Date | May 2022 | Marks available | 7 | Reference code | 22M.1.AHL.TZ2.11 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 2 |
Command term | State | Question number | 11 | Adapted from | N/A |
Question
A function is defined by , where .
A function is defined by , where .
The inverse of is .
A function is defined by , where .
Sketch the curve , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
Show that .
State the domain of .
Given that , find the value of .
Give your answer in the form , where .
Markscheme
-intercept A1
Note: Accept an indication of on the -axis.
vertical asymptotes and A1
horizontal asymptote A1
uses a valid method to find the -coordinate of the local maximum point (M1)
Note: For example, uses the axis of symmetry or attempts to solve .
local maximum point A1
Note: Award (M1)A0 for a local maximum point at and coordinates not given.
three correct branches with correct asymptotic behaviour and the key features in approximately correct relative positions to each other A1
[6 marks]
M1
Note: Award M1 for interchanging and (this can be done at a later stage).
EITHER
attempts to complete the square M1
A1
A1
OR
attempts to solve for M1
A1
Note: Award A1 even if (in ) is missing
A1
THEN
A1
and hence is rejected R1
Note: Award R1 for concluding that the expression for must have the ‘’ sign.
The R1 may be awarded earlier for using the condition .
AG
[6 marks]
domain of is A1
[1 mark]
attempts to find (M1)
(A1)
attempts to solve for M1
EITHER
A1
attempts to find their M1
A1
Note: Award all available marks to this stage if is used instead of .
OR
A1
attempts to solve their quadratic equation M1
A1
Note: Award all available marks to this stage if is used instead of .
THEN
(as ) A1
Note: Award A1 for
[7 marks]
Examiners report
Part (a) was generally well done. It was pleasing to see how often candidates presented complete sketches here. Several decided to sketch using the reciprocal function. Occasionally, candidates omitted the upper branches or forgot to calculate the y-coordinate of the maximum.
Part (b): The majority of candidates knew how to start finding the inverse, and those who attempted completing the square or using the quadratic formula to solve for y made good progress (both methods equally seen). Otherwise, they got lost in the algebra. Very few explicitly justified the rejection of the negative root.
Part (c) was well done in general, with some algebraic errors seen in occasions.