Skill Align QCE General Mathematics Paper 1 Combined Independent Practice Booklet - Free Online Pack 0
Original Skill Align Units 3&4 combined independent practice booklet; not a secure QCAA administration replica.
- Paper
- Paper 1 Combined Independent Practice Booklet - Free Online Pack 0
- Reading
- 5 minutes perusal
- Writing
- 90 minutes
- Assessment
- 57 marks
Any additional response paper must show your name and the relevant question number, and must be attached as directed by your school.
Independent practice conditions
- Answer all questions. Show relevant working, mathematical reasoning and conclusions in the spaces provided.
- You may use writing equipment, a QCAA-approved handheld scientific calculator and the QCAA formula book.
- During perusal time, read the questions and make notes only on planning paper supplied by your school.
- This is a combined independent practice booklet. Record each multiple-choice response by selecting A, B, C or D in this booklet; no separate multiple-choice response sheet is supplied.
- If you use additional response paper, show your name and the relevant question and part number, then attach it as directed by your school.
- This independent practice resource is not a secure QCAA examination paper. Complete the candidate details above so printed work can be identified.
Section 1
Questions 1–15 are multiple choice. In this combined independent practice booklet, select the best answer A, B, C or D for each question.
Question 1
1 mark- 3(:)30 pm
- 4(:)30 pm
- 6(:)30 pm
- 7(:)30 pm
Question 2
1 mark- 12% below the deseasonalised value
- 12% above the deseasonalised value
- 88% below trend
- 0.88% below trend
Question 3
1 mark- 4.80%
- 4.91%
- 5.20%
- 5.76%
Question 4
1 mark- A-B-C-D-E, weight 13
- A-C-D-E, weight 14
- A-B-D-E, weight 14
- A-C-E, weight 16
Question 5
1 mark- 15 percentage points
- 15% of all students
- 21 percentage points
- 129 percentage points
Question 6
1 mark- decreases by (4%)
- increases by (4%)
- decreases by (96%)
- decreases by (0.96) units
Question 7
1 mark- a strong positive association
- a weak positive association
- a strong negative association
- no association
Question 8
1 mark- 4500 km
- 5000 km
- 8000 km
- 9000 km
Question 9
1 mark- a_(BD)=a_(DB)=1
- a_(BD)=1,a_(DB)=0
- a_(BD)=0,a_(DB)=1
- a_(BD)=a_(DB)=0
Question 10
1 mark- B_(n+1)=1.004B_n-350
- B_(n+1)=0.996B_n-350
- B_(n+1)=1.04B_n-350
- B_(n+1)=B_n-1.004(350)
Question 11
1 mark- the greatest total duration
- the smallest total duration
- the fewest activities
- the most vertices
Question 12
1 mark- every vertex has even degree
- exactly two vertices have odd degree
- every vertex has odd degree
- it contains no cycle
Question 13
1 mark- 12
- 14
- 17
- 21
Question 14
1 mark- AUD 26000
- AUD 30000
- AUD 31200
- AUD 52000
Question 15
1 mark- 72% of response variation is explained by the linear model
- 28% of predictions are exact
- the gradient is 0.72
- the correlation coefficient is always 0.72
Section 2
Questions 16–25 are short response. Show relevant working, mathematical reasoning and conclusions.
Question 16
3 marksQuestion 17
4 marksQuestion 18
4 marksQuestion 19
4 marksQuestion 20
4 marksQuestion 21
4 marksQuestion 22
5 marksQuestion 23
4 marksQuestion 24
5 marksQuestion 25
5 marksWorked Solutions And Marking Guide
General marking principles
- Accept mathematically equivalent methods and forms. Apply follow-through where a later result correctly uses an earlier student value, unless that removes the mathematical demand being assessed.
Section 1 Question 1
Answer: 3(:)30 pm
Singapore is two hours behind Brisbane.
Mark allocation
- 1 mark: select 3(:)30 pm. Evidence: Singapore is two hours behind Brisbane.
Section 1 Question 2
Answer: 12% below the deseasonalised value
An index of 0.88 is 0.12 below 1.
Mark allocation
- 1 mark: select 12% below the deseasonalised value. Evidence: An index of 0.88 is 0.12 below 1.
Section 1 Question 3
Answer: 4.91%
(1+0.048 / 12)¹²-1approx0.0491.
Mark allocation
- 1 mark: select 4.91%. Evidence: (1+0.048 / 12)¹²-1approx0.0491.
