2013澳大利亚高考数学试题答案(8)

Question 12 (15 marks) Use a SEPARATE writing booklet.

(a) Using the substitution t = tanx π

2

1

2 , or otherwise, evaluate

dx . 0 4+5cos x

(b) The equation logx

e y log e

(1000 y)= 50

log e3 implicitly defines y as a function of x.

Show that y satisfies the differential equation

dy dx = y 50 y 1 1000

. (c) The diagram shows the region bounded by the graph y = ex, the x-axis and the

lines x = 1 and x = 3. The region is rotated about the line x = 4 to form a solid.

Find the volume of the solid.

Question 12 continues on page 9

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4

Question 12 (continued)

(d) The points P cp, c and Q

cq, c

, where p ≠q , lie on the rectangular

hyperbola with equation xy = c2.

The tangent to the hyperbola at P intersects the x-axis at A and the y-axis at B. Similarly, the tangent to the hyperbola at Q intersects the x-axis at C and the y- axis at D.

(i) Show that the equation of the tangent at P is x + p2y = 2cp. (ii) Show that A, B and O are on a circle with centre P. (iii) Prove that BC is parallel to PQ.

End of Question 12

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