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QUESTION 6 A loan of $10 000 has interest charged on a reducing balance at 6% p.a. compounding quarterly with quarterly repayments of $700. The balance after 6 months is (A) $8696.75 (B) $8891.75 (C) $8900.00 (D) $9794.00
QUESTION 7 What is the length of the minimum spanning tree for this network? All distances are in kilometres (km). (A) 22km (B) 14 km (C) 12km (D) 3 km
QUESTION 11 Determine the adjacency matrix that represents this graph. o1 H o oo o N — — O O Y — O —~ O o —~— O O n H (B) N — — O RIOI oo —~ — O w (A) H o oo o N oo o AN O Ao <o oo O wn = (D) o o — o N — — O — Y — O —~ O o —~ — O Ox n H= ©)
QUESTION 12 The activity times in the project network shown are in days. Key (for activity E) -Latest start time ; E 5 Earliest start time “Time to complete Finish The greatest float time for a non-critical activity in this network is (A) 2 days. (B) 4 days. (C) 5 days. (D) 12 days.
QUESTION 13 Which of the following is a planar graph with 5 vertices and 4 faces? ) L j ) % ©) J\ (D) ‘§‘
QUESTION 15 The graph shows the value of three different annuities over time. 3000 —e— Annuity 1 --%-- Annuity 2 2500 — 4 — Annuity 3 2000 1500 1000 Value of investment ($) 500 Number of terms Which of the following statements gives a plausible explanation for the different values after seven terms? (A) (B) ©) (D) Annuity 1 and Annuity 2 have higher regular deposits than Annuity 3. Annuity 1 and Annuity 2 have shorter interest terms than Annuity 3. Annuity 2 and Annuity 3 have a lower initial value than Annuity 1. Annuity 2 and Annuity 3 have higher interest rates than Annuity 1.
QUESTION 16 (3 marks) Use the graph to identify whether each of the following is a cycle an open walk an open trail or a closed trail. D a) AFCFB [1 mark] b) AFCEFBA [1 mark] c) ABCDEFA [1 mark]
QUESTION 20 (4 marks) Determine the monthly repayment on a $350 000 home loan over 25 years with 6.5% p.a. fixed interest compounded monthly.
QUESTION 21 (5 marks) This network shows the maximum number of supplies (in tonnes) that can be transported from a warehouse to a cyclone shelter along each road each day during an emergency. A 50 20 Warehouse Cyclone shelter 40 30 € a) Use the ‘maximum flow minimum cut’ theorem to determine the maximum amount of supplies that can reach the cyclone shelter each day. [3 marks] b) During a cyclone the intersection at vertex A is damaged and no longer allows for any supplies to pass through it. What is the new maximum amount of supplies each day that can reach the cyclone shelter? [2 marks]
QUESTION 25 (5 marks) A financial institution offers two investment options: Option 1: 7% p.a. compounding quarterly Option 2: 6.8% p.a. compounding monthly Use the effective interest rate formula to determine the option that will provide the better return.
QUESTION 5 (5 marks) A company has three tasks to allocate to three contractors. Each of the contractors has a quote recorded for each task shown in the table. The quotes are in thousands of dollars ($°000s). Contractor‘ 1PN Q ‘ Task 2 ‘ Task 3 Use a matrix method to determine the minimum cost if each contractor is allocated one task.
QUESTION 6 (7 marks) A company needs to complete the following project as quickly as possible. Each task can only be completed by a single employee and must be completed before that employee can start the next task. The owner believes that this project can be completed in minimal time with only three employees. Evaluate the reasonableness of this belief.
QUESTION 7 (7 marks) A couple saved for their retirement by making the same monthly payments for 20 years into an account that earned 4.2% p.a. compounded monthly. At the age of 65 the couple retired and used all their savings to purchase a perpetuity with an interest rate of 5.76% p.a. compounded monthly paying $3600 each month. How much did they save each month to prepare for their retirement?
QUESTION 2 How many faces does this planar graph have? (A) 3 (B) 4 C) 5 (D) 7
QUESTION 5 The coach of a four-person relay team is deciding on the order of the runners. Use the bipartite graph to determine which statement 1s correct. Simon first runner Balal second runner Ivan third runner Linh fourth runner (A) Linh should be the first runner. (B) Simon should be the second runner. (C) Ivan should be the third runner. (D) Balal should be the fourth runner.
