Band 2 – Page 6

NETWORKS, FUR2 2009 VCAA 3

The city of Robville contains eight landmarks denoted as vertices `N` to `U` on the network diagram below. The edges on this network represent the roads that link the eight landmarks.

Write down the degree of vertex `U`. (1 mark)
Steven wants to visit each landmark, but drive along each road only once. He will begin his journey at landmark `N`.
1. At which landmark must he finish his journey? (1 mark)
2. Regardless of which route Steven decides to take, how many of the landmarks (including those at the start and finish) will he see on exactly two occasions? (1 mark)
Cathy decides to visit each landmark only once.
1. Suppose she starts at `S`, then visits `R` and finishes at `T`.
  
  Write down the order Cathy will visit the landmarks. (1 mark)
2. Suppose Cathy starts at `S`, then visits `R` but does not finish at `T`.
  
  List three different ways that she can visit the landmarks. (1 mark)

Show Answers Only

`4`
1. `P\ text{(the other odd degree vertex)}`
2. `5`
1. `(SR)QPONU(T)`
2. `text(Other paths are)`
  `(SR)QPUTNO`
  `(SR)QPONTU`
  `(SR)TUNOPQ`
  `(SR)UTNOPQ`
  `text{(only 3 paths required)}`

Show Worked Solution

a. `4`

b.i. `P\ text{(the other odd degree vertex)}`

b.ii. `5 (N, T, R, P, U)`

c.i. `(SR)QPONU(T)`

c.ii. `text(Other paths are)`

`(SR)QPUTNO`

`(SR)QPONTU`

`(SR)TUNOPQ`

`(SR)UTNOPQ`

`text{(only 3 paths required)}`

NETWORKS, FUR2 2011 VCAA 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)
How many different ways are there to travel from Farnham to Carrie without passing through any town more than once? (1 mark)

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)

Another engineer decides to start and finish her road inspection at Dunlop.

If an assistant inspects two of the roads, this engineer can inspect the remaining six roads and visit each of the other five towns only once.

How many kilometres of road will the assistant need to inspect? (1 mark)

Show Answers Only

`200\ text(km)`
`6`
`text(Bredon)`
`240\ text(km)`

Show Worked Solution

a. `text{Farnham to Carrie (shortest)}`

`= 60 + 140`

`= 200\ text(km)`

b. `text(Different paths are)`

`FDC, FEDC, FEBC,`

`FEABC, FDEBC,`

`FDEABC`

`:. 6\ text(different ways)`

c. `text(A possible path is)\ DFEABCDEB\ text(and will finish)`

`text{at Bredon (the other odd-degree vertex).}`

d. `text(If the engineer’s path is)`

`DFEABCD,`

`text(Distance assistant inspects)`

`= 110 + 130`

`= 240\ text(km)`

NETWORKS, FUR2 2013 VCAA 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)
What is the shortest distance, in metres, from the entrance to picnic area `P3`? (1 mark)
A park ranger starts at the entrance and drives along every road in the park once.
1. At which picnic area will the park ranger finish? (1 mark)
2. What mathematical term is used to describe the route the park ranger takes? (1 mark)
A park cleaner follows a route that starts at the entrance and passes through each picnic area once, ending at picnic area `P1`.

Write down the order in which the park cleaner will visit the six picnic areas. (1 mark)

Show Answers Only

`3`
`1000\ text(m)`
1. `P4`
2. `text(Euler path)`
`E − P5 − P4 − P6 − P3 − P2 − P1`

Show Worked Solution

a. `3`

b. `text( Shortest distance)`

`= E – P1 – P3`

`= 600 + 400`

`= 1000\ text(m)`

c.i. `text(A route could be)`

`E − P1 − P2 − P3 − P4 − P5`

`− E − P6 − P1 − P3 − P6 − P4`

`:.\ text(Finish at)\ P4\ \ text{(the other odd degree vertex)}`

c.ii. `text(Euler path)`

d. `E − P5 − P4 − P6 − P3 − P2 − P1`

NETWORKS, FUR2 2015 VCAA 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)
What is the cheapest cost, in dollars, of installing cables between server `K` and server `N`? (1 mark)
An inspector checks the cables by walking along the length of each cable in one continuous path.

To avoid walking along any of the cables more than once, at which vertex should the inspector start and where would the inspector finish? (1 mark)
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.
1. The cheapest installation that will join the seven computer servers by cable in a connected network follows a minimum spanning tree.
  
  Draw the minimum spanning tree on the plan below. (1 mark)
2. 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)

Show Answers Only

`$300`
`$920`
`N\ text(and)\ P\ text{(or}\ P\ text(and)\ N)`
2. `$120`

Show Worked Solution

a. `$300`

b. `text(C)text(ost of)\ K\ text(to)\ N`

`= 440 + 480`

`= $920`

c. `N\ text(and)\ P\ text{(or}\ P\ text(and)\ N)`

MARKER’S COMMENT: Many students had difficulty finding the minimum spanning tree, often incorrectly excluding `PO` or `KL`.

d.i.

d.ii. `text(Disconnect)\ J – P\ text(and)\ O – P`

`text(Savings) = 200 + 400 = $600`

`text(Add in)\ M – N`

`text(C)text(ost) = $480`

`:.\ text(Net savings)`	`= 600 – 480`
	`= $120`

MATRICES, FUR2 2006 VCAA 1

A manufacturer sells three products, `A`, `B` and `C`, through outlets at two shopping centres, Eastown (`E`) and Noxland (`N`).

The number of units of each product sold per month through each shop is given by the matrix `Q`, where

`{:((qquadqquadqquad\ A,qquadquadB,qquad\ C)),(Q=[(2500,3400,1890),(1765,4588,2456)]{:(E),(N):}):}`

Write down the order of matrix `Q`. (1 mark)

The matrix `P`, shown below, gives the selling price, in dollars, of products `A`, `B`, `C`.

`P = [(14.50),(21.60),(19.20)]{:(A),(B),(C):}`

1. Evaluate the matrix `M`, where `M = QP`. (1 mark)
2. What information does the elements of matrix `M` provide? (1 mark)
Explain why the matrix `PQ` is not defined. (1 mark)

Show Answers Only

`2 xx 3`
1. `M = QP = [(135\ 320.5),(171\ 848.5)]`
2. `text(The total of selling products)\ A, B,and C`
  `text(at each of Eastown and Noxland.)`
`PQ\ text(is not defined because the number of)`
`text(columns in)\ P !=\ text(the number of rows in)\ Q.`

Show Worked Solution

a. `2 xx 3`

b.i.	`M`	`= QP`
		`= [(2500,3400,1890),(1765,4588,2456)][(14.50),(21.60),(19.20)]`
		`= [(135\ 320.5),(171\ 848.5)]`

b.ii. `text(The total revenue from selling products)\ A, B,`

`text(and)\ C\ text(at each of Eastown and Noxland.)`

c. `PQ\ text(is not defined because the number of)`

`text(columns in)\ P !=\ text(the number of rows in)\ Q.`

NETWORKS, FUR2 2010 VCAA 1

The members of one team are Kristy (`K`), Lyn (`L`), Mike (`M`) and Neil (`N`).

In one of the challenges, these four team members are only allowed to communicate directly with each other as indicated by the edges of the following network.

The adjacency matrix below also shows the allowed lines of communication.

`{:(quadKquadLquadMquadN),([(0,1,0,0),(1,0,1,0),(0,f,0,1),(0,g,1,0)]{:(K),(L),(M),(N):}):}`

Explain the meaning of a zero in the adjacency matrix. (1 mark)
Write down the values of `f` and `g` in the adjacency matrix. (1 mark)

Show Answers Only

`text(No direct communication is allowed.)`
`f = 1, g = 0`

Show Worked Solution

a. `text(No direct communication is allowed.)`

b. `f = 1, g = 0`

MATRICES, FUR2 2010 VCAA 1

In a game of basketball, a successful shot for goal scores one point, two points, or three points, depending on the position from which the shot is thrown.

`G` is a column matrix that lists the number of points scored for each type of successful shot.

`G = [(1),(2),(3)]`

In one game, Oscar was successful with

- 4 one-point shots for goal
- 8 two-point shots for goal
- 2 three-point shots for goal.

