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AusGrader has 11 QCAA Specialist Mathematics Alternative Sequence questions on Introduction to proof, all with instant AI grading and detailed marking feedback.

QCAA Specialist Mathematics Alternative Sequence — Introduction to proof Questions & Answers

Q6

2022

QCAA

Paper 1

1 mark

Q6

1 mark

Consider the following statement: 'If the angles are congruent, then the measures of the angles are equal'.

The contrapositive of this statement is

A

If the measures of the angles are equal, then the angles are congruent.

B

If the angles are not congruent, then the measures of the angles are equal.

C

If the measures of the angles are not equal, then the angles are not congruent.

D

If the angles are not congruent, then the measures of the angles are not equal.

Reveal Answer

A

If the measures of the angles are equal, then the angles are congruent.

This is the converse of the original statement, formed by swapping the hypothesis and conclusion ( $Q \implies P$ ) rather than negating and swapping them.

B

If the angles are not congruent, then the measures of the angles are equal.

This statement negates the hypothesis but keeps the conclusion the same ( $\neg P \implies Q$ ), which is not a valid logical transformation of the original statement.

C

If the measures of the angles are not equal, then the angles are not congruent.

Correct Answer

The contrapositive of a conditional statement ( $P \implies Q$ ) is formed by negating both the hypothesis and the conclusion, and then swapping them ( $\neg Q \implies \neg P$ ).

D

If the angles are not congruent, then the measures of the angles are not equal.

This is the inverse of the original statement, formed by negating both the hypothesis and conclusion without swapping their order ( $\neg P \implies \neg Q$ ).

Q15

2020

QCAA

Paper 2

5 marks

Q15b

A rational number can be expressed in the form $\frac{a}{b}$ where $a, b \in \mathbb{Z}$ and $b \neq 0$ .

Q15a

1 mark

Calculate $\frac{670}{3333}$

Express the result as a recurring decimal.

Reveal Answer

\frac{670}{3333} = 0.\overline{2010}

Marking Criteria

Descriptor	Marks
correctly expresses fraction as a recurring decimal	1

Q15b

2 marks

Show that the recurring decimal $8.7\overline{64}$ is a rational number.

Reveal Answer

\text{Let } x = 8.7\overline{64} \quad (1)

100x = 876.4\overline{64} \quad (2)

\text{By subtraction:}

99x = 867.7

x = \frac{867.7}{99}

8.7\overline{64} = \frac{8677}{990}

Marking Criteria

Descriptor	Marks
correctly determines an equation to represent $8.7\overline{64}$ without recurring decimals	1
expresses $8.7\overline{64}$ in the form $\frac{a}{b}$ where $a, b \in \mathbb{Z}$	1

Q15c

2 marks

Prove that the difference of any two rational numbers produces another rational number.

Reveal Answer

\text{Let the two rational numbers be } \frac{p}{q} \text{ and } \frac{r}{s} \text{ where } p, q, r, s \in \mathbb{Z} \text{ and } q, s \neq 0

\frac{p}{q} - \frac{r}{s} = \frac{ps - qr}{qs}

\text{Since } ps - qr \in \mathbb{Z} \text{ and } qs \in \mathbb{Z} \text{, the difference can be expressed in the form } \frac{a}{b} \text{ where } a, b \in \mathbb{Z} \text{ so the statement is proven.}

Marking Criteria

Descriptor	Marks
correctly expresses the difference of any two rational numbers as a single fraction	1
communicates that the difference forms a rational number by considering the definition	1

Q4

2020

QCAA

Paper 1

1 mark

Q4

1 mark

Consider the statement, $\forall n \in Z, (n-1)^2 - n$ is a prime number.
Which of the following values of $n$ provides a counterexample?

A

$n = -4$

B

$n = -1$

C

$n = 6$

D

$n = 9$

Reveal Answer

A

$n = -4$

Substituting $n = -4$ yields $(-4-1)^2 - (-4) = 25 + 4 = 29$ . Since 29 is a prime number, this does not provide a counterexample.

B

$n = -1$

Substituting $n = -1$ yields $(-1-1)^2 - (-1) = 4 + 1 = 5$ . Since 5 is a prime number, this does not provide a counterexample.

C

$n = 6$

Substituting $n = 6$ yields $(6-1)^2 - 6 = 25 - 6 = 19$ . Since 19 is a prime number, this does not provide a counterexample.

