Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: substitutes x=6 into hat y=12+3.5x; calculates 12+3.5(6)=33; states the predicted assessment score is 33. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Direct calculator evaluation of the fitted-line expression is acceptable when the substitution is shown.

Do not credit by itself: using the observed score 35 instead of the fitted value; omitting the substitution x=6

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: uses residual = observed - predicted; substitutes 35-33; states the residual is +2 score points. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Equivalent signed-deviation notation is acceptable if observed minus predicted is clear.

Do not credit by itself: reversing the subtraction to give -2; reporting 35 or 33 without a residual calculation

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: uses the positive fitted gradient to select r=+√0.81=+0.90; interprets r²=0.81 as approximately 81% of observed score variation accounted for by the fitted linear relationship; compares x=20 with the observed range 2leq xleq10; identifies extrapolation and explains that the fitted relationship may not continue outside the observed range.

Acceptable alternatives: The equivalent decimal r=0.9 and equivalent wording for variation accounted for by the model are acceptable.

Do not credit by itself: taking r=-0.90 despite the positive gradient; describing 81% of scores rather than variation; calling x=20 interpolation

Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: uses the growth multiplier 1.06 on M_n; adds 25 after applying growth; states M_(n+1)=1.06M_n+25 with M_0=420.

Acceptable alternatives: The equivalent shifted form M_n=1.06M_(n-1)+25, M_0=420, is acceptable.

Do not credit by itself: using 1.6 or 0.06 as the growth multiplier; adding 25 before applying the 6% growth; omitting M_0=420

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: substitutes M_0=420 into M_1=1.06M_0+25; calculates M_1=470.2; interprets the model output as about 470 members. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Either the model value 470.2 or the contextual estimate of about 470 members is acceptable when its meaning is stated.

Do not credit by itself: calculating only the 6% increase; rounding before adding 25; presenting 470.2 as an exact count of people

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: retains M_1=470.2 for the next iteration; calculates M_2=1.06(470.2)+25=523.412; interprets the result as about 523 members; states that membership is discrete and the assumed 6% rate may change. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: A correctly rounded whole-member estimate is acceptable after the unrounded model value has been established.

Do not credit by itself: using a rounded M_1 without showing the model value; treating 523.412 as an exact headcount; giving a generic limitation unrelated to discreteness or the fixed growth rate

Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: forms the viable route A-B-C-D-E; calculates its length as 4+2+3+5=14 km; verifies no alternative A-to-E route has a lower total. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Dijkstra's algorithm or an exhaustive, correctly totalled route comparison is acceptable.

Do not credit by itself: listing A-B-C-D-E without its total; selecting a route using the fewest edges rather than the least total weight

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: selects BC, CD, AB and DE without forming a cycle; connects all five sites using exactly four edges; calculates the selected-edge total 2+3+4+5=14 km. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Kruskal's or Prim's algorithm is acceptable if it produces the stated tree and total.

Do not credit by itself: including a cycle; leaving a site disconnected; reporting the shortest A-to-E route without constructing a spanning tree

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: recognises that both verified totals are 14 km; states that the shortest-path problem minimises one route from A to E; states that the minimum spanning tree connects all sites using the minimum total selected edge length; explains that the equal totals are coincidental and do not make the two optimisation problems equivalent.

Acceptable alternatives: Equivalent network-optimisation wording is acceptable when the one-route and all-sites objectives are both explicit.

Do not credit by itself: saying only that both answers are 14 km; claiming that a shortest path must also be a minimum spanning tree

Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: adds 80+110+140+90 to obtain the Year 1 total 420; adds 84+116+150+96 to obtain the Year 2 total 446; labels both annual totals with the correct year. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: A correct table or spreadsheet sum is acceptable when the four values assigned to each year are visible.

Do not credit by itself: combining all eight quarters into one total; swapping the Year 1 and Year 2 totals

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: uses the increase 446-420=26; divides by the original total 420 and multiplies by 100; rounds 6.190ldots% to 6.2%. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: An equivalent percentage-change calculation is acceptable if Year 1 is used as the base.

Do not credit by itself: dividing by 446; reporting the raw increase 26 as a percentage; failing to round to one decimal place

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: identifies Quarter 3 as the highest quarter in Year 1; identifies Quarter 3 as the highest quarter in Year 2; describes the repeated Quarter 3 peak as the observed seasonal feature; states that only two years of data are insufficient to assume the pattern will continue unchanged.

Acceptable alternatives: Equivalent wording about a repeated within-year Quarter 3 peak and the short data span is acceptable.

Do not credit by itself: describing only an overall increase between years; asserting that Quarter 3 will always be highest without acknowledging the two-year limitation

Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: uses the monthly interest multiplier 1.004 on P_n; subtracts 550 after applying interest; states P_(n+1)=1.004P_n-550 with P_0=18(,)000.

Acceptable alternatives: The equivalent shifted form P_n=1.004P_(n-1)-550, P_0=18(,)000, is acceptable.

Do not credit by itself: using 1.04 as the multiplier; subtracting the repayment before interest; omitting the initial balance

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: substitutes P_0=18(,)000 into the recurrence; calculates 1.004(18(,)000)-550=17(,)522; states the balance as AUD 17(,)522. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: A separated interest-plus-repayment calculation is acceptable if it is equivalent to the recurrence.

