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Date	May 2019	Marks available	1	Reference code	19M.1.AHL.TZ2.H_11
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 2
Command term	State	Question number	H_11	Adapted from	N/A

Question

Consider the functions $f$ and $g$ defined by $f\left( x \right) = {\text{ln}}\left| x \right|$ , $x \in \mathbb{R}$ \ $\left\{ 0 \right\}$ , and $g\left( x \right) = {\text{ln}}\left| {x + k} \right|$ , $x \in \mathbb{R}$ \ $\left\{ { - k} \right\}$ , where $k \in \mathbb{R}$ , $k > 2$ .

The graphs of $f$ and $g$ intersect at the point P .

Describe the transformation by which $f\left( x \right)$ is transformed to $g\left( x \right)$ .

[1]

State the range of $g$ .

[1]

Sketch the graphs of $y = f\left( x \right)$ and $y = g\left( x \right)$ on the same axes, clearly stating the points of intersection with any axes.

[6]

Find the coordinates of P.

[2]

The tangent to $y = f\left( x \right)$ at P passes through the origin (0, 0).

Determine the value of $k$ .

[7]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

translation $k$ units to the left (or equivalent) A1

[1 mark]

range is $\left( {g\left( x \right) \in } \right)\mathbb{R}$ A1

[1 mark]

correct shape of $y = f\left( x \right)$ A1

their $f\left( x \right)$ translated $k$ units to left (possibly shown by $x = - k$ marked on $x$ -axis) A1

asymptote included and marked as $x = - k$ A1

$f\left( x \right)$ intersects $x$ -axis at $x = - 1$ , $x = 1$ A1

$g\left( x \right)$ intersects $x$ -axis at $x = - k - 1$ , $x = - k + 1$ A1

$g\left( x \right)$ intersects $y$ -axis at $y = {\text{ln}}\,k$ A1

Note: Do not penalise candidates if their graphs “cross” as $x \to \pm \infty$ .

Note: Do not award FT marks from the candidate’s part (a) to part (c).

[6 marks]

at P ${\text{ln}}\left( {x + k} \right) = {\text{ln}}\left( { - x} \right)$

attempt to solve $x + k = - x$ (or equivalent) (M1)

$x = - \frac{k}{2} \Rightarrow y = {\text{ln}}\left( {\frac{k}{2}} \right)\,\,$ (or $y = {\text{ln}}\left| {\frac{k}{2}} \right|$ ) A1

P $\left( { - \frac{k}{2},\,\,{\text{ln}}\frac{k}{2}} \right)$ (or P $\left( { - \frac{k}{2},\,\,{\text{ln}}\left| {\frac{k}{2}} \right|} \right)$ )

[2 marks]

attempt to differentiate ${\text{ln}}\left( { - x} \right)$ or ${\text{ln}}\left| x \right|$ (M1)

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{1}{x}$ A1

at P, $\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{ - 2}}{k}$ A1

recognition that tangent passes through origin $\Rightarrow \frac{y}{x} = \frac{{{\text{d}}y}}{{{\text{d}}x}}$ (M1)

$\frac{{{\text{ln}}\left| {\frac{k}{2}} \right|}}{{ - \frac{k}{2}}} = \frac{{ - 2}}{k}$ A1

${\text{ln}}\left( {\frac{k}{2}} \right)\, = 1$ (A1)

$\Rightarrow k = 2{\text{e}}$ A1

Note: For candidates who explicitly differentiate ${\text{ln}}\left( {x} \right)$ (rather than ${\text{ln}}\left( { - x} \right)$ or ${\text{ln}}\left| x \right|$ , award M0A0A1M1A1A1A1.

[7 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » AHL 2.8—Transformations of graphs, composite transformations

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