UNSW School of Mathematics and Statistics
MATH2069 Mathematics 2A
Term 3, 2020 TEST 2a VERSION 2
• UNSW approved calculators are permitted
• Time Allowed: 40 minutes
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SINGLE PDF FILE.
1. [4 marks]
i) In the Argand plane, sketch the set
S = {z 2 C : 2  |z| < 4}.
ii) State if S is open, if it is connected, and if it is a domain.
iii) Find and sketch the image of S under the mapping w = 1z .
2. [3 marks] Using either
cos(z) =
1
2

eiz + eiz

or cos(x+ iy) = cos x cosh y i sin x sinh y,
solve cos z = 2, giving your answer in a+ ib form.
3. [3 marks]
i) Where is the function Log(3z + 2i) analytic?
ii) What is its derivative there?
4. [3 marks]
i) Show that if |z| = 6, then
28  |z2 z + 2i|  44.
ii) Hence, use the Estimation Lemma to prove thatI|z|=6 z + iz2 z + 2i dz
 3⇡.
5. [3 marks] Suppose that consists of the line segment from 2 to 1 + i. Find the
value of Z

(2z + 3z¯) dz.
Give your answer in Cartesian form.

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