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Complex Numbers, EXT2 N1 2023 HSC 7 MC

Which of the following statements about complex numbers is true?

  1. For all real numbers \(x, y, \theta\)  with  \(x \neq 0\),

\(\tan \theta=\dfrac{y}{x} \ \Rightarrow \ x+i y=r e^{i \theta}\), for some real number \(r\).

  1. For all non-zero complex numbers \(z_1\) and \(z_2\),

\(\operatorname{Arg}\left(z_1\right)=\theta_1\)  and  \(\operatorname{Arg}\left(z_2\right)=\theta_2 \ \Rightarrow \ \operatorname{Arg}\left(z_1 z_2\right)=\theta_1+\theta_2,\)

where \(\operatorname{Arg}\) denotes the principal argument.

  1. For all real numbers \(r_1, r_2, \theta_1, \theta_2\)  with  \(r_1, r_2>0\),

\(r_1 e^{i \theta_1}=r_2 e^{i \theta_2} \ \Rightarrow \ r_1=r_2\)  and  \(\theta_1=\theta_2 \text {. }\)

  1. For all real numbers \(x, y, r, \theta\)  with  \(r>0\)  and  \(x \neq 0\),

\(x+i y=r e^{i \theta} \ \Rightarrow \ \theta=\arctan  \Big(\dfrac{y}{x} \Big)\)

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\(A\)

Show Worked Solution

\(\text{Eliminating options by contradiction}\)

\(\text{Option}\ B:\)

\(\text{If}\ \ \theta_1= \pi\ \ \text{and}\ \ \theta_2=\dfrac{\pi}{2}, \ \theta_1 + \theta_2 = \dfrac{3\pi}{2} > \pi \)

\( -\pi < \operatorname{Arg}\left(z_1 z_2\right) < \pi\ \ \ \  \text{(Eliminate}\ B) \)

♦♦♦ Mean mark 18%.

\(\text{Option}\ C:\)

\(\text{If}\ \ \theta_1= \pi\ \ \text{and}\ \ \theta_2=3\pi, \ \operatorname{Arg}(e^{i\pi}) =  \operatorname{Arg}(e^{3i\pi}) \)

\( \text{However,}\ \ \theta_1 \neq \theta_2\ \ \ \  \text{(Eliminate}\ C) \)
 

\(\text{Option}\ D:\)

\(\text{If}\ \ x=y=-1, \ \theta=-\dfrac{3\pi}{4} \ \ (r>0) \)

\( \text{However,}\ \ \arctan\Big(\dfrac{-1}{-1}\Big)=\dfrac{\pi}{4} \ \ \  \text{(Eliminate}\ D) \)

\(\Rightarrow A\)

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)

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  1. `frac{1+sqrt3}{2} + i (frac{sqrt3 – 1}{2})`
  2. `2 \ text{cis} (frac{pi}{3})`
  3. `sqrt2 \ text{cis} (frac{pi}{12})`
  4. `frac{sqrt6-sqrt2}{4}`
Show Worked Solution
i.     `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})`

 

ii.   `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})`

 

iii.   `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})`

 

iv.  `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 2008 HSC 2b

  1. Write  `frac{1 + i sqrt3}{1 + i}`  in the form  `x + iy`, where `x` and `y` are real.  (2 marks)

  2. By expressing both  `1 + i sqrt3`  and  `1 + i`  in  modulus-argument form, write  `frac{1 + i sqrt3}{1 + i}`  in modulus-argument form.   (3 marks)

  3. Hence find  `cos frac{pi}{12}`  in surd form.  (1 mark)

  4. By using the result of part (ii), or otherwise, calculate  `(frac{1 + i sqrt3}{1 + i})^12`.   (1 mark)

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  1. `frac{1 + sqrt3}{2} – i ( frac{1 – sqrt3}{2} )`
  2. `sqrt2 (cos (frac{pi}{12}) + i sin (frac{pi}{12}))`
  3. `frac{sqrt2 + sqrt6}{4}`
  4. `-64`
Show Worked Solution
i.      `frac{1 + i sqrt3}{1 + i} xx frac{1 – i}{1 – i}` `= frac{(1 + i sqrt3)(1 – i)}{1 – i^2}`
    `= frac{1 – i + i sqrt3 – sqrt3 i^2}{2}`
    `= frac{1 + sqrt3}{2} – i ( frac{1 – sqrt3}{2} )`

 

ii.   `z_1 = 1 +  i sqrt3`

`| z_1 | = sqrt(1 + ( sqrt3)^2) = 2`

`text{arg} (z_1) = tan^-1 (sqrt3) = frac{pi}{3}`
  

`z_1 = 2 (cos frac{pi}{3} + i sin frac{pi}{3})`
 
`z_2 = 1 + i`

`| z_2 | = sqrt(1^2 + 1^2) = sqrt2`

`text{arg} (z_2) = tan^-1 (1) = frac{pi}{4}`

`z_2 = sqrt2 (cos frac{pi}{4} + i sin frac{pi}{4})`
 

`frac{1 + i sqrt3}{1 + i}` `= frac{z_1}{z_2}`
  `= frac{2}{sqrt2} ( cos ( frac{pi}{3} – frac{pi}{4} ) + i sin ( frac{pi}{3} – frac{pi}{4} ) )`
  `= sqrt2 ( cos (frac{pi}{12}) + i sin (frac{pi}{12}) )`

 

iii.  `text{Equating real parts of i and ii:}`

`sqrt2 cos (frac{pi}{12})` `= frac{1 + sqrt3}{2}`
`cos(frac{pi}{12})` `= frac{1 + sqrt3}{2 sqrt2} xx frac{sqrt2}{sqrt2}`
  `= frac{sqrt2 + sqrt6}{4}`

 

iv.     `(frac{1 + i sqrt2}{1 + i})^12` `= (sqrt2)^12 (cos (frac{pi}{12} xx 12) + i sin (frac{pi}{12} xx 12))`
    `= 64 (cos pi + i sin pi)`
    `= – 64`

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|}`?
 

 

 

  
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`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)  
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  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)

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

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

 

ii.   `u ­=` `z^2`
`­=` `4 text(cis)\ pi/2`
`­=` `4i`

COMMENT: 12a(iii) had the lowest mean mark (55%) of any part within Q12 and deserves attention.
iii.  `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
  1. `2`
  2. `(5 pi)/6`
  3. `(29 pi)/42`
Show Worked Solution
i.   `|\ z\ |` `= sqrt ((-sqrt3)^2 + 1^2)`
  `= 2`

 

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

 

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