Consider the equation  , where    is a complex number and are real numbers.

1. Explain why  .   (1 mark)
   

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2. Hence, or otherwise, find .   (2 marks)
   

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Let    and  .

1. Find  .   (1 mark)
   

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2. Find  .   (1 mark)
   

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Express    in the form  , where and are real numbers.  (2 marks)

The complex numbers    and    are given.

Find  , giving your answer in Cartesian form.  (2 marks)

Given the complex number  , where    and  is equivalent to

Let    and  .

Find, in the form  ,

1.     (1 mark)
   

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2.  .   (1 mark)
   

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Let    and  .  Find, in the form  ,

1.     (1 mark)
   

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2.     (1 mark)
   

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3.     (1 mark)
   

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Consider the complex numbers    and  .

1. Evaluate  .   (1 mark)
   

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2. Evaluate  .   (2 marks)
   

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Let    and  .

1. Find  .  (1 mark)
   

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2. Express    in the form  , where  and    are real numbers.  (2 marks)
   

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What is the value of  ?