QUESTION 7 Which cut (C; C C5or Cy) could be used to determine the maximum flow from the source to the sink in this network? Cnl C Z C\\3 C4 90 | Source Sink Not to scale (55 (A) C B) G C) G (D) C4
QUESTION 8 A basketball competition has six teams that have completed three rounds of competition as shown. - - - [ | Bls | — | - | v | v | = Mo | v | v | = v = s — | v = v e | - - | v | = S| v = | = | v The graph to represent this information has (A) 6 edges. (B) 9 edges. (C) 15 edges. (D) 18 edges.
QUESTION 12 Which statement is correct? (A) A minimum spanning tree must contain a loop. (B) A minimum spanning tree must contain a cycle. (C) Every network has only one minimum spanning tree. (D) A minimum spanning tree has one more vertex than the number of edges.
QUESTION 13 Based on this project network what is the minimum number of days required to complete the project? (A) 16 (B) 17 (©) 19 (D) 31
QUESTION 15 Which of the following investment options gives the best return? (A) 5.93% p.a. compounding daily (B) 5.95% p.a. compounding monthly (C) 5.97% p.a. compounding quarterly (D) 5.99% p.a. compounding six-monthly
QUESTION 17 (4 marks) Determine the monthly repayment on a reducing balance loan of $720 000 at 4.8% p.a. over 25 years. Give your answer to the nearest dollar.
QUESTION 19 (5 marks) The activity table for a project is shown. Activity Prerequisite Time activity (days) a) b) ©) Use the activity table to construct a network diagram including earliest and latest starting times. [3 marks] Determine the critical path. [1 mark] Determine the shortest completion time for the project. [1 mark]
QUESTION 22 (4 marks) Rosa borrowed $32 000 as a reducing balance loan at 4.8% p.a. compounding monthly with monthly repayments of $278. How much will Rosa owe after 2 months to the nearest cent?
QUESTION 23 (4 marks) The network diagram shows the flow of water from the tank (source) to the kitchen tap (sink). L L L L Not to scale Tank ] D . 15 E a) Explain which dashed line (L L L; or L) 1s not a valid cut. [1 mark] b) Calculate the capacity of each dashed line that is a valid cut. [3 marks]
QUESTION 2 (4 marks) All buildings in a school are connected by underground electricity cables indicated by the network. All measurements are in metres. Admin Class 1 Sports shed Class 4 Tuckshop Not to scale 60 The electricity cables need replacing and will cost $1200 per metre. The school wants to minimise costs by replacing the shortest length of cable necessary to connect all buildings. If the school has a budget of $155 000 evaluate whether they can afford this project.
QUESTION 3 (5 marks) Jo contributes $2500 per quarter to an annuity earning 3.6% p.a. compounding quarterly. At the end of 4 years Jo makes a one-off extra contribution of $10 000 and continues with the regular quarterly contributions. Determine the value of the annuity at the end of 6 years to the nearest dollar.
QUESTION 4 (6 marks) The graph shows two investment models. Model 1 is compounding annually while Model 2 increases linearly by $126 per year. Model 1 2750 __- Model 2 Amount ($) 2020 2038 Year Determine the difference between the two investment strategies in 2030 to the nearest dollar.
QUESTION 6 (7 marks) A tow truck company has three tow trucks (A B and C) and receives calls from three motorists (P Q and R) who have broken down. The network shows the distances in kilometres (km) from each of the tow trucks to each of the motorists where x represents the distance between Tow Truck C and Motorist P. Not to scale The minimum total distance travelled by the three tow trucks in order for each tow truck to visit exactly one motorist 1s 32 km. Use the Hungarian algorithm to determine the distance between Tow Truck C and Motorist P.
QUESTION 1 The float time in minutes for the non-critical activity of this project network is Watch the news 10 min ‘Collect Turn on 'Prepare 'Cook qp Eat @b ingredients oven ingredients food food 3 min 5 min 8 min 17 min 5 min (A) 10 (B) 20 ) 23 (D) 30
QUESTION 2 The total number of vertices in this graph is (A) (B) ©) (D) ~N O WD W
QUESTION 6 This semi-Eulerian graph can be changed to an Eulerian graph by (A) adding a loop to vertex B. (B) removing the loop at vertex A. (C) adding an edge between vertices A and D. (D) removing the edge between vertices B and C.
