- If `1/(root3(7+pi)) = (7+pi)^x`, find `x`. (1 mark)
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- Calculate the value of `1/(root3(7+pi))` to 3 significant figures. (1 mark)
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Functions, 2ADV F1 SM-Bank 50
Rationalise the denominator of `1/(4sqrt 3)`. (2 marks)
Vectors, EXT2 V1 SM-Bank 9
- Find the equation of line vector `underset ~r`, given it passes through `(1, 3, –2)` and `(2, –1, 2)`. (2 marks)
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- Determine if `underset ~r` passes through `(4, –9, 10)`. (1 mark)
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GRAPHS, FUR2 2019 VCAA 1
The graph below shows the membership numbers of the Wombatong Rural Women’s Association each year for the years 2008–2018.
- How many members were there in 2009? (1 mark)
-
- Show that the average rate of change of membership numbers from 2013 to 2018 was − 6 members per year. (1 mark)
- If the change in membership numbers continues at this rate, how many members will there be in 2021? (1 mark)
GEOMETRY, FUR2 2019 VCAA 1
The following diagram shows a cargo ship viewed from above.
The shaded region illustrates the part of the deck on which shipping containers are stored.
- What is the area, in square metres, of the shaded region? (1 mark)
Each shipping container is in the shape of a rectangular prism.
Each shipping container has a height of 2.6 m, a width of 2.4 m and a length of 6 m, as shown in the diagram below.
- What is the volume, in cubic metres, of one shipping container? (1 mark)
- What is the total surface area, in square metres, of the outside of one shipping container? (1 mark)
- One shipping container is used to carry barrels. Each barrel is in the shape of a cylinder.
Each barrel is 1.25 m high and has a diameter of 0.73 m, as shown in the diagram below.
Each barrel must remain upright in the shipping container
`qquad qquad`
What is the maximum number of barrels that can fit in one shipping container? (1 mark)
NETWORKS, FUR2 2019 VCAA 1
Fencedale High School has six buildings. The network below shows these buildings represented by vertices. The edges of the network represent the paths between the buildings.
- Which building in the school can be reached directly from all other buildings? (1 mark)
- A school tour is to start and finish at the office, visiting each building only once
- What is the mathematical term for this route? (1 mark)
- Draw in a possible route for this school tour on the diagram below. (1 mark)
MATRICES, FUR2 2019 VCAA 1
The car park at a theme park has three areas, `A, B` and `C`.
The number of empty `(E)` and full `(F)` parking spaces in each of the three areas at 1 pm on Friday are shown in matrix `Q` below.
`{:(qquad qquad qquad \ E qquad F),(Q = [(70, 50),(30, 20),(40, 40)]{:(A),(B),(C):}quad text(area)):}`
- What is the order of matrix `Q`? (1 mark)
- Write down a calculation to show that 110 parking spaces are full at 1 pm. (1 mark)
Drivers must pay a parking fee for each hour of parking.
Matrix `P`, below, shows the hourly fee, in dollars, for a car parked in each of the three areas.
`{:(qquad qquad qquad qquad qquad text{area}), (qquad qquad qquad A qquad quad quad B qquad qquad C), (Q = [(1.30, 3.50, 1.80)]):}`
- The total parking fee, in dollars, collected from these 110 parked cars if they were parked for one hour is calculated as follows.
`qquad qquad qquad P xx L = [207.00]`
where matrix `L` is a `3 xx 1` matrix.
Write down matrix `L`. (1 mark)
The number of whole hours that each of the 110 cars had been parked was recorded at 1 pm. Matrix `R`, below, shows the number of cars parked for one, two, three or four hours in each of the areas `A, B` and `C`.
`{:(qquadqquadqquadqquadquadtext(area)),(quad qquadqquadquad \ A qquad B qquad C),(R = [(3, 1, 1),(6, 10, 3),(22, 7,10),(19, 2, 26)]{:(1),(2),(3),(4):}\ text(hours)):}`
- Matrix `R^T` is the transpose of matrix `R`.
Complete the matrix `R^T` below. (1 mark)
`qquad R^T = [( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , ), ( , , , , , , , , )]`
- Explain what the element in row 3, column 2 of matrix `R^T` represents. (1 mark)
CORE, FUR2 2019 VCAA 2
The parallel boxplots below show the maximum daily temperature and minimum daily temperature, in degrees Celsius, for 30 days in November 2017.