Write a row matrix, `N`, that shows the number of each type of successful shot for goal that Oscar had in that game. (1 mark)
Matrix `P` is found by multiplying matrix `N` with matrix `G` so that `P = N xx G`

Evaluate matrix `P`. (1 mark)
In this context, what does the information in matrix `P` provide? (1 mark)

Show Answers Only

`N = [(4, 8, 2)]`
`P = [26]`
`text(The total points scored by Oscar in the game.)`

Show Worked Solution

a. `N = [(4, 8, 2)]`

b.	`P`	`= NG`
		`= [(4, 8, 2)][(1),(2),(3)]`
		`= [26]`

c. `text(The total points scored by Oscar in the game.)`

The diagram below shows the feeding paths for insects (`I`), birds (`B`) and lizards (`L`). The matrix `E` has been constructed to represent the information in this diagram. In matrix `E`, a 1 is read as "eat" and a 0 is read as "do not eat".

Referring to insects, birds or lizards
1. what does the 1 in column `B`, row `L`, of matrix `E` indicate? (1 mark)
2. what does the row of zeros in matrix `E` indicate? (1 mark)

The diagram below shows the feeding paths for insects (`I`), birds (`B`), lizards (`L`) and frogs (`F`).

The matrix `Z` has been set up to represent the information in this diagram.

Matrix `Z` has not been completed.

Complete the matrix `Z` above by writing in the seven missing elements. (1 mark)

Show Answers Only

`text(Birds eat lizards)`
`text(Insects, birds or lizards do not eat birds)`
`{:((qquadqquadquadI,B,L,F)),(Z = [(0,1,1,1),(0,0,0,0),(0,1,0,0),(0,1,1,0)]{:(I),(B),(L),(F):}):}`

Show Worked Solution

a.i. `text(The 1 represents that birds eat lizards.)`

a.ii. `text(It indicates that insects, birds or lizards DO NOT)`

`text(eat birds.)`

b.	`{:((qquadqquadquadI,B,L,F)),(Z = [(0,1,1,1),(0,0,0,0),(0,1,0,0),(0,1,1,0)]{:(I),(B),(L),(F):}):}`

MATRICES, FUR2 2013 VCAA 1

Five trout-breeding ponds, `P`, `Q`, `R`, `X` and `V`, are connected by pipes, as shown in the diagram below.

The matrix `W` is used to represent the information in this diagram.

`{:({:\ qquadqquadqquadPquadQquad\ Rquad\ Xquad\ V:}),(W = [(0,1,1,1,0), (1,0,0,1,0),(1,0,0,1,0),(1,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

In matrix `W`

• the 1 in column 1, row 2, for example, indicates that a pipe directly connects pond `P` and pond `Q`

• the 0 in column 1, row 5, for example, indicates that pond `P` and pond `V` are not directly connected by a pipe.

Find the sum of the elements in row 3 of matrix `W`. (1 mark)
In terms of the breeding ponds described, what does the sum of the elements in row 3 of matrix `W` represent? (1 mark)

The pipes connecting pond `P` to pond `R` and pond `P` to pond `X` are removed.

Matrix `N` will be used to show this situation. However, it has missing elements.

Complete matrix `N` below by filling in the missing elements in row 1 and column 1. (1 mark)

Show Answers Only

`2`
`text(The sum means that 2 other)`
`text(ponds connect directly to pond)\ R.`
`{:({:qquadqquadqquadPquadQquadRquadXquadV:}),(N = [(0,1,0,0,0),(1,0,0,1,0),(0,0,0,1,0),(0,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

Show Worked Solution

a. `1+ 0 + 0 +1+ 0 = 2`

b. `text(The sum means that 2 other ponds)`

`text(connect directly to pond)\ R.`

c.	`{:({:qquadqquadqquadPquadQquad\ Rquad\ Xquad\ V:}),(N = [(0,1,0,0,0),(1,0,0,1,0),(0,0,0,1,0),(0,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

MATRICES, FUR1 2009 VCAA 1 MC

`3[[2,1],[0,3]] + 2[[−1,0],[2,−7]]` equals

A. `[[4,3],[4,−5]]`

B. `6[[1,1],[2,−4]]`

C. `[[4,3],[4,2]]`

D. `5[[1,1],[2,−4]]`

E. `[[3,6],[7,4]]`

Show Answers Only

`A`

Show Worked Solution

`3[(2,1),(0,3)] + 2[(−1,0),(2,−7)]`

`= [(6,3),(0,9)] + [(−2,0),(4,−14)]`

`= [(4,3),(4,−5)]`

`=> A`

MATRICES, FUR1 2010 VCAA 1 MC

The order of the matrix `[(2, 2), (2, 2), (2, 2)]` is

A. `2 xx 2`

B. `2 xx 3`

C. `3 xx 2`

D. `4`

E. `6`

Show Answers Only

`C`

Show Worked Solution

`=> C`

MATRICES, FUR1 2013 VCAA 1 MC

`[(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)][(2), (0), (0), (2)] - 2× [(0), (-1), (-1), (0)] \ text (equals)`

Show Answers Only

`B`

Show Worked Solution

`I xx [(2),(0),(0),(2)] – 2[(0),(−1),(−1),(0)]`

`= [(2),(0),(0),(2)] – [(0),(−2),(−2),(0)] = [(2),(2),(2),(2)]`

`rArr B`

MATRICES, FUR1 2007 VCAA 2 MC

The number of tourists visiting three towns, Oldtown, Newtown and Twixtown, was recorded for three years.

The data is summarised in the table below.

The `3 xx 1` matrix that could be used to show the number of tourists visiting the three towns in the year 2005 is

A. `[(975, 1002, 1390)]`	B. `[(1002, 1081, 1095)]`

C. `[(975), (1002), (1390)]`	D. `[(1002), (1081), (1095)]`

E. `[(975, 1002, 1390), (2105, 1081, 1228), (610, 1095, 1380)]`

Show Answers Only

`D`

Show Worked Solution

`=> D`

MATRICES, FUR1 2011 VCAA 2 MC

If `A=[(0,1),(1,0)],\ B=[(1),(0)]` and `C= [(0),(1)]`, then `AB + 2C` equals

A. `[(0),(3)]`

B. `[(3),(0)]`

C. `[(1),(2)]`

D. `[(2),(0)]`

E. `[(2),(3)]`

Show Answers Only

`A`

Show Worked Solution

`AB + 2C`	`= [(0,1),(1,0)][(1),(0)] + 2[(0),(1)]`
	`= [(0),(1)] + [(0),(2)]`
	`= [(0),(3)]`

`=> A`

MATRICES, FUR1 2011 VCAA 1 MC

The matrix below shows the airfares (in dollars) that are charged by Zeniff Airlines to fly between Adelaide (`A`), Melbourne (`M`) and Sydney (`S`).

`{:(qquadqquadquadtext(from)),({:(qquadA,\ M,\ S):}),([(0,85,89),(85,0,99),(97,101,0)]):}{:(),(),(A),(M),(S):}{:(),(),(),(qquadtext(to)),():}`

The cost to fly from Melbourne to Sydney with Zeniff Airlines is

A. `$85`

B. `$89`

C. `$97`

D. `$99`

E. `$101`

Show Answers Only

`E`

Show Worked Solution

`=> E`

MATRICES, FUR1 2014 VCAA 1 MC

`[(0,0,0,0),(2,1,1,3),(0,0,0,0),(0,0,0,0)][(0,0,2,0),(0,0,1,0),(0,0,1,0),(0,0,3,0)]\ \ text(is equal to)`

Show Answers Only

`A`

Show Worked Solution

`=>A`

GRAPHS, FUR2 2012 VCAA 1

The cost, `C`, in dollars, of making `n` phones, is shown by the line in the graph below.