Section 1 Question 4
Answer: A-B-C-D-E, weight 13
Adding the edge weights gives 4+2+3+4=13, which is less than the alternatives.
Mark allocation
- 1 mark: select A-B-C-D-E, weight 13. Evidence: Adding the edge weights gives 4+2+3+4=13, which is less than the alternatives.
Section 1 Question 5
Answer: 15 percentage points
Compare the two conditional percentages: 72-57=15 percentage points.
Mark allocation
- 1 mark: select 15 percentage points. Evidence: Compare the two conditional percentages: 72-57=15 percentage points.
Section 1 Question 6
Answer: decreases by (4%)
The common ratio (0.96) retains (96%), so (4%) is lost each step.
Mark allocation
- 1 mark: select decreases by (4%). Evidence: The common ratio (0.96) retains (96%), so (4%) is lost each step.
Section 1 Question 7
Answer: a strong positive association
The coefficient is positive and close to 1.
Mark allocation
- 1 mark: select a strong positive association. Evidence: The coefficient is positive and close to 1.
Section 1 Question 8
Answer: 5000 km
((45) / (360)) × 40000=5000 km.
Mark allocation
- 1 mark: select 5000 km. Evidence: ((45) / (360)) × 40000=5000 km.
Section 1 Question 9
Answer: a_(BD)=a_(DB)=1
Adjacency is symmetric for an undirected edge.
Mark allocation
- 1 mark: select a_(BD)=a_(DB)=1. Evidence: Adjacency is symmetric for an undirected edge.
Section 1 Question 10
Answer: B_(n+1)=1.004B_n-350
Apply the interest multiplier, then subtract the repayment.
Mark allocation
- 1 mark: select B_(n+1)=1.004B_n-350. Evidence: Apply the interest multiplier, then subtract the repayment.
Section 1 Question 11
Answer: the greatest total duration
The longest-duration path determines the earliest completion time.
Mark allocation
- 1 mark: select the greatest total duration. Evidence: The longest-duration path determines the earliest completion time.
Section 1 Question 12
Answer: every vertex has even degree
An Eulerian circuit returns to its starting vertex and requires all degrees even.
Mark allocation
- 1 mark: select every vertex has even degree. Evidence: An Eulerian circuit returns to its starting vertex and requires all degrees even.
Section 1 Question 13
Answer: 14
Ordered values are 9, 12, 14, 17, 21.
Mark allocation
- 1 mark: select 14. Evidence: Ordered values are 9, 12, 14, 17, 21.
Section 1 Question 14
Answer: AUD 31200
600000 × 0.052=31200.
Mark allocation
- 1 mark: select AUD 31200. Evidence: 600000 × 0.052=31200.
Section 1 Question 15
Answer: 72% of response variation is explained by the linear model
The coefficient of determination measures explained response variation.
Mark allocation
- 1 mark: select 72% of response variation is explained by the linear model. Evidence: The coefficient of determination measures explained response variation.
Section 2 Question 16
(a) 12(:)20 pm.
Singapore is two hours behind Brisbane.
(b) 2(:)10 pm.
Add 1 hour 50 minutes to 12:20 pm.
Mark allocation
- Part a (1 mark): Evidence: Singapore is two hours behind Brisbane. Required result: 12(:)20 pm.
- Part b (2 marks): Evidence: Add 1 hour 50 minutes to 12:20 pm. Required result: 2(:)10 pm.
Section 2 Question 17
(a) Bus: 54 / 75=72%. Non-bus: 36 / 60=60%.
Evidence: use the two group totals as conditional denominators and report both percentages.
(b) The bus-user early-arrival rate is 12 percentage points higher, so the data support an association between bus use and early arrival.
Evidence: calculate 72-60=12 percentage points and state the direction without claiming causation.
Mark allocation
- Part a (2 marks): Evidence: use the two group totals as conditional denominators and report both percentages. Required result: Bus: 54 / 75=72%. Non-bus: 36 / 60=60%.
- Part b (2 marks): Evidence: calculate 72-60=12 percentage points and state the direction without claiming causation. Required result: The bus-user early-arrival rate is 12 percentage points higher, so the data support an association between bus use and early arrival.
Section 2 Question 18
(a) 40,000cos(60^circ)=20,000 km.
Evidence: scale the equatorial circumference by cos(60^circ).
(b) ((36) / (360)) × 20000=2000 km.
