How many VCAA General Mathematics questions cover Discrete mathematics?

AusGrader has 183 VCAA General Mathematics questions on Discrete mathematics, all with instant AI grading and detailed marking feedback.

VCAA General Mathematics Discrete mathematics

5 sample questions with marking guides and sample answers

2023

QCAA

Paper 1

1 mark

Determine the 6th term of the arithmetic sequence that begins 3, 9, …

729

2023

QCAA

Paper 1

1 mark

In January 2022, 40 fish were released into a new dam that has the capacity to support 10 000 fish. It is predicted that the dam will reach its capacity in January 2030 if the fish population doubles every year.
Which sequence rule models the prediction?

$t_n = t_1 r^{(n-1)}$ , where $t_1 = 40, r = 2, n = 8$

$t_n = t_1 r^{(n-1)}$ , where $t_1 = 40, r = 2, n = 9$

$t_n = t_1 + (n-1)d$ , where $t_1 = 40, d = 2, n = 8$

$t_n = t_1 + (n-1)d$ , where $t_1 = 40, d = 2, n = 9$

2025

SCSA

Paper 1

10 marks

When people become ill, their body responds with a change to their white blood cell count (WBCC) measured in '000s cells per microlitre (cells/μL).

The table below shows a person's white blood cell count after contracting an illness.
Let $n$ represent the number of hours after contracting an illness.

Time ( $n$ hours)	1	2	3	4	5	6
WBCC '000s (cells/μL)	8.4	8.7	9

Q1e

After an illness is diagnosed, a person receives the appropriate medication to aid their recovery. An indirect effect of the medication will be an overall reduction in the person's white blood cell count.

A person becomes ill and is given medication. Their white blood cell count ('000s cells/μL) follows the recursive rule $T_m = -0.5T_{m-1} + 20, \quad T_0 = 20$ , where $m$ is the number of hours after the person is given the medication.

Q1a

1 mark

The person's white blood cell count is increasing at a constant rate each hour. Show that this rate is 300 cells/μL.

$8700 - 8400 = 300$

Therefore the constant rate is 300 cells/ $\mu$ L.

Marking Criteria

Descriptor	Marks
correctly shows that the constant rate is 300 cells/ $\mu$ L	1

Q1b

1 mark

Complete the table above.

Time ( $n$ hours)	1	2	3	4	5	6
WBCC '000s (cells/ $\mu$ L)	8.4	8.7	9	9.3	9.6	9.9

Descriptor	Marks
determines all correct entries	1

Q1c

2 marks

Determine a simplified rule for the $n^{\text{th}}$ term to model the person's white blood cell count after contracting an illness.

$T_n = 8.4 + (n - 1)(0.3)$
$T_n = 8.1 + 0.3n$

Marking Criteria

Descriptor	Marks
uses the rule for an arithmetic sequence and substitutes correct $a$ and $d$ values	1
correctly simplifies the rule	1

Q1d

2 marks

Determine the white blood cell count in the person's body after 12 hours, if the white blood cell count maintains the same rate of increase.

$T_{12} = 8.1 + 0.3(12)$
$T_{12} = 11.7$

Therefore 11 700 cells/ $\mu$ L.

Marking Criteria

Descriptor	Marks
correctly determines the twelfth term	1
states correct white blood cell count in cells/ $\mu$ L	1

Q1e (i)

2 marks

Determine the white blood cell count in this person at the end of the second hour.

$T_1 = (-0.5)(20) + 20 = 10$
$T_2 = (-0.5)(10) + 20 = 15$

Therefore the white blood cell count in the body is 15 000 cells/ $\mu$ L at the end of the second hour.

Marking Criteria

Descriptor	Marks
determines correct value for $T_1$	1
states correct white blood cell count in cells/ $\mu$ L	1

Q1e (ii)

2 marks

Determine if the given medication, in the long run, can ensure this person's white blood cell count does not fall below 13 000 cells per microlitre.

$x = -0.5x + 20 \Rightarrow x = \frac{20}{1.5} = \frac{200}{15} = \frac{40}{3} = 13\frac{1}{3} \approx 13.3$

i.e. approximately 13 300 cells/ $\mu$ L, which is greater than 13 000

Therefore the given medication will ensure the person's white blood cell count does not fall below 13 000 cells per microlitre.

