3 pi over denominator 4 end fraction close parentheses plus straight i space sin open parentheses negative fraction numerator 3 pi over denominator 4 end fraction space close parentheses close parentheses

(iv)
fraction numerator 6.25 left parenthesis cos space 3.05 plus straight i space sin space 3.05 space right parenthesis over denominator 0.5 left parenthesis cos space 1.42 plus straight i space sin space 1.42 space right parenthesis end fraction

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7a
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3 marks

If the complex number  z subscript 1  is represented by a fixed point in an Argand diagram, then the value of the modulus  vertical line z minus z subscript 1 vertical line  gives the distance in the Argand diagram from  z subscript 1  to any other complex number  z.

It follows from this that the locus (i.e., set of points) in an Argand diagram for which an equation of the form

vertical line z minus z subscript 1 vertical line equals k

is true (where k greater or equal than 0  is a real number constant), is a circle of radius k with its centre at  z subscript 1.

On an Argand diagram, sketch the locus for which the equation

vertical line z minus left parenthesis 1 minus 3 i right parenthesis vertical line equals 4

is true.

7b
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2 marks

Use your sketch from part (a) to write down a complex number z  that satisfies each of the following inequalities:

(i)
vertical line z minus left parenthesis 1 minus 3 i right parenthesis vertical line less than 4
(ii)
vertical line z minus left parenthesis 1 minus 3 i right parenthesis vertical line greater than 4

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8a
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3 marks

If the complex number  z subscript 1  is represented by a fixed point in an Argand diagram, then the value of the modulus  vertical line z minus z subscript 1 vertical line  gives the distance in the Argand diagram from  z subscript 1  to any other complex number  z.

It follows from this that the locus (i.e., set of points) in an Argand diagram for which an equation of the form

vertical line z minus z subscript 1 vertical line equals vertical line z minus z subscript 2 vertical line

is true, is the perpendicular bisector of the line segment connecting the points z subscript 1 and  z subscript 2.

On an Argand diagram, sketch the locus for which the equation

vertical line z minus left parenthesis 3 plus 5 i right parenthesis vertical line equals vertical line z minus left parenthesis negative 3 minus i right parenthesis vertical line

is true.

8b
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2 marks

Use your sketch from part (a) to write down a complex number z  that satisfies each of the following inequalities:

(i)
vertical line z minus left parenthesis 3 plus 5 i right parenthesis vertical line less than vertical line z minus left parenthesis negative 3 minus i right parenthesis vertical line
(ii)
vertical line z minus left parenthesis 3 plus 5 i right parenthesis vertical line greater than vertical line z minus left parenthesis negative 3 minus i right parenthesis vertical line

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9a
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3 marks

If the complex number  z subscript 1  is represented by a fixed point in an Argand diagram, then the value of  arg left parenthesis z minus z subscript 1 right parenthesis  gives the angle measured (in radians!) from the half-line that starts at  z subscript 1 and goes right in the same direction as the positive real axis, to the line segment connecting z subscript 1 and z:

q9a-8-2-easy-cie-a-level-maths-pure

As with the argument of a complex number, the positive direction is defined to be anticlockwise from the line parallel to the positive real axis.

It follows from this that the locus (i.e., set of points) in an Argand diagram for which an equation of the form

arg left parenthesis z minus z subscript 1 right parenthesis equals k

is true (where k is a real number constant), is the half-line starting at  z subscript 1  and going off at a direction that makes an angle of k   radians measured anticlockwise from the line going to the right from  z subscript 1  parallel to the positive real axis.  Note that  z subscript 1  is not included in the locus.  If the value of  k  is negative, it means that the angle is measured clockwise instead.

On an Argand diagram, sketch the locus for which the equation

arg space left parenthesis z minus left parenthesis negative 2 plus i right parenthesis right parenthesis equals pi over 4

is true.

9b
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2 marks

Use your sketch from part (a) to write down a complex number z  that satisfies each of the following inequalities:

(i)  0 less than arg left parenthesis z minus left parenthesis negative 2 plus i right parenthesis right parenthesis less than pi over 4              (ii)   pi over 4 less than arg left parenthesis z minus left parenthesis negative 2 plus i right parenthesis right parenthesis less than pi over 2

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1a
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3 marks

The solutions to the quadratic equation z squared minus 8 z plus 25 equals 0 are z subscript 1 and z subscript 2.

