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CORE, FUR2 2019 VCAA 7

Phil is a builder who has purchased a large set of tools.

The value of Phil’s tools is depreciated using the reducing balance method.

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

 

  1. Use recursion to show that the value of the tools after two years, , is $48 600.   (1 mark)

  2. What is the annual percentage rate of depreciation used by Phil?   (1 mark)

  3. Phil plans to replace these tools when their value first falls below $20 000.

     

    After how many years will Phil replace these tools?   (1 mark)

  4. Phil has another option for depreciation. He depreciates the value of the tools by a flat rate of 8% of the purchase price per annum.

     

    Let be the value of the tools after years, in dollars.

     

    Write down a recurrence relation, in terms of and , that could be used to model the value of the tools using this flat rate depreciation.   (1 mark)

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a.  
 
 

  
b. 

 
c. 

 
d. 

CORE, FUR2 2019 VCAA 6

The total rainfall, in millimetres, for each of the four seasons in 2015 and 2016 is shown in Table 5 below.

  1. The seasonal index for winter is shown in Table 6 below.

     

    Use the values in Table 5 to find the seasonal indices for summer, autumn and spring.

  2. Write your answers in Table 6, rounded to two decimal places.   (2 marks)

     


  3. The total rainfall for each of the four seasons in 2017 is shown in Table 7 below.

     


    Use the appropriate seasonal index from Table 6 to deseasonalise the total rainfall for winter in 2017.

     

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

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a.  

 

b.  
   

CORE, FUR2 2019 VCAA 5

The scatterplot below shows the atmospheric pressure, in hectopascals (hPa), at 3 pm (pressure 3 pm) plotted against the atmospheric pressure, in hectopascals, at 9 am (pressure 9 am) for 23 days in November 2017 at a particular weather station.
 

A least squares line has been fitted to the scatterplot as shown.

The equation of this line is

pressure 3 pm = 111.4 + 0.8894 × pressure 9 am

  1. Interpret the slope of this least squares line in terms of the atmospheric pressure at this weather station at 9 am and at 3 pm.   (1 mark)

  2. Use the equation of the least squares line to predict the atmospheric pressure at 3 pm when the atmospheric pressure at 9 am is 1025 hPa.
  3. Round your answer to the nearest whole number.   (1 mark)

  4. Is the prediction made in part b. an example of extrapolation or interpolation?   (1 mark)

  5. Determine the residual when the atmospheric pressure at 9 am is 1013 hPa.
  6. Round your answer to the nearest whole number.   (1 mark)

  7. The mean and the standard deviation of pressure 9 am and pressure 3 pm for these 23 days are shown in Table 4 below.

    1. Use the equation of the least squares line and the information in Table 4 to show that the correlation coefficient for this data, rounded to three decimal places, is  = 0.966   (1 mark)

    2. What percentage of the variation in pressure 3 pm is explained by the variation in pressure 9 am?
    3. Round your answer to one decimal place.   (1 mark)

  1. The residual plot associated with the least squares line is shown below.
     

    1. The residual plot above can be used to test one of the assumptions about the nature of the association between the atmospheric pressure at 3 pm and the atmospheric pressure at 9 am.
    2. What is this assumption?   (1 mark)

    3. The residual plot above does not support this assumption.
    4. Explain why.   (1 mark)

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a.   

 

b.  
   

 

c. 

 

d.  
   
   
   
   

 

e.i.   

   
   
   

 

e.ii.  
 
   

 

f.i.  
 

 

f.ii. 

CORE, FUR2 2019 VCAA 4

The relative humidity (%) at 9 am and 3 pm on 14 days in November 2017 is shown in Table 3 below.

A least squares line is to be fitted to the data with the aim of predicting the relative humidity at 3 pm (humidity 3 pm) from the relative humidity at 9 am (humidity 9 am).

  1. Name the explanatory variable.   (1 mark)

  2. Determine the values of the intercept and the slope of this least squares line.
  3. Round both values to three significant figures and write them in the appropriate boxes provided.   (1 mark)

humidity 3 pm = 
 
   +  
 
   × humidity 9 am  (1 mark)
  1. Determine the value of the correlation coefficient for this data set.
  2. Round your answer to three decimal places.   (1 mark)

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a. 
 

b. 


 

c.    

CORE, FUR2 2019 VCAA 3

The five-number summary for the distribution of minimum daily temperature for the months of February, May and July in 2017 is shown in Table 2.

The associated boxplots are shown below the table.

Explain why the information given above supports the contention that minimum daily temperature is associated with the month. Refer to the values of an appropriate statistic in your response.  (2 marks)

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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.
 

