f: x |→ ex + 2 , x ∈ ℜ
g : x |→ ln x , x > 0
(a) State the range of f.
(b) Find fg(x) , giving your answer in its simplest form.
(c) Find the exact value of x for which f(2x +3) = 6
(d) Find f−1 , the inverse function of f, stating its domain.
(e) On the same axes sketch the curves with equation y = f(x) and y = f-1(x) coordinates of all the points where the curves cross the axes.
(ii) (1 - 10x)/(2x - 1)5 giving your answer in its simplest form.
(b) Given that y = 3 tan 2y, find dy/dx, in terms of x.7 (a)(i) Product Rule/Chain Rule
Given that f(x) = R cos(2x + α) , where R > 0 and 0 < α < 90°,
(a) find the value of R and the value of α.
(b) Hence solve the equation
f(x) = 7 cos 2x - 24 sin 2x = 12.5
for 0 ≤ x < 180° , giving your answers to 1 decimal place.
(c) Express 14 cos2x - 14 sinx cosx in the form a cos 2x + b sin 2x + c, where a, b, and c are constants to be found.
(d) Hence, using your answers to parts (a) and (c), deduce the maximum value of
14 cos2x - 14 sinx cos
8 (a) Rcos( ) method
The curve passes through the points P(−1.5, 0) and Q(0, 5P) as shown.
On separate diagrams, sketch the curve with equation
(a) y = |f(x)|
(b) y = f(|x|)
(c) y = 2f(3x)
Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.C3 Edexcel Core Mathematics June 2012 Question 10
(b) Hence show that
4cosec22θ cosec2θ = sec2θ
(c) Hence or otherwise solve, for 0 < θ < π,
4cosec22θ cosec2θ = 4
giving your answers in terms of π.
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