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.
7. The function f is defined by
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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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