Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Identify the initial value and constant weekly change.
2. Obtains or justifies: t_1 = 120, t_(n+1) = t_n + 7.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Calculate 120 + 5 × 7 = 155.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: 155 units.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Translate the constant and percentage changes into explicit sequence rules.
2. Obtains or justifies: A_n = 900 + 50(n - 1); G_n = 900(1.04)^(n - 1).

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Substitute n = 4 into both models.
2. Obtains or justifies: A_4 = 1050; G_4 is approximately 1012.3776.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Compare the calculated values, identify the larger result, and evaluate the constant-growth assumption.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: The arithmetic model is larger at period 4. Neither constant numerical growth nor constant percentage growth is guaranteed indefinitely because capacity and conditions can change.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (1 mark)

Award 1 mark for a correct answer with or without working. Award 0.5-mark partial credit for an incorrect answer that shows some correct relevant working; otherwise award 0 marks.

Indicative checkpoints (0.5-mark increments allowed):

1. Obtains or justifies: M: 0.388% per month and 12 periods. Q: 1.183% per quarter and 4 periods.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (1 mark)

Award 1 mark for a correct answer with or without working. Award 0.5-mark partial credit for an incorrect answer that shows some correct relevant working; otherwise award 0 marks.

Indicative checkpoints (0.5-mark increments allowed):

1. Obtains or justifies: M: 4.75% effective p.a.; Q: 4.81% effective p.a.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Compare like-for-like effective rates and identify a relevant model limitation.
2. Obtains or justifies: Account Q has the greater effective annual rate. Fees, access restrictions, risk or a rate change could alter the practical choice.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Multiply the opening balance by the monthly rate.
2. Obtains or justifies: Interest = AUD 96.75.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Rearrange B_1 = (1 + i)B_0 - R, then retain the interest-before-payment order.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: R = AUD 680.00 and B_(n+1) = 1.0045B_n - 680.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Apply one further complete monthly cycle to B_1.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: B_2 = AUD 20,330.88.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (1 mark)

Award 1 mark for a correct answer with or without working. Award 0.5-mark partial credit for an incorrect answer that shows some correct relevant working; otherwise award 0 marks.

Indicative checkpoints (0.5-mark increments allowed):

1. Obtains or justifies: At x = 5, y-hat = 20 MWh; at x = 12, y-hat = 36.8 MWh.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Compare each input with the stated data domain.
2. Obtains or justifies: The x = 5 prediction is interpolation within the observed domain. The x = 12 prediction is extrapolation and is less reliable because the fitted relationship may not continue beyond x = 8.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: State the model value at zero, then test zero against the data range and context.
2. Obtains or justifies: The intercept predicts 8 MWh at x = 0, but x = 0 lies outside the observed domain and may be impossible or irrelevant in this study.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use the site-specific denominators first, then pool the successes and cases by program.
2. Carries the setup through the required calculation or representation for part a.
3. Obtains or justifies: High-success sites: A 90% and B 90%. Low-success sites: A 10% and B 10%. Pooled: A 82% and B 18%.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (4 marks)