1. Calculate the gradient of the line, `C`, drawn above. (1 mark)
2. Write an equation for the cost, `C`, in dollars, of making `n` phones. (1 mark)
The revenue, `R`, in dollars, obtained from selling `n` phones is given by `R = 150n`
.
1. Draw this line on the graph above. (1 mark)
2. How many phones would need to be sold to obtain $54 000 in revenue? (1 mark)
Determine the number of phones that would need to be made and sold to break even. (1 mark)

Show Answers Only

1. `50`
2. `C = 20\ 000 + 50n`
1. `text(See Worked Solutions)`
2. `360`
`200`

Show Worked Solution

a.i. `text{Using (0, 20 000) and (300, 35 000)}`

`text(Gradient)`	`= (y_2 – y_1)/(x_2 – x_1)`
	`= (35\ 000 – 20\ 000)/(300 – 0)`
	`= 50`

a.ii. `C = 50n + 20\ 000`

b.i.

b.ii.	`54\ 000`	`= 150n`
	`:. n`	`= (54\ 000)/150`
		`= 360`

`:. 360\ text(planes need to be sold.)`

c. `text(Breakeven occurs when)`

`text(Revenue)`	`=\ text(C)text(osts)`
`150n`	`= 50n + 20\ 000`
`100n`	`= 20\ 000`
`:. n`	`= 200\ text(phones)`

CORE*, FUR2 2013 VCAA 1

Hugo is a professional bike rider.

The value of his bike will be depreciated over time using the flat rate method of depreciation.

The graph below shows his bike’s initial purchase price and its value at the end of each year for a period of three years.

What was the initial purchase price of the bike? (1 mark)
1. Show that the bike depreciates in value by $1500 each year. (1 mark)
2. Assume that the bike’s value continues to depreciate by $1500 each year.
  
  Determine its value five years after it was purchased. (1 mark)

The unit cost method of depreciation can also be used to depreciate the value of the bike.

In a two-year period, the total depreciation calculated at $0.25 per kilometre travelled will equal the depreciation calculated using the flat rate method of depreciation as described above.

Determine the number of kilometres the bike travels in the two-year period. (1 mark)

Show Answers Only

`$8000`
1. `text(See Worked Solutions)`
2. `$500`
`12\ 000\ text(km)`

Show Worked Solution

a. `$8000`

b.i. `text(Value after 1 year) = $6500`

`:.\ text(Annual depreciation)`	`= 8000 – 6500`
	`= $1500`

b.ii. `text(Value after)\ n\ text(years)`

`= 8000 – 1500n`

`:.\ text(After 5 years,)`

`text(Value)`	`= 8000 – 1500 xx 5`
	`= $500`

c. `text(After 2 years,)`

`text(Depreciation)`	`= 2 xx 1500`
	`= $3000`

`:.\ text(Distance travelled)`

`= 3000/0.25`

`= 12\ 000\ text(km)`

GRAPHS, FUR2 2013 VCAA 2

Students at the camp can participate in two different watersport activities: canoeing and surfing.

The cost of canoeing is $30 per hour and the cost of surfing is $20 per hour.

The budget allows each student to spend up to $200, in total, on watersport activities.

The way in which a student decides to spend the $200 is described by the following inequality.

30 × hours canoeing + 20 × hours surfing ≤ 200

Hillary wants to spend exactly two hours canoeing during the camp.

Calculate the maximum number of hours she could spend surfing. (1 mark)
Dennis would like to spend an equal amount of time canoeing and surfing.

If he spent a total of $200 on these activities, determine the maximum number of hours he could spend on each activity. (1 mark)

Show Answers Only

`7`
`4\ text(hours)`

Show Worked Solution

a. `text(Hillary spends 2 hrs canoeing)`

`text(Let)\ x = text(hours spent surfing)`

`30 xx 2 + 20x`	`= 200`
`20x`	`= 140`
`x`	`= 7`

`:.\ text(Maximum hours surfing = 7)`

b. `text(Let)\ t = text(time spent canoeing and surfing)`

`30 xx t + 20 xx t`	`= 200`
`50t`	`= 200`
`t`	`= 4\ text(hours)`

`:.\ text(Dennis could spend a maximum)`

`text(of 4 hours on each activity.)`

GRAPHS, FUR2 2013 VCAA 1

The distance-time graph below shows the first two stages of a bus journey from a school to a camp.

At what constant speed, in kilometres per hour, did the bus travel during stage 1 of the journey? (1 mark)
For how many minutes did the bus stop during stage 2 of the journey? (1 mark)

The third stage of the journey is missing from the graph.

During stage 3, the bus continued its journey to the camp and travelled at a constant speed of 60 km/h for one hour.

Draw a line segment on the graph above to represent stage 3 of the journey. (1 mark)
Find the average speed of the bus over the three hours.
Write your answer in kilometres per hour. (1 mark)

The distance, `D` km, of the bus from the school, `t` hours after departure is given by

`D = {(100t 0 ≤ t ≤ 1.5), (150 1.5 ≤ t ≤ 2), (60t + k 2≤ t ≤ 3):}`

Determine the value of `k`. (1 mark)

Show Answers Only

`text(100 km/hr)`
`text(30 minutes)`
`text(See Worked Solutions)`
`70\ text(km/hr)`
`30`

Show Worked Solution

a. `text(100 km/hr)`

b. `text(30 minutes)`

d. `text(Average speed over 3 hours)`

♦ Mean mark of parts (c) and (d) consumed was 46%.

`= (text(Total distance))/3`

`= 210/3`

`= 70\ text(km/hr)`

e. `text(When)\ t = 2,`

`D = 150,\ text(and)`

`D = 60 xx 2 + k`

`:. 120 + k`	`= 150`
`:. k`	`= 30`

GRAPHS, FUR2 2015 VCAA 1

Ben is flying to Japan for a school cultural exchange program.

The graph below shows the cost of a particular flight to Japan, in dollars, on each day in February.

What is the cost, in dollars, of this flight to Japan on 19 February? (1 mark)
On how many days in February is the cost of this flight to Japan more than $1000? (1 mark)

Show Answers Only

`$1200`
`8`

Show Worked Solution

a. `$1200`

b. `8`

Algebra, MET2 2014 VCAA 1

The population of wombats in a particular location varies according to the rule `n(t) = 1200 + 400 cos ((pi t)/3)`, where `n` is the number of wombats and `t` is the number of months after 1 March 2013.

Find the period and amplitude of the function `n`. (2 marks)
Find the maximum and minimum populations of wombats in this location. (2 marks)
Find `n(10)`. (1 mark)
Over the 12 months from 1 March 2013, find the fraction of time when the population of wombats in this location was less than `n(10)`. (2 marks)

Show Answers Only

`text(Period) = text(6 months);\ text(Amplitude) = 400`
`text(Max) = 1600;\ text(Min) = 800`
`1000`
`1/3`

Show Worked Solution

a. `text(Period) = (2pi)/n = (2pi)/(pi/3) = 6\ text(months)`

MARKER’S COMMENT: Expressing the amplitude as [800,1600] in part (a) is incorrect.

`text(A)text(mplitude) = 400`

b. `text(Max:)\ 1200 + 400 = 1600\ text(wombats)`

`text(Min:)\ 1200 – 400 = 800\ text(wombats)`

c. `n(10) = 1000\ text(wombats)`

`text(Solve)\ n(t)`

`= 1000\ text(for)\ t ∈ [0,12]`

`t= 2,4,8,10`

`text(S)text(ince the graph starts at)\ \ (0,1600),`

♦ Mean mark 48%.

`=> n(t) < 1000\ \ text(for)`

`t ∈ (2,4)\ text(or)\ t ∈ (8,10)`

`:.\ text(Fraction)`	`= ((4 – 2) + (10 – 8))/12`
	`= 1/3\ \ text(year)`

GRAPHS, FUR2 2006 VCAA 1

Harry operates a mobile pet care service. The call-out fee charged depends on the distance he has to travel to tend to a pet. The call-out fees for distances up to 30 km are shown on the graph below.

According to this graph
1. what is the call-out fee to travel a distance of 20 km? (1 mark)
2. what is the maximum distance travelled for a call-out fee of $10? (1 mark)

A call-out fee of $50 is charged to travel distances of more than 30 km but less than or equal to 40 km.

Draw this information on the graph above. (1 mark)

Show Answers Only

1. `$30`
2. `5\ text(km)`

Show Worked Solution

a.i. `$30`

a.ii. `text(5 km)`

Calculus, MET2 2015 VCAA 1

Let `f: R -> R,\ \ f(x) = 1/5 (x - 2)^2 (5 - x)`. The point `P(1, 4/5)` is on the graph of `f`, as shown below.