- Use the information in the boxplots to complete the following sentences.
For November 2017
i. | the interquartile range for the minimum daily temperature was |
|
°C (1 mark) |
ii. | the median value for maximum daily temperature was |
|
°C higher than the |
median value for minimum daily temperature (1 mark) |
iii. | the number of days on which the maximum daily temperature was less than the median value for |
minimum daily temperature was |
|
(1 mark) |
- The temperature difference between the minimum daily temperature and the maximum daily temperature in November 2017 at this location is approximately normally distributed with a mean of 9.4 °C and a standard deviation of 3.2 °C.
Determine the number of days in November 2017 for which this temperature difference is expected to be greater than 9.4 °C. (1 mark)
GRAPHS, FUR1 2019 VCAA 1 MC
GEOMETRY, FUR1 2019 VCAA 1 MC
The four bases of a baseball field form four corners of a square of side length 27.43 m, as shown in the diagram below.
A player ran from home base to first base, then to second base, then to third base and finally back to home base.
The minimum distance, in metres, that the player ran is
- 27.43
- 54.86
- 82.29
- 109.72
- 164.58
NETWORKS, FUR1 2019 VCAA 1 MC
MATRICES, FUR1 2019 VCAA 1 MC
Consider the following four matrix expressions.
`[(8), (12)]+[(4), (2)] qquad qquad qquad qquad qquad qquad quad [(8), (12)]+[(4, 0),(0, 2)]`
`[(8, 0),(12, 0)] + [(4), (2)] qquad qquad qquad qquad qquad \ [(8, 0),(12, 0)] + [(4, 0),(0, 2)]`
How many of these four matrix expressions are defined?
- 0
- 1
- 2
- 3
- 4
Calculus, SPEC1 2019 VCAA 5
The graph of `f(x) = cos^2(x) + cos(x) + 1` over the domain `0 <= x <= 2pi` is shown below.
- i. Find `f'(x)`. (1 mark)
- ii. Hence, find the coordinates of the turning points of the graph in the interval `(0, 2pi)`. (2 marks)
- Sketch the graph of `y = 1/(f(x))` on the set of axes above. Clearly label the turning points and endpoints of this graph with their coordinates. (3 marks)
Complex Numbers, EXT2 N1 2019 HSC 11e
Let `z = -1 + i sqrt 3`.
- Write `z` in modulus-argument form. (2 marks)
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- Find `z^3`, giving your answer in the form `x + iy`, where `x` and `y` are real numbers. (2 marks)
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Calculus, EXT2 C1 2019 HSC 11d
Find `int 6/(x^2-9) dx`. (3 marks)
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Complex Numbers, EXT2 N1 2019 HSC 11a
Let `z = 1 + 3i` and `w = 2 - i`.
- Find `z + bar w`. (1 mark)
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- Express `z/w` in the form `x + iy`, where `x` and `y` are real numbers. (2 marks)
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Complex Numbers, EXT2 N1 2019 HSC 1 MC
What is the value of `(3 - 2i)^2`?
- `5 - 12i`
- `5 + 12i`
- `13 - 12i`
- `13 + 12i`
Networks, STD1 N1 2019 HSC 1 MC
Calculus, EXT1* C1 2019 HSC 12c
The number of leaves, `L(t)`, on a tree `t` days after the start of autumn can be modelled by
`L(t) = 200\ 000e^(-0.14t)`
- What is the number of leaves on the tree when `t = 31`? (1 mark)
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- What is the rate of change of the number of leaves on the tree when `t = 31`? (2 marks)
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- For what value of `t` are there 100 leaves on the tree? (2 marks)
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Calculus, 2ADV C2 2019 HSC 11b
Differentiate `x^2 sin x`. (2 marks)
Trigonometry, 2ADV T1 2019 HSC 11a
Measurement, STD2 M1 2019 HSC 1 MC
Which of the following shapes has a perimeter of 12 cm?
