In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^2 + 1 is divisible by 3'. (i) Prove that the proposition P is true. (ii) Write down the contrapositive of the proposition P. (iii) Write down the converse of the proposition P and state, with reasons, whether this converse is true or false.

SSCE HSC Mathematics Extension 2 Geometry proofs using vectors: In the set of integers, let P be

In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^2 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

Question 15

In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^2 + 1 is divisible by 3'. (i) Prove that the proposition P is true. (ii) Wri... show full transcript

Worked Solution & Example Answer:In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^2 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

Step 1

(i) Prove that the proposition P is true.

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Answer

To prove that the proposition P is true, we will assume that $k + 1$ is divisible by 3. This implies that there exists an integer $j$ such that:

$k + 1 = 3j$

From here, we can derive:

$k = 3j - 1$

Now, we will calculate $k^2 + 1$ :

$k^2 + 1 = (3j - 1)^2 + 1$ $= 9j^2 - 6j + 1 + 1$ $= 9j^2 - 6j + 2$

Notice that we can factor this as follows:

$= 3(3j^2 - 2j) + 2$

Now, since the only case when $k + 1$ is divisible by 3 is when $j$ is an integer, we find that $k^2 + 1$ is also divisible by 3, confirming that proposition P is true.

Step 2

(ii) Write down the contrapositive of the proposition P.

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Answer

The contrapositive of the proposition P is:

'If $k^2 + 1$ is not divisible by 3, then $k + 1$ is not divisible by 3.'

Step 3

(iii) Write down the converse of the proposition P and state, with reasons, whether this converse is true or false.

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Answer

The converse of the proposition P is:

'If $k^2 + 1$ is divisible by 3, then $k + 1$ is divisible by 3.'

To determine if this converse is true, consider the case when $k^2 + 1$ is divisible by 3. This does not necessarily mean that $k + 1$ must also be divisible by 3, as shown through a counter-example:

Let $k = 1$ $k = 1$ :
- Then $k^2 + 1 = 1^2 + 1 = 2$ , which is not divisible by 3.
- However, if we consider $k = 2$ , then $k^2 + 1 = 4 + 1 = 5$ , and $k + 1 = 3$ , which is divisible by 3.
Upon checking multiple integers, we find scenarios of contradicting cases. Thus, the converse of proposition P is false.

In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^2 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

Worked Solution & Example Answer:In the set of integers, let P be the proposition: 'If k + 1 is divisible by 3, then k^2 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

(i) Prove that the proposition P is true.

(ii) Write down the contrapositive of the proposition P.

(iii) Write down the converse of the proposition P and state, with reasons, whether this converse is true or false.

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