The tangent at `P` cuts the y-axis at `S` and the x-axis at `Q.`

Write down the derivative `f prime (x)` of `f (x)`. (1 mark)
1. Find the equation of the tangent to the graph of `f` at the point `P(1, 4/5)`. (1 mark)
2. Find the coordinates of points `Q` and `S`. (2 marks)
Find the distance `PS` and express it in the form `sqrt b/c`, where `b` and `c` are positive integers. (2 marks)

Find the area of the shaded region in the graph above. (3 marks)

Show Answers Only

`−3/5(x – 4)(x – 2)`
1. `y = – 9/5x + 13/5`
2. `S (0, 13/5)`
  
  `Q (13/9, 0)`
`sqrt 106/5`
`108/5\ text(units²)`

Show Worked Solution

a.	`f(x)`	`= 1/5 (x – 2)^2 (5 – x)`
	`fprime(x)`	`=1/5 xx 2(x-2)(5-x) – 1/5 (x-2)^2`
		`= −3/5(x – 4)(x – 2)`

b.i. `text(Solution 1)`

`y = −9/5x + 13/5qquad[text(CAS:tangentLine)\ (f(x),x,1)]`

`text(Solution 2)`

`m_text(tan) = -9/5,\ \ text(through)\ \ (1, 4/5)`

`y-4/5`	`=-9/5 (x-1)`
`y`	`=-9/5 x +13/5`

b.ii. `text(At)\ S,\ \ x=0`

`y = −9/5 xx 0 + 13/5 = 13/5`

`:. S(0,13/5)`

`text(At)\ Q,\ \ y=0`

`0`	`= −9/5x + 13/5`
`x`	`=13/5 xx 5/9=13/9`

`:. Q(13/9,0)`

c. `P(1, 4/5),\ \ S(0,13/5)`

`text(dist)\ PS`	`=sqrt((x_2-x_1)^2 + (y_2-y_1)^2)`
	`= sqrt((1 – 0)^2 + (4/5 – 13/5)^2)`
	`= (sqrt106)/5`

d. `text(Find intersection pts of)\ SQ\ text(and)\ f(x),`

MARKER’S COMMENT: Many students complicated their answer by splitting up the area.

`text{Solve (using technology):}`

`1/5 (x – 2)^2 (5 – x) = −9/5x + 13/5`

`x = 1,7`

`:.\ text(Area)`	`= int_1^7(f(x) – (−9/5x + 13/5))dx`
	`= 108/5\ text(u²)`

GEOMETRY, FUR2 2007 VCAA 1-3

Question 1

Tessa is a student in a woodwork class.

The class will construct geometrical solids from a block of wood.

Tessa has a piece of wood in the shape of a rectangular prism.

This prism, `ABCDQRST`, shown in Figure 1, has base length 24 cm, base width 28 cm and height 32 cm.

On the front face of Figure 1, `ABRQ`, Tessa marks point `W` halfway between `Q` and `R` as shown in Figure 2 below. She then draws line segments `AW` and `BW` as shown.

Determine the length, in cm, of `QW`. (1 mark)
Calculate the angle `WAQ`. Write your answer in degrees, correct to one decimal place. (1 mark)
Calculate the angle `AWB` correct to one decimal place. (1 mark)
What fraction of the area of the rectangle `ABRQ` does the area of the triangle `AWB` represent? (1 mark)

Question 2

Tessa carves a triangular prism from her block of wood.

Using point `V`, halfway between `T` and `S` on the back face, `DCST`, of Figure 1, she constructs the triangular prism shown in Figure 3.

Show that, correct to the nearest centimetre, length `AW` is 34 cm. (1 mark)
Using length `AW` as 34 cm, find the total surface area, in cm², of the triangular prism `ABCDWV` in Figure 3. (2 marks)

Question 3

Tessa's next task is to carve the right rectangular pyramid `ABCDY` shown in Figure 4 below.

She marks a new point, `Y`, halfway between points `W` and `V` in Figure 3. She uses point `Y` to construct this pyramid.

Calculate the volume, in cm³, of the pyramid `ABCDY` in Figure 4. (1 mark)
Show that, correct to the nearest cm, length `AY` is 37 cm. (2 marks)
Using `AY` as 37 cm, demonstrate the use of Heron's formula to calculate the area, in cm², of the triangular face `YAB`. (2 marks)

Show Answers Only

Question 1

`12\ text(cm)`
`20.6^@`
`41.2^@\ text{(1 d.p.)}`
`1/2`

Question 2

`34\ text(cm)`
`3344\ text(cm²)`

Question 3

`7168\ text(cm³)`
`37\ text(cm)`
`420\ text(cm²)`

Show Worked Solution

`text(Question 1)`

a.	`QW`	`= 1/2 xx QR`
		`= 1/2 xx 24`
		`= 12\ text(cm)`

`tan\ /_ WAQ`	`= 12/32`
`:. /_ WAQ`	`= tan^-1\ 3/8`
	`= 20.55…`
	`= 20.6^@\ text{(1 d.p.)}`

c.	`/_ AWB`	`= 2 xx /_ WAG`
		`= 2 xx 20.6`
		`= 41.2^@\ text{(1 d.p.)}`

d.	`text(Area)\ ABRQ`	`= 24 xx 32`
		`= 768\ text(cm²)`
	`text(Area)\ Delta AWB`	`= 2 xx text(Area)\ Delta AQW`
		`= 2 xx 1/2 xx 12 xx 32`
		`= 384\ text(cm²)`

`:.\ text(Fraction of area)`

`= 384/768`

`= 1/2`

`text(Question 2)`

`text(Using Pythagoras in)\ Delta AWX,`

`AW`	`= sqrt (12^2 + 32^2)`
	`= sqrt 1168`
	`= 34.176…`
	`= 34\ text{cm (nearest cm) … as required.}`

b. `text(Total S.A. of triangular prism)`

`= text(area of base) + text(area of sides) + text(area of ends)`

`= 24 xx 28 + 2 xx (34 xx 28) + 2 xx (1/2 xx 24 xx 32)`

`= 672 + 1904 + 768`

`= 3344\ text(cm²)`

`text(Question 3)`

a.	`V`	`= 1/3 xx b xx h`
		`= 1/3 xx 24 xx 28 xx 32`
		`= 7168\ text(cm³)`

`text(Using Pythagoras in)\ Delta ABC,`

`AC`	`= sqrt (24^2 + 28^2)`
	`= 36.878…`

`text(Let)\ Z\ text(be the midpoint of)\ AC`

`:. AZ = (36.878…)/2 = 18.439…`

`text(Using Pythagoras in)\ Delta AYZ,`

`AY`	`= sqrt (32^2 + (18.439…)^2)`
	`= sqrt (1364)`
	`= 36.9…`
	`= 37\ text{cm (nearest cm) … as required}`

c. `text(Using Heron’s formula,)`

`s`	`= (37 + 37 + 24)/2=49`

`:.\ text(Area of)\ Delta YAB`

`= sqrt (49 (49 – 24) (49 – 37) (49 – 37))`

`= sqrt (49 xx 25 xx 12 xx 12)`

`= 420\ text(cm²)`

MATRICES, FUR2 2008 VCAA 1

Two subjects, Biology and Chemistry, are offered in the first year of a university science course.

The matrix `N` lists the number of students enrolled in each subject.

`N = [(460),(360)]{:(text(Biology)),(text(Chemistry)):}`

The matrix `P` lists the proportion of these students expected to be awarded an `A`, `B`, `C`, `D` or `E` grade in each subject.