Award up to 4 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 2.0 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Compare conditional and pooled results and link the reversal to unequal group composition.
2. Carries the setup through the required calculation or representation for part b.
3. Addresses the requested accuracy, unit, comparison or interpretation in: Explain why the pooled comparison is misleading and identify the confounding variable.
4. Obtains or justifies: Within each site type the programs have equal success rates. The pooled difference arises because Program A has far more cases at high-success sites and Program B has far more at low-success sites. Site type is the confounding variable, so the pooled association should not be interpreted as a program effect.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Plot every supplied pair at the correct coordinate and describe direction, form and strength.
2. Obtains or justifies: A correctly scaled and labelled scatterplot of (1, 14), (2, 24), (3, 25), (4, 39), (5, 35), (6, 51), (7, 49), (8, 66). The association is positive, approximately linear, and strong.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Enter the paired data in corresponding lists, calculate linear regression, and report the coefficient order correctly.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: y-hat = 7.714 + 6.702x; r = 0.965.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Report the calculator's coefficient of determination, name both variables in the explained-variation statement, and compare the prediction input with the observed domain.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: r-squared = 0.932. About 93.2% of the variation in solar energy output is explained by its fitted linear relationship with daily sunshine duration. A prediction at x = 10 is extrapolation beyond the observed range 1 to 8, so the fitted pattern may not continue and the result is less reliable.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part d (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use observed minus fitted for the named residual, plot residual against the explanatory variable with a zero reference line, then use random scatter or systematic curvature to assess the linear model.
2. Carries the setup through the required calculation or representation for part d.
3. Obtains or justifies: The residual is 4.48 MWh, so the observed solar energy output is 4.48 MWh above the fitted value. Residual points: (1, -0.42), (2, 2.88), (3, -2.82), (4, 4.48), (5, -6.23), (6, 3.07), (7, -5.63), (8, 4.67). A linear model is reasonably appropriate because the residuals are scattered on both sides of zero without a systematic curved pattern.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Add within rows and then combine the row totals.
2. Obtains or justifies: Urban: 71; regional: 61; overall: 132.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use each row total as the denominator and check that each row sums to 100%.
2. Obtains or justifies: Urban: Yes 64.8%, No 35.2%; regional: Yes 41.0%, No 59.0%.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use a common percentage scale so the conditional distributions can be compared.
2. Obtains or justifies: Two correctly labelled 100% bars, partitioned at the calculated Yes and No percentages, with a legend or direct labels.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part d (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Compare the conditional distributions and name a plausible confounder.
2. Obtains or justifies: The higher urban Yes percentage indicates an association between location and response. A confounding factor such as service access, age profile or sampling method could explain part of the difference.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Average each consecutive block of three values and place the result at the middle month.
2. Carries the setup through the required calculation or representation for part a.
3. Obtains or justifies: month 2: 90; month 3: 93.6667; month 4: 99.6667; month 5: 104; month 6: 112.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Plot time in order, align each average with its centre month, and distinguish the two series.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: A time plot containing all seven raw points and the five centred averages at months 2 to 6, clearly distinguished and correctly scaled.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Interpret both the smoothing and the limited length of the investigation.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: The centred averages show an overall upward movement, with a small mid-series interruption. Seven observations are too few to establish a stable long-term trend or seasonality.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: For each year, express every quarter as a proportion of that year's mean; then average the three corresponding quarterly proportions and check the index mean.
2. Obtains or justifies: Annual means: 93.05, 101.05, 109.05. Seasonal indices: Q1 0.776, Q2 1.089, Q3 1.312, Q4 0.823; mean = 1.000.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Divide the observed total by the matching unrounded seasonal index and explain that the seasonal component has been removed.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: Using the unrounded index 0.775752, deseasonalised value = 84.8 / 0.775752 = 109.3. This estimates the underlying trend-cycle level after removing the typical Q1 seasonal effect.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Regress deseasonalised value on time using unrounded indices, substitute the future time, multiply by the matching unrounded seasonal index, round both forecasts to one decimal place, and evaluate the stability assumption.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: Trend line: y-hat = 89.408 + 1.791t. Trend forecast at t = 13: 112.7. Reseasonalised Q1 forecast: 87.4. Both forecasts are rounded to one decimal place. The long-term trend and seasonal pattern are assumed to remain reasonably stable.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Represent constant and percentage growth correctly.
2. Carries the setup through the required calculation or representation for part a.
3. Obtains or justifies: A_n = 210 + 13(n - 1); G_n = 210(1.042)^(n - 1).