Evidence: use the longitude fraction of the latitude-circle circumference and include kilometres.
Mark allocation
- Part a (2 marks): Evidence: scale the equatorial circumference by cos(60^circ). Required result: 40,000cos(60^circ)=20,000 km.
- Part b (2 marks): Evidence: use the longitude fraction of the latitude-circle circumference and include kilometres. Required result: ((36) / (360)) × 20000=2000 km.
Section 2 Question 19
(a) AUD 96000.
105600 / 1.10=96000.
(b) AUD 96350.
102500 × 0.94=96350.
Mark allocation
- Part a (2 marks): Evidence: 105600 / 1.10=96000. Required result: AUD 96000.
- Part b (2 marks): Evidence: 102500 × 0.94=96350. Required result: AUD 96350.
Section 2 Question 20
(a) AUD 47444.
48000(1.0055)-820=47444.
(b) B_(n+1)=1.0055B_n-820, B_0=48000.
Apply interest, then subtract the repayment.
Mark allocation
- Part a (2 marks): Evidence: 48000(1.0055)-820=47444. Required result: AUD 47444.
- Part b (2 marks): Evidence: Apply interest, then subtract the repayment. Required result: B_(n+1)=1.0055B_n-820, B_0=48000.
Section 2 Question 21
(a) The points lie close to a straight increasing pattern.
There is little curvature or irregular scatter over the six observations.
(b) 35.2.
10+2.8(9)=35.2.
Mark allocation
- Part a (2 marks): Evidence: There is little curvature or irregular scatter over the six observations. Required result: The points lie close to a straight increasing pattern.
- Part b (2 marks): Evidence: 10+2.8(9)=35.2. Required result: 35.2.
Section 2 Question 22
(a) AUD 560000.
22400 / 0.04=560000.
(b) 3.66%.
(1.003)¹²-1approx0.0366.
Mark allocation
- Part a (2 marks): Evidence: 22400 / 0.04=560000. Required result: AUD 560000.
- Part b (3 marks): Evidence: (1.003)¹²-1approx0.0366. Required result: 3.66%.
Section 2 Question 23
(a) AB,BC,CE,CD.
These four low-weight edges connect all five vertices without a cycle.
(b) 16.
2+3+5+6=16.
Mark allocation
- Part a (2 marks): Evidence: These four low-weight edges connect all five vertices without a cycle. Required result: AB,BC,CE,CD.
- Part b (2 marks): Evidence: 2+3+5+6=16. Required result: 16.
Section 2 Question 24
(a) Each additional $100 of advertising is associated with an increase of about $1,900 in predicted weekly sales.
Evidence: interpret one x-unit as $100, one y-unit as $1000, and retain association language.
(b) hat y=7.2+1.9(7)=20.5, or $20,500.
Evidence: substitute x=7 and convert the response unit.
(c) About 97% of the variation in weekly sales is explained by the linear model over the observed domain 2le xle10.
Evidence: interpret explained response variation and give the numeric advertising domain.
Mark allocation
- Part a (2 marks): Evidence: interpret one x-unit as $100, one y-unit as $1000, and retain association language. Required result: Each additional $100 of advertising is associated with an increase of about $1,900 in predicted weekly sales.
- Part b (1 mark): Evidence: substitute x=7 and convert the response unit. Required result: hat y=7.2+1.9(7)=20.5, or $20,500.
- Part c (2 marks): Evidence: interpret explained response variation and give the numeric advertising domain. Required result: About 97% of the variation in weekly sales is explained by the linear model over the observed domain 2le xle10.
Section 2 Question 25
(a) 17 days.
Path S-A-C-T has duration 5+7+5=17, the greatest shown.
(b) Float =3 days; it can be delayed three days without delaying the project.
Float equals latest start minus earliest start.
Mark allocation
- Part a (2 marks): Evidence: Path S-A-C-T has duration 5+7+5=17, the greatest shown. Required result: 17 days.
- Part b (3 marks): Evidence: Float equals latest start minus earliest start. Required result: Float =3 days; it can be delayed three days without delaying the project.
