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Question 1. [5 marks] Find the real and imaginary parts of the complex number
(1 + i)4 (1 − i)4
+
.
(1 − i)3 (1 + i)3
1
√
Question 2. [5 marks] Find all the fifth roots of 2 − 2 2i and plot them in the complex plane.
1
Question 2. [5 marks] Suppose the complex numbers z1 and z2 satisfy
Re z1 > 0, Re z2 > 0.
Prove that
Arg(z1 z2 ) = Arg z1 + Arg z2 .
Give a counter-example to the above if the condition on the real parts of zi are removed.
(Note: This is essentially problem 6 on p. 24 of [BC].)
1
Question 4. [5 marks] Let z = x + iy. Use the Cauchy-Schwarz inequality to prove that
p
1
√ (|x| + |y|) ≤ x2 + y 2 ≤ |x| + |y|.
2
Find the conditions characterizing when equality holds in the above.
p
(Hint: x2 + y 2 = |x + iy|, and see Problem 4, p. 13 of [BC].)
1
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