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QUESTION 11 (7 marks) The vertices of a regular hexagon are positioned on the circumference of a unit circle as shown on the Argand plane. Im(2) Re(2) Consider the complex number w as shown on the plane. a) Determine w expressing your answer in the form 7 cis(6). b) Convert w into Cartesian form. Each vertex of the hexagon is a solution of an equation of the form z” = @ where z € C. c) State the value of n. d) State the value of a. e) Verify that w satisfies the equation z ” = a using the results from 11¢) and 11d). [1 mark] [2 marks] [1 mark] [1 mark] [2 marks]
QUESTION 18 (6 marks) Consider the polynomial P(z) =2 +az” +bz+c where a b ce Rand z e C. Two of the roots of P(z) are also roots of z* + 2% + 22 + 2 +1. The remaining root of P(z)is z=2. 2 Given z° —1=(z~- 1)(2 +2+z20 4z + 1) determine a possible expression for P(z). Leave your answer in expanded form.
QUESTION 15 (4 marks) Consider the equation z>=1where z € C. a) Sketch the solutions to z>=1on the Argand diagram. Im(z) Re(2) The solutions to z°=1 can be expressed in the form z = a + bi where a b € R. b) Determine the largest possible positive value of ab. [2 marks] [2 marks]
QUESTION 10 The Argand diagram that represents the solutions to z* =16 cis 3 (CS zeECis (A) (B) Im(z) (C) (D)
QUESTION 18 (5 marks) Consider the complex solutions to the following equation where 0 < arg(z) < z. (z+1)(2!4-2 47) —zila 474_73 +2°-z)=1-2 Let w be the solution with the maximum possible real part and w be the solution with the maximum possible imaginary part. 4 Show that Mi EZ. WwW
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