Diagnostic Checklist
| Topic | Questions | Marks | Marks Lost | Action |
|---|---|---|---|---|
| Time Zones | Q1 | 1 | ___ | Convert departure and arrival times through UTC, track the calendar date before applying the destination offset, and explain any International Date Line adjustment. |
| Time Series | Q2 | 1 | ___ | Use an odd centred moving mean or median, derive actual-to-trend ratios before averaging a seasonal index, and match the index to the forecast period when reseasonalising. |
| Loans and Investments | Q3 | 1 | ___ | Match interest and payment periods before selecting a present-value, future-value, perpetuity or financial recurrence model; retain unrounded values until the final financial decision. |
| Graphs and Networks | Q4 | 1 | ___ | Audit vertex degrees and connectivity, apply the relevant Euler, planar, shortest-path or spanning-tree condition, and justify the operational route consequence. |
| Two-way Tables and Association | Q5 | 1 | ___ | Calculate conditional percentages using each group total as the denominator, compare percentage-point differences, and separate association from causation. |
| Growth and Decay in Sequences | Q6 | 1 | ___ | Identify whether the sequence has a constant difference or constant ratio, apply the corresponding arithmetic or geometric rule, and check the requested term or threshold in context. |
| Bivariate Data | Q7 | 1 | ___ | Enter every ordered pair, report the least-squares equation, r and R², calculate residual as observed minus predicted, and check domain before interpreting a prediction. |
| Earth Geometry | Q8 | 1 | ___ | Choose the correct great-circle or latitude-circle circumference, apply the angular fraction with the cosine latitude factor where required, and retain kilometres throughout. |
| Graphs and Networks | Q9 | 1 | ___ | Audit vertex degrees and connectivity, apply the relevant Euler, planar, shortest-path or spanning-tree condition, and justify the operational route consequence. |
| Loans and Investments | Q10 | 1 | ___ | Match interest and payment periods before selecting a present-value, future-value, perpetuity or financial recurrence model; retain unrounded values until the final financial decision. |
| Project Networks | Q11 | 1 | ___ | Complete forward and backward event-time scans, identify zero-float activities and the critical path, then calculate and interpret float from earliest and latest times. |
| Graphs and Networks | Q12 | 1 | ___ | Audit vertex degrees and connectivity, apply the relevant Euler, planar, shortest-path or spanning-tree condition, and justify the operational route consequence. |
| Time Series | Q13 | 1 | ___ | Use an odd centred moving mean or median, derive actual-to-trend ratios before averaging a seasonal index, and match the index to the forecast period when reseasonalising. |
| Loans and Investments | Q14 | 1 | ___ | Match interest and payment periods before selecting a present-value, future-value, perpetuity or financial recurrence model; retain unrounded values until the final financial decision. |
| Bivariate Data | Q15 | 1 | ___ | Enter every ordered pair, report the least-squares equation, r and R², calculate residual as observed minus predicted, and check domain before interpreting a prediction. |
| Time Zones | Q16 | 3 | ___ | Convert departure and arrival times through UTC, track the calendar date before applying the destination offset, and explain any International Date Line adjustment. |
| Two-way Tables and Association | Q17 | 4 | ___ | Calculate conditional percentages using each group total as the denominator, compare percentage-point differences, and separate association from causation. |
| Earth Geometry | Q18 | 4 | ___ | Choose the correct great-circle or latitude-circle circumference, apply the angular fraction with the cosine latitude factor where required, and retain kilometres throughout. |
| Time Series | Q19 | 4 | ___ | Use an odd centred moving mean or median, derive actual-to-trend ratios before averaging a seasonal index, and match the index to the forecast period when reseasonalising. |
| Loans and Investments | Q20 | 4 | ___ | Match interest and payment periods before selecting a present-value, future-value, perpetuity or financial recurrence model; retain unrounded values until the final financial decision. |
| Sequences | Q21 | 4 | ___ | Identify whether the sequence has a constant difference or constant ratio, apply the corresponding arithmetic or geometric rule, and check the requested term or threshold in context. |
| Loans and Investments | Q22 | 5 | ___ | Match interest and payment periods before selecting a present-value, future-value, perpetuity or financial recurrence model; retain unrounded values until the final financial decision. |
| Graphs and Networks | Q23 | 4 | ___ | Construct a minimum spanning tree by any valid method, verify that every vertex is connected without a cycle, and total the selected edge weights. |
| Bivariate Data Analysis | Q24 | 5 | ___ | Enter every ordered pair, report the least-squares equation, r and R², calculate residual as observed minus predicted, and check domain before interpreting a prediction. |
| Project Networks | Q25 | 5 | ___ | Complete forward and backward event-time scans, identify zero-float activities and the critical path, then calculate and interpret float from earliest and latest times. |