C3 June 2006 Q7

EdexcelOld spec12 marksDifferentiation Functions (including |mod|)

7. For the constant \(k\), where \(k \gt 1\), the functions f and g are defined by\[\begin{aligned}&\mathrm{f} : x \mapsto \ln(x + k), &&x \gt -k,\\&\mathrm{g} : x \mapsto |2x - k|, &&x \in \mathbb{R}.\end{aligned}\]

(a) On separate axes, sketch the graph of f and the graph of g.
On each sketch state, in terms of \(k\), the coordinates of points where the graph meets the coordinate axes. (5)

(b) Write down the range of f. (1)

The curve \(C\) has equation \(y = \mathrm{f}(x)\). The tangent to \(C\) at the point with \(x\)-coordinate 3 is parallel to the line with equation \(9y = 2x + 1\).

(d) Find the value of \(k\). (4)

Sketch: log curve with vertical asymptote at x = -k, crossing the x-axis at (1 - k, 0) and the y-axis at (0, ln k) — Mark scheme (a)

Sketch: V-shaped graph through (0, k) with vertex on the x-axis at (k/2, 0) — Mark scheme (a)

Scheme	Marks
Log graph: Shape	B1
Intersection with \(-\)ve \(x\)-axis	dB1
\((0, \ln k)\), \((1 - k, 0)\)	B1
Mod graph: V shape, vertex on \(+\)ve \(x\)-axis	B1
\((0, k)\) and \(\left(\dfrac{k}{2}, 0\right)\)	B1
	(5)

Scheme	Marks
\(\mathrm{f}(x) \in \mathbb{R}\), \(-\infty \lt \mathrm{f}(x) \lt \infty\), \(-\infty \lt y \lt \infty\)	B1
	(1)

Mark scheme (c)

Scheme	Marks
\(\mathrm{fg}\left(\dfrac{k}{4}\right) = \ln\left\{k + \left\|\dfrac{2k}{4} - k\right\|\right\}\) or \(\mathrm{f}\left(\left\|-\dfrac{k}{2}\right\|\right)\)	M1
\(= \ln\left(\dfrac{3k}{2}\right)\)	A1
	(2)

Mark scheme (d)

Scheme	Marks
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1}{x + k}\)	B1
Equating (with \(x = 3\)) to grad. of line; \(\quad\dfrac{1}{3 + k} = \dfrac{2}{9}\)	M1; A1
\(k = 1\tfrac{1}{2}\)	A1ft
	(4)
	(12 marks)