`{:((qquadqquadqquadA,qquad\ B,qquadquadC,qquad\ D,qquadE)),(P = [(0.05,0.125,0.175,0.45,0.20)]):}`

Write down the order of matrix `P`. (1 mark)
Let the matrix `R = NP`.
1. Evaluate the matrix `R`. (1 mark)
2. Explain what the matrix element `R_24` represents. (1 mark)
Students enrolled in Biology have to pay a laboratory fee of $110, while students enrolled in Chemistry pay a laboratory fee of $95.
1. Write down a clearly labelled row matrix, called `F`, that lists these fees. (1 mark)
2. Show a matrix calculation that will give the total laboratory fees, `L`, paid in dollars by the students enrolled in Biology and Chemistry. Find this amount. (1 mark)

Show Answers Only

`1 xx 5`
1. `[(23,57.5,80.5,207,92),(18,45,63,162,72)]`
2. `text(See Worked Solutions)`
1. `text(See Worked Solutions)`
2. `text(See Worked Solutions)`

Show Worked Solution

a. `1 xx 5`

b.i.	`R`	`= NP`
		`= [(460), (360)] [0.05 quad 0.125 quad 0.175 quad 0.45 quad 0.20]`
		`= [(23,57.5,80.5,207,92),(18,45,63,162,72)]`

b.ii. `R_24\ text(represents the number of chemistry students)`

`text(expected to get a)\ D.`

c.i.

`{:(qquad qquad qquad B quad qquad C),(F = [(110, 95)]):}`

c.ii. `text(Total laboratory fees)`

`L = [(110, 95)] [(460), (360)] = [84\ 800]`

MATRICES, FUR2 2012 VCAA 1

Matrix `F` below shows the ﬂight connections for an airline that serves four cities, Anvil (`A`), Berga (`B`), Cantor (`C`) and Dantel (`D`).

`{:(qquad qquad qquad qquad quad text(from)),((qquad qquad quad\ A,B,C,D)),(F = [(0\ ,1\ ,0\ ,0),(1,0,1,0),(0,0,0,1),(0,1,0,0)]):}{:(),(),(A),(B),(C),(D):}{:(),(qquad text(to)):}`

In this matrix, the `1` in column `C` row `B`, for example, indicates that, using this airline, you can ﬂy directly from Cantor to Berga. The `0` in column `C` row `D`, for example, indicates that you cannot ﬂy directly from Cantor to Dantel.

Complete the following sentence.

On this airline, you can ﬂy directly from Berga to ________ and ________. (1 mark)
List the route that you must follow to ﬂy from Anvil to Cantor. (1 mark)
Evaluate the matrix product `G = KF`, where `K = [1,1,1,1]`. (1 mark)
In the context of the problem, what information does matrix `G` contain? (1 mark)

Show Answers Only

`text(Anvil and Dantel)`
`text(Anvil – Berga – Dantel – Cantor)`
`[1,2,1,1]`
`text(Matrix)\ G\ text(contains the information of the total)`
`text(amount of direct flights out of each of 4 cities,)`
`text(that go to another city in the same group.)`

Show Worked Solution

a. `text(Anvil and Dantel)`

b. `text(Anvil – Berga – Dantel – Cantor)`

c.	`G`	`= KF`
		`= [1,1,1,1][(0,1,0,0),(1,0,1,0),(0,0,0,1),(0,1,0,0)]`
		`= [1,2,1,1]`

d. `text(Matrix)\ G\ text(contains the information of the total)`

`text(amount of direct flights out of each of 4 cities,)`

`text(that go to another city in the same group.)`

CORE, FUR2 2008 VCAA 1

In a small survey, twenty-five Year 8 girls were asked what they did (walked, sat, stood, ran) for most of the time during a typical school lunch time.

Their responses are recorded below.

`{:(text(sat),text(stood),text(sat),text(ran),text(sat)),(text(walked ),text(walked ),text(sat),text(walked ),text(ran)),(text(sat),text(walked),text(walked ),text(walked),text(ran)),(text(walked),text(ran),text(walked),text(ran),text(walked)),(text(ran),text(sat),text(ran),text(ran),text(walked)):}`

Use the data to

complete the following frequency table (1 mark)
determine the percentage of Year 8 girls who ran for most of the time during a typical school lunch time. (1 mark)

Show Answers Only

`text(32%)`

Show Worked Solution

b. `text(Percentage who ran)`

`=8/25 xx 100text(%)`

`= 32text(%)`

MATRICES, FUR2 2015 VCAA 1

Students in a music school are classified according to three ability levels: beginner (`B`), intermediate (`I`) or advanced (`A`).

Matrix `S_0`, shown below, lists the number of students at each level in the school for a particular week.

`S_0 = [(20), (60), (40)] {:(B), (I), (A):}`

How many students in total are in the music school that week? (1 mark)

The music school has four teachers, David (`D`), Edith (`E`), Flavio (`F`) and Geoff (`G`).

Each teacher will teach a proportion of the students from each level, as shown in matrix `P` below.

`{:(qquadqquadqquad{:(Dquad,Eqquad,Fqquad,G):}),(P = [(0.25,0.5,0.15,0.1)]):}`

The matrix product, `Q = S_0P`, can be used to find the number of students from each level taught by each teacher.

1. Complete matrix `Q`, shown below, by writing the missing elements in the shaded boxes. (1 mark)
2. How many intermediate students does Edith teach? (1 mark)

The music school pays the teachers $15 per week for each beginner student, $25 per week for each intermediate student and $40 per week for each advanced student.

These amounts are shown in matrix `C` below.

`{:(qquad qquad quad{:(B quad, I quad,A):}),(C = [(15, 25, 40)]):}`

The amount paid to each teacher each week can be found using a matrix calculation.

1. Write down a matrix calculation in terms of `Q` and `C` that results in a matrix that lists the amount paid to each teacher each week. (1 mark)
2. How much is paid to Geoff each week? (1 mark)

Show Answers Only

`120`
1. `[(5, 10, 3, 2), (15, 30, 9, 6), (10, 20, 6, 4)]`
2. `30`
1. `C xx Q`
2. `$340`

Show Worked Solution

a. `20 + 60 + 40 = 120`

b.i.	`Q`	`= S_0 P`
		`= [(20), (60), (40)] [(0.25, 0.5, 0.15, 0.1)]`
		`= [(5, 10, 3, 2), (15, 30, 9, 6), (10, 20, 6, 4)]`

b.ii. `text(Edith teaches 30 intermediate students.)`

c.i. `C xx Q`

c.ii. `CQ = [(15, 25, 40)] [(5, 10, 3, 2), (15, 30, 9, 6), (10, 20, 6, 4)]`

`:.\ text(Geoff’s pay)`

`= 15 xx 2 + 25 xx 6 + 40 xx 4`

`= $340`

CORE, FUR2 2009 VCAA 1

Table 1 shows the number of rainy days recorded in a high rainfall area for each month during 2008.

The dot plot below displays the distribution of the number of rainy days for the 12 months of 2008.

Circle the dot on the dot plot that represents the number of rainy days in April 2008. (1 mark)
For the year 2008, determine
1. the median number of rainy days per month (1 mark)
2. the percentage of months that have more than 10 rainy days. Write your answer correct to the nearest per cent. (1 mark)

Show Answers Only

1. `15.5`
2. `text(92%)`

Show Worked Solution

b.i.	`text(Median)`	`= text{(6th + 7th)}/2`
		`=(15+16)/2`
		`=15.5`

b.ii. `text(Months with more than 10 rainy days)`

`=11/12 xx text(100%)`

`=91.66…`

`=92text(%)\ \ text{(nearest %)}`

CORE, FUR2 2010 VCAA 1

Table 1 shows the percentage of women ministers in the parliaments of 22 countries in 2008.

What proportion of these 22 countries have a higher percentage of women ministers in their parliament than Australia? (1 mark)
Determine the median, range and interquartile range of this data. (2 marks)

The ordered stemplot below displays the distribution of the percentage of women ministers in parliament for 21 of these countries. The value of Canada is missing.

Complete the stemplot above by adding the value for Canada. (1 mark)
Both the median and the mean appropriate measures of centre for this distribution.

Explain why. (1 mark)

Show Answers Only

`0.5`
`text(Median= 28, Range = 56, IQR = 17)`
`1 | 246`
`text(S)text(ince the distribution is approximately)`

`text(symmetric, the median and mean will be)`

`text(appropriate measures of the centre.)`

Show Worked Solution

a. `11/22 = 0.5`

b. `text(22 data points,)`

`text(Median)`	`=\ text{(11th + 12th)}/2`
	`= (32 + 24)/2`
	`= 28`

`text(Range)`	`= 56 – 0`
	`= 56`

`Q_1=21 and Q_3=38`

`text(IQR)`	`= 38 – 21`
	`= 17`

c. `1 | 2 quad 4 quad 6`

♦♦ Part (d) was “poorly answered”.
MARKER’S COMMENT: The use of “symmetric” gained a mark while “evenly distributed” was deemed too vague.

d. `text(S)text(ince the distribution is approximately)`

`text(symmetric, the median and mean will be)`

`text(appropriate measures of the centre.)`

CORE, FUR2 2013 VCAA 1

A development index is used as a measure of the standard of living in a country.