D

$n = 9$

Correct Answer

Substituting $n = 9$ yields $(9-1)^2 - 9 = 64 - 9 = 55$ . Since 55 is a composite number ( $5 \times 11$ ), it proves the statement false and serves as a valid counterexample.

Q16

2020

QCAA

Paper 1

6 marks

Q16

6 marks

Prove that $\sqrt{2}$ is irrational using proof by contradiction.

Reveal Answer

Assume that $\sqrt{2}$ is a rational number.
$\therefore \sqrt{2}=\frac{a}{b},\ a,b\in\mathbb{Z},\ b\ne 0$
where $a$ and $b$ only have 1 as a common factor.
$\frac{a^2}{b^2}=2$
$a^2=2b^2\ \ldots (1)$
$a^2$ is divisible by 2 and therefore $a$ is divisible by 2
So $a=2k$ , $k\in\mathbb{Z}$
From (1),
$(2k)^2=2b^2$
$4k^2=2b^2$
$b^2=2k^2$
So $b^2$ is divisible by 2 and therefore $b$ is divisible by 2
Both $a$ and $b$ have a common factor of 2.
This contradicts the assumption that $\sqrt{2}$ is a rational number.
It is proven that $\sqrt{2}$ is irrational.

Marking Criteria

Descriptor	Marks
correctly communicates assumption and subsequent ratio and common factor statement	1
expresses $a^2$ as a multiple of 2	1
communicates that both $a^2$ and $a$ are divisible by 2	1
expresses $b^2$ as a multiple of 2	1
communicates that both $b^2$ and $b$ are divisible by 2	1
communicates a suitable conclusion based on the contradiction of the assumption	1

Q2

2020

QCAA

Paper 1

1 mark

Q2

1 mark

If the statement $A \Rightarrow B$ is true, which of the following statements must always be true?

A

$B \Rightarrow A$

B

$\neg A \Rightarrow \neg B$

C

$\neg (B \Rightarrow A)$

D

$\neg B \Rightarrow \neg A$

Reveal Answer

A

$B \Rightarrow A$

Incorrect. This is the converse of the original statement, which is not logically equivalent to $A \Rightarrow B$ . Just because $A$ implies $B$ does not mean $B$ implies $A$ .

B

$\neg A \Rightarrow \neg B$

Incorrect. This is the inverse of the original statement, which is not logically equivalent to $A \Rightarrow B$ . The inverse shares the same truth value as the converse, not the original conditional.

C

$\neg (B \Rightarrow A)$

Incorrect. This states that the converse is false, but the converse can sometimes be true (such as when both $A$ and $B$ are true). Thus, it is not guaranteed to always be true.

D

$\neg B \Rightarrow \neg A$

Correct Answer

Correct. This is the contrapositive of the original statement, $\neg B \Rightarrow \neg A$ . A conditional statement and its contrapositive are always logically equivalent and share the exact same truth value.

Q13

2022

QCAA

Paper 1

6 marks

Q13

Consider the mathematical statement
$\exists x \in R \text{ such that } x^2 + 1 = 0$

Q13c

Consider the proposition $\forall x \in Z$ .
$x \geq 3 \Rightarrow x^2 \geq 9$

Q13a

1 mark

Write the statement in words.

Reveal Answer

There exists a value of $x$ that is an element of the set of real numbers such that the sum of the square of $x$ and one equals zero.

Marking Criteria

Descriptor	Marks
correctly writes $\exists, \in$ and $R$ in a suitable sentence	1

Q13b

2 marks

Determine whether the statement is true or false. Provide mathematical reasoning to support your response.

Reveal Answer

False
$x^2 + 1 = 0$
$x^2 = -1$
The solutions to this equation are not real numbers.

Marking Criteria

Descriptor	Marks
correctly identifies that the statement is false	1
correctly provides mathematical reasoning to support that the statement is false	1

Q13c

1 mark

State the converse of the proposition using mathematical notation.

Reveal Answer

The converse is
$x^2 \geq 9 \Rightarrow x \geq 3 \ \forall \ x \in Z$

Marking Criteria

Descriptor	Marks
correctly states the converse	1

Q13d

2 marks

Use a counter example to disprove the converse stated in Question 13c).

Reveal Answer

Consider $x = -4$
$(-4)^2 \geq 9$
$16 \geq 9$ is true but $-4 \not\geq 3$ so the converse is false.

Marking Criteria

Descriptor	Marks
identifies a suitable counter example value of $x$	1
uses mathematical reasoning to disprove the converse	1

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