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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 2022 HSC 16a

A square in the Argand plane has vertices

          and  .

The complex numbers and lie on the square and form the vertices of an equilateral triangle, as shown in the diagram.
 
 
                 
 
Find the exact value of the complex number .  (4 marks)

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♦♦♦ Mean mark 21%.


 

 
 

 

 
 
 
 

 

Complex Numbers, EXT2 N2 2020 HSC 14a

Let be a complex number and let 

The diagram shows points and which represent and , respectively, in the Argand plane.
 


 

  1. Explain why triangle    is an equilateral triangle.   (2 marks)

  2. Prove that  .  (3 marks)

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i.

 

    
 
   

 

 

ii.    
 
 
 
 
 

COMMENT: The identity to prove suggests the addition of 2 cubes is a possible strategy.