The bar chart below displays the development index for 153 countries in four categories: low, medium, high and very high.

How many of these countries have a very high development index? (1 mark)
What percentage of the 153 countries has either a low or medium development index?

Write your answer, correct to the nearest percentage. (1 mark)

Show Answers Only

`31`
`text(61%)`

Show Worked Solution

a. `31\ \ text{(from graph)}`

b. `text(Number of countries with low or high index)`

`=45 + 49 = 94`

`:.\ text(Percentage)`	`=94/153 xx 100text(%)`
	`=61.43…`
	`=61 text(%)\ \ text{(nearest %)}`

CORE, FUR2 2014 VCAA 1

The segmented bar chart below shows the age distribution of people in three countries, Australia, India and Japan, for the year 2010.

Write down the percentage of people in Australia who were aged 0 – 14 years in 2010.

Write your answer, correct to the nearest percentage. (1 mark)
In 2010, the population of Japan was 128 000 000.

How many people in Japan were aged 65 years and over in 2010? (1 mark)
From the graph above, it appears that there is no association between the percentage of people in the 15 – 64 age group and the country in which they live.

Explain why, quoting appropriate percentages to support your explanation. (1 mark)

Show Answers Only

`text(19%)`
`29\ 440\ 000`
`text(The percentages of people in the 15 – 64 age group)`
`text(in each country is: Australia 67%, India 64%, and)`
`text(Japan 64%.)`
`text(S)text(ince the percentages are very close no matter which)`
`text(country this age group belonged to, there is no association.)`

Show Worked Solution

a. `text(19%)`

b. `text(From the graph, 23% of Japan’s population is 65 or over.)`

`:.\ text(The number of people 65 or over)`

`=128\ 000\ 000 xx 23/100`

`= 29\ 440\ 000`

c. `text(The percentages of people in the 15 – 64 age group)`

♦ Mean mark 41%.

`text(in each country is: Australia 67%, India 64%, and)`

`text(Japan 64%.)`

`text(S)text(ince the percentages are very close no matter which)`

`text(country this age group belonged to, there is no association.)`

NETWORKS, FUR1 2015 VCAA 1 MC

In the graph above, the number of vertices of odd degree is

Show Answers Only

`C`

Show Worked Solution

`text{Two vertices have degree 3 (all others are even)}`

`=> C`

Financial Maths, STD2 F1 SM-Bank 3 MC

$6000 is invested in an account that earns simple interest at the rate of 3.5% per annum.

The total interest earned in the first four years is

$70
$84
$210
$840

Show Answers Only

`D`

Show Worked Solution

`P = 6000,\ \ r = 3.5text(%),\ \ n = 4`

`I`	`= Prn`
	`= 6000 xx 3.5/100 xx 4`
	`= 840`

`=> D`

Financial Maths, STD2 F4 SM-Bank 2 MC

Sally purchased an electronic game machine on hire purchase. She paid $140 deposit and then $25.50 per month for two years.

The total amount that Sally paid is

A. $191

B. $446

C. $612

D. $752

Show Answers Only

`D`

Show Worked Solution

`text(Total paid)`	`= 140 + 25.50 xx 2 xx12`
	`= $752`

`=>D`

PATTERNS, FUR1 2015 VCAA 1 MC

The first four terms in a geometric sequence are: `5, 10, 20, 40, …`

The fifth term in this sequence is

A. `45`

B. `50`

C. `60`

D. `80`

E. `100`

Show Answers Only

`D`

Show Worked Solution

`text(GP where)\ \ \ a`	`=5, and`
`r`	`=t_2/t_1=10/5=2`

`t_n`	`=ar^(n-1)`
`t_5`	`= 5 xx 2^4`
	`= 80`

`=> D`

PATTERNS, FUR1 2006 VCAA 1 MC

Which one of the following sequences shows the first five terms of an arithmetic sequence?

A. `1, 3, 9, 27, 81\ …`

B. `1, 3, 7, 15, 31\ …`

C. `– 10, – 5, 5, 10, 15\ …`

D. `– 4, – 1, 2, 5, 8\ …`

E. `1, 3, 8, 15, 24\ …`

Show Answers Only

`D`

Show Worked Solution

`text(Only D has a consistent common difference)`

`text(between all values.)`

`rArr D`

PATTERNS, FUR1 2007 VCAA 1 MC

For the geometric sequence

`24, 6, 1.5\ …`

the common ratio of the sequence is

A. `– 18`

B. `0.25`

C. `0.5`

D. `4`

E. `18`

Show Answers Only

`B`

Show Worked Solution

`r=t_2/t_1=6/24=0.25`

`rArr B`

CORE, FUR1 2007 VCAA 11-13 MC

The following information relates to Parts 1, 2 and 3.

The time series plot below shows the revenue from sales (in dollars) each month made by a Queensland souvenir shop over a three-year period.

Part 1

This time series plot indicates that, over the three-year period, revenue from sales each month showed

A. no overall trend.

B. no correlation.

C. positive skew.

D. an increasing trend only.

E. an increasing trend with seasonal variation.

Part 2

A three median trend line is fitted to this data.

Its slope (in dollars per month) is closest to

A. `125`

B. `146`

C. `167`

D. `188`

E. `255`

Part 3

The revenue from sales (in dollars) each month for the first year of the three-year period is shown below.

If this information is used to determine the seasonal index for each month, the seasonal index for September will be closest to

A. `0.80`

B. `0.82`

C. `1.16`

D. `1.22`

E. `1.26`

Show Answers Only

`text (Part 1:)\ E`

`text (Part 2:)\ C`

`text (Part 3:)\ E`

Show Worked Solution

`text (Part 1)`

`text(The time series plot clearly shows an increasing)`

`text(trend and a seasonal spike and drop over the)`

`text(Summer months.)`

`rArr E`

`text (Part 2)`

♦ Mean mark 37%.
MARKERS’ COMMENT: A common error was to incorrectly use the 6 month and 30 month data points as the median.

`text(From the graph, the median of the bottom third)`

`text(of data points is)\ \ (6.5, 3000).`

`text(From the graph, the median of the top third of)`

`text(data points is)\ \ (30.5, 7000).`

`:.\ text(Gradient of the three median line)`

`=(7000 – 3000)/(30.5 – 6.5)`

`=166.66…`

`rArr C`

`text (Part 3)`

`text(Average monthly sales)\ `	`= (43\ 872)/12`
	`=3656`

`:.\ text(Seasonal index for September)`

`=4597/3656`

`=1.257…`

`rArr E`

CORE, FUR1 2007 VCAA 1-2 MC

The dot plot below shows the distribution of the number of bedrooms in each of 21 apartments advertised for sale in a new high-rise apartment block.

Part 1

The mode of this distribution is

A. `1`

B. `2`

C. `3`

D. `7`

E. `8`

Part 2

The median of this distribution is

A. `1`

B. `2`

C. `3`

D. `4`

E. `5`

Show Answers Only

`text (Part 1:)\ A`

`text (Part 2:)\ B`

Show Worked Solution

`text (Part 1)`

`rArr A`

`text (Part 2)`

`text(The median of 21 data points is the 11th value.)`

`:.\ text(Median) = 2`

`rArr B`

PATTERNS, FUR1 2008 VCAA 2 MC

For an examination, 8600 examination papers are to be printed at a rate of 25 papers per minute. After one hour, the number of examination papers that still need to be printed is

A. `1600`

B. `2500`

C. `6100`

D. `7100`

E. `8575`

Show Answers Only

`D`

Show Worked Solution

`text(Printing rate = 25 per minute)`

`:.\ text(In 1 hour, exam papers printed)`

`= 60xx25`

`= 1500`

`:.\ text(Papers still to be printed)`

`= 8600-1500`

`= 7100`

`=>D`

Algebra, MET2 2015 VCAA 6 MC

For the polynomial `P(x) = x^3 - ax^2 - 4x + 4,\ \ P(3) = 10,` the value of `a` is

A. `− 3`

B. `− 1`

C. `1`

D. `3`

E. `10`

Show Answers Only

`C`

Show Worked Solution

`P(3)`	`= 3^3 – 3^2*a -4(3)+4`
`10`	`=27-9a-12+4`
`9a`	`=9`
`:. a`	`= 1`

`=> C`

CORE*, FUR2 2014 VCAA 1

The adult membership fee for a cricket club is $150.

