(i)     Find the value of $c$.

(ii)     Show that $b = \frac{\pi }{6}$.

(iii)     Find the value of $a$.

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

(i)     Find $w$.

(ii)     Hence or otherwise, find the maximum positive rate of change of $g$.

(i)     valid approach     (M1)

eg$\,\,\,\,\,$$\frac{{5 + 17}}{2}$

$c = 11$    A1     N2

(ii)     valid approach     (M1)

eg$\,\,\,\,\,$period is 12, per $= \frac{{2\pi }}{b},{\text{ }}9 - 3$

$b = \frac{{2\pi }}{{12}}$    A1

$b = \frac{\pi }{6}$     AG     N0

(iii)     METHOD 1

valid approach     (M1)

eg$\,\,\,\,\,$$5 = a\sin \left( {\frac{\pi }{6} \times 3} \right) + 11$, substitution of points

$a =  - 6$     A1     N2

METHOD 2

valid approach     (M1)

eg$\,\,\,\,\,$$\frac{{17 - 5}}{2}$, amplitude is 6

$a =  - 6$     A1     N2

[6 marks]

(i)     $k = 2.5$     A1     N1

(ii)     $g(x) =  - 6\sin \left( {\frac{\pi }{6}(x - 2.5)} \right) + 11$     A2     N2

[3 marks]

(i)     METHOD 1 Using $g$

recognizing that a point of inflexion is required     M1

eg$\,\,\,\,\,$sketch, recognizing change in concavity

evidence of valid approach     (M1)

eg$\,\,\,\,\,$$g''(x) = 0$, sketch, coordinates of max/min on ${g'}$

$w = 8.5$ (exact)     A1     N2

METHOD 2 Using $f$

recognizing that a point of inflexion is required     M1

eg$\,\,\,\,\,$sketch, recognizing change in concavity

evidence of valid approach involving translation     (M1)

eg$\,\,\,\,\,$$x = w - k$, sketch, $6 + 2.5$

$w = 8.5$ (exact)     A1     N2

(ii)     valid approach involving the derivative of $g$ or $f$ (seen anywhere)     (M1)

eg$\,\,\,\,\,$$g'(w),{\text{ }} - \pi \cos \left( {\frac{\pi }{6}x} \right)$, max on derivative, sketch of derivative

attempt to find max value on derivative     M1

eg$\,\,\,\,\,$$- \pi \cos \left( {\frac{\pi }{6}(8.5 - 2.5)} \right),{\text{ }}f'(6)$, dot on max of sketch

3.14159

max rate of change $= \pi$ (exact), 3.14     A1     N2

[6 marks]