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Calculus, MET2 2016 VCAA 14 MC

A rectangle is formed by using part of the coordinate axes and a point  , where    on the parabola  
 

Which one of the following is the maximum area of the rectangle?

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♦♦ Mean mark 37%.

 

 
 

 

Calculus, MET2 2011 VCAA 19 MC

A part of the graph of    is shown below. Zoe finds the approximate area of the shaded region by drawing rectangles as shown in the second diagram.

met2-2011-vcaa-19-mc

Zoe's approximation is   more than the exact value of the area.

The value of is closest to

A.   

B.   

C.   

D.   

E.   

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♦ Mean mark 43%.

       
 

 

 
 

NETWORKS, FUR2 2016 VCAA 3

A new skateboard park is to be built in Beachton.

This project involves 13 activities, to .

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


 

  1. Determine the earliest start time for activity .   (1 mark)

  2. The minimum completion time for the skateboard park is 15 days.

     

    Write down the critical path for this project.   (1 mark)

  3. Which activity has a float time of two days?   (1 mark)

  4. The completion times for activities and can each be reduced by one day.

     

    The cost of reducing the completion time by one day for these activities is shown in the table below.
     

     

       

     

    What is the minimum cost to complete the project in the shortest time possible?   (1 mark)

  5. The original skateboard park project from part (a), before the reduction of time in any activity, will be repeated at another town named Campville, but with the addition of one extra activity.

     

    The new activity, , will take six days to complete and has a float time of one day.

     

    Activity will finish at the same time as the project.

     

     i.  Add activity to the network below.   (1 mark) 


      
    ii.  What is the latest start time for activity ?   (1 mark)

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a.   
   

  
b.  

♦♦ Mean mark part (c) 37%, part (d) 21%.
MARKER’S COMMENT: In part (d), cannot be crashed, therefore shortest duration is 14 days. Activity is cheapest to reduce.
  

c.  
  

d.  


  

e.i.   

♦♦ Mean mark part (e)(i) 21%, (e)(ii) 27%.
MARKER’S COMMENT: In (e)(ii), activity must have an arrow on it.
  

e.ii.   
   
   

NETWORKS, FUR2 2016 VCAA 2

The suburb of Alooma has a skateboard park with seven ramps.

The ramps are shown as vertices , , , , , and on the graph below.
 

 

The tracks between ramps and and between ramps and are rough, as shown on the graph above.

  1. Nathan begins skating at ramp and follows an Eulerian trail.

     

    At which ramp does Nathan finish?   (1 mark)

  2. Zoe begins skating at ramp and follows a Hamiltonian path.

     

    The path she chooses does not include the two rough tracks.

     

    Write down a path that Zoe could take from start to finish.   (1 mark)

  3. Birra can skate over any of the tracks, including the rough tracks.

     

    He begins skating at ramp and will complete a Hamiltonian cycle.

     

    In how many ways could he do this?   (1 mark) 

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Show Worked Solution

a.  

“XYZWVZUYTU`


  

b.  


  

c.  

♦♦ Mean mark 31%.

NETWORKS, FUR2 2016 VCAA 1

A map of the roads connecting five suburbs of a city, Alooma (), Beachton (), Campville (), Dovenest () and Easyside (), is shown below.
 


  

  1. Starting at Beachton, which two suburbs can be driven to using only one road?   (1 mark)

A graph that represents the map of the roads is shown below.
 


 

One of the edges that connects to vertex is missing from the graph.

  1.  i. Add the missing edge to the graph above.   (1 mark)

    (Answer on the graph above)
  2. ii. Explain what the loop at represents in terms of a driver who is departing from Dovenest.   (1 mark)

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a.   

b.i. 

b.ii.

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a.  

 

b.i.   

 

b.ii. 

♦♦ Mean mark 30%.

GRAPHS, FUR2 2016 VCAA 3

A company produces two types of hockey stick, the ‘Flick’ and the ‘Jink’.

Let be the number of Flick hockey sticks that are produced each month.

Let be the number of Jink hockey sticks that are produced each month.

Each month, up to 500 hockey sticks in total can be produced.

