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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 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 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 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 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 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 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 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 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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