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Date November 2015 Marks available 3 Reference code 15N.2.sl.TZ0.3
Level SL only Paper 2 Time zone TZ0
Command term Find Question number 3 Adapted from N/A

Question

Let \(f(x) = 2\ln (x - 3)\), for \(x > 3\). The following diagram shows part of the graph of \(f\).

Find the equation of the vertical asymptote to the graph of \(f\).

[2]
a.

Find the \(x\)-intercept of the graph of \(f\).

[2]
b.

The region enclosed by the graph of \(f\), the \(x\)-axis and the line \(x = 10\) is rotated \(360\)° about the \(x\)-axis. Find the volume of the solid formed.

[3]
c.

Markscheme

valid approach     (M1)

eg\(\;\;\;\)horizontal translation \(3\) units to the right

\(x = 3\) (must be an equation)     A1     N2

[2 marks]

a.

valid approach     (M1)

eg\(\;\;\;f(x) = 0,{\text{ }}{e^0} = x - 3\)

\(4,{\text{ }}x = 4,{\text{ }}(4,{\text{ }}0)\)     A1     N2

[2 marks]

b.

attempt to substitute either their correct limits or the function into formula involving \({f^2}\)      (M1)

eg\(\;\;\;\int_4^{10} {{f^2},{\text{ }}\pi \int {{{\left( {2\ln (x - 3)} \right)}^2}{\text{d}}x} } \)

\(141.537\)

volume = \(142\)     A2     N3

[3 marks]

Total [7 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 6 - Calculus » 6.5 » Volumes of revolution about the \(x\)-axis.
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