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Complex Numbers, SPEC2 2022 VCAA 2

Two complex numbers and are given by   and  , where .

  1.  i. Given that  , show that  .   (2 marks)

  2. ii. One set of possible values for and is    and  .
  3.     Hence, or otherwise, find the other set of possible values.   (1 mark)

  4. Plot and label the points representing    and    on the Argand diagram below.
     

  1. The ray given by    passes through the midpoint of the line interval that joins the points    and  .
  2. Find, in radians, the value of and plot this ray on the Argand diagram in part b.   (2 marks)

  3. The line interval that joins the points    and    cuts the circle    into a major and a minor segment.
  4. Find the area of the minor segment, giving your answer correct to two decimal places.   (2 marks)

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a.i. 

a.ii. 

b.   
       

c.   

d.   

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a.i. 

 

 

 
 
Mean mark (a) 54%.

 
a.ii.


 

b.   
       


c.    
♦ Mean mark (c) 39%.

d.   

♦ Mean mark (d) 40%.

Complex Numbers, SPEC1 2019 VCAA 7

  1. Show that  .   (1 mark)

  2. Find  , expressing your answer in the form  , where  ,  .   (2 marks)

  3. Find the integer values of    for which    is real.   (1 mark)

  4. Find the integer values of    for which  , where    is a real number.   (1 mark)

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a. 

   

b.   
   
   
   

 

c.   

 

d. 

Complex Numbers, SPEC2 2012 VCAA 2

  1.  Given that  , show that  .   (2 marks)

  2. Express    in polar form.   (1 mark)

  3. i.  Write down    in polar form.   (1 mark)

  4. ii. On the Argand diagram below, shade the region defined by

.   (2 marks)


 

  1.  Find the area of the shaded region in part c.   (2 marks) 
  2. i.  Find the value(s) of such that  , where  .   (3 marks)

  3. ii. Find    for the value(s) of found in part i.   (1 mark)

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  2. i. 

     

    ii. 

  5. i. 

     

    ii. 

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a.   
 
   

 

 

b.i.   
   

 

b.ii.   
   

 

c.   

 

d. 

Mean mark 51%.


  

 


 

♦ Mean mark (e)(i) 34%.

e.i.   
 
 
 
 

 

e.ii. 

♦ Mean mark (e)(ii) 36%.

 
   
   

 

Complex Numbers, SPEC1 2018 VCAA 2

  1. Show that  .   (1 mark)

  2. Evaluate  , giving your answer in the form  , where  .   (3 marks)

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a.  
   
 
     
 

 

b.