The inequalities below represent constraints on the number of each hockey stick that can be produced each month.

Constraint 1 Constraint 2
Constraint 3 Constraint 4
  1. Interpret Constraint 4 in terms of the number of Flick hockey sticks and the number of Jink hockey sticks produced each month.  (1 mark)

There is another constraint, Constraint 5, on the number of each hockey stick that can be produced each month.

Constraint 5 is bounded by Line , shown on the graph below.

The shaded region of the graph contains the points that satisfy constraints 1 to 5.

  1. Write down the inequality that represents Constraint 5.  (1 mark)

The profit, , that the company makes from the sale of the hockey sticks is given by

  1. Find the maximum profit that the company can make from the sale of the hockey sticks.  (1 mark)
  2. The company wants to change the selling price of the Flick and Jink hockey sticks in order to increase its maximum profit to $42 000.

    All of the constraints on the numbers of Flick and Jink hockey sticks that can be produced each month remain the same.

    The profit, , that is made from the sale of hockey sticks is now given by

     

    The profit made on the Flick hockey sticks is dollars per hockey stick.

    The profit made on the Jink hockey sticks is dollars per hockey stick.

    The maximum profit of $42 000 is made by selling 400 Flick hockey sticks and 100 Jink hockey sticks.

    What are the values of and ?  (2 marks)

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a.  

♦♦♦ Mean mark 25%.

 

b.  

 

c.  

 

 

d.  

♦♦♦ Mean mark 7%.
MARKER’S COMMENT: Very few students were able to identify the need for the sliding line concept and execute it correctly in this question.

 

MATRICES, FUR2 2016 VCAA 3

A travel company is studying the choice between air (), land (), sea () or no () travel by some of its customers each year.

Matrix , shown below, contains the percentages of customers who are expected to change their choice of travel from year to year.

Let be the matrix that shows the number of customers who choose each type of travel years after 2014.

Matrix below shows the number of customers who chose each type of travel in 2014.

Matrix below shows the number of customers who chose each type of travel in 2015.

  1. Find the values missing from matrix .   (1 mark)

  2. Write a calculation that shows that 478 customers were expected to choose air travel in 2015.   (1 mark)

  3. Consider the customers who chose sea travel in 2014.
  4. How many of these customers were expected to choose sea travel in 2015?   (1 mark)

  5. Consider the customers who were expected to choose air travel in 2015.
  6. What percentage of these customers had also chosen air travel in 2014?
  7. Round your answer to the nearest whole number.   (1 mark)

In 2016, the number of customers studied was increased to 1360.
Matrix , shown below, contains the number of these customers who chose each type of travel in 2016.

  1. The company intends to increase the number of customers in the study in 2017 and in 2018.
  2. The matrix that contains the number of customers who are expected to choose each type of travel in 2017 () and 2018 () can be determined using the matrix equations shown below.


 

    1. The element in the fourth row of matrix is – 80.
    2. Explain this number in the context of selecting customers for the studies in 2017 and 2018.   (1 mark)

    3. Determine the number of customers who are expected to choose sea travel in 2018.
    4. Round your answer to the nearest whole number.   (2 marks)

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  5.  


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a.   
 

b.   

♦♦ Mean mark part (b) and (c) was 35% and 45% respectively.

 

c.  

 

 

d.  

♦♦♦ Mean mark 17%.


 

 
 

 

e.i.  

♦ Mean mark 14%.
MARKER’S COMMENT: “80 people chose not to travel” was a common answer that received no marks.

 

e.ii.  
   
   

 

 
 

 

GEOMETRY, FUR2 2016 VCAA 5

A golf course has a sprinkler system that waters the grass in the shape of a sector, as shown in the diagram below. 
 

A sprinkler is positioned at point and can turn through an angle of 100°.

The shaded area on the diagram shows the area of grass that is watered by the sprinkler.

  1. If 147.5 m² of grass is watered, what is the maximum distance, metres, that the water reaches from ?

     

    Round your answer to the nearest metre.  (1 mark)

  2. Another sprinkler can water a larger area of grass.

     

    This sprinkler will water a section of grass as shown in the diagram below.
     

