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Complex Numbers, SPEC1 2024 VCAA 1

Consider the function with rule  ,  where  .

  1. Verify that    is a factor of .   (1 mark)

  2. Hence or otherwise, solve the equation  .   
  3. Give your answers in Cartesian form.  (2 marks)

  4. Plot the solutions of    on the Argand diagram below.  (1 mark)

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a.  


 


 

b.   


 

c.   
         

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a.  


 


 

b.   


 

c.   
         

♦ Mean mark (c) 49%.

Complex Numbers, SPEC2 2023 VCAA 2

Let .

  1. Verify that is a root of  .   (1 marks)
  1. List the other roots of    in polar form.   (1 mark)
  1. On the Argand diagram below, plot and label the points that represent all the roots of  .   (2 marks)

  1.  i. On the Argand diagram below, sketch the ray that originates at the real root of    and passes through the point represented by .   (1 mark)

     

  1. ii. Find the equation of this ray in the form , where , and is measured in radians in terms of .   (1 mark)

  2. Verify that the equation    can be expressed in the form 
  3.     .   (1 mark)
  4.   i. Express  in the form , where .   (1 mark)

  5. ii. Given that    satisfies  ,

use De Moivre's theorem to show that

      1. .   (2 marks)

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a.   


 

b.   

c.   
          

d.i.   
          

d.ii.


 

e.     
     
     

 

f.i.   
 

f.ii. 

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a.   


 

b.   

c.   
          

d.i.   
          

d.ii.


 

e.     
     
     

 

f.i.   
 

f.ii. 

Complex Numbers, SPEC2 2019 VCAA 2

  1. Show that the solutions of  , where  , are  .   (1 mark)

  2. Plot the solutions of    on the Argand diagram below.   (1 mark)

     

     

Let  , where  , represent the circle of minimum radius that passes through the solutions of  .

    1. Find    and  .   (2 marks)

    2. Find the cartesian equation of the circle  .   (1 mark)

    3. Sketch the circle on the Argand diagram in part a.ii. Intercepts with the coordinate axes do not need to be calculated or labelled.   (1 mark)

  2. Find all values of  , where  , for which the solutions of    satisfy the relation  .   (2 marks)

  3. All complex solutions of    have non-zero real and imaginary parts.

     

    Let    represent the circle of minimum radius in the complex plane that passes through these solutions, where  .

     

    Find    and    in terms of    and  .   (2 marks)

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  2.   
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a.i.   
   
   
   

 

a.ii.   

 

b.i.   

 

 

b.ii.   
 
 

 

b.iii.   

 

c.   


 

 

d.   

 

Complex Numbers, SPEC1 2012 VCAA 3

Consider the equation 

  1. Given that    is a root of the equation, find the other two roots in the form  , where     (3 marks)

  2. Plot all of the roots clearly on the Argand diagram below.   (1 mark)

     

     


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  1.  

  2.  

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a.   


 


 

 
 
 
 

 

♦♦ Mean mark part (b) 33%.

 

b.   

Complex Numbers, SPEC2 2017 VCAA 4

  1. Express    in polar form.  (1 mark)

  2. Show that the roots of    are    and  .  (1 mark)

  3. Express the roots of    in terms of  .  (1 mark)

  4. Show that the cartesian form of the relation    is    (2 marks)

  5. Sketch the line represented by    and plot the roots of    on the Argand diagram below.  (2 marks)

     
       

  6. The equation of the line passing through the two roots of    can be expressed as  , where  .

     

    Find in terms of .  (1 mark)

  7. Find the area of the major segment bounded by the line passing through the roots of    and the major arc of the circle given by  .  (2 marks)

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  5.  
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a.   

 

 
 

 

 

b.   

 

♦♦ Mean mark part (c) 31%.

c.   
 

 

d.
    
   
 
 

 

 

e.   

 

f.   

♦♦♦ Mean mark 1%!


 

  

 
 

 

♦♦ Mean mark 31%.

g.   
   

 

 

 

Complex Numbers, SPEC2-NHT 2017 VCAA 2

One root of a quadratic equation with real coefficients is  .

    1. Write down the other root of the quadratic equation.   (1 mark)

    2. Hence determine the quadratic equation, writing it in the form  .   (2 marks)

  1. Plot and label the roots of     on the Argand diagram below.   (3 marks)

  

 

  1. Find the equation of the line that is the perpendicular bisector of the line segment joining the origin and the point  . Express your answer in the form  .   (2 marks)

  1. The three roots plotted in part b. lie on a circle.

     

    Find the equation of this circle, expressing it in the form  ,  where  .   (3 marks)

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a.i.   

 

 

a.ii.   
 
 
 
 

 

b.   
 

 

 
   
 

 

   

 

c.   

 


 

 
 
 

 

d.