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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%.

Vectors, EXT2 V1 2022 HSC 9 MC

Let and be two distinct points in three-dimensional space. Let be the midpoint of .

Let be the set of all points such that  . 

Let be the set of all points such that .

The intersection of and is the circle .

What is the radius of the circle ?

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. 

  

 
   
 
   
   

 


♦♦♦ Mean mark 20%.

Complex Numbers, EXT2 N2 2020 SPEC2 2

Two complex numbers, and , are defined as    and  .

  1. Express the relation    in the cartesian form  , where  .  (3 marks)

  2. Plot the points that represent and and the relation on the Argand diagram below.  (2 marks)
     
         
     
  3. State a geometrical interpretation of the graph of    in relation to the points that represent and .  (1 mark)

  4.  i. Sketch the ray given by    on the Argand diagram in part b.  (1 mark)

  5. ii. In Cartesian form, write down the function that describes the ray  .  (1 mark)

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  2.  
  4.  i.
  5. ii.
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a.   

 

b.   

 

c.   

 

d.i.   

 

d.ii.   

Complex Numbers, EXT2 N2 2011 HSC 4a

Let    and    be real numbers with  . Let    be a complex number such that

     

  1. Prove that    (2 marks)

  2. Hence, describe the locus of all complex numbers    such that    (1 mark)

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i.   
 
 
 
 
 

  

♦ Mean mark part (ii) 44%.

ii.