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Evaluate both rules at n = 6.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: A_6 = 275; G_6 is approximately 257.9633.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (4 marks)

Award up to 4 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 2.0 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Compare results and evaluate the modelling assumptions.
2. Carries the setup through the required calculation or representation for part c.
3. Addresses the requested accuracy, unit, comparison or interpretation in: Identify which model is larger at period 6 and give one reason actual growth may follow neither model.
4. Obtains or justifies: The arithmetic model is larger at period 6. Changing resources, capacity or participation can prevent either constant-change assumption from continuing.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Three ratio steps separate terms 2 and 5; take the positive cube root and back-divide once.
2. Carries the setup through the required calculation or representation for part a.
3. Obtains or justifies: r³ = 1.259712, so r = 1.08 and t_1 = 320.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Multiply successively by the recovered ratio and preserve the increasing curved pattern of the discrete points.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: Table values: 320.0, 345.6, 373.2, 403.1, 435.4, 470.2. Plot the six ordered pairs using term number horizontally.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use the geometric nth-term and finite-sum formulas with the recovered first term and ratio.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: t_n = 320(1.08)^(n - 1), and S_6 = 2347.5.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use the initial value and add the common difference successively.
2. Carries the setup through the required calculation or representation for part a.
3. Obtains or justifies: t_1 = 36, t_(n+1) = t_n + 4; table values 36, 40, 44, 48, 52.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Plot term number horizontally and term value vertically without presenting the sequence as continuous data.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: The plotted points are (1, 36), (2, 40), (3, 44), (4, 48), (5, 52); the points are discrete.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use the arithmetic nth-term rule and 12(first term + twelfth term) / 2.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: t_n = 36 + 4(n - 1); t_12 = 80; S_12 = 696.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Subtract the equations to isolate the multiplier, then substitute to find the constant.
2. Obtains or justifies: 70 = 82a + c and 62.8 = 70a + c, giving a = 0.6 and c = 20.8.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Continue the reconstructed recurrence through five updates from u_0.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: u_5 = 54.33.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Solve the equilibrium equation and use the multiplier magnitude and initial position to classify convergence.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: L = c / (1 - a) = 52. Because |a| < 1, the sequence converges to 52, approaching from above for the supplied initial value.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Apply the recurrence for the first two end-of-month deposits.
2. Obtains or justifies: A_1 = AUD 230.00; A_2 = AUD 460.94.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Recognise the accumulated deposits as a geometric series.
2. Obtains or justifies: A_n = 230[(1.0041)^n - 1] / (1.0041 - 1).

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Evaluate the annuity expression at n = 12.
2. Obtains or justifies: A_12 = AUD 2,823.10.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part d (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use payment = principal x effective rate and identify the constant-return assumption.
2. Carries the setup through the required calculation or representation for part d.
3. Obtains or justifies: Annual withdrawal = AUD 1,062.50. The effective return must remain sufficient and fees, inflation and investment risk are ignored.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use the common interest multiplier and each plan's end-of-month payment.
2. Carries the setup through the required calculation or representation for part a.
3. Obtains or justifies: S_0 = A_0 = 32000; S_(n+1) = 1.0055S_n - 980; A_(n+1) = 1.0055A_n - 1160.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Iterate both recurrences for exactly six complete cycles and compare balances on the same date.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: Standard balance = AUD 27,109.18; accelerated balance = AUD 26,014.22; accelerated balance is AUD 1,094.96 lower.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Accumulate monthly interest from each opening balance and distinguish cash paid from interest avoided.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: The accelerated plan saves AUD 14.96 in interest during the six months. Its earlier principal reductions lower later interest, so the balance benefit includes both extra payments and avoided interest.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Subtract the year-1 value from cost, then express repeated equal depreciation.
2. Obtains or justifies: Annual depreciation = AUD 3,600.00 and V_n = 36000 - 3600n.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Divide the year-1 value by cost, then convert retained proportion to a depreciation percentage.
2. Obtains or justifies: Retained proportion = 0.82, so depreciation rate = 18% and R_n = 36000(0.82)^n.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Evaluate each recovered model at the same year.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: Straight-line value = AUD 18,000.00; reducing-balance value = AUD 13,346.63.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part d (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: The lower book value corresponds to greater accumulated depreciation; subtract the values.
2. Carries the setup through the required calculation or representation for part d.
3. Obtains or justifies: The reducing-balance method records more accumulated depreciation; the book-value difference is AUD 4,653.37.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Convert both nominal annual rates using the matching compounding frequency.
2. Obtains or justifies: M: 0.400% per month, 12 periods per year. Q: 1.225% per quarter, 4 periods per year.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Compound each periodic rate for one year before comparing the offers.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: M: 4.907% effective p.a.; Q: 4.991% effective p.a. Offer Q is greater.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use the selected periodic rate for all compounding periods over three years, then evaluate the product conditions.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: Future value = AUD 10,415.88. This assumes the advertised rate and compounding arrangement remain unchanged and ignores fees and tax.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Subtract the smallest entry in each row from every entry in that row.
2. Obtains or justifies: Row minima: 2, 3, 1, 4. Row-reduced matrix: A [7, 0, 5, 6]; B [3, 1, 0, 4]; C [4, 7, 0, 7]; D [3, 2, 5, 0].

