Complex numbers assignment PDF

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Practise questions for complex numbers...

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WTW124 (2021)

Assignment 12 [Total = 10]

Surname: First names: Student Number:

Last digit: α =

The University of Pretoria commits itself to produce academic work of integrity. I affirm that I am aware of and have the read the Rules and Policies of the University, more specifically the Disciplinary Procedure and the Tests and Examination Rules, which prohibit any unethical, dishonest or improper conduct during tests, assignments, examinations and/or any other forms of assessment. I am aware that no student or any other person may assist or attempt to assist another student, or obtain help, or attempt to obtain help from another student or any other person during tests, assessments, assignments, examinations and/or any other forms of assessment. You must complete all questions. Only selected questions will be marked for a total of 10 marks. NB: This must be STRICTLY INDIVIDUAL WORK. If we detect plagiarism or two identical answers, both students will get ZERO for the assignment. These problems must NOT be discussed on any online platform, including Discord. 1. (a) Find all the complex number(s) that satisfies the equations below: (i) (2 + 3i)z + 3i = 2. (ii) z 2 − 2 = 3iz. (b) Use induction to prove the following statement: The complex number (1 + 2i)n has an odd real part for any integer n > 2. (c) Let α be the last digit of your student number. Sketch the following sets in the Argand plane (aka complex plane): (i) A = {z ∈ C : |z| 6 α + 2, Im(z) ≥ α + 1}.

(ii) B = {z ∈ C : |z + 2i| = |z − 2|}.

[5] 2. Let z =

√ 3 2

+ 21 i and w =

√ 2 2

+

√ 2 i. 2

(a) Find zw in polar form, with principal argument. ). ) and tan( 5π ), sin( 5π (b) Use (a) to find the exact values cos( 5π 12 12 12 π (c) Use multiplication or division of z and w to find the exact values of cos( π12 ), sin( 12 ) and π ). tan( 12

[5] 3. (a) Let a = x + yi and b = s + ti with x, y, s, t real numbers. Adopt the proof of Theorem 6.4.3 and assume that Exercise 6.3 No. 7 is true to prove that e2a = e2a−b . eb NB: Do not use Theorem 6.4.3 (b) Given that

√ √ √  3 1  3−5 5 3+1 + i (1 + 5i) = + i, 2 2 2 2

Show that π 6

 √  5√ 3 + 1 . = arctan (5) − arctan 3−5 [5]...