Write down the value of $q$;

Write down the value of $h$;

Write down the value of $k$.

Find $\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x}$.

Hence, find the area of the region enclosed by the graphs of $h$ and ${h^{ - 1}}$.

Let $d$ be the vertical distance from a point on the graph of $h$ to the line $y = x$. There is a point ${\text{P}}(a,{\text{ }}b)$ on the graph of $h$ where $d$ is a maximum.

Find the coordinates of P, where $0.111 < a < 3.31$.

$q = 2$     A1     N1

Note:     Accept $q = 1$, $h = 0$, and $k = 3 - \ln (2)$, 2.31 as candidate may have rewritten $g(x)$ as equal to $3 + \ln (x) - \ln (2)$.

[1 mark]

$h = 0$     A1     N1

Note:     Accept $q = 1$, $h = 0$, and $k = 3 - \ln (2)$, 2.31 as candidate may have rewritten $g(x)$ as equal to $3 + \ln (x) - \ln (2)$.

[1 mark]

$k = 3$     A1     N1

Note:     Accept $q = 1$, $h = 0$, and $k = 3 - \ln (2)$, 2.31 as candidate may have rewritten $g(x)$ as equal to $3 + \ln (x) - \ln (2)$.

[1 mark]

2.72409

2.72     A2     N2

[2 marks]

recognizing area between $y = x$ and $h$ equals 2.72     (M1)

eg$\,\,\,\,\,$

recognizing graphs of $h$ and ${h^{ - 1}}$ are reflections of each other in $y = x$     (M1)

eg$\,\,\,\,\,$area between $y = x$ and $h$ equals between $y = x$ and ${h^{ - 1}}$

$2 \times 2.72\int_{0.111}^{3.31} {\left( {x - {h^{ - 1}}(x)} \right){\text{d}}x = 2.72}$

5.44819

5.45     A1     N3

[??? marks]

valid attempt to find $d$     (M1)

eg$\,\,\,\,\,$difference in $y$-coordinates, $d = h(x) - x$

correct expression for $d$     (A1)

eg$\,\,\,\,\,$$\left( {\ln \frac{1}{2}x + 3} \right)(\cos 0.1x) - x$

valid approach to find when $d$ is a maximum     (M1)

eg$\,\,\,\,\,$max on sketch of $d$, attempt to solve $d’ = 0$

0.973679

$x = 0.974$     A2     N4 

substituting their $x$ value into $h(x)$     (M1)

2.26938

$y = 2.27$     A1     N2

[7 marks]