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Complex Numbers, EXT2 N2 2024 HSC 14c

For the complex numbers \(z\) and \(w\), it is known that  \(\arg \left(\dfrac{z}{w}\right)=-\dfrac{\pi}{2}\).  

Find \(\left|\dfrac{z-w}{z+w}\right|\).   (2 marks)

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\(\abs{\dfrac{z-w}{z+w}}=1\)

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\(\arg \left(\dfrac{z}{w}\right)=\arg \, z-\arg \, w=-\dfrac{\pi}{2}\)

\(\text{Graphically, \(\arg \, z\) is a \(90^{\circ}\) anticlockwise rotation from \(\arg \, w\).}\)

♦ Mean mark 49%.

\(\abs{z-w}=\abs{z+w} \ \text{(diagonals of rectangle)}\)

\(\therefore \abs{\dfrac{z-w}{z+w}}=1\)

Complex Numbers, EXT2 N2 2018 HSC 11d

The points `A`, `B` and `C` on the Argand diagram represent the complex numbers `u`, `v` and `w` respectively.

The points `O`, `A`, `B` and `C` form a square as shown on the diagram.
 

 
It is given that  `u = 5 + 2i`.

  1.  Find  `w`.  (1 mark)

  2.  Find  `v`.  (1 mark)

  3.  Find  `text(arg)(w/v)`.  (1 mark)

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  1. `−2 + 5i`
  2. `3 + 7i`
  3. `pi/4`
Show Worked Solution
i.    `w` `= iu`
    `= i(5 + 2i)`
    `= −2 + 5i`

 

ii.    `v` `= u + w`
    `= 5 + 2i + (−2 + 5i)`
    `= 3 + 7i`

 

iii.    `text(arg)(w/v)` `= text(arg)(w) – text(arg)(v)`
    `= pi/4\ \ (text(diagonal of square bisects corner))`

Complex Numbers, EXT2 N2 2017 HSC 13e

The points  `A, B, C`  and  `D`  on the Argand diagram represent the complex numbers  `a, b, c`  and  `d` respectively. The points form a square as shown on the diagram.
 


 

By using vectors, or otherwise, show that  `c = (1 + i) d - ia`.  (2 marks)

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`text(Proof)\ \ text{(See Worked Solutions)}`

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♦ Mean mark 49%.
`c – d` `= i (d – a)\ \ \ text{(rotation of}\ pi/2 text{)}`
`:. c ` `= d + id – ia`
  `= (1 + i) d – ia\ text(… as required.)`

Complex Numbers, EXT2 N2 2010 HSC 2d

Let  `z = cos theta + i sin theta`  where  `0 < theta < pi/2`.

On the Argand diagram the point `A` represents  `z`, the point `B` represents  `z^2`  and the point `C` represents  `z + z^2`.
 

Complex Numbers, EXT2 2010 HSC 2d
 

Copy or trace the diagram into your writing booklet.

  1. Explain why the parallelogram  `OACB`  is a rhombus.   (1 mark)

  2. Show that  `text(arg)\ (z + z^2) = (3theta)/2`.   (1 mark)

  3. Show that  `| z + z^2 | = 2 cos  theta/2`.   (2 marks)

  4. By considering the real part of  `z + z^2`, or otherwise deduce that

  5.  

        `cos theta + cos 2theta = 2 cos  theta/2 cos  (3theta)/2`.   (1 mark) 

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  1. `text(See Worked Solutions.)`
  2. `text(See Worked Solutions.)`
  3. `text(See Worked Solutions.)`
  4. `text(See Worked Solutions.)`
Show Worked Solution

i.   `z = cos theta + i sin theta`

Complex Numbers, EXT2 2010 HSC 2d Answer 

`|\ OA\ |=|\ z\ |=1`

`|\ OB\ | =|\ z^2\ |=|\ z\ |^2 = 1`

`:.\ OACB\ text(is a parallelogram with a pair of adjacent sides equal.)`

`:.\ OACB\ text(is a rhombus.)` 

 

ii.  `text(arg)\ z^2 = 2\ text(arg)\ z = 2 theta`

`∠BOA = 2 theta − theta = theta`
 

`text(S)text(ince)\ OACB\ text(is a rhombus then)\ CO\ text(bisects)\ ∠BOA`

`:.∠COA = theta/2`

`:.\ text(arg)(z + z^2) = theta + theta/2 = (3theta)/2`

 

iii.  `OC = |\ z + z^2\ |`

`text(Join)\ \ AB\ \ text(so that it meets)\ \ OC\ \ text(at)\ \ M`

`AB ⊥ OC, and OM=OC\ \ \ text{(diagonals of a rhombus)}`

♦♦ Mean mark part (iii) 26%.

`text(In)\ \ Delta OAM:`

`cos\ theta/2` `=(OM)/(OA)`
  `=OM`
`:.OC` `=2 xx OM`
  `=2 cos\ theta/2`

 

iv.    `z + z^2` `= cos theta + i sin theta + (cos theta + i sin theta)^2`
    `= cos theta + cos 2theta + i(sin theta + sin 2theta)\ \ \ \ text{(De Moivre)}`
`:.\ text(Re)(z+z^2)=cos theta + cos 2theta`

♦♦ Mean mark part (iv) 44%.

`text{Using parts (ii) and (iii),}`

`z + z^2=2 cos\ theta/2(cos (3theta)/2+ i sin (3theta)/2)`
 

`text(Equating real parts:)`

`cos theta + cos 2theta = 2 cos  theta/2 cos\ (3theta)/2`

Complex Numbers, EXT2 N2 2011 HSC 2b

On the Argand diagram, the complex numbers  `0, 1 + i sqrt 3 , sqrt 3 + i`  and  `z`  form a rhombus.
 


 

  1. Find  `z`  in the form  `a + ib`, where  `a`  and  `b`  are real numbers.  (1 mark)

  2. An interior angle, `theta`, of the rhombus is marked on the diagram.

     

    Find the value of `theta.`  (2 marks)

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  1. `(1 + sqrt 3) + i(1 + sqrt 3)`
  2. `(5 pi)/6`
Show Worked Solution
i.   `z` `= 1 + i sqrt 3 + sqrt 3 + i`
  `= (1 + sqrt 3) + i (1 + sqrt 3)`

 

ii.   `text(arg)\ z = tan^-1 ((1 + sqrt 3)/(1 + sqrt 3)) = pi/4`

`text(arg)\ (sqrt 3 + i) = tan^-1 (1/sqrt 3) = pi/6`

`text(Difference) = pi/4 – pi/6 = pi/12`
 

`=>\ text(Opposite angles of a rhombus are equal)`

`=>\ text(The diagonals of a rhombus bisect the angles)`

`:.theta` `= pi – 2 xx pi/12`
  `= (5 pi)/6\ \ text{(angle sum of triangle)`