Junior members are offered a discount of $30 off the adult membership fee.

Write down the discount for junior members as a percentage of the adult membership fee. (1 mark)

Adult members of the cricket club pay $15 per match in addition to the membership fee of $150.

If an adult member played 12 matches, what is the total this member would pay to the cricket club? (1 mark)

If a member does not pay the membership fee by the due date, the club will charge simple interest at the rate of 5% per month until the fee is paid.

Michael paid the $150 membership fee exactly two months after the due date.

Calculate, in dollars, the interest that Michael will be charged. (1 mark)

The cricket club received a statement of the transactions in its savings account for the month of January 2014.

The statement is shown below.

1. Calculate the amount of the withdrawal on 17 January 2014. (1 mark)
2. Interest for this account is calculated on the minimum balance for the month and added to the account on the last day of the month.
  
  What is the annual rate of interest for this account?
  
  Write your answer, correct to one decimal place. (1 mark)

Show Answers Only

`text(20%)`
`$330`
`$15`
1. `$17\ 000`
2. `3.5`

Show Worked Solution

a. `text(Discount for junior members)`

`= 30/150 xx 100text(%)`

`= 20text(%)`

b.	`text(Match Payments)`	`= 12 xx 15=$180`

`:.\ text(Total paid to the club)`

`= 150 + 180`

`= $330`

c.	`I`	`= (PrT)/100`
		`= (150 xx 5 xx 2)/100`
		`= $15`

d.i. `text(Withdrawal on 17-Jan)`

`= 59\ 700 – 42\ 700`

`= $17\ 000`

d.ii. `text(Minimum Jan balance) = $42\ 700`

`47\ 200 xx r xx 1/12`	`= 125.12`
`:. r`	`= (125.12 xx 12)/(42\ 700)`
	`= 0.0351…`

`:.\ text(Annual interest rate) = 3.5text(%)`

Algebra, STD2 A1 SM-Bank 13

If `(y-3)/3 =5`, find `y`. (1 mark)

Show Answers Only

`18`

Show Worked Solution

`(y3)/3`	`= 5`
`y3`	`= 15`
`y`	`= 18`

Algebra, STD2 A1 SM-Bank 3

Find the value of `r` given `r/7-4 = 3`. (1 mark)

Show Answers Only

`49`

Show Worked Solution

`r/7-4`	`= 3`
`r/7`	`= 7`
`:.r`	`= 49`

Algebra, STD2 A1 SM-Bank 2

If `A = P(1 + r)^n`, find `A` given `P = $300`, `r = 0.12` and `n = 3` (give your answer to the nearest cent). (2 marks)

Show Answers Only

`$421.48\ \ text{(nearest cent)}`

Show Worked Solution

`A`	`= P(1 + r)^n`
	`= 300(1 + 0.12)^3`
	`= 300(1.12)^3`
	`= 421.478…`
	`= $421.48\ \ text{(nearest cent)}`

Harder Ext1 Topics, EXT2 2015 HSC 13c

A small spherical balloon is released and rises into the air. At time `t` seconds, it has radius `r` cm, surface area `S = 4 pi r^2` and volume `V = 4/3 pi r^3`.

As the balloon rises it expands, causing its surface area to increase at a rate of `((4 pi)/3)^(1/3)\ \text(cm)^2 text(s)^-1`. As the balloon expands it maintains a spherical shape.

By considering the surface area, show that
1. `(dr)/(dt) = 1/(8 pi r) (4/3 pi)^(1/3).` (2 marks)
Show that
1. `(dV)/(dt) = 1/2 V^(1/3).` (2 marks)
When the balloon is released its volume is `8000\ text(cm³)`. When the volume of the balloon reaches `64000\ text(cm³)` it will burst.
How long after it is released will the balloon burst? (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`1\ \ text(hour)`

Show Worked Solution

(i)	`(dS)/(dt)`	`= (dS)/(dr) xx (dr)/(dt)`
	`((4 pi)/3)^(1/3)`	`=8 pi r xx (dr)/(dt)`
	`(dr)/(dt)`	`=((4 pi)/3)^(1/3) xx 1/(8 pi r)`
		`=1/(8 pi r) (4/3 pi)^(1/3)`

(ii) `(dV)/(dt) = (dV)/(dr) xx (dr)/(dt)`

`(dV)/(dt) =`	`4 pi r^2 xx ((4 pi)/3)^(1/3) xx 1/(8 pi r)`
`=`	`r/2 xx ((4 pi)/3)^(1/3)`
`=`	`1/2 (4/3 pi r^3)^(1/3)`
`=`	`1/2 V^(1/3)`

(iii) `text(Find)\ \ t\ \ text(when)\ \ V=64\ 000`

`text(When)\ \ t = 0,\ \ V = 8000`

`(dV)/(dt)`	`=1/2 V^(1/3)`
`(dt)/(dV)`	`=2/V^(1/3)`
`int_0^t\ dt`	`= int_8000^64000 2/V^(1/3)\ dV`
`:.t`	`= [3V^(2/3)]_8000^64000`
	`= 3(1600 – 400)`
	`= 3600\ \ text(seconds)`
	`= 1\ \ text(hour)`

Conics, EXT2 2015 HSC 13a

The hyperbolas `H_1:\ \ x^2/a^2 - y^2/b^2 = 1` and `H_2:\ \ x^2/a^2 - y^2/b^2 = -1` are shown in the diagram.

Let `P(a sec theta, b tan theta)` lie on `H_1` as shown on the diagram.

Let `Q` be the point `(a tan theta, b sec theta)`.

Verify that the coordinates of `Q(a tan theta, b sec theta)` satisfy the equation for `H_2.` (1 mark)
Show that the equation of the line `PQ` is `bx + ay = ab (tan theta + sec theta).` (2 marks)
Prove that the area of `Delta OPQ` is independent of `theta.` (3 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i) `text(Substitute)\ \ Q(a tan theta, b sec theta)\ \ text(into)`

`\ \ x^2/a^2 – y^2/b^2 = -1`

`text(LHS)`	`=(a^2 tan^2 theta)/a^2 – (b^2 sec^2 theta)/b^2`
	`=tan^2 theta – sec^2 theta\ \ \ \ \ \ (sec^2 theta = tan^2 theta +1)`
	`=-1`

`:. Q\ \ text(lies on)\ \ H_2`

(ii)	`m_(PQ)`	`=(b tan theta-b sec theta)/(a sec theta- a tan theta)`
		`=(-b(sec theta – tan theta))/(a(sec theta – tan theta))`
		`=-b/a`

`:. text(Equation of)\ \ PQ`

`(y – b tan theta)`	`=-b/a (x – a sec theta)`
`ay – ab tan theta`	`=-bx + ab sec theta`
`bx + ay`	`=ab (tan theta + sec theta)`

♦ Mean mark 47%.

