2006 Mathematics Paper1
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- Words: 579
- Pages: 22
2006-CE-MATH 1-2
2006-CE-MATH 1-3
2006-CE-MATH 1-4
2006-CE-MATH 1-5
2006-CE-MATH 1-6
2006-CE-MATH 1-7
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)
___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________
2006-CE_MATH 1-8
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)
_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________
(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)
_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________
2006-CE-MATH 1-9
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)
________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________
(b) Find the equation of CM.
Hence, or otherwise, find the coordinates of C.
(3 marks)
________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________
2006-CE-MATH 1-10
(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) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________
2006-CE-MATH 1-11
2006-CE-MATH 1-12
2006-CE-MATH 1-13
2006-CE-MATH 1-14
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2006-CE-MATH 1-15
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2006-CE-MATH 1-16
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2006-CE-MATH 1-17
2006-CE-MATH 1-18
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2006-CE-MATH 1-19
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2006-CE-MATH 1-20
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2006-CE-MATH 1-21
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2006-CE-MATH 1-3
2006-CE-MATH 1-4
2006-CE-MATH 1-5
2006-CE-MATH 1-6
2006-CE-MATH 1-7
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)
___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________
2006-CE_MATH 1-8
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)
_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________
(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)
_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________
2006-CE-MATH 1-9
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)
________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________
(b) Find the equation of CM.
Hence, or otherwise, find the coordinates of C.
(3 marks)
________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________
2006-CE-MATH 1-10
(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) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________
2006-CE-MATH 1-11
2006-CE-MATH 1-12
2006-CE-MATH 1-13
2006-CE-MATH 1-14
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2006-CE-MATH 1-15
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2006-CE-MATH 1-16
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2006-CE-MATH 1-17
2006-CE-MATH 1-18
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2006-CE-MATH 1-19
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2006-CE-MATH 1-20
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2006-CE-MATH 1-21
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