QUESTION 7 This matrix was obtained after applying the Hungarian algorithm to determine the optimal allocation of three people Elandra (E) Farid (F) and Grace (G) to three tasks: legal (L) monitoring (M) and verification (V). L M V E|0 0 7 F|0 3 8 G|1 O O The optimal allocation is (A) EtoV FtoMand Gto L. (B) EtoV FtoLandGtoM. (C) EtoM FtoLand Gto V. (D) EtoM FtoVand GtoL.
QUESTION 9 Identify the graph that is a spanning tree. (A) o @) wn ~e ©) o &) = (B) (D) )—U w2 o 2 ( = ( =
QUESTION 17 (4 marks) An investment of $50 000 that compounds interest monthly is modelled by the recurrence relation A . 1=1.003754 where Ay =50 000. a) What would be the advertised interest rate per annum compounding monthly? [2 marks] b) How many months would it take for the value of the investment to exceed $51 000? [2 marks]
QUESTION 20 (4 marks) The table summarises the distances in kilometres (km) between three flower stores and three delivery locations: A B and C. Use the Hungarian algorithm to determine the minimum total distance needed to deliver flowers to all locations if each store delivers flowers to only one location. A B | C Store 1 24
QUESTION 21 (5 marks) The paths connecting various landmarks in a park are shown. D P Key B Bus stop C Coffee shop D Duck pond P Playground ¢ R Rose garden R W Water feature B a) Identify one cycle that passes through the rose garden and the playground. [1 mark] b) Identify whether the graph is Eulerian or semi-Eulerian. Justify your response. [2 marks] c) Construct an adjacency matrix from the graph using the vertex order listed in the key. [2 marks]
QUESTION 25 (5 marks) A couple borrow money to complete home renovations. Their bank has loaned the amount at 2.4% p.a. compounding monthly with repayments of $993.14 each month for 15 years. a) Determine the amount of money borrowed. [3 marks] b) Write a recurrence relation for the amount owing after n» months. [2 marks]
QUESTION 2 (5 marks) Tam deposits a fixed amount at the end of each month into an account paying 8.6% p.a. compounding monthly. From an initial zero balance she accumulates $51 343.85 in four years. A financial planner has advised Tam that she would have been at least $3000 better off if she had instead deposited half of the fixed amount at the end of each fortnight into an account paying 7.9% p.a. compounding fortnightly. Evaluate the reasonableness of this advice.
QUESTION 4 (5 marks) The table shows the current road length (in kilometres) between six towns. v [ vern [ o | e [ R | A T [ 6 s | [ % e | e | s e | e e | s w Cwew | T s e The government plans to build a direct road between Manon and Farra. Use a network diagram to determine the length of the direct road if it is to be 4 km shorter than the length of the current shortest road route between Manon and Farra.
QUESTION 5 (5 marks) The map details the length (in metres) of paths between nine key locations in a theme park. The annual cost to maintain the paths is $214 per metre. The theme park manager believes at least $138 000 can be saved each year if some paths are removed while still allowing visitors to access every key location using paths. Roller- coaster Ferris 170 Wheel Car park Eatlng area 05 T 6 60 Toilet Bumper Stunt 170 cars h show 185 165 65 Water slide Laser Not to scale show Evaluate the reasonableness of the manager’s belief.
QUESTION 3 The duration in minutes of all activities in a project are shown. Activity pe yTefTe[s|;rpeyy The critical path for the project is PRSV. What is the earliest completion time for the project if it starts at 11:00 am? (A) 12:30 pm (B) 1:10pm (C) 1:30pm (D) 2:10 pm
QUESTION 8 Activities P and Q are the critical activities for a project. Activity Prerequisite activity
QUESTION 11 An annuity with an initial zero balance has $500 deposited at the end of every month. The annuity earns 4.8% p.a. interest compounding monthly. At the end of the fourth month the balance is closest to (A) $2002 (B) $2008 (C) $2012 (D) $2014
QUESTION 12 A reducing balance loan with an initial balance of $6000 is modelled by the recurrence relation Anal = [ + “) A — 400 where n is the number of months. The loan balance at the end of two months is closest to (A) $5100 (B) $5200 (C) $5215 (D) $5230
QUESTION 13 Four athletes Eoin (E) Fedir (F) Gede (G) and Hana (H) compete in three events: javelin (J) long jump (L) and sprints (S). Which bipartite graph represents this information? “ok J ok J F L F L G S G S H H (C) (D) oo o- - mraoantt n ron n
QUESTION 17 (4 marks) Terome paid a $50 000 deposit on a house valued at $570 000 and borrowed the remainder as a reducing balance loan at 6.6% p.a. compounding monthly. Determine the monthly repayment required to pay off the loan over 25 years.