Work out the values of z subscript 1 and z subscript 2, giving your answers in the form p plus-or-minus q i where p and q are integers.

1b
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3 marks

On an Argand diagram, show the complex numbers z subscript 1 comma space z subscript 2 comma space z subscript 1 plus z subscript 2 and z subscript 1 minus z subscript 2 in position vector form.

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2a
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2 marks

z equals 4 minus 3 i

Show that z squared equals 7 minus 24 i

2b
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4 marks

Find, showing your working:

(i)
vertical line z squared vertical line
(ii)
arg left parenthesis z squared right parenthesis, giving your answer in radians to 2 decimal places.
2c
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1 mark

Show z and z squared on an Argand diagram.

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3a
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2 marks

The complex numbers z subscript 1 and z subscript 2 are such that z subscript 2 equals negative 2 plus p i and z subscript 1 over z subscript 2 equals 1 minus 3 i.

Show that z subscript 1 equals left parenthesis 3 p minus 2 right parenthesis plus left parenthesis p plus 6 right parenthesis i.

3b
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3 marks

Given that vertical line z subscript 1 vertical line equals square root of 130,  find the two possible values of p.

3c
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2 marks

Given additionally that arg left parenthesis z subscript 1 right parenthesis equals tan to the power of negative 1 end exponent open parentheses begin inline style 9 over 7 end style close parentheses,  find the precise value of p.

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4a
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3 marks

Express the complex number negative 8 plus 15 i in the form  r left parenthesis cos theta plus i space sin theta space right parenthesis,  where r is a positive real number and theta is given in radians correct to 2 decimal places.  The value for theta should be given in the interval  negative pi less than theta less or equal than pi.

4b
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2 marks

A complex number has modulus 8 and argument  straight pi over 3.  Express the number in the form  a plus b i, where a and b are real numbers.

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5a
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4 marks

table row z equals cell 12 minus 5 i space space end cell row w equals cell 3 plus 4 i space end cell end table

(i)
Find  z w, giving your answer in the form  a plus b i  where a and b are real numbers.
(ii)
Calculate the modulus and argument of each of the complex numbers z comma space w spaceand z w, and show that these satisfy the standard results

vertical line z subscript 1 z subscript 2 vertical line equals vertical line z subscript 1 vertical line vertical line z subscript 2 vertical line space space space space space and space space space space space space arg left parenthesis z subscript 1 z subscript 2 right parenthesis equals arg left parenthesis z subscript 1 right parenthesis plus arg left parenthesis z subscript 2 right parenthesis.

 

5b
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3 marks
(i)
Using your answers to part (a) and the standard results
open vertical bar z subscript 1 over z subscript 2 close vertical bar equals fraction numerator vertical line z subscript 1 vertical line over denominator vertical line z subscript 2 vertical line end fraction space space space space space space space space space space and space space space space space space space space arg space open parentheses z subscript 1 over z subscript 2 close parentheses equals arg left parenthesis z subscript 1 right parenthesis minus arg left parenthesis z subscript 2 right parenthesis

       calculate the values of open vertical bar z over w close vertical bar and arg open parentheses straight z over straight w close parentheses.

(ii)
Hence express z over w  in the form  r left parenthesis cos space theta plus i space sin space theta space right parenthesis,  where r is a positive real number and theta is given in radians correct to 2 decimal places.

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6
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4 marks

z subscript 1 equals 12 open parentheses cos pi over 6 plus straight i space sin straight pi over 6 close parentheses

z subscript 2 equals 3 open parentheses cos fraction numerator 5 pi over denominator 12 end fraction plus straight i space sin space fraction numerator 5 straight pi over denominator 12 end fraction close parentheses

Work out

(i)
z subscript 1 z subscript 2

(ii)
z subscript 1 over z subscript 2

giving your answers in modulus-argument form.

