smc-1049-50-Powers

Complex Numbers, EXT2 N1 2024 HSC 11e

Write the number \(\sqrt{3}+i\) in modulus-argument form. (2 marks)

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Hence, or otherwise, write \((\sqrt{3}+i)^7\) in exact Cartesian form. (2 marks)

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i. \(2 \text{cis}\left(\dfrac{\pi}{6}\right)=2\left(\cos \left(\dfrac{\pi}{6}\right)+i \sin \left(\dfrac{\pi}{6}\right)\right)\)

ii. \(-64 \sqrt{3}-64 i\)

Show Worked Solution

i. \(z=\sqrt{3}+i\)

\(|z|=\sqrt{3+1}=2\)

\(\arg (z)=\tan ^{-1}\left(\dfrac{1}{\sqrt{3}}\right)=\dfrac{\pi}{6}\)

\(z=2 \text{cis}\left(\dfrac{\pi}{6}\right)=2\left(\cos \left(\dfrac{\pi}{6}\right)+i \sin \left(\dfrac{\pi}{6}\right)\right)\)

ii.	\((\sqrt{3}+i)^7\)	\(=2^7\left(\cos \left(\dfrac{7 \pi}{6}\right)+i \sin \left(\dfrac{7 \pi}{6}\right)\right)\)
		\(=128\left(-\dfrac{\sqrt{3}}{2}-\dfrac{1}{2} i\right)\)
		\(=-64 \sqrt{3}-64 i\)

Complex Numbers, EXT2 N1 EQ-Bank 14

Let `z = sqrt3-3 i`

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

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Find the smallest integer `n`, such that `z^n + (overset_z)^n = 0`. (3 marks)

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a. `2 sqrt3 text{cis} (frac{-pi}{3})`

b. `3`

Show Worked Solution

a.	`z`	`= sqrt3-3 i`
	`\|z\|`	`= sqrt((sqrt3)^2 + 3^2) = 2 sqrt3`

`tan theta= frac{3}{sqrt3}=sqrt3\ \ =>\ \ `theta= frac{pi}{3}`

`text{arg} (z)=-frac{pi}{3}`

`therefore z = 2 sqrt3 \ text{cis} (frac{-pi}{3})`

b. `z^n + (overset_z)^n = 0`

`[2 sqrt3 \ cos (frac{-pi}{3}) + i sin (frac{-pi}{3})]^n + [ 2 sqrt3 \ cos (frac{-pi}{3})-i sin (frac{-pi}{3}) ]^n = 0`

`(2 sqrt3)^n [cos (frac{-n pi}{3}) + i sin (frac{-n pi}{3}) + cos (frac{-n pi}{3})-i sin (frac{-n pi}{3}) = 0`

`2 \ cos (frac{-n pi}{3})`	`= 0`
`cos (frac{n pi}{3})`	`= 0`
`frac{n pi}{3}`	`= frac{pi}{2} + k pi \ , \ k = 0, ± 1, ± 2, …`
`frac{n}{3}`	`= frac{(2k + 1)}{2}`
`n`	`= frac{3 (2k + 1)}{2}`

`text{Numerator will always be odd ⇒ no solution exists}`

Complex Numbers, EXT2 N1 2005 HSC 2b

Let `beta = 1-i sqrt3`.

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

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Express `beta^5` in modulus-argument form. (2 marks)

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Hence express `beta^5` in the form `x+iy`. (1 mark)

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a. `2 \ text{cis} (-frac{pi}{3})`

b. `32 \ text{cis} (frac{pi}{3})`

c. `16 + i 16 sqrt3`

Show Worked Solution

a. `beta = 1-i sqrt3`

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

`tan theta`	`= frac{sqrt3}{1} = sqrt3`
`theta`	`= frac{pi}{3}`
`text{arg} (beta)`	`= -frac{pi}{3}`

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

b.	`beta^5`	`= 2^5 \ text{cis} (-frac{pi}{3} xx5)`
		`= 32 \ text{cis} (-frac{5pi}{3} + 2 pi)`
		`= 32 \ text{cis} (frac{pi}{3})`

c.	`beta^5`	`= 32 ( cos (frac{pi}{3}) + i sin (frac{pi}{3}) )`
		`= 32 ( frac{1}{2} + i frac{sqrt3}{2})`
		`= 16 + i 16 sqrt3`

Complex Numbers, EXT2 N1 2019 HSC 11e

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

Write `z` in modulus-argument form. (2 marks)

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Find `z^3`, giving your answer in the form `x + iy`, where `x` and `y` are real numbers. (2 marks)

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i. `z = 2 text(cis) (2 pi)/3`

ii. `8 + 0i`

Show Worked Solution

i. `|\ z\ |= -1 + i sqrt 3= sqrt((-1)^2 + (sqrt 3)^2)= 2`

	`tan theta`	`= -sqrt 3`
	`text(arg)(z)`	`= (2 pi)/3`
	`:. z`	`= 2 text(cis) (2 pi)/3`

ii. `z^3 = 2^3 [cos(3 xx (2 pi)/3) + i sin (3 xx (2 pi)/3)]\ \ \ text{(by De Moivre)}`

`= 8(cos 2 pi + i sin 2 pi)`

`= 8(1 + 0i)`

`= 8 + 0i`

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.

