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Complex Numbers, SPEC1 2018 VCAA 2

  1. Show that  .   (1 mark)

  2. Evaluate  , giving your answer in the form  , where  .   (3 marks)

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a.  
   
 
     
 

 

b.   
   
   

 

   
 

 

 
 
 
 
 

Calculus, MET1 2018 VCAA 8

Let  , where is a positive real constant.

  1.  Show that  .   (1 mark)

  2. Find the value of for which the graphs of    and    have exactly one point of intersection.   (3 marks)

Let  . The diagram below shows sections of the graphs of  and for  .
 


 

Let be the area of the region bounded by the curves  and the line  .

  1. Write down a definite integral that gives the value of .   (1 mark)

  2. Using your result from part a., or otherwise, find the value of such that  .   (3 marks)

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a.   

 

 

b.  

♦♦♦ Mean mark 7%.
MARKER’S COMMENT: Most students found the correct quadratic equation but solved for .

 

 

 

c.   
   
   

♦♦ Mean mark 29%.

 

d.   
 
 
 
 
 
 
 

Probability, MET1 2018 VCAA 6

Two boxes each contain four stones that differ only in colour.

Box 1 contains four black stones.

Box 2 contains two black stones and two white stones.

A box is chosen randomly and one stone is drawn randomly from it.

Each box is equally likely to be chosen, as is each stone.

  1.  What is the probability that the randomly drawn stone is black?  (2 marks)

  2.  It is not known from which box the stone has been drawn.
  3. Given that the stone that is drawn is black, what is the probability that it was drawn from Box 1?  (2 marks)

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a.   
   
   

 

b.   
   
   

Statistics, STD2 S1 SM-Bank 2 MC

A dataset has the following five-number summary.

If the range of the dataset is 8, what is the minimum value of the dataset?

  1.  2
  2.  3
  3.  4
  4.  7
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Statistics, STD2 S1 SM-Bank 1 MC

A survey asked the following question for students born in Australia:

"Which State or Territory were you born in?"

How would the responses be classified?

  1. Categorical, ordinal
  2. Categorical, nominal
  3. Numerical, discrete
  4. Numerical, continuous
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Statistics, STD2 S4 SM-Bank 1

A student claimed that as time spent swimming training increases, the time to run a 1 kilometre time trial decreases.

After collecting and analysing some data, the student found the correlation coefficient, , to be – 0.73.

What does this correlation indicate about the relationship between the time a student spends swimming training and their 1 kilometre run time trial times.  (1 mark)

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Algebra, STD2 A4 SM-Bank 6 MC

A computer application was used to draw the graphs of the equations

  and 

Part of the screen is shown.

What is the solution when the equations are solved simultaneously?

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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.

 

  1. In kilometers, what is the shortest distance between Farnham and Carrie?  (1 mark)

  2. 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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a.  

 

b.  

 

Probability, STD2 S2 2011 HSC 26a

The two spinners shown are used in a game.

2UG 2011 26a1

Each arrow is spun once. The score is the total of the two numbers shown by the arrows.
A table is drawn up to show all scores that can be obtained in this game.

2UG 2011 26a2

  1. What is the value of in the table?   (1 mark)

  2. What is the probability of obtaining a score less than 4?   (1 mark)

  3. On Spinner , a 2 is obtained. What is the probability of obtaining a score of 3?   (1 mark)

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i.   
 

ii.   
 

iii.   

Networks, STD2 N3 2018 FUR2 1

The graph below shows the possible number of postal deliveries each day between the Central Mail Depot and the Zenith Post Office.

The unmarked vertices represent other depots in the region.

The weighting of each edge represents the maximum number of deliveries that can be made each day.
 


 

  1.  Cut A, shown on the graph, has a capacity of 10.

     

     Two other cuts are labelled as Cut B and Cut C.

    1.  Write down the capacity of Cut B.  (1 mark)

    2.  Write down the capacity of Cut C. (1 mark)

  2.  Determine the maximum number of deliveries that can be made each day from the Central Mail  Depot to the Zenith Post Office.  (1 mark)

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  2.  
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a.i.   
   

