Alevel Excel数学 Mathemactics 05_C3_June_2006(2)

6.

(a) Using sin2 ? + cos2 ? ? 1, show that the cosec2 ? – cot2 ? ? 1.

(2)

(b) Hence, or otherwise, prove that

cosec4 ? – cot4 ? ? cosec2 ? + cot2 ?.

(2)

(c) Solve, for 90? < ? < 180?,

cosec4 ? – cot4 ? = 2 – cot ?.

(6)

7.

For the constant k, where k > 1, the functions f and g are defined by

f: x ? ln (x + k), x > –k, g: x ? ?2x – k?, x ? ?.

(a) On separate axes, sketch the graph of f and the graph of g.

On each sketch state, in terms of k, the coordinates of points where the graph meets the coordinate axes.

(5) (b) Write down the range of f.

(1)

?k?(c) Find fg?? in terms of k, giving your answer in its simplest form.

?4?(2)

The curve C has equation y = f(x). The tangent to C at the point with x-coordinate 3 is parallel to the line with equation 9y = 2x + 1.

(d) Find the value of k.

(4)

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8.

(a) Given that cos A = 3, where 270? < A < 360?, find the exact value of sin 2A. 4(5)

??????(b) (i) Show that cos ?2x?? + cos ?2x?? ? cos 2x.

3?3???(3)

Given that

??????y = 3 sin2 x + cos ?2x?? + cos ?2x??,

3?3??? TOTAL FOR PAPER: 75 MARKS

END

(ii) show that

dy = sin 2x. dx (4)

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