2006 Mathematics Paper1

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Page total Section A(2) (33 Marks) Answer ALL questions in this section and write your answers in the spaces provided.

10. Let f(x) = (x –a) (x – b) (x + 1) – 3 , where a and b are positive integers with a<b . It is given that f(1) = 1. (a)

(i) Prove that (a – 1) (b – 1) = 2 (ii) Write down the values of a and b.

(3 marks)

___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ (b)

Let g(x) = x3 – 6x2 – 2x +7 . Using the results of (a)(ii), find f(x) – g(x). Hence find the exact values of all the roots of the equation f(x) = g(x) .

(4 marks)

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11. In Figure 5, ABCDEF is a thin six-sided polygonal metal sheet, where all the measurements are correct to the nearest cm.

(a) Write down the maximum absolute error of the measurements.

(1 mark)

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(b) Find the least possible area of the metal sheet.

(3 marks)

_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ (c) The actual area of the metal sheet is x cm2 . Find the range of values of x.

(4 marks)

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12. In Figure 6, CM is the perpendicular bisector of AB, where C and M are points lying on the x-axis and AB respectively.

BD and CM intersect at K.

(a) Write down the coordinates of M.

(1 mark)

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(b) Find the equation of CM.

Hence, or otherwise, find the coordinates of C.

(3 marks)

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(c) (i)

Find the equation of BD.

(ii) Using the result of (c)(i), find the coordinates of K.

Hence find the ratio of the area of

△ AMC to the area of △AKC. (5 marks) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________

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