Moderate -0.5 This is a straightforward proof by contradiction requiring students to assume x+y is rational, manipulate the equation to show x would then be rational (contradicting the given), and conclude. The logic is direct with minimal steps, making it easier than average, though it does require understanding proof structure and basic algebraic manipulation of rationals.
Suppose \(x\) is an irrational number, and \(y\) is a rational number, so that \(y = \frac{m}{n}\), where \(m\) and \(n\) are integers and \(n \neq 0\).
Prove by contradiction that \(x + y\) is not rational. [4]
So x(cid:14) y(cid:32) , where p and q are integers
q
(cid:11)pn–mq(cid:12)
p m
(cid:159) x(cid:32) – (cid:32) which is rational
q n qn
Answer
Marks
x is irrational so this is a contradiction
E1
B1
B1
c
E1
Answer
Marks
[4]
m2.1
2.1
i3.1a
Answer
Marks
2.4
e
or stating that the difference of two
fractions is rational
Answer
Marks
Guidance
11
0
3
12 a
3
1
12 b
1
1
Question 11:
11 | Suppose x + y is rational
p
So x(cid:14) y(cid:32) , where p and q are integers
q
(cid:11)pn–mq(cid:12)
p m
(cid:159) x(cid:32) – (cid:32) which is rational
q n qn
x is irrational so this is a contradiction | E1
B1
B1
c
E1
[4] | m2.1
2.1
i3.1a
2.4 | e
or stating that the difference of two
fractions is rational
11 | 0 | 3 | 1 | 0 | 4 | 0
12 a | 3 | 1 | 0 | 0 | 4 | 0
12 b | 1 | 1 | 0 | 0 | 2 | 0
Suppose $x$ is an irrational number, and $y$ is a rational number, so that $y = \frac{m}{n}$, where $m$ and $n$ are integers and $n \neq 0$.
Prove by contradiction that $x + y$ is not rational. [4]
\hfill \mbox{\textit{OCR MEI Paper 2 Q11 [4]}}