A. | B. | ||
C. | D. |
NOT TO SCALE
Calculus, SPEC1 2011 VCAA 3
- Show that `f(x) = (2x^2 + 3)/(x^2 + 1)` can be written in the form `f(x) = 2 + 1/(x^2 + 1).` (1 mark)
- Sketch the graph of the relation `y = (2x^2 + 3)/(x^2 + 1)` on the axes below.
Label any asymptotes with their equations and label any intercepts with the axes, writing them as coordinates. (3 marks)
- Find the area enclosed by the graph of the relation `y = (2x^2 + 3)/(x^2 + 1)`, the `x`-axis, and the lines `x = -1` and `x = 1.` (3 marks)
Calculus, SPEC2 2017 VCAA 2
A helicopter is hovering at a constant height above a fixed location. A skydiver falls from rest for two seconds from the helicopter. The skydiver is subject only to gravitational acceleration and air resistance is negligible for the first two seconds. Let downward displacement be positive.
- Find the distance, in metres, fallen in the first two seconds. (2 marks)
- Show that the speed of the skydiver after two seconds is 19.6 ms–1. (1 mark)
After two seconds, air resistance is significant and the acceleration of the skydiver is given by `a = g -0.01v^2`.
- Find the limiting (terminal) velocity, in ms–1, that the skydiver would reach. (1 mark)
- i. Write down an expression involving a definite integral that gives the time taken for the skydiver to reach a speed of 30 ms–1. (2 marks)
- ii. Hence, find the time, in seconds, taken to reach a speed of 30 ms–1, correct to the nearest tenth of a second. (1 mark)
- Write down an expression involving a definite integral that gives the distance through which the skydiver falls to reach a speed of 30 ms–1. Find this distance, giving your answer in metres, correct to the nearest metre. (3 marks)
Calculus, SPEC2 2018 VCAA 3
Part of the graph of `y = 1/2 sqrt(4x^2 - 1)` is shown below.
The curve shown is rotated about the `y`-axis to form a volume of revolution that is to model a fountain, where length units are in metres.
- Show that the volume, `V` cubic metres, of water in the fountain when it is filled to a depth of `h` metres is given by `V = pi/4(4/3h^3 + h)`. (2 marks)
- Find the depth `h` when the fountain is filled to half's its volume. Give your answer in metres, correct to two decimal places. (2 marks)
The fountain is initially empty. A vertical jet of water in the centre fills the fountain at a rate of 0.04 cubic metres per second and, at the same time, water flows out from the bottom of the fountain at a rate of `0.05 sqrt h` cubic metres per second when the depth is `h` metres.
- i. Show that `(dh)/(dt) = (4-5sqrt h)/(25 pi (4h^2 + 1))`. (2 marks)
- ii. Find the rate, in metres per second, correct to four decimal places, at which the depth is increasing when the depth is 0.25 m. (1 mark)
- Express the time taken for the depth to reach 0.25 m as a definite integral and evaluate this integral correct to the nearest tenth of a second. (2 marks)
- After 25 seconds the depth has risen to 0.4 m.
Using Euler's method with a step size of five seconds, find an estimate of the depth 30 seconds after the fountain began to fill. Give your answer in metres, correct to two decimal places. (2 marks) - How far from the top of the fountain does the water level ultimately stabilise? Give your answer in metres, correct to two decimal places. (2 marks)
Mechanics, SPEC1 2016 VCAA 1
A taut rope of length `1 2/3` m suspends a mass of 20 kg from a fixed point `O`. A horizontal force of `P` newtons displaces the mass by 1 m horizontally so that the taut rope is then at an angle of `theta` to the vertical.
- Show all the forces acting on the mass on the diagram below. (1 mark)
- Show that `sin (theta) = 3/5`. (1 mark)
- Find the magnitude of the tension force in the rope in newtons. (2 marks)
Networks, STD2 N2 2011 FUR2 1
Aden, Bredon, Carrie, Dunlop, Enwin and Farnham are six towns.
The network shows the road connections and distances between these towns in kilometres.
- In kilometers, what is the shortest distance between Farnham and Carrie? (1 mark)
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- How many different ways are there to travel from Farnham to Carrie without passing through any town more than once? (1 mark)
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Algebra, MET2 2018 VCAA 1 MC
Let `f: R -> R,\ f(x) = 4 cos ((2 pi x)/3) + 1`.