 

a.ii.   
   

♦ Mean mark part (b) 32%.
COMMENT: Review carefully! Most common incorrect answer was 9.

 

b.  

Probability, MET1 2018 VCAA 4

Let be a normally distributed random variable with a mean of 6 and a variance of 4. Let be a random variable with the standard normal distribution.

  1.  Find  .  (1 mark)

  2.  Find    such that  .  (1 mark)

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a.   


 

♦ Mean mark part (b) 41%.

b.   
   
   

 

Graphs, MET1 2018 VCAA 3

Let  .

  1. Solve the equation    for  .   (2 marks)

  2. Sketch the graph of the function  on the axes below. Label the endpoints and local minimum point with their coordinates.   (3 marks)

 

 

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  2.  
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a.
 

 

 

b.   

GRAPHS, FUR2 2018 VCAA 3

Robert wants to hire a geologist to help him find potential gold locations.

One geologist, Jennifer, charges a flat fee of $600 plus 25% commission on the value of gold found.

The following graph displays Jennifer’s total fee in dollars.
 


 

Another geologist, Kevin, charges a total fee of $3400 for the same task.

  1. Draw a graph of the line representing Kevin’s fee on the axes above.  (1 mark)

     

    (answer on the axes above.)

  2. For what value of gold found will Kevin and Jennifer charge the same amount for their work?  (1 mark)
  3. A third geologist, Bella, has offered to assist Robert.
  4. Below is the relation that describes Bella’s fee, in dollars, for the value of gold found.


  5. The step graph below representing this relation is incomplete.

     

    Complete the step graph by sketching the missing information.  (2 marks)
     

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a.  

 

b.    

♦ Mean mark 41%.

 

 

c.  

GRAPHS, FUR2 2018 VCAA 2

The weight of gold can be recorded in either grams or ounces.

The following graph shows the relationship between weight in grams and weight in ounces.
 


 

The relationship between weight measured in grams and weight measured in ounces is shown in the equation

     weight in grams = × weight in ounces
 

  1. Show that    (1 mark)
  2. Robert found a gold nugget weighing 0.2 ounces.

     

    Using the equation above, calculate the weight, in grams, of this gold nugget.  (1 mark)

  3. Last year Robert sold gold to a buyer at $55 per gram.
    The buyer paid Robert a total of $12 474.

     

    Using the equation above, calculate the weight, in ounces, of this gold.   (1 mark)

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a.    

Mean mark 53%.
MARKER’S COMMENT: Giving the equation and writing “solve” did not receive a mark in part (a).

 

 

b.  
   

 

c.  
   

 

 

GRAPHS, FUR2 2018 VCAA 1

The following chart displays the daily gold prices (dollars per gram) for the month of November 2017.
 

  1. Which day in November had the lowest gold price?  (1 mark)
  2. Between which two consecutive days did the greatest increase in gold price occur?  (1 mark)
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a.   
 

b.   

GEOMETRY, FUR1 2018 VCAA 02 MC

A triangle    is shown in the diagram below.


 

The length of the side    is 18 cm.

The length of the side    is 26 cm.

The angle    is 30°.

The area of triangle , in square centimetres, is closest to

  1.  117
  2.  162
  3.  171
  4.  234
  5.  468
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  ²

 

GEOMETRY, FUR2 2018 VCAA 1

Tennis balls are packaged in cylindrical containers.

Frank purchases a container of tennis balls that holds three standard tennis balls, stacked one on top of the other.

This container has a radius of 3.4 cm and a height of 20.4 cm, as shown in the diagram below.
 

  1. What is the diameter, in centimetres, of this container?   (1 mark)
  2. What is the total outside surface area of this container, including both ends?

     

    Write your answer in square centimetres, rounded to one decimal place.  (1 mark)

A standard tennis ball is spherical in shape with a radius of 3.4 cm.

