(a)     On the same set of axes draw, on graph paper, the graphs, for $0 \leqslant x \leqslant \frac{{5\pi }}{4}$. Use a scale of $1$ cm to $\frac{\pi }{8}$ on your $x$-axis and $5$ cm to $1$ unit on your $y$-axis.

(b)     Show that the $x$-intercepts of the graph $y = {{\text{e}}^{ - x}}$ sin 4x are $\frac{{n\pi }}{4}$ , $n = 0$, $1$, $2$, $3$, $4$, $5$.

(c)     Find the $x$-coordinates of the points at which the graph of $y = {{\text{e}}^{ - x}}\sin 4x$ meets the graph of $y = {{\text{e}}^{ - x}}$ . Give your answers in terms of $\pi$.

(d)     (i)     Show that when the graph of $y = {{\text{e}}^{ - x}}\sin 4x$ meets the graph of $y = {{\text{e}}^{ - x}}$ , their gradients are equal.

  (ii)     Hence explain why these three meeting points are not local maxima of the graph $y = {{\text{e}}^{ - x}}\sin 4x$ .

(e)     (i)     Determine the $y$-coordinates, ${y_1}$ , ${y_2}$ and ${y_3}$, where ${y_1} > {y_2} > {y_3}$ , of the local maxima of $y = {{\text{e}}^{ - x}}\sin 4x$ for $0 \leqslant x \leqslant \frac{{5\pi }}{4}$ . You do not need to show that they are maximum values, but the values should be simplified.

  (ii)     Show that ${y_1}$ , ${y_2}$ and ${y_3}$ form a geometric sequence and determine the common ratio $r$ .

(a)

     A3

Note: Award A1 for each correct shape,

   A1 for correct relative position.

[3 marks]

(b)     ${{\text{e}}^{ - x}}\sin \left( {4x} \right) = 0$     (M1)

$\sin \left( {4x} \right) = 0$     A1

$4x = 0$, $\pi$, $2\pi$, $3\pi$, $4\pi$, $5\pi$     A1

$x = 0$, $\frac{\pi }{4}$, $\frac{2\pi }{4}$, $\frac{3\pi }{4}$, $\frac{4\pi }{4}$, \(\frac{5\pi }{4}\)     AG

[3 marks]

(c)     ${{\text{e}}^{ - x}} = {{\text{e}}^{ - x}}\sin \left( {4x} \right) = 0$ or reference to graph

$\sin 4x = 1$,      M1

$\sin 4x = 1$, $frac{\pi }{2}$, $frac{5\pi }{2}$, $frac{9\pi }{2}$     A1

Although the final question on the paper it had parts accessible even to the weakest candidates. The vast majority of candidates earned marks on part (a), although some graphs were rather scruffy. Many candidates also tackled parts (b), (c) and (d). In part (b), however, as the answer was given, it should have been clear that some working was required rather than reference to a graph, which often had no scale indicated. In part d(i), although the functions were usually differentiated correctly, it was often the case that only one point was checked for the equality of the gradients. In part e(i) many candidates who got this far were able to determine the $y$-coordinates of the local maxima numerically using a GDC, and that was given credit. Only the exact values, however, could be used in part e(ii).