2020HSCShort answer

Q15 (a) (i)

2 marks

Question

In the set of integers, let $P$ be the proposition: “If $k+1$ is divisible by $3$ , then $k^3+1$ is divisible by $3”.

(i) Prove that the proposition $P$ is true.

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct proof	2
Factors $k^3+1$ <br>OR<br>Writes $k+1=3j$ , or equivalent merit	1

Sample answer:

Suppose $k+1$ is divisible by 3, then $k+1=3j$ , for some integer $j$ .

k^3+1=(k+1)(k^2-k+1)

=3j(k^2-k+1)

Hence $k^3+1$ is divisible by 3.

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Syllabus alignment

Working MathematicallyProof

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

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