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. (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 3D space: 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^3 + 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^3 + 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^3 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

Step 1

Prove that the proposition P is true.

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Answer

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

$k + 1 = 3j$

We can express k as:

$k = 3j - 1$

Next, we calculate k^3 + 1:

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

Expanding this using the binomial theorem gives:

$(3j - 1)^3 + 1 = 27j^3 - 27j^2 + 9j - 1 + 1 = 27j^3 - 27j^2 + 9j$

It is evident that:

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

Since 9j^3 - 9j^2 + 3j is an integer, this shows that k^3 + 1 is divisible by 3. Therefore, we conclude that the proposition P is true.

Step 2

Write down the contrapositive of the proposition P.

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Answer

The contrapositive of the proposition P is:

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

Step 3

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

To evaluate whether this converse is true, consider a counterexample. Let k = 2:

For k = 2, we have:

k + 1 = 3, which is divisible by 3.
k^3 + 1 = 2^3 + 1 = 8 + 1 = 9, which is also divisible by 3.

Now, consider k = 1:

k + 1 = 2, which is not divisible by 3.
k^3 + 1 = 1^3 + 1 = 1 + 1 = 2, which is not divisible by 3.

This demonstrates that the converse is false in general since it does not hold for all integer values of k.

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' - 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^3 + 1 is divisible by 3' - HSC - SSCE Mathematics Extension 2 - Question 15 - 2020 - Paper 1

Prove that the proposition P is true.

Write down the contrapositive of the proposition P.

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

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