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Subtract each column minimum, then choose one zero in every row and every column.
2. Obtains or justifies: Column minima after row reduction: 3, 0, 0, 0. Reduced matrix: A [4, 0, 5, 6]; B [0, 1, 0, 4]; C [1, 7, 0, 7]; D [0, 2, 5, 0]. Independent zeros can be selected at A-2, B-1, C-3 and D-4.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Return to the original costs and add the four entries selected by the independent-zero assignment.
2. Obtains or justifies: Assign A to 2, B to 1, C to 3 and D to 4. Minimum total cost = 13.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Compare viable routes; A-B-C-D-E has length 1 + 2 + 3 + 4 = 10.
2. Carries the setup through the required calculation or representation for part a.
3. Obtains or justifies: A-B-C-D-E, with total length 10.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (4 marks)

Award up to 4 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 2.0 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Remove the unavailable edge, compare the remaining complete routes, and justify the selected alternative.
2. Carries the setup through the required calculation or representation for part b.
3. Addresses the requested accuracy, unit, comparison or interpretation in: Edge C-D closes. Find the new shortest route from A to E and explain why the original route can no longer be used.
4. Obtains or justifies: A-B-D-E, with total length 12. The original route requires the closed edge C-D, so it is infeasible.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (4 marks)

Award up to 4 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 2.0 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Select the least available edge that joins a new vertex without forming a cycle.
2. Carries the setup through the required calculation or representation for part a.
3. Addresses the requested accuracy, unit, comparison or interpretation in: Use Prim's algorithm to list the edges in one minimum spanning tree.
4. Obtains or justifies: AB, BC, CD and DE.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (4 marks)

Award up to 4 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 2.0 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Add the four selected weights and justify exclusion by the cut or cycle property.
2. Carries the setup through the required calculation or representation for part b.
3. Addresses the requested accuracy, unit, comparison or interpretation in: Find the total weight and explain why the three longer diagonal edges are excluded.
4. Obtains or justifies: Total weight = 18. The path edges connect all five vertices without a cycle, while each diagonal is heavier than the corresponding connection already available.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Start A and B at zero; for each later activity use the largest completion time among its immediate predecessors.
2. Obtains or justifies: EST(C) = 3, EST(D) = 3, EST(E) = 5, EST(F) = 10; minimum duration = 13 days.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Begin from the project finish and subtract durations, using the smallest allowable successor start at a merge.
2. Obtains or justifies: LST(A) = 0, LST(B) = 0, LST(C) = 6, LST(D) = 3, LST(E) = 5, LST(F) = 10.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Calculate LST minus EST for every activity; the zero-float activities form the critical path.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: Floats: A 0, B 0, C 3, D 0, E 0, F 0 day(s). Critical path: A-B-D-E-F; project duration: 13 days.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (4 marks)

Award up to 4 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 2.0 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Allocate flow on complete source-to-sink paths and verify capacity and conservation.
2. Carries the setup through the required calculation or representation for part a.
3. Addresses the requested accuracy, unit, comparison or interpretation in: Give a feasible flow of value 9 from S to T.
4. Obtains or justifies: Send 4 units on S-A-T and 5 units on S-B-T. All capacities and flow conservation conditions are satisfied.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (4 marks)