    1. Write a calculation that shows that the volume, rounded to one decimal place, of one standard tennis ball is 164.6 cm³.  (1 mark)
    2. Write a calculation that shows that the volume, rounded to one decimal place, of the cylindrical container that can hold three standard tennis balls is 740.9 cm³. (1 mark)
    3. How much unused volume, in cubic centimetres, surrounds the tennis balls in this container?

       

      Round your answer to the nearest whole number.  (1 mark)

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a.   
   
   

 

b.   
   
   
   

 

c.i.   
   
   

 

c.ii.   
   
   
   

 

c.iii.   
   
   
   

NETWORKS, FUR2 2018 VCAA 3

At the Zenith Post Office all computer systems are to be upgraded.

This project involves 10 activities, to .

The directed network below shows these activities and their completion times, in hours.
 

  1. Determine the earliest starting time, in hours, for activity .   (1 mark)

  2. The minimum completion time for the project is 15 hours.

     

    Write down the critical path.   (1 mark)

  3. Two of the activities have a float time of two hours.

     

    Write down these two activities.   (1 mark)

  4. For the next upgrade, the same project will be repeated but one extra activity will be added.
    This activity has a duration of one hour, an earliest starting time of five hours and a latest starting time of 12 hours.

     

    Complete the following sentence by filling in the boxes provided.   (1 mark)

     

    The extra activity could be represented on the network above by a directed edge from the

   end of activity   
 
   to the start of activity   
 
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  3.  
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a.  

 

 
b. 
 

c.  

♦ Mean mark 45%.

 


 


 

d.  

♦♦ Mean mark 25%.
 

NETWORKS, FUR2 2018 VCAA 1

The graph below shows the possible number of postal deliveries each day between the Central Mail Depot and the Zenith Post Office.

The unmarked vertices represent other depots in the region.

The weighting of each edge represents the maximum number of deliveries that can be made each day.
 


 

  1.  Cut A, shown on the graph, has a capacity of 10.

     

     Two other cuts are labelled as Cut B and Cut C.
     i.  Write down the capacity of Cut B.   (1 mark)

    ii.  Write down the capacity of Cut C.   (1 mark)

  1. Determine the maximum number of deliveries that can be made each day from the Central Mail   Depot to the Zenith Post Office.   (1 mark)

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  1.  i. 
    ii.  
  2.  
Show Worked Solution
a.i.   
   

 

a.ii.   
   

♦ Mean mark part (b) 32%.
COMMENT: Review carefully! Most common incorrect answer was 9.

 

b.  

MATRICES, FUR2 2018 VCAA 3

The Hiroads company has a contract to maintain and improve 2700 km of highway.

Each year sections of highway must be graded , resurfaced or sealed .

The remaining highway will need no maintenance that year.

Let be the state matrix that shows the highway maintenance schedule for the th year after 2018.

The maintenance schedule for 2018 is shown in matrix below.

 

 

The type of maintenance in sections of highway varies from year to year, as shown in the transition matrix , below.
 


 

  1. Of the length of highway that was graded in 2018, how many kilometres are expected to be resurfaced the following year?   (1 mark)

  2. Show that the length of highway that is to be graded in 2019 is 460 km by writing the appropriate numbers in the boxes below.   (1 mark)

     

 
 
 
 

 

The state matrix describing the highway maintenance schedule for the nth year after 2018 is given by


 

  1. Complete the state matrix, , below for the highway maintenance schedule for 2019 (one year after 2018).   (1 mark)

     



     

  2. In 2020, 1536 km of highway is expected to require no maintenance
  3. Of these kilometres, what percentage is expected to have had no maintenance in 2019?
  4. Round your answer to one decimal place.   (1 mark)

  5. In the long term, what percentage of highway each year is expected to have no maintenance ?
  6. Round your answer to one decimal place.   (1 mark)

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a.   
   

♦ Mean mark part (a) 48%.

 

b.   
   

 

c.   
   