  1. Use the information in the boxplots to complete the following sentences.
  2. For November 2017
  3.    i. the interquartile range for the minimum daily temperature was _____ °C   (1 mark)

  4.  ii. the median value for maximum daily temperature was _____ °C higher than the median value for minimum daily temperature   (1 mark)

  5. iii. the number of days on which the maximum daily temperature was less than the median value for minimum daily temperature was _____    (1 mark)

  1. 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.
  2. Determine the number of days in November 2017 for which this temperature difference is expected to be greater than 9.4 °C.  (1 mark)

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a.i. 
 

a.ii.   
 

 

 

a.iii. 

   
 

b.   
   

CORE, FUR2 2019 VCAA 1

Table 1 shows the day number and the minimum temperature, in degrees Celsius, for 15 consecutive days in May 2017.
 

  1. Which of the two variables in this data set is an ordinal variable?   (1 mark)

The incomplete ordered stem plot below has been constructed using the data values for days 1 to 10.

  1. Complete the stem plot above by adding the data values for days 11 to 15.   (1 mark)

  2. The ordered stem plot below shows the maximum temperature, in degrees Celsius, for the same 15 days.

     

     

    Use this stem plot to determine

  3.  i. the value of the first quartile    (1 mark)

  4. ii. the percentage of days with a maximum temperature higher than 15.3 °C.   (1 mark)

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a.   

b.  

 
c.i.   

 
 

c.ii.   
   

GRAPHS, FUR1 2019 VCAA 7 MC

The feasible region for a linear programming problem is shaded in the diagram below.
 


 

The equation of the objective function for this problem is of the form
 

,    where    and 
 

The dotted line passing through the points and in the diagram has the same slope as the objective function for this problem.

The minimum value of the objective function can be determined by calculating its value at

  1. point only.
  2. point only.
  3. point only.
  4. any point along the line segment  .
  5. any point along the line segment  .
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GRAPHS, FUR1 2019 VCAA 5 MC

Giovanni makes and sells toy train sets.

Each toy train set costs $40.00 to make and sells for $65.00

Giovanni also has a fixed cost for making a batch of toy train sets.

Last Sunday, Giovanni sold a batch of 35 toy train sets for a profit of $250.

Giovanni’s fixed cost for this batch of train sets is

  1.    $250
  2.    $625
  3.    $875
  4.  $1400
  5.  $2275
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GRAPHS, FUR1 2019 VCAA 4 MC

The graph below shows a straight line that crosses the -axis at  , passes through the point    and crosses the -axis at  .
 


 

What is the value of ?

  1.  14
  2.  16
  3.  18
  4.  20
  5.  22
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GRAPHS, FUR1 2019 VCAA 3 MC

A shop sells two types of discs: compact discs (CDs) and digital video discs (DVDs).

CDs are sold for $7.00 each and DVDs are sold for $13.00 each.

Bonnie bought a total of 16 discs for $178.00

How many DVDs did Bonnie buy?

  1.   3
  2.   5
  3.   7
  4.   9
  5. 11
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GEOMETRY, FUR1 2019 VCAA 6 MC

All cities in China are in the same time zone.

Wuhan (31° N, 114° E) and Chengdu (31° N, 104° E) are both cities in China.

On one day in January, the sun set in Wuhan at 5.50 pm.

Assuming that 15° of longitude equates to a one-hour time difference, the time the sun set in Chengdu on the same day is

  1. 4.20 pm
  2. 5.10 pm
  3. 5.50 pm
  4. 6.30 pm
  5. 7.20 pm
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GEOMETRY, FUR1 2019 VCAA 5 MC

In triangle  , the point    lies on the line  , as shown in the diagram below.
 


 

The length of the line    is equal to the length of the line  .

Which one of the following statements is true?

  1. The angle    is equal to the angle  .
  2. The angle    is equal to the angle  .
  3. The sum of the angle    and the angle    is equal to 180°.
  4. The sum of the angle    and the angle    is equal to 180°.
  5. The sum of the angle    and the angle    is equal to the angle  .
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GEOMETRY, FUR1 2019 VCAA 4 MC

Triangle , shown below, has side lengths of 3 cm, 4 cm and 5 cm.
 


 

Four other triangles have the following side lengths:

    • Triangle has side lengths of 3 cm, 6 cm and 8 cm.
    • Triangle has side lengths of 4 cm, 8 cm and 12 cm.
    • Triangle has side lengths of 6 cm, 8 cm and 10 cm.
    • Triangle has side lengths of 9 cm, 12 cm and 15 cm.

