2013澳大利亚高考数学试题答案(13)

Question 15 (15 marks) Use a SEPARATE writing booklet.

(a) The Argand diagram shows complex numbers w and z with arguments φ and θ

respectively, where φ<θ. The area of the triangle formed by 0, w and z is A.

Show that = 4 iA.

(b) The polynomial P(x) = ax4 + bx3 + cx2 + e has remainder –3 when divided by x – 1. The polynomial has a double root at x = –1.

(i) Show that 4a

+2c = 92

. (ii) Hence, or otherwise, find the slope of the tangent to the graph y = P(x)

when x = 1.

(c) Eight cars participate in a competition that lasts for four days. The probability

that a car completes a day is 0.7. Cars that do not complete a day are eliminated.

(i) Find the probability that a car completes all four days of the competition. (ii) Find an expression for the probability that at least three cars complete all

four days of the competition.

Question 15 continues on page 15

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