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QUESTION 6 Givenz=2—2iand w=-3 + i calculate z> — (A) 3-9i (B) 3-7i (C) 11-9i (D) 11-7i
QUESTION 10 The subset of the complex plane that represents arg|z +i — 1| +% =0forzeCis (A) Im(z) (B) Im(z) 5 Re(z) 5 Re(2) ©) Im(z) (D) Im(z) Re(z) Re(z)
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 function P(z) —22% + az® +62% +az + b where abeZ One of the roots of P(z) 1Sz =—i Determine the possible value/s for a and b such that all remaining roots of P(z) have an imaginary component.
QUESTION 8 Letu=1+iandv=-12+5i Re(u5 — |v|) 1S A) 17 (B) -4 <€) 8 D) 9
QUESTION 16 (6 marks) Consider the identity cos(40) = Acos? (0)+ Bsin? (0)+ C where A Band C € Z a) Determine the values of 4 B and C using De Moivre’s theorem. [5 marks] b) State an appropriate method of verifying your results from 16a). [1 mark]
QUESTION 2 When the polynomial P(z) = z° —iz* — z—i is divided by z — i the remainder is A) -2 (B) 0 ©) 2i (D) 4
QUESTION 6 The subset of the complex plane that represents |z| = |z — 2| for z € C is (A) Im(2) (B) Im(z) ©) Im(2) (D) Im(2) A
QUESTION 13 (6 marks) Use z =a+bi and w = ¢+ di where a b ¢ d € R to prove ’z — w‘z = ‘Z‘Z + |w|2 — 2Re(zv_v)
QUESTION 8 =2 The imaginary part of [Cis (%D 1s (A) (B) ©) (D) —-6.83 -0.71 0.71 1.17
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 1 Let z=a+3i and w= -3+ bi where a b € R. If z = w then (A) a=-3 b=-3 B) a=-3 b=3 (C) a=3 b=-3 (D) a=3 b=3
QUESTION 12 (6 marks) Given zy=a +bi zy=c+di Va b c d € R and z # 0 prove the identity _|Zl‘ _|Zz| A D
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 1 A solution of the equation 22 = ai whereae R isz=-2-2i. The other solution is (A) -8i (B) —2+2i C) 2+2i (D) 8i
QUESTION 18 (5 marks) Consider the polynomials P(z) = 2+ (i- a)22 —2biz +3i and O(z) = z—2i where a be R. P(z Given @ has a remainder of a — bi evaluate the reasonableness that (z — (a —bi )) 1s a factor of P(z).
QUESTION 8 Given z = a + bi where a b € R a < 0 and b > 0 the Argand plane that represents a possible position of w where w = iZ7 is (A) (B) Im(z) Im(z) ° © 0 Re(z) 0 Re(z) (©) (D) Im(z) Im(z) 0 Re(z) 5 Re(z)
QUESTION 18 (7 marks) The following complex equations share at least one common solution. z3—8=0 kz* + 2kz3 — 2kz®> — 6z — 24k =0 Given k € R determine the value of k.
QUESTION 1 Letz = cis (Z?n) Im (z*%) is (A) —0.95 (B) 0.09 (C) 031 (D) 0.81
QUESTION 7 Consider the line that passes through the origin as shown in the complex plane below. Im(z) Re(z) Not drawn to scale The subset of the complex plane that best defines this line is (A) |z—3+2i| =36 (B) |z+3-2i| =36 (C) arg(z—(3—2i))=123.7° (D) arg(z—3+ 2i) = 146.3°
QUESTION 8 Which of the following is a solution of the equation z2 + 2iz — 5 = 0? (A) 1-—2i (B) 1+ 2i C) 2—i (D) 2+
QUESTION 11 (6 marks) a) Use the binomial theorem to expand (cos(8) + i sin(@))3. Express your result in simplified form. [2 marks] b) Use De Moivre’s theorem to expand (cos(8) + isin(6))3. [1 mark] ¢) Use your results from 11a) and 11b) to prove cos(38) = 4 cos>(0) — 3 cos(8). [3 marks]
QUESTION 3 One solution of z? — z7 — 7z-2=0 isz=-2. Which equation could be used to determine the remaining solutions? (A) z7-3z-1=0 (B) z7-3z+1=0 (C) z7-z-1=0 (D) z>-z+1=0
QUESTION 6 The shaded region defined as {2 :|z+2-i|< 5} a {z : Re(z)< hz € C is best represented by (A) (B) (D)
QUESTION 13 (5 marks) z| Given z € C where z # 0 prove —— = | zl ZZ
QUESTION 3 Given that 2i is a root of z’- pz-—q=9 where p g € R determine the values of p and g. (A) p=-4 and q=-4 (B) p=-4 andq=4 (C) p=0 and gq=-—4 (D) p=0 and q=4
QUESTION 10 The Argand diagram that represents the solutions to z* =16 cis 3 (CS zeECis (A) (B) Im(z) (C) (D)
QUESTION 12 (7 marks) Consider the complex number z = -—3 + 2i. a) Determine z using the binomial theorem. Leave your answer in the form a+bi where a be R. b) Convert z into the form of rcis(@) where —z <0 < z. c) Use the result from Question 12b) to determine z using De Moivre’s theorem. Leave your answer in the form of rcis(@) where -a <0 < a. d) Evaluate the reasonableness of your results from Questions 12a) and 12c) noting that the two methods to determine z* should produce the same result. [2 marks] [1 mark] [2 marks] [2 marks]
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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