     


    The section of grass that is watered is 4.5 m wide at all points.

     

    Water can reach a maximum of 12 m from the sprinkler at .

     

    What is the area of grass that this sprinkler will water?

     

    Round your answer to the nearest square metre.  (2 marks)

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  2. ²²
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♦ Mean mark 48%.
a.   
 
 
   
 
   
♦♦♦ Mean mark 23%.

 

b.   
   
   
    ²²

GEOMETRY, FUR2 2016 VCAA 3

A golf tournament is played in St Andrews, Scotland, at location 56° N, 3° W.

  1. Assume that the radius of Earth is 6400 km.

     

    Find the shortest great circle distance to the equator from St Andrews.

     

    Round your answer to the nearest kilometre.  (1 mark)

  2. The tournament begins on Thursday at 6.32 am in St Andrews, Scotland.

     

    Many people in Melbourne will watch the tournament live on television.

     

    Assume that the time difference between Melbourne (38° S, 145° E) and St Andrews (56° N, 3° W) is 10 hours.

     

    On what day and at what time will the tournament begin in Melbourne?  (1 mark) 

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a.   

♦ Mean mark 42%.
MARKER’S COMMENT: Longitude figures were not relevant here!

 

b.  

♦♦ Mean mark 37%.
MARKER’S COMMENT: The time difference between the two cities here is given! Longitudinal difference calculations are not required.

CORE, FUR2 2016 VCAA 7

 

Ken has borrowed $70 000 to buy a new caravan.

He will be charged interest at the rate of 6.9% per annum, compounding monthly.

  1. For the first year (12 months), Ken will make monthly repayments of $800.

    1. Find the amount that Ken will owe on his loan after he has made 12 repayments.   (1 mark)
    2. What is the total interest that Ken will have paid after 12 repayments?   (1 mark)
  2. After three years, Ken will make a lump sum payment of $L in order to reduce the balance of his loan.
  3. This lump sum payment will ensure that Ken’s loan is fully repaid in a further three years.
  4. Ken’s repayment amount remains at $800 per month and the interest rate remains at 6.9% per annum, compounding monthly.
  5. What is the value of Ken’s lump sum payment, $L?
  6. Round your answer to the nearest dollar.   (2 marks)

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a.i.

a.ii.

b. 

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a.i.  

♦ Mean mark 46%.

 

♦♦ Mean mark 26%.

a.ii.   
   
   

  
b.   

♦♦♦ Mean mark 20%.
MARKER’S COMMENT: Correct input tables allowed for a method mark if answers were calculated incorrectly.

   

  

 

  

 

CORE*, FUR2 2016 VCAA 6

Ken’s first caravan had a purchase price of $38 000.

After eight years, the value of the caravan was $16 000.

  1. Show that the average depreciation in the value of the caravan per year was $2750.   (1 mark)

  2. Let be the value of the caravan years after it was purchased.

     

    Assume that the value of the caravan has been depreciated using the flat rate method of depreciation.

     

    Write down a recurrence relation, in terms of and , that models the value of the caravan.   (1 mark)

  3. The caravan has travelled an average of 5000 km in each of the eight years since it was purchased.

     

    Assume that the value of the caravan has been depreciated using the unit cost method of depreciation.

     

    By how much is the value of the caravan reduced per kilometre travelled?   (1 mark)

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a.    


♦♦ Mean mark (a) 39%.
MARKER’S COMMENT: A “show that” question should include an equation
  

b.   
 

♦♦ Mean mark (b) 33%.
MARKER’S COMMENT: A lack of attention to detail and careless errors were common!
  

c.   
   

 


♦♦ Mean mark (c) 29%.
  

CORE, FUR2 2016 VCAA 5

Ken has opened a savings account to save money to buy a new caravan.

The amount of money in the savings account after years, , can be modelled by the recurrence relation shown below.

