SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Complex Numbers, EXT2 N1 2023 HSC 1 MC

Which of the following is equal to \((a+i b)^3\)?

  1. \( (a^3-3 a b^2)+i (3 a^2 b+b^3) \)
  2. \( (a^3+3 a b^2)+i (3 a^2 b+b^3) \)
  3. \( (a^3-3 a b^2)+i (3 a^2 b-b^3) \)
  4. \( (a^3+3 a b^2)+i(3 a^2 b-b^3)\)
Show Answers Only

\(C\)

Show Worked Solution
\((a+i b)^3\) \(=a^3+3a^2ib+3a(ib)^2+(ib)^3\)  
  \(=a^3+3a^2ib-3ab^2-ib^3\)  
  \(=(a^3-3ab^2)+i(3a^2b-b^3) \)  

 
\(\Rightarrow C\)

Complex Numbers, EXT2 N1 SM-Bank 7

Calculate  `i^(1!) + i^(2!) + i^(3!) + …+ i^(100!)`

giving your answer in the form  `x+iy`  (3 marks)

Show Answers Only

`95+i`

Show Worked Solution

`i^(1!) =i^1 =  i`

`i^(2!) =i^2= −1`

`i^(3!) = i^6=-1`

`i^(4!) = i^24 = 1`

`i^(5!) = i^(24 xx 5)= 1^5 = 1`

`=> i^(n!) = 1\ \ text(for)\ \ n >= 4`

`:.\ text(Sum)` `= i – 1 – 1 + 97`
  `= 95 + i`

Complex Numbers, EXT2 N1 2019 HSC 8 MC

Let  `z`  be a complex number such that  `z^2 = -i bar z`.

Which of the following is a possible value for `z`?

  1. `1/2 - sqrt 3/2 i`
  2. `1/2 + sqrt 3/2 i`
  3. `sqrt 3/2 - 1/2 i`
  4. `sqrt 3/2 + 1/2 i`
Show Answers Only

`C`

Show Worked Solution

`text(Solution 1)`

`text(Consider)\ C:`

`z` `=sqrt3/2 – 1/2 i`  
`barz` `=sqrt3/2 + 1/2 i`  
`-i barz` `=1/2-sqrt3/2 i`  

 

`z^2` `=(sqrt3/2 – 1/2 i)^2`  
  `=3/4 – 2 sqrt3/2 * 1/2 i -1/4`  
  `=1/2 – sqrt3/2 i`  

 
`=>C`

 

`text(Solution 2)`

`text(Let)\ \ text(arg)(z)` `= theta`
`text(arg)(z^2)` `= 2 theta`

 

`text(arg)(-i bar z)` `=text(arg)(i bar z) – pi`
  `= text(arg) (bar z) – pi + pi/2`
  `= -theta – pi/2`

 

`:. 2 theta` `= -theta – pi/2`
`3 theta` `= -pi/2`
`theta` `= -pi/6`

 
`=>   C`

Proof, EXT2 P1 2018 HSC 9 MC

It is given that  `a`, `b` are real and  `p`, `q` are purely imaginary.

Which pair of inequalities must always be true?

  1. `a^2p^2 + b^2q^2 <= 2abpq,qquada^2b^2 + p^2q^2 <= 2abpq`
  2. `a^2p^2 + b^2q^2 <= 2abpq,qquada^2b^2 + p^2q^2 >= 2abpq`
  3. `a^2p^2 + b^2q^2 >= 2abpq,qquada^2b^2 + p^2q^2 <= 2abpq`
  4. `a^2p^2 + b^2q^2 >= 2abpq,qquada^2b^2 + p^2q^2 >= 2abpq`
Show Answers Only

`B`

Show Worked Solution

`a, b ->\ text(real)qquad\ p, q ->\ text(purely imaginary)`

`=> ab, pq, (ab – pq)\ text(are real)`

`=> ap, bq, (ap – bq)\ text(are purely imaginary.)`
 

`(ab – pq)^2` `>= 0`
`a^2b^2 + p^2q^2` `>= 2abpq\ \ (text(Eliminate A and C))`

 

`(ap – bq)^2` `<= 0`
`a^2p^2 + b^2q^2` `<= 2abpq`

 
`=>B`