2013澳大利亚高考数学试题答案(12)

Question 14 (continued)

(c) (i) Given a positive integer n, show that sec 2n

θ=

k∑n

=0

n k

tan2 k

θ. (ii) Hence, by writing sec8θ as sec6θ sec2θ, find

sec 8θdθ. (d) A triangle has vertices A, B and C. The point D lies on the interval AB such that

AD = 3 and DB = 5. The point E lies on the interval AC such that AE = 4, DE = 3 and EC = 2.

B5

D 3

A 3 4

NOT TO SCALE

2 C

(i) Prove that ABC and AED are similar. (ii) Prove that BCED is a cyclic quadrilateral. (iii) Show that CD = 21 .

(iv) Find the exact value of the radius of the circle passing through the points

B, C, E and D.

End of Question 14

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