1
Complex Numbers
1a.
[5 marks]
Solve
.
1b.
[3 marks]
Show that
.
1c.
[9 marks]
Let
.
Find the modulus and argument of
in terms of
. Express each answer in its simplest form.
1d.
[5 marks]
Hence find the cube roots of
in modulus-argument form.
2.
[4 marks]
In the following Argand diagram the point A represents the complex number
and the point
B represents the complex number
. The shape of ABCD is a square. Determine the complex
numbers represented by the points C and D.
3a.
[6 marks]
Find three distinct roots of the equation
giving your answers in modulus-
argument form.

2
3b.
[3 marks]
The roots are represented by the vertices of a triangle in an Argand diagram.
Show that the area of the triangle is
.
4a.
[6 marks]
(i) Use the binomial theorem to expand
.
(ii)
Hence use De Moivre’s theorem to prove
(iii) State a similar expression for
in terms of
and
.
4b.
[4 marks]
Let
, where
is measured in degrees, be the solution of
which
has the smallest positive argument.

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- Fall '18
- Complex number, Use De Moivre’s theorem