Skill Align HSC Mathematics Standard 1 - Free Online Pack 0
Original Skill Align simulation paper aligned with the 2026 HSC Mathematics Standard 1 examination format.
- Paper
- HSC Mathematics Standard 1
- Reading
- 10 minutes
- Writing
- 2 hours
- Assessment
- 80 marks
NESA-approved calculators may be used. A reference sheet is provided separately with this paper as “Mathematics_Standard_1_and_2_Reference_Sheet.pdf”. Resource: https://www.nsw.gov.au/education-and-training/nesa/curriculum/mathematics/mathematics-standard-stage-6-2017
Section I
Attempt Questions 1-10. Allow about 15 minutes for this section. Select the best answer for each question.
Question 1
1 mark- A courier's preferred route
- The mass of a parcel
- The time taken for a delivery
- The number of parcels delivered in one shift
Question 2
1 mark- 0.35 km / h
- 21 km / h
- 35 km / h
- 56.7 km / h
Question 3
1 mark- 0.35 km
- 3.5 km
- 35 km
- 350 km
Question 4
1 mark- A
- B
- C
- E
Question 5
1 mark- AUD 36
- AUD 60
- AUD 108
- AUD 180
Question 6
1 mark- 5 / 66
- 5 / 33
- 25 / 144
- 10 / 33
Question 7
1 mark- 5.82 x 10⁻³
- 5.82 x 10⁻⁴
- 58.2 x 10⁻⁵
- 0.582 x 10⁻³
Question 8
1 mark- Billing
- Delivery
- Returns
- Product
Question 9
1 mark- 12000(1.048)³
- 12000(1.004)³
- 12000(1.004)³⁶
- 12000(1.048)³⁶
Question 10
1 mark- 4.5π m
- 9π m
- 18π m
- 81π m
Section II
Attempt Questions 11-28. Allow about 1 hour and 45 minutes for this section. Show relevant mathematical reasoning and/or calculations.
Question 11
3 marksQuestion 12
2 marksQuestion 13
3 marksQuestion 14
5 marksQuestion 15
8 marksQuestion 16
3 marksQuestion 17
2 marksQuestion 18
3 marksQuestion 19
4 marksQuestion 20
7 marksQuestion 21
3 marksQuestion 22
4 marksQuestion 23
2 marksQuestion 24
5 marksQuestion 25
4 marksQuestion 26
4 marksQuestion 27
4 marksQuestion 28
4 marksWorked Solutions And Marking Guide
Section I Question 1
Answer: The number of parcels delivered in one shift
A parcel count is numerical and takes whole-number values.
Section I Question 2
Answer: 21 km / h
Eighteen minutes is 0.3 hours, so speed is 6.3 / 0.3 = 21 km / h.
Section I Question 3
Answer: 3.5 km
7(50000) = 350000 cm = 3.5 km.
Section I Question 4
Answer: C
Vertex C meets edges AC, BC, CD and CE.
Section I Question 5
Answer: AUD 60
Profit is 15(12) - [48 + 6(12)] = 180 - 120 = 60.
Section I Question 6
Answer: 5 / 33
The probability is (5 / 12)(4 / 11) = 5 / 33.
Section I Question 7
Answer: 5.82 x 10⁻⁴
Move the decimal point four places to the right to obtain 5.82.
Section I Question 8
Answer: Billing
Billing has 30 of 120 enquiries, so its relative frequency is 0.25.
Section I Question 9
Answer: 12000(1.004)³⁶
The monthly rate is 0.048 / 12 = 0.004 and there are 36 months.
Section I Question 10
Answer: 9π m
Circumference equals π times the diameter.
Section II Question 11
(a) 17.5%.
The reduction is 14 700 kWh, and 14700 / 84000 x 100 = 17.5%.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Represents the quantities in part a with a valid successive percentage change relationship.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Substitutes the values stated in part a consistently into that relationship.
Part a (1 mark)
Obtains the correct final result 17.5%.
Section II Question 12
(a) 2 hours 45 minutes.
From 13:47 to 15:47 is 2 hours, then to 16:32 is 45 minutes.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
From 13:47 to 15:47 is 2 hours.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result 2 hours 45 minutes.
Section II Question 13
(a) 16 minutes.
30 - 14 = 16.
(b) 22 minutes.
The total is 110 and 110 / 5 = 22.
Mark allocation
- Part a: accept the correct answer unless the prompt explicitly requires supporting reasoning.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Gives the correct result 16 minutes.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
The total is 110 and 110 / 5 = 22.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
Obtains the correct final result 22 minutes.
Section II Question 14
(a) A-C-D-E, 15 minutes.
The route totals 6 + 4 + 5 = 15 minutes.
(b) It takes 19 minutes.