(iii) `text(Area)\ \ Delta OPQ = 1/2 xx QP xx d,\ \ text(where)`

`d= text(Perpendicular distance from)\ \ O\ \ text(to)\ \ QP`

`d`	`= \|\ (-ab(tan theta + sec theta))/sqrt (a^2 + b^2)\ \|`
	`= (ab (tan theta + sec theta))/sqrt (a^2 + b^2)`

`text(Distance)\ \ QP`

`QP^2`	`=(a sec theta – a tan theta)^2 + (b tan theta – b sec theta)^2`
	`=a^2(sec theta – tan theta)^2+b^2(sec theta – tan theta)^2`
	`=(a^2+b^2)(sec theta – tan theta)^2`
`QP`	`=sqrt(a^2+b^2) *\|\ sec theta – tan theta\ \|`
	`=sqrt(a^2+b^2)(sec theta – tan theta),\ \ \ \ \ (sec theta>=tan theta\ \ text(for)\ \ 0<=theta<=90^@)`

`text(Area)\ \ Delta OPQ`	`=1/2 sqrt(a^2+b^2)(sec theta – tan theta)*(ab (tan theta + sec theta))/sqrt(a^2 + b^2)`
	`= (ab)/2 xx (sec theta – tan theta) (sec theta + tan theta)`
	`= (ab)/2 xx (sec^2 theta – tan^2 theta)`
	`= (ab)/2`

`:.\ text(Area)\ \ Delta OPQ\ \ text(is independent of)\ \ theta.`

Conics, EXT2 2015 HSC 11d

Sketch `(x^2)/25 + y^2/16 = 1` indicating the coordinates of the foci. (2 marks)

Show Answers Only

Show Worked Solution

`(x^2)/25 + y^2/16 = 1`

`=>a = 5,\ \ \ b = 4`

`text(Using)\ \ b^2`	`=a^2(1-e^2)`
`16`	`= 25 (1 – e^2)`
`e^2`	`=1 – 16/25`
	`=9/25`
`:. e`	`=3/5`

`text(Foci coordinates) = (+-ae,0)`

`:.S(3, 0),\ \ \ S prime(– 3, 0)`

Polynomials, EXT2 2015 HSC 11c

Find `A, B` and `C` such that

`1/(x(x^2 + 2)) = A/x + (Bx + C)/(x^2 + 2).` (2 marks)

Show Answers Only

`A = 1/2,\ \ \ B = -1/2,\ \ \ C = 0`

Show Worked Solution

`1/(x(x^2 + 2))`	`= A/x + (Bx + C)/(x^2 + 2)`
`1`	`= A (x^2 + 2) + (Bx + C) x`
`1`	`= (A+B)x^2 + Cx + 2A`

`2A = 1,\ \ =>A=1/2`

`C=0`

`A + B = 0,\ \ =>B=-1/2`

`:.A = 1/2,\ \ \ \ B = -1/2,\ \ \ \ C = 0`

Conics, EXT2 2006 HSC 3b

The diagram shows the graph of the hyperbola

`x^2/144 - y^2/25 = 1.`

Find the coordinates of the points where the hyperbola intersects the `x`-axis. (1 mark)
Find the coordinates of the foci of the hyperbola. (2 marks)
Find the equations of the directrices and the asymptotes of the hyperbola. (2 marks)

Show Answers Only

`(12, 0),\ (– 12, 0)`
`(13, 0),\ (– 13, 0)`
`text(Directrices:)\ \ x = +- 144/13;\ \ text(Asymptotes:)\ \ y = +- 5/12 x`

Show Worked Solution

(i) `text(Intersection when)\ \ y=0`

`x^2/144 + 0`	`=1`
`x`	`= +-12`

`:. xtext(-axis intersection at)\ \ (12, 0),\ (– 12, 0)`

(ii) `a=12,\ \ b = 5`

`text(Using)\ \ \ b^2`	`=a^2(e^2-1)`
`25`	`= 144 (e^2 – 1)`
`25/144`	`= e^2 – 1`
`e^2`	`= 169/144`
`e`	`= 13/12`

`:.S(ae,0)-=(13,0)`

`:. S′(– ae,0)-=(– 13,0)`

(iii)	`text(Directrices when)\ \ \ x`	`=+-a/e`
		`=+-144/13`
	`text(Asymptotes when)\ \ \ y`	`=+-b/a x`
		`=+- 5/12 x`

Complex Numbers, EXT2 N1 2007 HSC 2a

Let `z = 4 + i` and `w = bar z`. Find, in the form `x + iy`,

`w` (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
`w - z` (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
`z/w`. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`4 – i`
`-2i`
`15/17 + 8/17 i`

Show Worked Solution

i. `z = 4 + i,\ \ w = bar z = 4 – i`

ii. `w – z`	`= 4 – i – (4 + i)`
	`= -2i`

iii. `z/w`	`=(4 + i)/(4 – i) xx (4+i)/(4+i)`
	`=(4 + i)^2/(16+1)`
	`=(16+8i-1)/17`
	`=15/17 + 8/17 i`

Calculus, EXT2 C1 2015 HSC 6 MC

Which expression is equal to `int x^2 sin x\ dx`

`-x^2 cos x - int 2 x cos x\ dx`
`-2x cos x + int x^2 cos x\ dx`
`-x^2 cos x + int 2x cos x\ dx`
`-2x cos x - int x^2 cos x\ dx`

Show Answers Only

`C`

Show Worked Solution

`u`	`= x^2,`	`\ \ \ \ u′`	`= 2x`
`v′`	`= sin x,`	`v`	`= -cos x`

`int uv′\ dx`	`=uv-int u′v\ dx`
	`= x^2 (-cos x) – int 2x (-cos x) dx`
	`= -x^2 cos x + int 2x cos x\ dx`

`=> C`

Functions, EXT1′ F2 2015 HSC 4 MC

The polynomial `x^3 + x^2 - 5x + 3` has a double root at `x = alpha.`

What is the value of `alpha ?`

`-5/3`
`-1`
`1`
`5/3`

Show Answers Only

`C`

Show Worked Solution

`P prime (x)`	`= 3x^2 + 2x – 5`
	`= (3x + 5) (x – 1)`

`:. text(Only possible double roots occur when)`

`x=1\ \ text(or)\ \ x=-5/3`

`P(1)=1+1-5+3=0`

`:. x = 1\ \ text(is double root.)`

`=> C`

Functions, EXT1′ F1 2015 HSC 3 MC

Which graph best represents the curve `y = (x - 1)^2 (x + 3)^5?`

Show Answers Only

`A`

Show Worked Solution

`text(Quick elimination)`

`text(At)\ \ x=0,\ \ y>0\ \ \ \ =>\ text{NOT (B) or (C)}`

`(x – 1)^2\ \ text(gives a turning point at)\ \ x = 1,`

`(x + 3)^5\ \ text(gives a change in concavity at)\ \ x = -3.`

`=> A`

Complex Numbers, EXT2 N1 2015 HSC 2 MC

What value of `z` satisfies `z^2 = 7 - 24i?`

`4 - 3i`
`-4 - 3i`
`3 - 4i`
`-3 - 4i`

Show Answers Only

`A`

Show Worked Solution

`(4 – 3i)^2`	`= 16 – 24i + 9i^2`
	`= 7 – 24i`

`=> A`

Polynomials, EXT2 2014 HSC 1 MC

What are the values of `a`, `b` and `c` for which the following identity is true?

`(5x^2 − x + 1)/(x(x^2 + 1)) = a/x + (bx + c)/(x^2 + 1)`

`a = 1`, `b = 6`, `c = 1`
`a = 1`, `b = 4`, `c = 1`
`a = 1`, `b = 6`, `c = −1`
`a = 1`, `b = 4`, `c = −1`

Show Answers Only

`D`

Show Worked Solution

`text(Using partial fractions:)`

`(5x^2 − x + 1)/(x(x^2 + 1))`	`= a/x + (bx + c)/(x^2 + 1)`
`5x^2 − x + 1`	`= a(x^2 + 1) + (bx + c)x`
	`= (a + b)x^2 + cx + a`

`:.c = −1,\ \ a = 1`

`a + b`	`= 5`
`1 + b`	`= 5`
`b`	`= 4`

`=> D`

Calculus, EXT2 C1 2013 HSC 1 MC

Which expression is equal to `int tan x\ dx?`

`sec^2 x + c`
`-ln (cos x) + c`
`(tan^2 x)/2 + c`
`ln (sec x + tan x) + c`

Show Answers Only

`B`

Show Worked Solution

`int tan x\ dx =`	`int (sin x)/(cos x)\ dx`
`=`	`-ln (cos x) + c`

`=> B`

b.i.	`W`	`=PQ`
		`= [(6.80,5.30,6.20),(7.30,4.90,6.15)][(8),(11),(3)]`
		`= [(131.30),(130.75)]`