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Complex Numbers, EXT2 N1 2004 HSC 2b

Let  `alpha = 1 + i sqrt3`  and  `beta = 1 + i`.

  1. Find  `frac{alpha}{beta}`, in the form  `x + i y`.   (1 mark)

  2. Express `alpha` in modulus-argument form.   (3 marks)

  3. Given that `beta` has the modulus-argument form
     
         `beta = sqrt2 (cos frac{pi}{4} + i sin frac{pi}{4})`.
     
    find the modulus-argument form of  `frac{alpha}{beta}`.   (1 mark)

  4. Hence find the exact value of  `sin frac{pi}{12}`   (1 mark)

Show Answers Only

a.    `frac{1+sqrt3}{2} + i (frac{sqrt3 – 1}{2})`

b.    `2 \ text{cis} (frac{pi}{3})`

c.    `sqrt2 \ text{cis} (frac{pi}{12})`

d.    `frac{sqrt6-sqrt2}{4}`

Show Worked Solution
a.     `frac{alpha}{beta}` `= frac{1 + i sqrt3}{1 + i} xx frac{1 – i}{1 – i}`
    `= frac{(1 + i sqrt3)(1 – i)}{1^2 – i^2}`
    `= frac{1 – i + i sqrt3 – i^2 sqrt3}{2}`
    `= frac{1+sqrt3}{2} + i (frac{sqrt3 – 1}{2})`

 

b.     `alpha` `= 1 + i sqrt3`
  `| alpha |` `= sqrt(1^2 + (sqrt3)^2) = 2`

`text{arg} \ (alpha) = tan^-1 (frac{sqrt3}{1}) = frac{pi}{3}`

`therefore \ alpha = 2 text{cis} (frac{pi}{3})`

 

c.     `beta` `= sqrt2 text{cis} (frac{pi}{4})`
  `frac{alpha}{beta}` `= frac{2}{sqrt2} \ text{cis} (frac{pi}{3} – frac{pi}{4})`
    `= sqrt2 text{cis}  (frac{pi}{12})`

 

d.    `text{Equating imaginary parts of i and ii:}`

`sqrt2 \ sin \ (frac{pi}{12})` `= frac{sqrt3 – 1}{2}`
`sin (frac{pi}{12})` `= frac{sqrt3 – 1}{2 sqrt2} xx frac{sqrt2}{sqrt2}`
  `= frac{sqrt6 – sqrt2}{4}`

Complex Numbers, EXT2 N1 2020 HSC 4 MC

The diagram shows the complex number `z` on the Argand diagram.
 


 

Which of the following diagrams best shows the position of  `frac{z^2}{|z|}`?
 

 

 

  
Show Answers Only

`A`

Show Worked Solution

`text{Let} \ \ z = r\ text(cis)\ theta`

`z^2` `= r^2  text(cis)\ (2 theta)`
`|z|` `= r`
`therefore  frac{z^2}{|z|}` `= frac{r^2 \ text(cis)\ (2 theta)}{r}`
  `= r\ text(cis)\ (2 theta)`

 
`text{On Argand diagram, it lies on the dotted line`

`text{(modulus the same) with an argument that is}`

`text{doubled.}`
  

`=> \ A`

Complex Numbers, EXT2 N1 2017 HSC 11a

Let  `z = 1 - sqrt 3 i`  and  `w = 1 + i`.

  1. Find the exact value of the argument of `z`.  (1 mark)
  2. Find the exact value of the argument of  `z/w`.  (2 marks)  
Show Answers Only
  1. `-pi/3`
  2. `-(7 pi)/12`
Show Worked Solution

i.  `z = 1 – i sqrt 3`

`text(arg)\ z = – pi/3`

 

ii.  `w = 1 + i`

`text(arg)\ w = pi/4`

`text(arg)\ (z/w)` `= text(arg)\ (z) – text(arg)\ (w)`
  `= – pi/3 – pi/4`
  `= -(7 pi)/12`

Complex Numbers, EXT2 N1 2015 HSC 12a

The complex number `z` is such that `|\ z\ |=2`  and  `text(arg)(z) = pi/4.`

Plot each of the following complex numbers on the same half-page Argand diagram.

  1.  `z`   (1 mark)

  2.  `u = z^2`   (1 mark)

  3.  `v = z^2-bar z`   (1 mark)

Show Answers Only

a.    `text(See Worked Solutions)`

b.    `text(See Worked Solutions)`

c.    `text(See Worked Solutions)`

Show Worked Solution

a.    `z ­=2 text(cis) pi/4=sqrt 2 (1 + i)`
 

b.    `u=z^2=4 text(cis)\ pi/2=4i`
 

COMMENT: Mean mark (c) 55%.
c.    `v ­` `=z^2-bar z`
  `=4i-sqrt 2 (1-i)`
  `=- sqrt 2 + (4 + sqrt 2) i`

Complex Numbers, EXT2 N1 2015 HSC 11b

Consider the complex numbers  `z = -sqrt 3 + i`  and  `w = 3 (cos\ pi/7 + i sin\ pi/7).`

  1. Evaluate  `|\ z\ |.`   (1 mark)

  2. Evaluate  `text(arg)(z).`   (1 mark)

  3. Find the argument of  `z/w.`   (1 mark)

Show Answers Only

a.    `2`

b.    `(5 pi)/6`

c.    `(29 pi)/42`

Show Worked Solution
a.   `|\ z\ |` `= sqrt ((-sqrt3)^2 + 1^2)`
  `= 2`

 

b.   `text(arg)\ (z) ­=` `tan^-1 (1- sqrt 3)`
`­=` `pi – pi/6`
`­=` `(5 pi)/6`

 

c.   `text(arg) (z/w) ­=` `text(arg)\ z – text(arg)\ w`
`­=` `(5 pi)/6 – pi/7`
`­=` `(29 pi)/42`

Complex Numbers, EXT2 N1 2013 HSC 3 MC

The Argand diagram below shows the complex number  `z.`
 


 

Which diagram best represents  `z^2?`

Show Answers Only

`D`

Show Worked Solution

`text(Consider)\ \ z\ \ text(in polar form:)`

`|\ z\ |` `< 1`
`:.|\ z^2\ |` `< |\ z\ |`

 

`text(arg) (z^2) = 2text(arg) (z)`

`=>  D`