The route total is 7 + 7 + 5 = 19, which is 4 minutes longer.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
The route totals 6 + 4 + 5 = 15 minutes.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Represents the quantities in part a with a valid minimum route relationship.
Part a (1 mark)
Obtains the correct final result A-C-D-E, 15 minutes.
Part b (1 mark)
The route total is 7 + 7 + 5 = 19, which is 4 minutes longer.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
States a conclusion supported by the preceding work: It takes 19 minutes.
Section II Question 15
(a) The probabilities are frac13 and ((13) / (35)), so drawing at least one orange token is more likely.
For one green and one white, add frac7(15)frac5(14)+frac5(15)frac7(14)=frac13. For at least one orange, use the complement 1-((12) / (15))((11) / (14))=1-((22) / (35))=((13) / (35)). Since ((13) / (35))>frac13, at least one orange is more likely.
(b) Expected orange results 16; conditional probability frac7(12).
The orange probability is 3 / 15, so the expected number is 80(3 / 15)=16. There are 12 non-orange tokens, of which 7 are green, so P(greenmidnot orange)=7 / 12.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
For one green and one white, add frac7(15)frac5(14)+frac5(15)frac7(14)=frac13.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
For at least one orange, use the complement 1-((12) / (15))((11) / (14))=1-((22) / (35))=((13) / (35)).
Part a (1 mark)
Since ((13) / (35))>frac13, at least one orange is more likely.
Part a (1 mark)
Represents the quantities in part a with a valid multi-stage probability relationship.
Part a (1 mark)
Obtains the correct final result The probabilities are frac13 and ((13) / (35)), so drawing at least one orange token is more likely.
Part b (1 mark)
The orange probability is 3 / 15.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
The expected number is 80(3 / 15)=16.
Part b (1 mark)
Obtains the correct final result Expected orange results 16; conditional probability frac7(12).
Section II Question 16
(a) 30 L / min.
720 / 24 = 30.
(b) 55 minutes.
1.65 kL = 1650 L, and 1650 / 30 = 55.
Mark allocation
- Part a: accept the correct answer unless the prompt explicitly requires supporting reasoning.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Gives the correct result 30 L / min.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
1.65 kL = 1650 L, and 1650 / 30 = 55.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
Obtains the correct final result 55 minutes.
Section II Question 17
(a) 31.25%.
30 / 96 x 100 = 31.25%.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Represents the quantities in part a with a valid percentage from a chart relationship.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result 31.25%.
Section II Question 18
(a) AUD 9,751.44.
Use 8400(1.038)⁴, which is approximately 9751.44.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Use 8400(1.038)⁴, which is approximately 9751.44.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Represents the quantities in part a with a valid compound growth relationship.
Part a (1 mark)
Obtains the correct final result AUD 9,751.44.
Section II Question 19
(a) 18 m by 11.25 m.
Multiply each plan length by 250 and convert centimetres to metres.
(b) 202.5 square metres.
18 x 11.25 = 202.5.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Multiply each plan length by 250 and convert centimetres to metres.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result 18 m by 11.25 m.
Part b (1 mark)
18 x 11.25 = 202.5.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
Obtains the correct final result 202.5 square metres.
Section II Question 20
(a) The interval rates are 9, 5, -2 and 6 percentage points per hour; the greatest positive rate occurs from hour 0 to hour 2.
From 0 to 2 hours, (42-24) / 2=9. From 2 to 4 hours, (52-42) / 2=5. From 4 to 8 hours, (44-52) / 4=-2. From 8 to 12 hours, (68-44) / 4=6 percentage points per hour. The greatest is 9.
(b) The overall average rate is ((11) / (3)), or approximately 3.67, percentage points per hour; one linear model is unsuitable because the interval gradients change and one interval decreases.
The net change is 68-24=44 percentage points over 12 hours, so the overall average is 44 / 12=11 / 3. A single linear rule requires a constant gradient, but the displayed interval rates are 9,5,-2,6.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
From 0 to 2 hours, (42-24) / 2=9.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
From 2 to 4 hours, (52-42) / 2=5.
Part a (1 mark)
From 4 to 8 hours, (44-52) / 4=-2.
Part a (1 mark)
Obtains the correct final result The interval rates are 9, 5, -2 and 6 percentage points per hour; the greatest positive rate occurs from hour 0 to hour 2.
Part b (1 mark)
The net change is 68-24=44 percentage points over 12 hours.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
The overall average is 44 / 12=11 / 3.
Part b (1 mark)
States a conclusion supported by the preceding work: The overall average rate is ((11) / (3)), or approximately 3.67, percentage points per hour; one linear model is unsuitable because the interval gradients change and one interval decreases.
Section II Question 21
(a) 750 fish.
Use 180 / N = 36 / 150, giving N = 180(150) / 36 = 750.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Use 180 / N = 36 / 150, giving N = 180(150) / 36 = 750.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Represents the quantities in part a with a valid capture-recapture relationship.
