(a)     Sketch the curve \(y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1\) , \(0 < x < 4\) showing clearly the coordinates of the points of intersection with the x-axis and the coordinates of any local maxima and minima.

(b)     Find the values of x for which \(\left| {\ln x} \right| > \left| {\cos x} \right| + 0.1\), \(0 < x < 4\) .

(a)

     A1

Note: Award A1 for shape.

x-intercepts 0.354, 1.36, 2.59, 2.95     A2

Note: Award A1 for three correct, A0 otherwise.

maximum = (1.57, 0.352) = \(\left( {\frac{\pi }{2},0.352} \right)\)     A1

minimum = (1, – 0.640) and (2.77, – 0.0129)     A1

(b)     \(0 < x < 0.354,{\text{ }}1.36 < x < 2.59,{\text{ }}2.95 < x < 4\)     A2

Note: Award A1 if two correct regions given.

[7 marks]

Solutions to this question were extremely disappointing with many candidates doing the sketch in degree mode instead of radian mode. The two adjacent intercepts at 2.59 and 2.95 were often missed due to an unsatisfactory window. Some sketches were so small that a magnifying glass was required to read some of the numbers; candidates would be well advised to draw sketches large enough to be easily read.