 

d.   

 
 

 

e. 

 

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 months,  , can be modelled by the recurrence relation shown below.

 

  1. How many dollars does Julie initially invest?   (1 mark)
  2. Recursion can be used to calculate the balance of the account after one month.
    1. Write down a calculation to show that the balance in the account after one month, , is  $12 074.40.   (1 mark)
    2. After how many months will the balance of Julie’s account first exceed $12 300?   (1 mark)
  3. A rule of the form    can be used to determine the balance of Julie's account after months.
    1. Complete this rule for Julie’s investment after months by writing the appropriate numbers in the boxes provided below.   (1 mark)

    2. Balance = 
       
      		  × 
       
      		  
    3. What would be the value of    if Julie wanted to determine the value of her investment after three years?   (1 mark)

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a.   
 

b.i.  
   
   

 

b.ii.  
 
 

 

 
c.i.  

 
c.ii. 

CORE, FUR2 2018 VCAA 3

Table 3 shows the yearly average traffic congestion levels in two cities, Melbourne and Sydney, during the period 2008 to 2016. Also shown is a time series plot of the same data.

The time series plot for Melbourne is incomplete.

  1. Use the data in Table 3 to complete the time series plot above for Melbourne.   (1 mark)

  2. A least squares line is used to model the trend in the time series plot for Sydney. The equation is

       

  1.   i. Draw this least squares line on the time series plot.   (1 mark)

  2.  ii. Use the equation of the least squares line to determine the average rate of increase in percentage congestion level for the period 2008 to 2016 in Sydney.   (1 mark)

    iii. Use the least squares line to predict when the percentage congestion level in Sydney will be 43%.   (1 mark)

The yearly average traffic congestion level data for Melbourne is repeated in Table 4 below.

  1. When a least squares line is used to model the trend in the data for Melbourne, the intercept of this line is approximately –1514.75556
  2. Round this value to four significant figures.   (1 mark)

  3. Use the data in Table 4 to determine the equation of the least squares line that can be used to model the trend in the data for Melbourne. The variable year is the explanatory variable.
  4. Write the values of the intercept and the slope of this least squares line in the appropriate boxes provided below.
  5. Round both values to four significant figures.   (2 marks)

congestion level = 
 
  + 
 
  × year
  1. Since 2008, the equations of the least squares lines for Sydney and Melbourne have predicted that future traffic congestion levels in Sydney will always exceed future traffic congestion levels in Melbourne.

     

    Explain why, quoting the values of appropriate statistics.   (2 marks)

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a.   

 

b.i.   

 

b.ii.  

♦ Mean mark (b)(ii) 36%.
COMMENT: Major problems caused by part (b)(ii). Review!

 

 

b.iii.   
 
 
   

 

c. 

 

d.   

 

e.   

♦♦♦ Mean mark 18%.

 

CORE, FUR2 2018 VCAA 2

The congestion level in a city can be recorded as the percentage increase in travel time due to traffic congestion in peak periods (compared to non-peak periods).

This is called the percentage congestion level.

The percentage congestion levels for the morning and evening peak periods for 19 large cities are plotted on the scatterplot below.
 

  1. Determine the median percentage congestion level for the morning peak period and the evening peak period.

     

    Write your answers in the appropriate boxes provided below.   (2 marks)

Median percentage congestion level for morning peak period
%
Median percentage congestion level for evening peak period
%

A least squares line is to be fitted to the data with the aim of predicting evening congestion level from morning congestion level.

The equation of this line is.

evening congestion level = 8.48 + 0.922 × morning congestion level

  1. Name the response variable in this equation.   (1 mark)

  2. Use the equation of the least squares line to predict the evening congestion level when the morning congestion level is 60%.   (1 mark)

  3. Determine the residual value when the equation of the least squares line is used to predict the evening congestion level when the morning congestion level is 47%.
  4. Round your answer to one decimal place?   (2 marks)

  5. The value of the correlation coefficient is 0.92
  6. What percentage of the variation in the evening congestion level can be explained by the variation in the morning congestion level?
  7. Round your answer to the nearest whole number.   (1 mark)

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a.   