Part a (1 mark)
Obtains the correct final result 750 fish.
Section II Question 22
(a) 30π square metres.
(75 / 360)π(12)² = 30π.
(b) 5π metres.
(75 / 360)(2π)(12) = 5π.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
(75 / 360)π(12)² = 30π.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result 30π square metres.
Part b (1 mark)
(75 / 360)(2π)(12) = 5π.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
Obtains the correct final result 5π metres.
Section II Question 23
(a) AUD 5,100.
The taxable amount is 30000, so tax is 0.17(30000) = 5100.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
The taxable amount is 30000.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result AUD 5,100.
Section II Question 24
(a) AUD 4,500.
The total loss is 31500 over 7 years.
(b) AUD 24,000.
42000 - 4(4500) = 24000.
(c) 75%.
31500 / 42000 = 0.75.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
- Part c: accept the correct answer unless the prompt explicitly requires supporting reasoning.
Detailed marking criteria
Part a (1 mark)
The total loss is 31500 over 7 years.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result AUD 4,500.
Part b (1 mark)
42000 - 4(4500) = 24000.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
Obtains the correct final result AUD 24,000.
Part c (1 mark)
Gives the correct result 75%.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Section II Question 25
(a) 111 square metres.
14(9) - 5(3) = 126 - 15 = 111.
(b) AUD 3,552.
111(32) = 3552.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
14(9) - 5(3) = 126 - 15 = 111.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result 111 square metres.
Part b (1 mark)
111(32) = 3552.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
Obtains the correct final result AUD 3,552.
Section II Question 26
(a) 0.157.
0.35(0.30) + 0.65(0.08) = 0.157.
(b) Approximately 0.669.
The joint probability is 0.105, so use 0.105 / 0.157.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Represents the quantities in part a with a valid conditional probability relationship.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result 0.157.
Part b (1 mark)
The joint probability is 0.105.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
Obtains the correct final result Approximately 0.669.
Section II Question 27
(a) A strong negative association.
Charge generally falls as operating hours increase.
(b) The point near (10, 86); it has an extreme x-value and does not follow the trend.
Its horizontal separation gives it high leverage.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
Charge generally falls as operating hours increase.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
States a conclusion supported by the preceding work: A strong negative association.
Part b (1 mark)
Its horizontal separation gives it high leverage.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
States a conclusion supported by the preceding work: The point near (10, 86); it has an extreme x-value and does not follow the trend.
Section II Question 28
(a) L(n+1) = 1.006L(n) - 240.
The existing balance is multiplied by 1.006 before the repayment is subtracted.
(b) AUD 6,096.79.
L(1) = 6500(1.006) - 240 = 6299. Then L(2) = 6299(1.006) - 240 = 6096.794, which rounds to AUD 6,096.79.
Mark allocation
- Part a: award one mark for each distinct evidence statement in the criteria.
- Part b: award one mark for each distinct evidence statement in the criteria.
Detailed marking criteria
Part a (1 mark)
The existing balance is multiplied by 1.006 before the repayment is subtracted.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part a (1 mark)
Obtains the correct final result L(n+1) = 1.006L(n) - 240.
Part b (1 mark)
L(1) = 6500(1.006) - 240 = 6299.
Acceptable alternatives: Accept an equivalent mathematically valid method that demonstrates the same assessable decision.
Part b (1 mark)
Obtains the correct final result AUD 6,096.79.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Algebra and Modelling | Q5, Q7 | 2 | ___ | Review the listed algebra and modelling items, their worked solutions and the evidence-specific marking criteria. |
| Financial Mathematics | Q9, Q18(a), Q23(a), Q24(a), Q24(b), Q24(c), Q28(a), Q28(b) | 15 | ___ | Review the listed financial mathematics items, their worked solutions and the evidence-specific marking criteria. |
| Measurement and Trigonometry | Q3, Q10, Q19(a), Q19(b), Q22(a), Q22(b), Q25(a), Q25(b) | 14 | ___ | Review the listed measurement and trigonometry items, their worked solutions and the evidence-specific marking criteria. |
| Networks | Q4, Q14(a), Q14(b) | 6 | ___ | Review the listed networks items, their worked solutions and the evidence-specific marking criteria. |
| Rates and Ratios | Q2, Q11(a), Q12(a), Q16(a), Q16(b), Q20(a), Q20(b) | 16 | ___ | Review the listed rates and ratios items, their worked solutions and the evidence-specific marking criteria. |
| Statistics and Probability | Q1, Q6, Q8, Q13(a), Q13(b), Q15(a), Q15(b), Q17(a), Q21(a), Q26(a), Q26(b), Q27(a), Q27(b) | 27 | ___ | Review the listed statistics and probability items, their worked solutions and the evidence-specific marking criteria. |