Do not credit by itself: subtracting AUD 550 without adding interest; using 4% instead of 0.4%; omitting the AUD context

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: uses the unrounded first balance 17(,)522; calculates 1.004(17(,)522)-550=17(,)042.088; rounds the balance to AUD 17(,)042.09; explains that monthly interest offsets part of each repayment, so the two-repayment fall is not exactly AUD 1(,)100. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: An equivalent month-by-month schedule is acceptable if interest is applied before each repayment and rounding occurs only at the end.

Do not credit by itself: subtracting AUD 1100 directly from the opening balance; rounding before the second interest calculation; attributing the difference to fees not stated in the model

Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: calculates 4+5+3=12 days for S-A-C-T; calculates 6+4+3=13 days for S-B-D-T; labels each duration with its corresponding complete chain. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: A forward-pass table is acceptable if it establishes the same two complete-chain durations.

Do not credit by itself: adding edges from different chains; reporting only the longer duration without the duration of each chain

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: uses the complete-chain durations 12 and 13 days; selects the longer chain S-B-D-T; states the project duration is 13 days.

Acceptable alternatives: Equivalent critical-path or forward-pass working is acceptable if it identifies S-B-D-T and 13 days.

Do not credit by itself: selecting the shorter 12-day chain; identifying a single activity rather than a complete critical path

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: increases the A-C duration from 5 to 7 days; calculates the delayed upper chain as 4+7+3=14 days; compares 14 days with the unchanged lower-chain duration 13 days; states the new critical path S-A-C-T and project duration 14 days. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: A revised forward pass is acceptable if it shows the upper path at 14 days and the lower path at 13 days.

Do not credit by itself: adding two days to the whole project without checking both paths; retaining S-B-D-T as critical after the upper path becomes longer

Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: counts two incident edges at P and states deg(P)=2; counts three incident edges at each of Q and R and states deg(Q)=deg(R)=3; counts two incident edges at S and states deg(S)=2.

Acceptable alternatives: An adjacency list or row-sum method is acceptable if each undirected edge is counted once at each endpoint.

Do not credit by itself: counting an undirected edge only at one endpoint; confusing degree with the total number of vertices

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: states that an Euler circuit requires every non-isolated vertex to have even degree; identifies Q and R as the two odd-degree vertices; concludes that no Euler circuit exists.

Acceptable alternatives: Equivalent use of Euler's circuit theorem is acceptable.

Do not credit by itself: claiming connectivity alone guarantees an Euler circuit; giving no conclusion after identifying the odd vertices

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: states that a connected graph with exactly two odd-degree vertices has an Euler trail; identifies Q and R as the required start and end vertices; gives a valid trail such as Q-P-R-S-Q-R; verifies that every edge is used exactly once.

Acceptable alternatives: The reverse trail R-Q-S-R-P-Q, or any other valid edge-once trail from one odd vertex to the other, is acceptable.

Do not credit by itself: giving a route that repeats or omits an edge; starting and ending at even-degree vertices; calling the open trail a circuit

Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: selects the first three observations 120,135,150; calculates 120+135+150=405; divides by 3 to obtain the centred moving average 135. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: A technology mean calculation is acceptable if the selected three observations are identified.

Do not credit by itself: dividing by the six observations; using only two observations; reporting the total 405 rather than the mean

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: uses 135,150,132 to calculate 139; uses 150,132,126 to calculate 136; uses 132,126,144 to calculate 134. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Equivalent spreadsheet or calculator means are acceptable when the three windows and the order 139, 136, 134 are clear.

Do not credit by itself: using non-overlapping windows; failing to drop the oldest observation when the window moves; listing the averages in the wrong order

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: compares the moving averages 135,139,136,134 with the original observations; identifies an underlying level around the mid-130s; states that the smoothed series is fairly stable rather than showing a strong sustained trend; explains that smoothing suppresses short local peaks and falls such as 150 and 120. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Equivalent wording about a stable underlying level and reduced local variation is acceptable.

Do not credit by itself: claiming that smoothing preserves every peak; describing only one original value without interpreting the smoothed pattern

Part a (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: forms the route A-C-D; calculates its time as 4+2=6 minutes; verifies it is shorter than alternatives from A to D. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Dijkstra's algorithm or an exhaustive route comparison is acceptable.

Do not credit by itself: using the fewest edges without comparing weights; giving 6 minutes without the A-C-D route

Part b (3 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 3: forms the route D-E-F; calculates its time as 3+2=5 minutes; compares 5 minutes with the direct D-F edge of 7 minutes. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Dijkstra's algorithm or a correct direct comparison is acceptable.

Do not credit by itself: choosing the direct 7-minute edge without comparing the route via E; giving 5 minutes without the D-E-F route

Part c (4 marks)

Award one mark for each distinct evidence statement demonstrated, to a maximum of 4: uses the verified shortest section A-C-D=6 minutes; uses the verified shortest section D-E-F=5 minutes; combines the sections as A-C-D-E-F and confirms that the route passes through D; calculates the constrained minimum total 6+5=11 minutes. Apply consequential marking when a correct later method uses an earlier incorrect value, unless that value makes the task materially easier.

Acceptable alternatives: Equivalent section-by-section shortest-path working is acceptable if the complete route A-C-D-E-F and 11 minutes are stated.

Do not credit by itself: giving an unconstrained A-to-F route that bypasses D; adding incompatible partial routes; omitting the complete route or total time