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Edexcel Core Mathematics C3 June 2009

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C3 Mathematics Edexcel June 2009 Question 5

Figure 2 shows a sketch of part of the curve with equation y = f(x), x ∈ ℝ.
The curve meets the coordinate axes at the points A(0,1–k) and B(½ln k, 0)
where k is a constant and k > 1, as shown in Figure 2.
On separate diagrams, sketch the curve with equation
(a) y = |f(x)|
(b) y = f-1(x)

Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes.
Given that f(x) = e2x - k
(c) state the range of f,
(d) find f-1(x)
(e) write down the domain of f-1.

C3 Mathematics Edexcel June 2009 Question 6

6. (a) Use the identity cos(A + B) = cos A cos B - sin A sin B to show that
cos 2A = 1 - 2 sin2 A
The curves C1 and C2 have equations
C1: y = 3 sin 2x
C2: y = 4 sin2 x - 2 cos 2x
(b) Show that the x-coordinates of the points where C1 and C2 intersect satisfy the equation
4 cos 2x + 3 sin 2x = 2
(c) Express 4cos 2x + 3 sin 2x in the form R cos (2x – α), where R > 0 and 0 < α < 90°,
giving the value of α to 2 decimal places.
(d) Hence find, for 0 ≤ x ≤ 180°, all the solutions of
4 cos 2x + 3 sin 2x = 2
giving your answers to 1 decimal place.

C3 Mathematics Edexcel June 2009 Question 7

7. The function f is defined by

f(x) = 1 - 2/(x + 4) + (x - 8)/(x - 2)(x + 4), x ∈ ℝ , x ≠ -4, x ≠ 2
(a) Show that f(x) = (x - 3)/(x - 2)

The function g is defined by
g(x) = (ex - 3)(ex - 2)
(b) Differentiate g(x) to show that g'(x) = ex/(ex - 2)
(c) Find the exact values of x for which g'(x) = 1

8. (a) Write down sin 2x in terms of sin x and cos x
(b) Find, for 0 < x < π, all the solutions of the equation
cosec x - 8cos x = 0
giving your answers to 2 decimal places.

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