Mean mark 56%.
COMMENT: This question was surprisingly poorly answered. Review carefully!

 


 

b.   
 

c.   
   

 
d.  

Mean mark part (d) 53%.
COMMENT: Many students had problems at a number of stages in this part.

 
 
 

 

e.   
 
   

 

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).
  1. How many variables in this data set are categorical variables?  (1 mark)

  2. How many variables in this data set are ordinal variables  (1 mark)

  3. Name the large UK cities with a medium level of traffic congestion.  (1 mark)

  4. Use the data in Table 1 to complete the following two-way frequency table, Table 2.  (2 marks)

     

     


     

  5. 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.
 


 

  1. Describe the shape of the distribution of the increase in travel time for the 23 cities.  (1 mark)

  2. The data value 52 is below the upper fence and is not an outlier.
  3. Determine the value of the upper fence.  (1 mark)

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a.   
 

b.   
 

c.   
 

d.   

 

e.   
   
   

 
f.   
 

g.   
 

 

MATRICES, FUR1 2018 VCAA 3 MC

Five people, India (), Jackson (), Krishna (), Leanne () and Mustafa (), competed in a table tennis tournament.

Each competitor played every other competitor once only.

Each match resulted in a winner and a loser.

The matrix below shows the tournament results.
 


 

A 1 in the matrix shows that the competitor named in that row defeated the competitor named in that column.

For example, the 1 in the fourth row shows that Leanne defeated Jackson.

A 0 in the matrix shows that the competitor named in that row lost to the competitor named in that column.

There is an error in the matrix. The winner of one of the matches has been incorrectly recorded as a 0.

This match was between

  1. India and Mustafa.
  2. India and Krishna.
  3. Krishna and Leanne.
  4. Leanne and Mustafa.
  5. Jackson and Mustafa.
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CORE, FUR1 2018 VCAA 17-18 MC

The value of an annuity investment, in dollars, after years, , can be modelled by the recurrence relation shown below.

 
Part 1

What is the value of the regular payment added to the principal of this annuity investment?

  1.   $34.00
  2. $156.40
  3. $466.00
  4. $500.00
  5. $656.40

 
Part 2

Between the second and third years, the increase in the value of this investment is closest to

  1.   $656
  2.   $658
  3.   $661
  4. $1315
  5. $1975
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CORE, FUR1 2018 VCAA 15 MC

The table below shows the monthly profit, in dollars, of a new coffee shop for the first nine months of 2018.

Using four-mean smoothing with centring, the smoothed profit for May is closest to

  1. $2502
  2. $3294
  3. $3503
  4. $3804
  5. $4651
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CORE, FUR1 2018 VCAA 3-5 MC

The pulse rates of a population of Year 12 students are approximately normally distributed with a mean of 69 beats per minute and a standard deviation of 4 beats per minute.
 

Part 1

A student selected at random from this population has a standardised pulse rate of 

This student’s actual pulse rate is

  1.  59 beats per minute.
  2.  63 beats per minute.
  3.  65 beats per minute.
  4.  73 beats per minute.
  5.  79 beats per minute.
     

Part 2

Another student selected at random from this population has a standardised pulse rate of 

The percentage of students in this population with a pulse rate greater than this student is closest to

  1.   2.5%
  2.   5%
  3.  16%
  4.  68%
  5.  84%
     

Part 3

A sample of 200 students was selected at random from this population.

The number of these students with a pulse rate of less than 61 beats per minute or greater than 73 beats per minute is closest to

  1.  19
  2.  37
  3.  64
  4.  95
  5. 190
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Show Worked Solution

 

 


 

 


 

 


 

 

 

 

CORE, FUR1 2018 VCAA 1-2 MC

The dot plot below displays the difference in travel time between the morning peak and the evening peak travel times for the same journey on 25 days.
 