The triangles that are similar to triangle are

  1. triangle and triangle .
  2. triangle , triangle and triangle .
  3. triangle and triangle .
  4. triangle and triangle .
  5.  triangle and triangle .
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GEOMETRY, FUR1 2019 VCAA 2 MC

Town B is located on a bearing of 060° from Town A.

The diagram that could illustrate this is

A.   B.  
C.   D.  
E.      
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Networks, STD2 N2 SM-Bank 34

The following diagram shows the distances, in metres, along a series of cables connecting a main server to seven points, to , in a computer network.
 


 

Calculate the minimum length of cable, in metres, required to ensure that each of the seven points is connected to the main server directly or via another point.   (2 marks)

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NETWORKS, FUR1 2019 VCAA 5 MC

The following diagram shows the distances, in metres, along a series of cables connecting a main server to seven points, to , in a computer network.
 


 

The minimum length of cable, in metres, required to ensure that each of the seven points is connected to the main server directly or via another point is

  1. 175
  2. 203
  3. 208
  4. 221
  5. 236
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NETWORKS, FUR1 2019 VCAA 4 MC

Two graphs, labelled Graph 1 and Graph 2, are shown below.
 


 

Which one of the following statements is not true?

  1. Graph 1 and Graph 2 are isomorphic.
  2. Graph 1 has five edges and Graph 2 has six edges.
  3. Both Graph 1 and Graph 2 are connected graphs.
  4. Both Graph 1 and Graph 2 have three faces each.
  5. Neither Graph 1 nor Graph 2 are complete graphs.
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NETWORKS, FUR1 2019 VCAA 3 MC

The flow of water through a series of pipes is shown in the network below

The numbers on the edges show the maximum flow through each pipe in litres per minute.
 

 
The capacity of Cut , in litres per minute, is

  1. 11
  2. 13
  3. 14
  4. 16
  5. 17
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MATRICES, FUR1 2019 VCAA 7 MC

The communication matrix below shows the direct paths by which messages can be sent between two people in a group of six people, to .
 


 

A 1 in the matrix shows that the person named in that row can send a message directly to the person named in that column.

For example, the 1 in row 4, column 2 shows that can send a message directly to .

In how many ways can get a message to by sending it directly to one other person?

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
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MATRICES, FUR1 2019 VCAA 6 MC

A water park is open from 9 am until 5 pm.

There are three activities, the pool , the slide and the water jets , at the water park.

Children have been found to change their activity at the water park each half hour, as shown in the transition matrix, , below.
 


 

A group of children has come to the water park for the whole day.

The percentage of these children who are expected to be at the slide at closing time is closest to

  1.  14%
  2.  20%
  3.  24%
  4.  25%
  5.  62%
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MATRICES, FUR1 2019 VCAA 5 MC


 

Which one of the following systems of simultaneous linear equations best represents the matrix equation above?

A.   B.  
   
   
   
           
C.   D.  
   
   
   
           
E.        
       
       
       
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MATRICES, FUR1 2019 VCAA 4 MC

Stella completed a multiple-choice test that had 10 questions.

Each question had five possible answers, and .

For question number one, Stella chose the answer .

Stella chose each of the nine remaining answers, in order, by following the transition matrix, , below
 


 

What answer did Stella choose for question number six?

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CORE, FUR1 2019 VCAA 22 MC

A machine is purchased for $30 000

It produces 24 000 items each year.

The value of the machine is depreciated using a unit cost method of depreciation.

After three years, the value of the machine is $18 480.

A rule for the value of the machine after units are produced, , is

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CORE, FUR1 2019 VCAA 20 MC

Consider the following amortisation table for a reducing balance loan.
 


 

The annual interest rate for this loan is 3.6%.

Interest is calculated immediately before each payment.

For this loan, the repayments are made

  1. weekly.
  2. fortnightly
  3. monthly.
  4. quarterly.
  5. yearly
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CORE, FUR1 2019 VCAA 15-16 MC

The time series plot below shows the monthly rainfall at a weather station, in millimetres, for each month in 2017.
 


 

Part 1

The median monthly rainfall for 2017 was closest to

  1.  53 mm
  2.  82 mm
  3.  96 mm
  4. 103 mm
  5. 111 mm

 
Part 2

If seven-mean smoothing is used to smooth this time series plot, the number of smoothed data points would be

  1.  3
  2.  5
  3.  6
  4.  8
  5. 10
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CORE, FUR1 2019 VCAA 13-14 MC

The time, in minutes, that Liv ran each day was recorded for nine days.

These times are shown in the table below.
  


  

The time series plot below was generated from this data.
  