Marking Criteria

Descriptor	Marks
determines correct steady-state value	1
correctly justifies medication will ensure the person's white blood cell count does not fall below 13 000 cells per microlitre	1

2025

SCSA

Paper 2

13 marks

Josh lives in Sydney and Simon lives in Perth. They each decide to drive to Alice Springs to meet before driving together to Darwin. Simon and Josh will both leave for Alice Springs on Saturday, 2 January.

Due to traffic in Sydney, Josh can only drive 250 km on the first day. After this he plans to drive 3 km more each day than the previous day.

Let $n$ represent the number of days spent driving.

Q6c

The distance Simon drives each day can be represented by the rule $T_n = 220(1.06)^{n-1}$ .

Q6e

The approximate distance from Sydney to Alice Springs is 2770 km and from Perth to Alice Springs is 2550 km.

Q6a

2 marks

Write a recursive rule to model the distance Josh drives each day.

Descriptor	Marks
states correct rule	1
states correct value for $T_1$	1

Q6b

1 mark

Calculate how far Josh is away from Sydney by the end of day four.

Descriptor	Marks
calculates correct distance	1

Q6c

2 marks

Interpret what the numbers 220 and 1.06 represent in the context of the question.

The 220 represents the distance travelled in kilometres by Simon by the end of day one.
1.06 represents a 6% increase in distance travelled each day.

Marking Criteria

Descriptor	Marks
correctly interprets what the 220 represents in the context of the question	1
correctly interprets what the 1.06 represents in the context of the question	1

Q6d

3 marks

Determine the day and date when Simon drove 330.8 km.

\begin{align*} 330.8 &= 220(1.06)^{n-1}\\ n &= 8 \end{align*}

Therefore Simon travels 330.8 km on day 8 which is Saturday, 9 January.

Marking Criteria

Descriptor	Marks
solves correctly for $n$ or states $T_8 = 330.8$	1
states correct day	1
states correct date	1

Q6e

3 marks

Determine who is the first to arrive in Alice Springs. Justify your answer.

Josh $\rightarrow$ total distance after 10 days = 2635 kilometres
total distance after 11 days = 2915 kilometres
Therefore Josh will arrive on the $11^{\text{th}}$ day.

Simon $\rightarrow$ total distance after 9 days = 2528.09 kilometres
total distance after 10 days = 2899.77 kilometres
Therefore Simon will arrive on the $10^{\text{th}}$ day.

Simon will arrive in Alice springs first.

Marking Criteria

Descriptor	Marks
determines Simon's correct day of arrival	1
determines Josh's correct day of arrival	1
concludes who will arrive first with justification based on the day of arrival	1

Q6f

2 marks

After Josh and Simon meet in Alice Springs, they drive together to Darwin. The distance they travel each day can be represented by the rule $T_{n+1} = aT_n + b$ .

Given they travel 380 km on day one, 309 km on day two and 269.95 km on day three, determine the value of $a$ and $b$ .

$309 = a(380) + b$ ...(1)
$269.95 = a(309) + b$ ...(2)

$a = 0.55, \; b = 100$

Marking Criteria

Descriptor	Marks
determines the correct value for $a$	1
determines the correct value for $b$	1

Q18

2020

QCAA

Paper 1

4 marks

Q18

Exhibition organisers believe that the number of attendees increases each day as an arithmetic sequence. The organisers know that 353 people attended the first day and 439 people attended the third day.

Q18a

2 marks

Determine the common difference.

Descriptor	Marks
Correctly provides mathematical reasoning to support the answer	1
Correctly determines the common difference	1

Q18b

2 marks

Use the result from 18a) to predict the number of people who will attend the sixth day.

Find $t_6$
$t_6 = t_1 + 5d$
$= 353 + 5 \times 43$
$= 568$

They would expect 568 people to attend the sixth day.

Marking Criteria

Descriptor	Marks
Substitutes into an appropriate rule	1
Determines value	1

VCAA General Mathematics Discrete mathematics

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