Proof

WMA14Topic 4.12 questions

Proof by contradiction.

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Original practice questions written in the style of the exam.

Question 15 marks
Use proof by contradiction to prove that there are no positive integers $x$ and $y$ such that $x^2 - y^2 = 10$ .
Mark scheme
- B1 Assumes there exist positive integers $x$ , $y$ with $x^2 - y^2 = 10$ (1)

10

M1 Factorises:

(x - y)(x + y) = 10

(1)

M1 Considers the factor pairs,

x - y = 1, x + y = 10

or

x - y = 2, x + y = 5

(1)

M1 Shows each pair gives non-integer

x

(e.g.

x = \tfrac{11}{2}

,

x = \tfrac{7}{2}

), or notes

x - y

and

x + y

must have the same parity (1)

A1 Contradiction stated, so no such positive integers exist (1)