Date | November 2021 | Marks available | 4 | Reference code | 21N.2.AHL.TZ0.10 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Find | Question number | 10 | Adapted from | N/A |
Question
Consider the function f(x)=x2-x-122x-15, x∈ℝ, x≠152.
Find the coordinates where the graph of f crosses the
x-axis.
y-axis.
Write down the equation of the vertical asymptote of the graph of f.
The oblique asymptote of the graph of f can be written as y=ax+b where a, b∈ℚ.
Find the value of a and the value of b.
Sketch the graph of f for -30≤x≤30, clearly indicating the points of intersection with each axis and any asymptotes.
Express 1f(x) in partial fractions.
Hence find the exact value of 3∫01f(x)d, expressing your answer as a single logarithm.
Markscheme
Note: In part (a), penalise once only, if correct values are given instead of correct coordinates.
attempts to solve (M1)
and A1
[2 marks]
Note: In part (a), penalise once only, if correct values are given instead of correct coordinates.
A1
[1 mark]
A1
Note: Award A0 for .
Award A1 in part (b), if is seen on their graph in part (d).
[1 mark]
METHOD 1
attempts to expand (M1)
A1
equates coefficients of (M1)
A1
METHOD 2
attempts division on M1
M1
A1
A1
METHOD 3
A1
M1
equates coefficients of : (M1)
A1
METHOD 4
attempts division on M1
A1
M1
A1
[4 marks]
two branches with approximately correct shape (for ) A1
their vertical and oblique asymptotes in approximately correct positions with both branches showing correct asymptotic behaviour to these asymptotes A1
their axes intercepts in approximately the correct positions A1
Note: Points of intersection with the axes and the equations of asymptotes are not required to be labelled.
[3 marks]
attempts to split into partial fractions: (M1)
A1
A1
[3 marks]
attempts to integrate and obtains two terms involving ‘ln’ (M1)
A1
A1
A1
Note: The final A1 is dependent on the previous two A marks.
[4 marks]