Sketch the graph of $y = f(x)$, stating the coordinates of any maximum and minimum points and points of intersection with the x-axis.

Solve the inequality $\ln x \leqslant {{\text{e}}^{\cos x}},{\text{ }}0 < x \leqslant 10$.

A correct graph shape for $0 < x \leqslant 10$     A1

maxima (3.78, 0.882) and (9.70, 1.89)     A1

minimum (6.22, –0.885)     A1

x-axis intercepts (1.97, 0), (5.24, 0) and (7.11, 0)     A2

Note:     Award A1 if two x-axis intercepts are correct.

[5 marks]

$0 < x \leqslant 1.97$     A1

$5.24 \leqslant x \leqslant 7.11$     A1

[2 marks]

Part (a) was reasonably well done although more care was required when showing correct endpoint behaviour. A number of sketch graphs suggested the existence of either a vertical axis intercept or displayed an open circle on the vertical axis. A large number of candidates did not state the coordinates of the various key features correct to three significant figures. A large number of candidates did not locate the maximum near $x = 10$. Most candidates were able to locate the x-axis intercepts and the minimum. A few candidates unfortunately sketched a graph from a GDC set in degrees.

In part (b), a number of candidates identified the correct critical values but used incorrect inequality signs. Some candidates attempted to solve the inequality algebraically.