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7
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4 marks

Given that  vertical line z minus 3 vertical line equals vertical line z minus 4 i vertical line:

(i)
On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.
(ii)
Shade the region of your diagram that satisfies the inequality vertical line z minus 3 vertical line greater or equal than vertical line z minus 4 i vertical line.

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8
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4 marks

Given that arg left parenthesis z plus 1 right parenthesis equals pi over 4:

(i)
On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.
(ii)
Shade the region of your diagram that satisfies the inequality  0 less or equal than arg left parenthesis z plus 1 right parenthesis less or equal than pi over 4.

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9a
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4 marks

f left parenthesis z right parenthesis equals z cubed minus 5 z squared plus 3 z minus 119.

Show that 7 and negative 1 plus 4 i are roots of the cubic equation f left parenthesis z right parenthesis equals 0, and write down the third root of the equation.

9b
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2 marks

Verify that there is a constant c such that all three roots of the cubic equation f left parenthesis z right parenthesis equals 0 satisfy

vertical line z minus 2 vertical line equals c.

9c
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4 marks
(i)
Draw an Argand diagram showing the locus of points representing all complex numbers z for which vertical line z minus 2 vertical line equals c.
Mark the points corresponding to the three roots of the cubic equation f left parenthesis z right parenthesis equals 0.
(ii)
Shade the region of your diagram that satisfies the inequality vertical line z minus 2 vertical line less or equal than c.

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1a
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4 marks

The solutions to the cubic equation z cubed minus 11 z squared plus 36 z minus 26 equals 0 are z subscript 1 comma space z subscript 2 and z subscript 3, where  z subscript 3 element of straight real numbers.

Given that z subscript 1 equals 5 plus i, work out the other two solutions to the equation.

1b
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4 marks
(i)
On an Argand diagram, show the complex numbers z subscript 1 comma space z subscript 2 comma negative 3 z subscript 3 and z subscript 1 minus z subscript 2 as position vectors.
(ii)
Describe the geometric transformation that maps z subscript 1 to z subscript 2.

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2
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6 marks

z equals negative 2 plus 15 over 4 i

Find, showing your working:

(i)
z squared
(ii)
vertical line z squared vertical line
(iii)
arg left parenthesis z squared right parenthesis, giving your answer in radians to 2 decimal places.

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3a
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3 marks

The complex numbers z subscript 1 and z subscript 2 are such that z subscript 1 equals 5 plus p i and  z subscript 1 over z subscript 2 equals negative 1 plus i.

Find z subscript 2 in the form  a plus b i, giving the real numbers a and b in terms of p.

3b
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3 marks

Given that vertical line z subscript 2 vertical line equals square root of 73,  find the possible values of p.

3c
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2 marks

Given additionally that arg left parenthesis z subscript 2 right parenthesis equals 2.78  in radians to 2 decimal places, determine the exact value of  Im left parenthesis z subscript 2 right parenthesis.

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4a
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3 marks

Express the complex number negative 4 minus 15 over 2 space i  in the form  r left parenthesis cos space theta plus i space sin space theta space right parenthesis,  where r is a positive real number and theta is given in radians correct to 2 decimal places.  The value for theta should be given in the interval  negative pi less than theta less or equal than pi.

4b
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2 marks

A complex number has modulus square root of 12 and argument  negative fraction numerator 5 pi over denominator 6 end fraction. Express the number in the form  a plus b i, where a and b are real numbers.

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5
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4 marks

For the general complex numbers

z subscript 1 equals r subscript 1 left parenthesis cos space theta subscript 1 plus straight i space sin space theta subscript 1 right parenthesis space space space space space space space and space space space space space space space z subscript 2 equals r subscript 2 left parenthesis cos space theta subscript 2 plus straight i space sin space theta subscript 2 right parenthesis

given in modulus-argument form, use algebra and the appropriate trigonometric compound angle formulae to prove the results

vertical line z subscript 1 z subscript 2 vertical line equals vertical line z subscript 1 vertical line vertical line z subscript 2 vertical line space space space space space space space space space space and space space space space space space space space arg left parenthesis z subscript 1 z subscript 2 right parenthesis equals arg left parenthesis z subscript 1 right parenthesis plus arg left parenthesis z subscript 2 right parenthesis.