The period of this function is
- 1
- 2
- 3
- 4
- 5
GRAPHS, FUR2 2018 VCAA 1
GEOMETRY, FUR1 2018 VCAA 1 MC
NETWORKS, FUR1 2018 VCAA 1 MC
NETWORKS, FUR2 2018 VCAA 2
In one area of the town of Zenith, a postal worker delivers mail to 10 houses labelled as vertices `A` to `J` on the graph below.
- Which one of the vertices on the graph has degree 4? (1 mark)
For this graph, an Eulerian trail does not currently exist.
- For an Eulerian trail to exist, what is the minimum number of extra edges that the graph would require. (1 mark)
- The postal worker has delivered the mail at `F` and will continue her deliveries by following a Hamiltonian path from `F`.
Draw in a possible Hamiltonian path for the postal worker on the diagram below. (1 mark)
MATRICES, FUR2 2018 VCAA 1
A toll road is divided into three sections, `E, F` and `G`.
The cost, in dollars, to drive one journey on each section is shown in matrix `C` below.
`C = [(3.58),(2.22),(2.87)]{:(E),(F),(G):}`
- What is the cost of one journey on section `G`? (1 mark)
- Write down the order of matrix `C`. (1 mark)
- One day Kim travels once on section `E` and twice on section `G`.
His total toll cost for this day can be found by the matrix product `M xx C`.
Write down the matrix `M`. (1 mark)
CORE, FUR2 2018 VCAA 4
Julie deposits some money into a savings account that will pay compound interest every month.
The balance of Julie’s account, in dollars, after `n` months, `V_n` , can be modelled by the recurrence relation shown below.
`V_0 = 12\ 000, qquad V_(n + 1) = 1.0062 V_n`
- How many dollars does Julie initially invest? (1 mark)
- Recursion can be used to calculate the balance of the account after one month.
- Write down a calculation to show that the balance in the account after one month, `V_1`, is $12 074.40. (1 mark)
- After how many months will the balance of Julie’s account first exceed $12 300 (1 mark)
- A rule of the form `V_n = a xx b^n` can be used to determine the balance of Julie's account after `n` months.
- Complete this rule for Julie’s investment after `n` months by writing the appropriate numbers in the boxes provided below. (1 mark)
- Complete this rule for Julie’s investment after `n` months by writing the appropriate numbers in the boxes provided below. (1 mark)
balance = |
|
× |
|
`n` |
-
- What would be the value of `n` if Julie wanted to determine the value of her investment after three years? (1 mark)
CORE, FUR2 2018 VCAA 1
The data in Table 1 relates to the impact of traffic congestion in 2016 on travel times in 23 cities in the United Kingdom (UK).
The four variables in this data set are:
- city — name of city
- congestion level — traffic congestion level (high, medium, low)
- size — size of city (large, small)
- increase in travel time — increase in travel time due to traffic congestion (minutes per day).
- How many variables in this data set are categorical variables? (1 mark)
- How many variables in this data set are ordinal variables (1 mark)
- Name the large UK cities with a medium level of traffic congestion. (1 mark)
- Use the data in Table 1 to complete the following two-way frequency table, Table 2. (2 marks)
- What percentage of the small cities have a high level of traffic congestion? (1 mark)
Traffic congestion can lead to an increase in travel times in cities. The dot plot and boxplot below both show the increase in travel time due to traffic congestion, in minutes per day, for the 23 UK cities.
- Describe the shape of the distribution of the increase in travel time for the 23 cities. (1 mark)
- The data value 52 is below the upper fence and is not an outlier.
Determine the value of the upper fence. (1 mark)
Functions, 2ADV F2 SM-Bank 1
- Draw the graph `y = ln x`. (1 mark)
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- Explain how the above graph can be transformed to produce the graph
`y = 3ln(x + 2)`
and sketch the graph, clearly identifying all intercepts. (3 marks)
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Calculus, EXT1 C1 EQ-Bank 12
A tank is initially full. It is drained so that at time `t` seconds the volume of water, `V`, in litres, is given by
`V = 50(1 - t/80)^2` for `0 <= t <= 100`
- How much water was initially in the tank? (1 mark)
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- After how many seconds was the tank one-quarter full? (1 mark)
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- At what rate was the water draining out the tank when it was one-quarter full? (2 marks)
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Trigonometry, EXT1 T2 SM-Bank 3
Show that
`sin(8x + 3x) + sin(8x - 3x) = 2sin(8x)cos(3x)`. (1 mark)
Calculus, 2ADV C1 2008 HSC 6b
The graph shows the velocity of a particle, `v` metres per second, as a function of time, `t` seconds.