 

Part 1

The percentage of days when there was five minutes difference in travel time between the morning peak and the evening peak travel times is

  1.    0%
  2.    5%
  3.  20%
  4.  25%
  5.  28%
     

Part 2

The median difference in travel time is

  1.  3.0 minutes.
  2.  3.5 minutes.
  3.  4.0 minutes.
  4.  4.5 minutes.
  5.  5.0 minutes.
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Show Worked Solution

 
 

 

 

Networks, STD2 N3 SM-Bank 31

Murray is building a new garage. The project involves activities to .
 

 

The network diagram shows these activities and their completion times in days.

  1. Which TWO activities immediately precede activity ?  (1 mark)

  2. By completing the diagram shown, calculate the minimum time required to build the new garage.  (2 marks)

  3. Hence, what is the float time for activity ?  (1 mark)

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a.   

 

b.   

 

c.   

Networks, STD2 N2 SM-Bank 28

In central Queensland, there are four petrol stations , , and . The table shows the length, in kilometres, of roads connecting these petrol stations.
 


 

  1. Construct a network diagram to represent the information in the table.  (2 marks)

  2. A petrol tanker needs to refill each station. It starts at Station and visits each station.

     

    Calculate the shortest distance that can be travelled by the petrol tanker. In your answer, include the order the petrol stations are refilled.  (2 marks)

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  1.  
Show Worked Solution
a.   

 

b.   

 

Networks, STD2 N2 SM-Bank 27 MC

This diagram shows the possible paths (in km) for laying gas pipes between various locations.
 

 
Gas is be supplied from one location. Any one of the locations can be the source of the supply.

What is the minimum total length of length of the pipes required to provide gas to all the locations?

  1.  46 km
  2.  48 km
  3.  50 km
  4.  52 km
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Show Worked Solution


  


  

 

 

Measurement, STD2 M2 SM-Bank 2

Bert is in Moscow, which is three hours behind of Coordinated Universal Time (UTC).

Karen is in Sydney, which is eleven hours ahead of UTC.

  1. Bert is going to ring Karen at 9 pm on Tuesday, Moscow time. What day and time will it be in Sydney when he rings?  (1 mark)

  2. Karen is going to fly from Sydney to Moscow. Her flight will leave on Wednesday at 8 am, Sydney time, and will take 15 hours. What day and time will it be in Moscow when she arrives?  (2 marks)

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a.   

 

 

b.   
   

 

 

Measurement, STD2 M7 SM-Bank 4

Blood pressure is measured using two numbers: systolic pressure and diastolic pressure. If the measurement shows 120 systolic and 80 diastolic, it is written as 120/80.

The bars on the graph show the normal range of blood pressure for people of various ages.

  1. What is the normal range of blood pressure for a 53-year-old?  (2 marks)

  2. Ralph, aged 53, had a blood pressure reading of 173 over 120. A doctor prescribed Ralph a medication to reduce his blood pressure to be within the normal range. To check that the medication was being effective, the doctor measured Ralph's blood pressure for 10 weeks and recorded the following results.  
     

     
    With reference to the data provided, comment on the effectiveness of the medication during the 10-week period in returning Ralph’s blood pressure to the normal range.  (3 marks)

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a.   

       

b.   

   

Show Worked Solution

a.   

 

b.   

     

L&E, 2ADV E1 SM-Bank 2

The population of Indian Myna birds in a suburb can be described by the exponential function

where is the time in months.

  1.  What will be the population after 2 years?  (1 mark)

  2.  Draw a graph of the population.  (2 marks)

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  2.  

Show Worked Solution

i.   

 
 
 

 

ii.

Calculus, 2ADV C1 SM-Bank 3

The displacement metres from the origin at time seconds of a particle travelling in a straight line is given by

     when   

  1.  Calculate the velocity when  .  (1 mark)

  2.  When is the particle stationary?  (2 marks)

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  1.  
  2.  
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i.    

 

 

 

ii.