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6
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4 marks

z subscript 1 equals 9 open parentheses cos pi over 6 plus straight i space sin pi over 6 close parentheses

z subscript 2 equals 4 open parentheses cos fraction numerator 4 straight pi over denominator 3 end fraction space plus straight i space sin space fraction numerator 4 pi over denominator 3 end fraction close parentheses 

Work out 

(i)
z subscript 1 z subscript 2
(ii)
z subscript 1 over z subscript 2

giving your answers in modulus-argument form with theta in the interval  negative pi less than theta less or equal than pi.

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7
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4 marks

Given that vertical line z plus 1 minus 2 i vertical line equals vertical line z minus 7 plus 4 i vertical line:

(i)
On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.
(ii)
Shade the region of your diagram that satisfies the inequality vertical line z plus 1 minus 2 i vertical line greater than vertical line z minus 7 plus 4 i vertical line                                     .

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8a
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4 marks

On an Argand diagram, sketch the loci (i.e., sets of points) for which each of the following equations is true:

(i)
arg left parenthesis z plus 2 minus 2 i right parenthesis equals pi over 4
(ii)
vertical line z minus 3 minus 2 i vertical line equals 5
8b
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2 marks

Shade the region of your diagram that satisfies both of the following inequalities:

0 less or equal than arg left parenthesis z plus 2 minus 2 i right parenthesis less or equal than pi over 4 space space space space space and space space space space vertical line z minus 3 minus 2 i vertical line less or equal than 5

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9a
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4 marks

You are given that z equals negative 7 satisfies the cubic equation

z cubed plus 9 z squared plus 27 z plus 91 equals 0

Find the other two roots of the cubic equation.

9b
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2 marks

Verify that there is a constant c such that all three roots of the cubic equation satisfy

vertical line z plus 3 vertical line equals c.

9c
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3 marks

Draw an Argand diagram showing the locus of points representing all complex numbers z for which  vertical line z plus 3 vertical line equals c.

Mark the points corresponding to the three roots of the cubic equation.

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1a
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4 marks

f left parenthesis z right parenthesis equals 2 z cubed minus 13 z squared plus 60 z minus 100

Given that 5 over 2 is a root of the equation  f left parenthesis z right parenthesis equals 0,  use algebra to solve f left parenthesis z right parenthesis equals 0 completely.

1b
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2 marks

Show all three solutions on an Argand diagram.

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2
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7 marks

For a general complex number z equals x plus i y,  where x comma space y element of straight real numbers and  z not equal to 0:

(i)
Show that begin inline style 1 over z to the power of asterisk times end style equals begin inline style fraction numerator z over denominator vertical line z vertical line squared end fraction end style
(ii)
Show that arg open parentheses begin inline style fraction numerator 1 over denominator straight z asterisk times end fraction end style close parentheses equals negative arg left parenthesis z to the power of asterisk times right parenthesis.
(iii)
Given that x less than 0, write down in radians the principal argument of  z plus z to the power of asterisk times.

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3a
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6 marks

The complex numbers z subscript 1 and z subscript 2 are such that z subscript 1 equals 1 plus p i and  z subscript 1 over z subscript 2 equals negative 2 plus i.

Given that vertical line z subscript 2 vertical line equals square root of 58, find the possible values of p.  Be sure to show clear algebraic working.

3b
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2 marks

Given additionally that arg left parenthesis z subscript 2 right parenthesis equals 2.09  in radians to 2 decimal places, find the exact value of Im left parenthesis z subscript 2 superscript asterisk times right parenthesis.

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4a
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3 marks

z equals left parenthesis square root of 3 minus 3 right parenthesis minus left parenthesis square root of 3 plus 3 right parenthesis i

Express z in the form  r left parenthesis cos space theta plus i space sin space theta space right parenthesis, where r is a positive real number and theta is given as an exact value in radians.  The value for theta should be given in the interval  negative pi less than theta less or equal than pi.