- What is the initial velocity of the particle? (1 mark)
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- When is the velocity of the particle equal to zero? (1 mark)
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- When is the acceleration of the particle equal to zero? (1 mark)
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Complex Numbers, EXT2 N2 2018 HSC 11d
The points `A`, `B` and `C` on the Argand diagram represent the complex numbers `u`, `v` and `w` respectively.
The points `O`, `A`, `B` and `C` form a square as shown on the diagram.
It is given that `u = 5 + 2i`.
- Find `w`. (1 mark)
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- Find `v`. (1 mark)
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- Find `text(arg)(w/v)`. (1 mark)
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Complex Numbers, EXT2 N1 2018 HSC 11a
Let `z = 2 + 3i` and `w = 1 - i.`
- Find `zw`. (1 mark)
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- Express `barz - 2/w` in the form `x + iy`, where `x` and `y` are real numbers. (2 marks)
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Measurement, STD2 M7 2018 HSC 27a
Jenny used her mobile phone while she was overseas for one month.
Her mobile phone plan has a base monthly cost of $50. While overseas, she is also charged 33 cents per SMS message sent and 26 cents per MB of data used.
During her month overseas, Jenny sent 120 SMS messages and used 1400 MB of data.
What was her mobile phone bill for the month overseas? (2 marks)
Statistics, STD2 S1 2018 HSC 26d
The graph displays the mean monthly rainfall in Sydney and Perth.
- For how many months is the mean monthly rainfall higher in Perth than in Sydney? (1 mark)
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- For which of the two cities is the standard deviation of the mean monthly rainfall smaller? Justify your answer WITHOUT calculations. (1 mark)
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Linear Functions, 2UA 2018 HSC 2 MC
Functions, 2ADV F1 2018 HSC 1 MC
What is the value of `7^(-1.3)` correct to two decimal places?
(A) 0.07
(B) 0.08
(C) -12.54
(D) -12.55
Networks, STD2 N3 2008 FUR1 1 MC
Steel water pipes connect five points underground.
The directed graph below shows the directions of the flow of water through these pipes between these points.
The directed graph shows that water can flow from
A. point 1 to point 2.
B. point 1 to point 4.
C. point 4 to point 1.
D. point 4 to point 2.
Networks, FUR2 2016 VCE 1
A map of the roads connecting five suburbs of a city, Alooma (`A`), Beachton (`B`), Campville (`C`), Dovenest (`D`) and Easyside (`E`), is shown below.
- Starting at Beachton, which two suburbs can be driven to using only one road? (1 mark)
A graph that represents the map of the roads is shown below.
One of the edges that connects to vertex `E` is missing from the graph.
-
- Add the missing edge to the graph above. (1 mark)
(Answer on the graph above.)
- Explain what the loop at `D` represents in terms of a driver who is departing from Dovenest. (1 mark)
- Add the missing edge to the graph above. (1 mark)
Networks, STD2 N2 2011 FUR2 1
Aden, Bredon, Carrie, Dunlop, Enwin and Farnham are six towns.
The network shows the road connections and distances between these towns in kilometres.
- In kilometres, what is the shortest distance between Farnham and Carrie? (1 mark)
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- How many different ways are there to travel from Farnham to Carrie without passing through any town more than once? (1 mark)
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An engineer plans to inspect all of the roads in this network.
He will start at Dunlop and inspect each road only once.
- At which town will the inspection finish? (1 mark)
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Networks, STD2 N2 2017 FUR2 1
Bus routes connect six towns.
The towns are Northend (`N`), Opera (`O`), Palmer (`P`), Quigley (`Q`), Rosebush (`R`) and Seatown (`S`).
The graph below gives the cost, in dollars, of bus travel along these routes.
Bai lives in Northend (`N`) and he will travel by bus to take a holiday in Seatown (`S`).