`z` (1 mark)

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`u = z^2` (1 mark)

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`v = z^2-bar z` (1 mark)

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a. `text(See Worked Solutions)`

b. `text(See Worked Solutions)`

c. `text(See Worked Solutions)`

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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 2007 HSC 2b

Write ` 1 + i` in the form `r (cos theta + i sin theta).` (2 marks)

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Hence, or otherwise, find `(1 + i)^17` in the form `a + ib`, where `a` and `b` are integers. (3 marks)

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a. `sqrt 2 ( cos pi/4 + i sin pi/4)`

b. `256 + 256i`

Show Worked Solution

`\|\ 1+i\ \|`	`=sqrt(1^2+1^2)=sqrt2`
`text(arg)(1+i)`	`=pi/4`
`:. 1 + i =`	`sqrt 2 (cos pi/4 + i sin pi/4)`

b. `(1 + i)^17`	`=(sqrt 2)^17 (cos\ pi/4 + i sin\ pi/4)^17`
	`=2^8 sqrt 2 (cos (17 pi)/4 + i sin (17 pi)/4)\ \ \ \ text{(De Moivre)}`
	`=2^8 sqrt 2 (cos pi/4 + i sin pi/4)`
	`=2^8 sqrt2(1/sqrt2 + 1/sqrt2 i)`
	`=2^8 (1 + i)`
	`=256 + 256 i`

Complex Numbers, EXT2 N1 2015 HSC 5 MC

Given that `z = 1 -i`, which expression is equal to `z^3 ?`

`sqrt 2 (cos((-3 pi)/4) + i sin((-3 pi)/4))`
`2 sqrt 2 (cos((-3 pi)/4) + i sin((-3 pi)/4))`
`sqrt 2 (cos((3 pi)/4) + i sin((3 pi)/4))`
`2 sqrt 2 (cos((3 pi)/4) + i sin((3 pi)/4))`

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`B`

Show Worked Solution

`z`	`=1-i`
`\|\ 1-i\ \|`	`=sqrt2`
`text{arg}(z)`	`=-pi/4`

`z`	`=sqrt 2 (cos(-pi/4) + i sin(-pi/4))`
`:.z^3`	`=2 sqrt 2 (cos((-3 pi)/4) + i sin((-3 pi)/4))\ \ \ \ text{(De Moivre)}`

`=> B`

Complex Numbers, EXT2 N1 2014 HSC 4 MC

Given `z = 2(cos\ pi/3 + i sin\ pi/3)`, which expression is equal to `(bar {:z:})^(−1)`?

`1/2(cos\ pi/3 − i sin\ pi/3)`
`2(cos\ pi/3 − i sin\ pi/3)`
`1/2(cos\ pi/3 + i sin\ pi/3)`
`2(cos\ pi/3 + i sin\ pi/3)`

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`C`

Show Worked Solution

`z`	`= 2text(cis)(pi/3)`
`barz`	`= 2text(cis)(−pi/3)`
`(barz)^(−1)`	`= (2text(cis)(−pi/3))^(−1)`
	`= 2^(−1)text(cis)(pi/3)`
	`= 1/2text(cis)(pi/3)`
	`= 1/2(cos\ pi/3 + i sin\ pi/3)`

`=> C`

Complex Numbers, EXT2 N1 2013 HSC 3 MC

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

Which diagram best represents `z^2?`

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`D`

Show Worked Solution

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

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

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

`=> D`

Complex Numbers, EXT2 N1 2012 HSC 11d

Write `z = sqrt3-i` in modulus-argument form. (2 marks)

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Hence express `z^9` in the form `x + iy`, where `x` and `y` are real. (1 mark)

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a. `2\ text(cis)(-pi/6)`

b. `i512`

Show Worked Solution

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

`:.z = sqrt3 − i`	`= 2(sqrt3/2 − 1/2i)`
	`= 2(cos\ (-pi/6) + i\ sin\ (-pi/6))`
	`= 2\ text(cis)(-pi/6)`

b.	`z^9`	`= 2^9\ (cos\ (-pi/6) + i\ sin\ (-pi/6))^9`
		`= 2^9\ text(cis)(-(9pi)/6)\ \ \ \ text{(by De Moivre)}`
		`=512\ text(cis)(-(3pi)/2)`
		`= 512(0 + i)`
		`=i512`

Complex Numbers, EXT2 N1 2013 HSC 11a

Let `z = 2-i sqrt 3` and `w = 1 + i sqrt 3.`

Find `z + bar w.` (1 mark)

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Express `w` in modulus–argument form. (2 marks)

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Write `w^24` in its simplest form. (2 marks)

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a. `3-i\ 2 sqrt 3`

b. `2 text(cis) pi/3`

c. `2^24`

Show Worked Solution

a. `z = 2-i sqrt 3\ ,\ \ w = 1 + i sqrt 3`

`bar w = 1-i sqrt 3`

`z + bar w`	`= 2-i sqrt 3 + 1-i sqrt 3`
	`= 3-i\ 2 sqrt 3`

b. `|\ w\ |=sqrt(1^2 + (sqrt3)^2)=2`

`:.w`	`= 2 (1/2 + i sqrt 3/2)`
	`=2(cos\ pi/3 + i sin\ pi/3)`
	`= 2 text(cis) pi/3`

MARKER’S COMMENT: The directive “in its simplest form” required students to convert `text(cis)\ 8pi` to 1.

c. `w^24`	`= 2^24 text(cis)\ (24 xx pi/3)`
	`= 2^24\ text(cis)(8 pi)`
	`= 2^24`

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