2023TrialShort answer

Q11 (c)

2 marks

Question

Given $a\in\mathbb{Z}$ , prove that if $3a^2-4a+5$ is even, then $a$ is odd.

Try the question before revealing the answer.

Solution

Consider the equivalent contrapositive:

If $a$ even, then $3a^2-4a+5$ is odd.

Let $a=2b$ , $b\in\mathbb{Z}$ .

\begin{aligned} 3a^2-4a+5&=3(2b)^2-4(2b)+5\\ &=2(6b^2-4b+2)+1\\ &=2M+1,\text{ where }M=6b^2-4b+2 \end{aligned}

$M$ is an integer.

$\therefore\ 3a^2-4a+5$ is odd.

$\therefore$ original statement is true, since the equivalent contrapositive is true.

Supplied marking guidance · human reviewed

Compare this with your working, not only your final sentence.

Syllabus alignment

Proof

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.

Spot a problem? Send feedback