- Bai considers travelling by bus along the route Northend (`N`) – Opera (`O`) – Seatown (`S`).
How much would Bai have to pay? (1 mark)
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- If Bai takes the cheapest route from Northend (`N`) to Seatown (`S`), which other town(s) will he pass through? (1 mark)
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Networks, STD2 N2 2013 FUR2 1
The vertices in the network diagram below show the entrance to a wildlife park and six picnic areas in the park: `P1`, `P2`, `P3`, `P4`, `P5` and `P6`.
The numbers on the edges represent the lengths, in metres, of the roads joining these locations.
- In this graph, what is the degree of the vertex at the entrance to the wildlife park? (1 mark)
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- What is the shortest distance, in metres, from the entrance to picnic area `P3`? (1 mark)
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Networks, STD2 N2 2015 FUR1 5 MC
The graph below represents a friendship network. The vertices represent the four people in the friendship network: Kwan (K), Louise (L), Milly (M) and Narelle (N).
An edge represents the presence of a friendship between a pair of these people. For example, the edge connecting K and L shows that Kwan and Louise are friends.
Which one of the following graphs does not contain the same information?
Networks, STD2 N2 2009 FUR1 2 MC
Networks, STD2 N2 2015 FUR2 1
A factory requires seven computer servers to communicate with each other through a connected network of cables.
The servers, `J`, `K`, `L`, `M`, `N`, `O` and `P`, are shown as vertices on the graph below.
The edges on the graph represent the cables that could connect adjacent computer servers.
The numbers on the edges show the cost, in dollars, of installing each cable.
- What is the cost, in dollars, of installing the cable between server `L` and server `M`? (1 mark)
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- What is the cheapest cost, in dollars, of installing cables between server `K` and server `N`? (1 mark)
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- The computer servers will be able to communicate with all the other servers as long as each server is connected by cable to at least one other server.
- The cheapest installation that will join the seven computer servers by cable in a connected network follows a minimum spanning tree.
- The factory’s manager has decided that only six connected computer servers will be needed, rather than seven.
How much would be saved in installation costs if the factory removed computer server `P` from its minimum spanning tree network?
A copy of the graph above is provided below to assist with your working. (1 mark)
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- The cheapest installation that will join the seven computer servers by cable in a connected network follows a minimum spanning tree.
Networks, STD2 N2 2015 FUR1 1 MC
Networks, STD2 N2 2013 FUR1 1 MC
Networks, STD2 N2 2010 FUR1 2 MC
Networks, STD2 N2 2012 FUR1 1 MC
GRAPHS, FUR2 2017 VCAA 1
NETWORKS, FUR2 2017 VCAA 1
Bus routes connect six towns.
The towns are Northend (`N`), Opera (`O`), Palmer (`P`), Quigley (`Q`), Rosebush (`R`) and Seatown (`S`).
The graph below gives the cost, in dollars, of bus travel along these routes.
Bai lives in Northend (`N`) and he will travel by bus to take a holiday in Seatown (`S`).
- Bai considers travelling by bus along the route Northend (`N`) – Opera (`O`) – Seatown (`S`).
How much would Bai have to pay? (1 mark)
- If Bai takes the cheapest route from Northend (`N`) to Seatown (`S`), which other town(s) will he pass through? (1 mark)
- Euler’s formula, `v + f = e + 2`, holds for this graph.
Complete the formula by writing the appropriate numbers in the boxes provided below. (1 mark)
MATRICES, FUR2 2017 VCAA 2
Junior students at a school must choose one elective activity in each of the four terms in 2018.
Students can choose from the areas of performance (`P`), sport (`S`) and technology (`T`).
The transition diagram below shows the way in which junior students are expected to change their choice of elective activity from term to term.
- Of the junior students who choose performance (`P`) in one term, what percentage are expected to choose sport (`S`) the next term? (1 mark)
Matrix `J_1` lists the number of junior students who will be in each elective activity in Term 1.
`J_1 = [(300),(240),(210)]{:(P),(S),(T):}`
- 306 junior students are expected to choose sport (`S`) in Term 2.
Complete the calculation below to show this. (1 mark)
- In Term 4, how many junior students in total are expected to participate in performance (`P`) or sport (`S`) or technology (`T`)? (1 mark)
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