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Date None Specimen Marks available 5 Reference code SPNone.2.sl.TZ0.9
Level SL only Paper 2 Time zone TZ0
Command term Find and Write down Question number 9 Adapted from N/A

Question

Let \(h(x) = \frac{{2x - 1}}{{x + 1}}\) , \(x \ne - 1\) .

Find \({h^{ - 1}}(x)\) .

[4]
a.

(i)     Sketch the graph of h for \( - 4 \le x \le 4\) and \( - 5 \le y \le 8\) , including any asymptotes.

(ii)    Write down the equations of the asymptotes.

(iii)   Write down the x-intercept of the graph of h .

[7]
b(i), (ii) and (iii).

Let R be the region in the first quadrant enclosed by the graph of h , the x-axis and the line \(x = 3\).

(i)     Find the area of R.

(ii)    Write down an expression for the volume obtained when R is revolved through \({360^ \circ }\) about the x-axis.

[5]
c(i) and (ii).

Markscheme

\(y = \frac{{2x - 1}}{{x + 1}}\)

interchanging x and y (seen anywhere)     M1

e.g. \(x = \frac{{2y - 1}}{{y + 1}}\)

correct working     A1

e.g. \(xy + x = 2y - 1\)

collecting terms     A1

e.g. \(x + 1 = 2y - xy\) , \(x + 1 = y(2 - x)\)

\({h^{ - 1}}(x) = \frac{{x + 1}}{{2 - x}}\)     A1     N2

[4 marks]

a.


     A1A1A1A1     N4

Note: Award A1 for approximately correct intercepts, A1 for correct shape, A1 for asymptotes, A1 for approximately correct domain and range.

(ii) \(x = - 1\) , \(y = 2\)     A1A1     N2

(iii) \(\frac{1}{2}\)     A1     N1

[7 marks]

b(i), (ii) and (iii).

(i) \({\text{area}} = 2.06\)     A2     N2

(ii) attempt to substitute into volume formula (do not accept \(\pi \int_a^b {{y^2}{\rm{d}}x} \) )     M1

volume \( = \pi {\int_{\frac{1}{2}}^3 {\left( {\frac{{2x - 1}}{{x + 1}}} \right)} ^2}{\rm{d}}x\)     A2     N3

[5 marks]

c(i) and (ii).

Examiners report

[N/A]
a.
[N/A]
b(i), (ii) and (iii).
[N/A]
c(i) and (ii).

Syllabus sections

Topic 6 - Calculus » 6.5 » Areas under curves (between the curve and the \(x\)-axis).
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