Edexcel A-Level Mathematics AS P2.1.2 Proof By Exhaustion Questions

Practise Edexcel IAL Mathematics P2.1.2 by listing all permitted integer cases, simplifying each expression and concluding only after the complete exhaustion check.

Exam points

complete all table rows for possible a, b and c values before testing abc divisibility
split integers into residue cases such as 3p+1 and 3p+2, then simplify each case
conclude only after every allowed case gives the required multiple or non-multiple result

[Maximum number: 4]

Use proof by exhaustion to prove that if n is an integer then

5 n^{2}+n+12

is always even.

For n odd, let n=2k+1:

5n^2+n+12=5(2k+1)^2+2k+1+12

=20k^2+22k+18=2(10k^2+11k+9)

For n even, let n=2k:

5n^2+n+12=5(2k)^2+2k+12

=20k^2+2k+12=2(10k^2+k+6)

Both expressions are even, so $5n^2+n+12$ is even for all integers n.

M1: starts proof by considering odd or even n.
M1: considers both odd and even cases.
A1: both cases attempted with at least one correct even expression.
A1: complete proof with suitable conclusion.

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Edexcel A-Level Mathematics AS P2.1.2 Proof By Exhaustion Questions question 1