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Complex Numbers, EXT2 N2 2023 HSC 16a

Let be the complex number  .

  1. Show that  .   (2 marks)

The vertices of a triangle can be labelled and in anticlockwise or clockwise direction, as shown.
 

Three complex numbers and are represented in the complex plane by points and respectively.

  1. Show that if triangle is anticlockwise and equilateral, then  .   (2 marks)

  2. It can be shown that if triangle is clockwise and equilateral, then . (Do NOT prove this.)
  3. Show that if is an equilateral triangle, then

  (2 marks)

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i.   

 
   
   
   

 
ii.   

 
 
♦♦♦ Mean mark (ii) 18%.

iii.   


 

 
 
 

 

♦♦♦ Mean mark (iii) 11%.

Complex Numbers, EXT2 N2 2023 14a

Let be the complex number    and be the complex number  .

  1. By first writing and in Cartesian form, or otherwise, show that
  2.    .  (3 marks)

  3. The complex numbers and are represented in the complex plane by the vectors and respectively, where is the origin.
  4. Show that  .  (2 marks)

  5. Deduce that  .  (1 mark)

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i.   

 
   
   
   
   
   
   

 
ii.   


  

iii.   


 

 
   
   
   
   
♦♦ Mean mark (iii) 26%.