Date | May 2021 | Marks available | 9 | Reference code | 21M.1.AHL.TZ2.12 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 2 |
Command term | Prove | Question number | 12 | Adapted from | N/A |
Question
The following diagram shows the graph of for , with asymptotes at and .
Describe a sequence of transformations that transforms the graph of to the graph of for .
Show that where and .
Verify that for .
Using mathematical induction and the result from part (b), prove that for .
Markscheme
EITHER
horizontal stretch/scaling with scale factor
Note: Do not allow ‘shrink’ or ‘compression’
followed by a horizontal translation/shift units to the left A2
Note: Do not allow ‘move’
OR
horizontal translation/shift unit to the left
followed by horizontal stretch/scaling with scale factor A2
THEN
vertical translation/shift up by (or translation through A1
(may be seen anywhere)
[3 marks]
let and M1
and (A1)
A1
A1
so where and . AG
[4 marks]
METHOD 1
(or equivalent) A1
A1
A1
AG
METHOD 2
(or equivalent) A1
Consider
A1
A1
AG
METHOD 3
(or equivalent) A1
A1
A1
[3 marks]
let be the proposition that for
consider
when and so is true R1
assume is true, ie. M1
Note: Award M0 for statements such as “let ”.
Note: Subsequent marks after this M1 are independent of this mark and can be awarded.
consider :
(M1)
A1
M1
A1
Note: Award A1 for correct numerator, with factored. Denominator does not need to be simplified
A1
Note: Award A1 for denominator correctly expanded. Numerator does not need to be simplified. These two A marks may be awarded in any order
A1
Note: The word ‘arctan’ must be present to be able to award the last three A marks
is true whenever is true and is true, so
is true for for R1
Note: Award the final R1 mark provided at least four of the previous marks have been awarded.
Note: To award the final R1, the truth of must be mentioned. ‘ implies ’ is insufficient to award the mark.
[9 marks]