  1. How much money did Ken initially deposit into the savings account?  (1 mark)

  2. Use recursion to write down calculations that show that the amount of money in Ken’s savings account after two years, , will be $16 224.  (1 mark)

  3. What is the annual percentage compound interest rate for this savings account?  (1 mark)

  4. The amount of money in the account after years, , can also be determined using a rule.
    i.     Complete the rule below by writing the appropriate numbers in the boxes provided.   (1 mark)

     
     
    		 ­
     
  5. ii. How much money will be in Ken’s savings account after 10 years?   (1 mark)

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  4. i.
    ii.  
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a.   
   
b.   
 
   
 
   

MARKER’S COMMENT: (b) Stating is not using recursion as required here and did not gain a mark.
c.   
   

 

d.i.  
d.ii.   
   
   

♦ Mean mark (d)(ii) 47%.
MARKER’S COMMENT: Rounding to $22 203.70 lost a mark!

CORE, FUR2 2016 VCAA 4

The time series plot below shows the minimum rainfall recorded at the weather station each month plotted against the month number (1 = January, 2 = February, and so on).

Rainfall is recorded in millimetres.

The data was collected over a period of one year.
 

  1. Five-median smoothing has been used to smooth the time series plot above.

     

    The first four smoothed points are shown as crosses (×).

     

    Complete the five-median smoothing by marking smoothed values with crosses (×) on the time series plot above.   (2 marks)

The maximum daily rainfall each month was also recorded at the weather station.

The table below shows the maximum daily rainfall each month for a period of one year.

The data in the table has been used to plot maximum daily rainfall against month number in the time series plot below.
 

  1. Two-mean smoothing with centring has been used to smooth the time series plot above.

     

    The smoothed values are marked with crosses (×).

     

    Using the data given in the table, show that the two-mean smoothed rainfall centred on October is 157.25 mm.   (2 marks)

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a.   
♦ Mean mark of both Parts (a) and (b) was 49%.
MARKER’S COMMENT: Use the accurate table data when available. Reading values from the graph will cause inaccuracies.
b.   
   
 
   

 

CORE, FUR2 2016 VCAA 3

The data in the table below shows a sample of actual temperatures and apparent temperatures recorded at a weather station. A scatterplot of the data is also shown.

The data will be used to investigate the association between the variables apparent temperature and actual temperature.
 

  1. Use the scatterplot to describe the association between apparent temperature and actual temperature in terms of strength, direction and form.   (1 mark)

  2.  i. Determine the equation of the least squares line that can be used to predict the apparent temperature from the actual temperature.
  3. Write the values of the intercept and slope of this least squares line in the appropriate boxes provided below.
  4. Round your answers to two significant figures.   (3 marks)
     apparent temperature    
 
  
 
   actual temperature
  1. ii. Interpret the intercept of the least squares line in terms of the variables apparent temperature and actual temperature.   (1 mark)

  2. The coefficient of determination for the association between the variables apparent temperature and actual temperature is 0.97
  3. Interpret the coefficient of determination in terms of these variables.   (1 mark)

  4. The residual plot obtained when the least squares line was fitted to the data is shown below.
     
     
  5.  i. A residual plot can be used to test an assumption about the nature of the association between two numerical variables.
  6.     What is this assumption?   (1 mark)
  7. ii. Does the residual plot above support this assumption? Explain your answer.   (1 mark)
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a.   

b.i. 

b.ii.  `text(When actual temperature is 0°C, on average,)`

c. 

d.i. 

d.ii. 

Show Worked Solution

a.   
 

b.i.   


 

♦♦ Mean mark of part (b)(ii) – 28%.
MARKER’S COMMENT: “the predicted apparent temp is -1.7°C” also gained a mark.
b.ii.   
 

 

♦ Mean mark 49%.
IMPORTANT: Any mention of causality loses a mark!

c. 


  

d.i. 

♦ Mean mark of both parts of (d) was 46%.

d.ii. 

CORE, FUR2 2016 VCAA 2

A weather station records daily maximum temperatures.

  1. The five-number summary for the distribution of maximum temperatures for the month of February is displayed in the table below.