QUESTION 19 (4 marks) Ngarra compares two investment options and decides option A will provide the better return. ¢ Option A: 5.60% p.a. compounding monthly ¢ Option B: 5.62% p.a. compounding quarterly Use the effective annual rate of interest formula to evaluate the reasonableness of Ngarra’s decision.
QUESTION 20 (4 marks) Graphs 1| and 2 are shown. | 2 a) Show that Euler’s formula applies to graph 1. b) What feature in the drawing should be changed to represent graph 2 as planar? c) Redraw graph 2 to represent it as a planar graph. [2 marks] [1 mark] [1 mark]
QUESTION 21 (4 marks) The road network shows the number of vehicles per hour travelling from the airport to the town centre. cut X _ College Airport Town centre \ Showground a) Determine the capacity of cut X. b) Determine the maximum flow from the airport to the town centre. [1 mark] [2 marks]
QUESTION 23 (5 marks) A network graph is shown. ec a) State the degree of vertex E. [1 mark] b) State the number of edges joining D and E. [1 mark] c) Construct an adjacency matrix from the graph with the vertices in alphabetical order. [3 marks]
QUESTION 24 (5 marks) Jed is preparing and serving a meal of curry and rice with naan bread. The table shows the duration and prerequisites for all the tasks they must complete. Task description Duration (minutes) Prerequisite Assemble equipment 2 — Boil rice 20 Prepare curry ingredients 6 Simmer curry 40 Fry naan bread + Serve meal 2 |B Do Make naan bread 8 a) Construct a network diagram to show the sequence of tasks from start to finish labelling all tasks and durations. Use forward and backward scanning to show the earliest and latest starting times for all tasks. [3 marks] b) Determine the critical activities and minimum completion time for preparing and serving the meal. [2 marks]
QUESTION 1 (5 marks) A triathlon relay has three sections: swim (S) cycle (C) and run (R). The matrix shows the average number of minutes for three athletes Jane (J) Knox (K) and Levi (L) to complete each section. S C R J|40 56 66 K/ 36 60 72 L/}25 48 78 Use the Hungarian algorithm to predict the minimum total relay time if assigning each athlete to completing one section.
QUESTION 3 (5 marks) The diagram represents a network of 10 ski stations connected by chairlift cables. The length (km) of each cable is shown except for cable C which is closed for maintenance. When cable C reopens the minimum total cable length required to connect all stations will decrease by | km. Determine the length of cable C and the minimum total cable length required to connect all stations when cable C reopens.
QUESTION 5 (7 marks) At 9:00 am a security guard begins their patrol of the eight work sites represented in the network diagram starting and ending at site A. They drive at 40 km/h on the roads between sites and check every site once for 15 minutes. The length (km) of each road corresponds to the terms of the arithmetic sequence ¢ = ¢ + 2(n — 1) where ¢ = 1. Determine the earliest possible time the security guard can finish their patrol and identify the route they must follow. B ty C
QUESTION 7 (5 marks) Five years ago a retiree invested $100 000 in a compound interest account earning 3.8% p.a. compounding monthly. They now intend to use the balance of the account to begin a perpetuity that will return 4% p.a. compounding annually and pay them $6000 each year. Provide advice to the retiree about whether their compound interest investment is large enough to finance the perpetuity.
QUESTION 1 Identify a minimum spanning tree from the diagrams below. (A) (B)
QUESTION 2 The graph below shows the relationship between the amount owed and time taken to repay a reducing balance loan. 140 000 120 000 o 000e 000 00 c ..... 100 000 T 80 000 60 000 Amount owed ($) 40 000 20 000 0 20 40 60 80 100 120 140 160 180 Time (months) The owed amount will be reduced to half the original amount borrowed in (A) 60 months. (B) 81 months. (C) 112 months. (D) 162 months.