Award up to 4 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 2.0 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Calculate the cut capacity and invoke the maximum-flow minimum-cut result.
2. Carries the setup through the required calculation or representation for part b.
3. Addresses the requested accuracy, unit, comparison or interpretation in: Identify a cut that proves the flow is maximum.
4. Obtains or justifies: The cut immediately before T contains A-T with capacity 4 and B-T with capacity 5, so its capacity is 9. A feasible flow of 9 therefore equals the minimum cut and is maximum.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: For each leg use east = distance sin(bearing) and north = distance cos(bearing), then add corresponding components.
2. Obtains or justifies: Eastward component = 27.47 km; northward component = 5.99 km.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Apply Pythagoras' theorem to the total east and north components.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: Distance = 28.12 km.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Measure clockwise from north using atan2(east, north) and verify the result from the displacement signs.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: True bearing = 078 degrees. Both component signs place the displacement in the corresponding compass quadrant, confirming the atan2 bearing.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Multiply Earth's radius by cosine of the latitude.
2. Obtains or justifies: 5020 km.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Take the longitude fraction of the parallel circumference.
2. Carries the setup through the required calculation or representation for part b.
3. Obtains or justifies: 526 km.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Convert distance to time and evaluate the constant-speed spherical model.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: Approximately 45 minutes. Real routes, wind and speed changes are ignored.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Create a consistent triangle representation with every supplied quantity placed at the correct side or angle.
2. Obtains or justifies: A triangle with adjacent sides labelled 7.2 km and 9.3 km, their included angle labelled 52 degrees, and the opposite side labelled c.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Apply the cosine rule using the included angle.
2. Obtains or justifies: c = 7.48 km.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use one-half times the two known sides times sine of the included angle.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: 26.38 square kilometres.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Subtract the City A offset, add the first-flight duration in UTC, then apply the City B offset while retaining the date.
2. Obtains or justifies: Departure: Wed 14 Oct 09:20 UTC. City B arrival: Thu 15 Oct 02:40 local time.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Add the stopover in City B local time, then subtract twelve hours to return to UTC.
2. Obtains or justifies: City B departure: Thu 15 Oct 04:25 local time; Wed 14 Oct 16:25 UTC.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (3 marks)

Award up to 3 marks in 0.5-mark increments for the demonstrated checkpoints. A correct answer with no relevant working receives a maximum of 1.5 marks; incorrect or incomplete answers may receive partial credit for correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Add the second flight in UTC, apply UTC-8 to obtain the correct date across the International Date Line, and compare UTC departure and arrival instants for elapsed time.
2. Carries the setup through the required calculation or representation for part c.
3. Obtains or justifies: City C arrival: Wed 14 Oct 15:35 local time. Total elapsed time: 14 h 15 min. The calculation assumes the stated fixed offsets and scheduled flight and stopover durations apply without delay.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part a (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Connect the horizontal sight reference lines using parallel-line angle facts.
2. Obtains or justifies: The horizontal lines through the observer and each boat are parallel, so the alternate interior angles are equal.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part b (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Use tangent with platform height opposite and horizontal distance adjacent for each right triangle.
2. Obtains or justifies: P: 71.9 m; Q: 51.2 m.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.

Part c (2 marks)

Award 2 marks for a correct answer with relevant working. A correct answer with no working receives a maximum of 1.5 marks. Award partial credit in 0.5-mark increments for demonstrated correct progress.

Indicative checkpoints (0.5-mark increments allowed):

1. Establishes a valid setup consistent with: Subtract the two horizontal distances only after identifying the shared straight bearing and side.
2. Obtains or justifies: Distance PQ = 20.7 m. The boats and platform base are assumed collinear on the same side of the platform.

Acceptable alternatives: Accept mathematically equivalent working, notation and correctly rounded values that satisfy the task.

Do not credit by itself: Using an operation or denominator that does not match the supplied model. Reporting a numerical result without the requested unit, accuracy or contextual interpretation.