Part 1

Both three-median smoothing and five-median smoothing are being considered for this data.

Both of these methods result in the same smoothed value on day number

  1. 3
  2. 4
  3. 5
  4. 6
  5. 7

 
Part 2

A least squares line is to be fitted to the time series plot shown above.

The equation of this least squares line, with day number as the explanatory variable, is closest to

  1. day number = 23.8 + 2.29 × time
  2. day number = 28.5 + 1.77 × time
  3. time = 23.8 + 1.77 × day number
  4. time = 23.8 + 2.29 × day number
  5. time = 28.5 + 1.77 × day number
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CORE, FUR1 2019 VCAA 9-10 MC

A least squares line is used to model the relationship between the monthly average temperature and latitude recorded at seven different weather stations. The equation of the least squares line is found to be
 

 
Part 1

When the numbers in this equation are correctly rounded to three significant figures, the equation will be

 
Part 2

The coefficient of determination was calculated to be 0.893743

The value of the correlation coefficient, rounded to three decimal places, is

  1. − 0.945
  2. − 0.898
  3.  0.806
  4.  0.898
  5.  0.945
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CORE, FUR1 2019 VCAA 8 MC

Percy conducted a survey of people in his workplace. He constructed a two-way frequency table involving two variables.

One of the variables was attitude towards shorter working days (for, against).

The other variable could have been

  1. age (in years).
  2. sex (male, female).
  3. height (to the nearest centimetre).
  4. income (to the nearest thousand dollars).
  5. time spent travelling to work (in minutes).
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CORE, FUR1 2019 VCAA 7 MC

The volume of a cup of soup served by a machine is normally distributed with a mean of 240 mL and a standard deviation of 5 mL.

A fast-food store used this machine to serve 160 cups of soup.

The number of these cups of soup that are expected to contain less than 230 mL of soup is closest to

  1.     3
  2.     4
  3.     8
  4.   26
  5. 156
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CORE, FUR1 2019 VCAA 6 MC

The time taken to travel between two regional cities is approximately normally distributed with a mean of 70 minutes and a standard deviation of 2 minutes.

The percentage of travel times that are between 66 minutes and 72 minutes is closest to

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CORE, FUR1 2019 VCAA 1-3 MC

The histogram below shows the distribution of the population size of 48 countries in 2018.

Part 1

The number of these countries with a population size between 5 million and 20 million people is

  1.  11
  2.  17
  3.  23
  4.  34
  5.  35

 
Part 2

The shape of this histogram is best described as

  1. positively skewed with no outliers. 
  2. positively skewed with outliers.
  3. approximately symmetric.
  4. negatively skewed with no outliers.
  5. negatively skewed with outliers.

 
Part 3

The histogram below shows the population size for these 48 countries plotted on a scale.

Based on this histogram, the number of countries with a population size that is less than people is

  1. 1
  2. 5
  3. 7
  4. 8
  5. 48
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Part 1


 

Part 2


 

Part 3

Statistics, EXT1 S1 SM-Bank 11

Within a particular population, it is known that the percentage of left-handers is 17%.

A research project randomly selects 200 people from this population.

Assuming this sample proportion is normally distributed, what is the probability that the percentage of people that are left-handed in this sample is

  1. greater than 20%   (2 marks)

  2. less than 10%   (2 marks)

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i.   

 
 
   
 
     

 
   
   

 

 
   
   

 

ii.   
   

 

 
   

Functions, EXT1 F2 SM-Bank 2

The polynomial    has roots  .

  1. Find the value of  .  (1 mark)

  2. Find the value of  .  (1 mark)

  3. It is known that two of the roots are equal in magnitude but opposite in sign.

     

    Find the third root and hence find the value of .  (2 marks)

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i.   

 

ii.   

 

iii.   

 

Statistics, EXT1 S1 SM-Bank 6

After counting votes in an election, it is known that 40% of people voted for the Katter party.

A sample of 600 voting cards are taken and inspected. Assuming this sample proportion is approximately normally distributed, what is the probability that the percentage of voting cards inspected that chose the Katter party is less than 36%?   (3 marks)

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Statistics, EXT1 S1 SM-Bank 4

In an experiment, a pair of dice are rolled 70 times.

A success is recorded if the sum of the dice roll is 5 or less.

  1. What is the mean of this binomial distribution?

     

    Give your answer to one decimal place.  (3 marks)

  2. What is the standard deviation?

     

    Give your answer to one decimal place.  (1 mark)

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i.   

STRATEGY: A table (or array) can be a very efficient and error minimising strategy in questions like this.

 
 

 

ii.