 

  1. There are no outliers in this distribution.
  2.  i. Use the five-number summary above to construct a boxplot on the grid below.   (1 mark)

 

  1. ii. What percentage of days had a maximum temperature of 21°C, or greater, in this particular February?   (1 mark)

  2. The boxplots below display the distribution of maximum daily temperature for the months of May and July.
     

  3.   i. Describe the shapes of the distributions of daily temperature (including outliers) for July and for May.   (1 mark)

  4.  ii. Determine the value of the upper fence for the July boxplot.   (1 mark)

  5. iii. Using the information from the boxplots, explain why the maximum daily temperature is associated with the month of the year. Quote the values of appropriate statistics in your response.   (1 mark)

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a.i.   

a.ii.  

b.i.   
 

b.ii. 

b.iii.

Show Worked Solution
a.i.   

a.ii.  

MARKER’S COMMENT: Incorrect May descriptors included “evenly or normally distributed”, “bell shaped” and “symmetrically skewed.”
b.i.   
 

 

b.ii.   
   
   
   
♦♦ Mean mark (b)(iii) – 30%.
COMMENT: Refer to the difference in medians. Just quoting the numbers was not enough to gain a mark here.

b.iii.

NETWORKS, FUR1 2016 VCAA 6-7 MC

The directed graph below shows the sequence of activities required to complete a project.

All times are in hours.
 

 
Part 1

The number of activities that have exactly two immediate predecessors is

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4

 
Part 2

There is one critical path for this project.

Three critical paths would exist if the duration of activity

  1. I were reduced by two hours.
  2. E were reduced by one hour.
  3. G were increased by six hours.
  4. K were increased by two hours.
  5. F were increased by two hours. 
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♦ Mean mark of Part 2: 45%.

GRAPHS, FUR1 2016 VCAA 7 MC

Simon grows cucumbers and zucchinis.

Let be the number of cucumbers that are grown.

Let be the number of zucchinis that are grown.

For every two cucumbers that are grown, Simon grows at least three zucchinis.

An inequality that represents this situation is

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GEOMETRY, FUR1 2016 VCAA 7 MC

The diagram below shows a rectangular-based right pyramid, ABCDE.

In this pyramid, AB = DC = 24 cm, AD = BC = 10 cm and AE = BE = CE = DE = 28 cm.

The height, OE, of the pyramid, in centimetres, is closest to

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MATRICES, FUR1 2016 VCAA 8 MC

The matrix below shows the result of each match between four teams, A, B, C and D, in a bowling tournament. Each team played each other team once and there were no draws.
 


 

In this tournament, each team was given a ranking that was determined by calculating the sum of its one-step and two-step dominances. The team with the highest sum was ranked number one (1). The team with the second-highest sum was ranked number two (2), and so on.

Using this method, team C was ranked number one (1).

Team A would have been ranked number one (1) if the winner of one match had lost instead.

That match was between teams

  1. A and B.
  2. A and D.
  3. B and C.
  4. B and D.
  5. C and D.
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Show Worked Solution

♦ Mean mark 42%.


  

 

MATRICES, FUR1 2016 VCAA 7 MC

Each week, the 300 students at a primary school choose art (A), music (M) or sport (S) as an afternoon activity.

The transition matrix below shows how the students’ choices change from week to week.
 

 

Based on the information above, it can be concluded that, in the long term

  1. no student will choose sport.
  2. all students will choose to stay in the same activity each week.
  3. all students will have chosen to change their activity at least once.
  4. more students will choose to do music than sport.
  5. the number of students choosing to do art and music will be the same.
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CORE, FUR1 2016 VCAA 22 MC

The first three lines of an amortisation table for a reducing balance home loan are shown below.

The interest rate for this home loan is 4.8% per annum compounding monthly.

The loan is to be repaid with monthly payments of $1500.
 

 
The amount of payment number 2 that goes towards reducing the principal of the loan is

  1.  $486
  2.  $502
  3.  $504
  4.  $996
  5.  $998
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Show Worked Solution
 
 

 

CORE, FUR1 2016 VCAA 14-16 MC

The table below shows the long-term average of the number of meals served each day at a restaurant. Also shown is the daily seasonal index for Monday through to Friday.