4b
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2 marks

A complex number w is such that vertical line w vertical line equals square root of 2 space space vertical line z vertical line and a r g left parenthesis w right parenthesis equals 2 left parenthesis pi plus a r g left parenthesis z right parenthesis space right parenthesis.

Express w in the form  a plus b i, where a and b are real numbers.

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5
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5 marks

For the general complex numbers

z subscript 1 equals r subscript 1 left parenthesis cos space theta subscript 1 plus straight i space sin space theta subscript 1 right parenthesis space space space space space space space space space space space space and space space space space space space space space space space z subscript 2 equals r subscript 2 left parenthesis cos space theta subscript 2 plus straight i space sin space theta subscript 2 right parenthesis

given in modulus-argument form, where  z subscript 2 not equal to 0,  use algebra and the appropriate trigonometric compound angle formulae to prove the results

open vertical bar z subscript 1 over z subscript 2 close vertical bar equals fraction numerator vertical line z subscript 1 vertical line over denominator vertical line z subscript 2 vertical line end fraction space space space space space space space space space space space and space space space space space space space space space space arg open parentheses straight z subscript 1 over straight z subscript 2 close parentheses equals arg left parenthesis z subscript 1 right parenthesis minus arg left parenthesis z subscript 2 right parenthesis.

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6a
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3 marks

z subscript 1 equals negative 4 open parentheses cos pi over 3 divided by 3 minus straight i space sin straight pi over 3 divided by 3 close parentheses

z subscript 2 equals 2 open parentheses cos fraction numerator 2 pi over denominator 3 end fraction divided by 3 minus straight i space sin fraction numerator 2 pi over denominator 3 end fraction close parentheses

Re-express z subscript 1 and z subscript 2 in correct modulus-argument form with theta in the interval  negative pi less than theta less or equal than pi.

6b
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4 marks

Work out

(i)
z subscript 1 z subscript 2
(ii)
z subscript 2 over z subscript 1

giving your answers in modulus-argument form with theta in the interval  negative pi less than theta less or equal than pi.

6c
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4 marks
(i)
On an Argand diagram, show the complex numbers  z subscript 1 comma space z subscript 2 and z subscript 1 plus z subscript 2 as position vectors.
(ii)
Use your diagram to explain briefly why we speak of a ‘parallelogram rule’ for complex number addition.

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7
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5 marks

On an Argand diagram, shade the region which satisfies both of the following inequalities:

vertical line z minus 4 i vertical line greater than vertical line z vertical line space space space space space and space space space space vertical line z minus 4 i vertical line less than vertical line z plus 2 vertical line

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8
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7 marks

On an Argand diagram, shade the region which satisfies all three of the following inequalities:

vertical line z plus 4 minus 3 i vertical line less or equal than 3 space space
fraction numerator 3 pi over denominator 4 end fraction less or equal than arg left parenthesis z plus 1 minus 3 i right parenthesis less or equal than pi space space
vertical line z plus 8 vertical line greater than vertical line z plus 4 vertical line

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9a
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5 marks

f left parenthesis z right parenthesis equals z cubed plus left parenthesis 5 minus 7 i right parenthesis z squared minus left parenthesis 36 plus 30 i right parenthesis z minus left parenthesis 150 minus 30 i right parenthesis

Given that

z cubed plus left parenthesis 5 minus 7 i right parenthesis z squared minus left parenthesis 36 plus 30 i right parenthesis z minus left parenthesis 150 minus 30 i right parenthesis equals left parenthesis z plus left parenthesis 5 minus i right parenthesis right parenthesis left parenthesis z squared minus 6 i z minus 30 right parenthesis

find all three roots of the cubic equation f left parenthesis z right parenthesis equals 0. Be sure to show clear algebraic working.

9b
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2 marks

Verify that there is a constant c such that all three roots of the cubic equation f left parenthesis z right parenthesis equals 0 satisfy

vertical line z minus i vertical line equals c

9c
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3 marks

Draw an Argand diagram showing the locus of points representing all complex numbers z for which vertical line z minus i vertical line equals c.

Mark the points corresponding to the three roots of the cubic equation  f left parenthesis z right parenthesis equals 0.

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