Date | May 2022 | Marks available | 5 | Reference code | 22M.1.AHL.TZ2.9 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 2 |
Command term | Prove | Question number | 9 | Adapted from | N/A |
Question
Prove by contradiction that the equation has no integer roots.
Markscheme
METHOD 1 (rearranging the equation)
assume there exists some such that M1
Note: Award M1 for equivalent statements such as ‘assume that is an integer root of ’. Condone the use of throughout the proof.
Award M1 for an assumption involving .
Note: Award M0 for statements such as “let’s consider the equation has integer roots…” ,“let be a root of …”
Note: Subsequent marks after this M1 are independent of this M1 and can be awarded.
attempts to rearrange their equation into a suitable form M1
EITHER
A1
is even R1
which is not even and so cannot be an integer R1
Note: Accept ‘ which gives a contradiction’.
OR
A1
R1
is even which is not true and so cannot be an integer R1
Note: Accept ‘ is even which gives a contradiction’.
OR
A1
R1
is is not an integer and so cannot be an integer R1
Note: Accept ‘ is not an integer which gives a contradiction’.
OR
A1
R1
is not an integer and so cannot be an integer R1
Note: Accept is not an integer which gives a contradiction’.
THEN
so the equation has no integer roots AG
METHOD 2
assume there exists some such that M1
Note: Award M1 for equivalent statements such as ‘assume that is an integer root of ’. Condone the use of throughout the proof. Award M1 for an assumption involving and award subsequent marks based on this.
Note: Award M0 for statements such as “let’s consider the equation has integer roots…” ,“let be a root of …”
Note: Subsequent marks after this M1 are independent of this M1 and can be awarded.
let (and )
for all is a (strictly) increasing function M1A1
and R1
thus has only one real root between and , which gives a contradiction
(or therefore, contradicting the assumption that for some ), R1
so the equation has no integer roots AG
[5 marks]