 

Part 1

The seasonal index for Wednesday is 0.84

This tells us that, on average, the number of meals served on a Wednesday is

  1. 16% less than the daily average.
  2. 84% less than the daily average.
  3. the same as the daily average.
  4. 16% more than the daily average.
  5. 84% more than the daily average.

 

Part 2

Last Tuesday, 108 meals were served in the restaurant.

The deseasonalised number of meals served last Tuesday was closest to

  1.  

 

Part 3

The seasonal index for Saturday is closest to

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Show Worked Solution

 

 

 

CORE, FUR1 2016 VCAA 11-12 MC

The table below gives the Human Development Index (HDI) and the mean number of children per woman (children) for 14 countries in 2007.

A scatterplot of the data is also shown. 
 

 

Part 1

The scatterplot is non-linear.

A log transformation applied to the variable children can be used to linearise the scatterplot.

With HDI as the explanatory variable, the equation of the least squares line fitted to the linearised data is closest to

  1. log(children) = 1.1 – 0.0095 × HDI
  2. children = 1.1 – 0.0095 × log(HDI)
  3. log(children) = 8.0 – 0.77 × HDI
  4. children = 8.0 – 0.77 × log(HDI)
  5. log(children) = 21 – 10 × HDI

 

Part 2

There is a strong positive association between a country’s Human Development Index and its carbon dioxide emissions.

From this information, it can be concluded that

  1. increasing a country’s carbon dioxide emissions will increase the Human Development Index of the country.
  2. decreasing a country’s carbon dioxide emissions will increase the Human Development Index of the country.
  3. this association must be a chance occurrence and can be safely ignored.
  4. countries that have higher human development indices tend to have higher levels of carbon dioxide emissions.
  5. countries that have higher human development indices tend to have lower levels of carbon dioxide emissions.
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Show Worked Solution

 

CORE, FUR1 2016 VCAA 8 MC

Parallel boxplots would be an appropriate graphical tool to investigate the association between the monthly median rainfall, in millimetres, and the

  1. monthly median wind speed, in kilometres per hour.
  2. monthly median temperature, in degrees Celsius.
  3. month of the year (January, February, March, etc.).
  4. monthly sunshine time, in hours.
  5. annual rainfall, in millimetres.
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Show Worked Solution

♦ Mean mark 45%.

CORE, FUR1 2016 VCAA 6-7 MC

Part 1

The histogram below shows the distribution of the number of billionaires per million people for 53 countries.
 

Using this histogram, the percentage of these 53 countries with less than two billionaires per million people is closest to

 

Part 2

The histogram below shows the distribution of the number of billionaires per million people for the same 53 countries as in Part 1, but this time plotted on a scale.
 

Based on this histogram, the number of countries with one or more billionaires per million people is

  1.  
  2.  
  3.  
  4.  
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Show Worked Solution

 

♦ Mean mark 45%.
MARKER’S COMMENT: On a scale, 1 is plotted as 0 because , or 

CORE, FUR1 2016 VCAA 1-2 MC

The blood pressure (low, normal, high) and the age (under 50 years, 50 years or over) of 110 adults were recorded. The results are displayed in the two-way frequency table below.
 

     

Part 1

The percentage of adults under 50 years of age who have high blood pressure is closest to

  1.  11%
  2.  19%
  3.  26%
  4.  44%
  5.  58%

Part 2

The variables blood pressure (low, normal, high) and age (under 50 years, 50 years or over) are

  1. both nominal variables.
  2. both ordinal variables.
  3. a nominal variable and an ordinal variable respectively.
  4. an ordinal variable and a nominal variable respectively.
  5. a continuous variable and an ordinal variable respectively.
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Show Worked Solution

 
 

 

 

♦♦ Mean mark of Part 2: 31%.
MARKER’S COMMENT: Many students incorrectly identified the age (under 50, 50 or over) as nominal.

Number and Algebra, NAP-C1-24

Cecil is packing away his club's lawn bowls after a practice session.

One bag carries 4 lawn bowls, as shown below.

Cecil needs to pack away 16 bowls.

Which of these shows how Cecil could work out the number of bags he needs?

 
 
 
 
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