QUESTION 3 The graph below has six vertices. The vertex with the smallest degree is A) 1 (B) 2 ©C) 3 (D) 4
QUESTION 8 Activities A B C D E F G H and J must be completed for the project shown in the network. Activity times are in minutes. G 7 Start Finish The earliest starting time in minutes for Activity J on the network diagram above is A) 16 (B) 15 C) 13 (D) 11
QUESTION 9 The edges J R M P P Q on the network diagram above make (A) apath. (B) acycle. (C) aclosed trail. (D) an open walk.
QUESTION 13 For a personal loan of $350 a payment of $60 is made after interest is added each month. The following recurrence relation shows the remaining balance at the end of consecutive months. The recurrence relation is Anv = 1.02434 - 60 The remaining balance after two months is (A) $191.73 (B) $242.81 (C) $245.76 (D) $298.51
QUESTION 14 Identify the planar graph that has four faces. (A) g (B) j (C) B (D) ;
QUESTION 16 Determine the value of the cut made on the network diagram below. B 70 E<—Cut (A) 160 (B) 170 (C) 180 (D) 240
QUESTION 17 A reducing balance loan of $12 000 was taken out at 10.5% per annum with interest calculated monthly for five years. Determine the monthly repayment required to pay off the loan. (A) $152.93 (B) $162.18 (C) $25793 (D) $267.18
QUESTION 18 Which adjacency matrix below represents the same information that is given in the network diagram? T & To (B) To (A) oo o = UOllO _ o —~ © @ — N — o o o n = =2 woI > © © — © Do -~ o o _ o © © - N — o o o n = =2 woI To (D) To ©) b o = = O UOlOl - © — @ — N — o o o N == woxL J S e —~ 8 o Do o © = _ o © —~ O N — ©o o o n == WOL]
QUESTION 19 An investment of $10 000 is made at an interest rate of 7.5% per annum compounding half yearly. The number of years it will take for the value of the investment to exceed $18 000 is (A) 7 (B) 8 C) 15 (D) 16
QUESTION 21 (4 marks) To provide herself with a regular income at retirement Mary invests in an annuity worth $270 000 at 3.5% per annum compounding monthly for 20 years. Calculate how much income she will receive each month.
QUESTION 25 (3 marks) The connected weighted graph below shows the time taken in minutes and seconds to walk between places in a park. Rose garden 1 min 55 sec 1 min 50 sec Butterfly house 3 min 10 sec 2 min 30 sec 2 min 20 sec 4 min 55 sec Tlayground Garden shed 2 min 20 sec Cafe a) Identify which two places are joined by a network bridge. [1 mark] b) Calculate the earliest time that a gardener could arrive at the butterfly house if they leave the garden shed at 8:55 am. [2 marks]
QUESTION 30 (4 marks) The network diagram below illustrates the time taken in days for 10 different activities labelled A—K involved in the completion of a task. H 10 E 2 a) Identify the critical path for the diagram above. [1 mark] b) Calculate the latest starting time for Activity G. [1 mark] c) Calculate the float time for Activity E. [2 marks]
QUESTION 2 (4 marks) A cable company is calculating the cost of laying cable between five houses (A B C D and E). The unit cost of laying the cable is $2.50 per metre. The shortest distances (in metres) between the houses are shown below. a) Construct a network diagram to represent the distances between the houses in the space provided below. [1 mark] b) Determine the minimum cost to connect all of the houses. [3 marks]
QUESTION 4 (3 marks) You have a savings account with an interest rate of 2.4% per annum paid quarterly. Your bank makes you a new offer. You are entitled to either an increase in the interest rate of 0.5% per annum or to your current interest paid monthly. Determine which option is the most profitable.
QUESTION 5 (8 marks) Kate Luca and Marcel are carpenters. Each carpenter needs to build one piece of furniture: a bookcase a chair or a desk. As each carpenter works at a different rate the table below shows the total cost for each carpenter to build each piece of furniture. a) Use the Hungarian algorithm to determine who should build each piece of furniture to minimise the total cost. [5 marks] b) Evaluate the reasonableness of your solution to 5a). [3 marks]
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