2013澳大利亚高考数学试题答案(15)

Question 16 (15 marks) Use a SEPARATE writing booklet.

(a)

(i) Find the minimum value of P(x) =2x3 15x2 +24x +16, for x ≥0. (ii) Hence, or otherwise, show that for x ≥0,

( x +1 ) x 2 + ( x +4 )2

≥25x2 .(iii) Hence, or otherwise, show that for m ≥0 and n ≥0,

( m+n)2

+ ( m+n+ 4 )2

≥

100mn

m+n+1 .

(b) A small bead P of mass m can freely move along a string. The ends of the string

S. The bead undergoes uniform circular motion with radius r and constant angular velocity ω in a horizontal plane.

The forces acting on the bead are the gravitational force and the tension forces along the string. The tension forces along PS and PS′ have the same magnitude T.

The length of the string is 2a and SS′=2ae, where 0 <e <1. The horizontal plane through P meets SS′at Q. The midpoint of SS′is O and β=∠S′PQ.Theparameter θ is chosen so that OQ =a cosθ.

Question 16 continues on page 17

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