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CORE*, FUR2 2015 VCAA 3

Jane and Michael decide to set up an annual music scholarship.

To fund the scholarship, they invest in a perpetuity that pays interest at a rate of 3.68% per annum.

The interest from this perpetuity is used to provide an annual $460 scholarship.

  1. Determine the minimum amount they must invest in the perpetuity to fund the scholarship.   (1 mark)

  2. For how many years will they be able to provide the scholarship?   (1 mark)

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a.   

 

 

b.  

CORE*, FUR2 2015 VCAA 1

Jane and Michael have started a business that provides music at parties.

The business charges customers $88 per hour.

The $88 per hour includes a 10% goods and services tax (GST).

  1. Calculate the amount of GST included in the $88 hourly rate.  (1 mark)

Jane and Michael’s first booking was a party where they provided music for four hours.

  1. Calculate the total amount they were paid for this booking.  (1 mark)

After six months of regular work, Jane and Michael decided to increase the hourly rate they charge by 12.5%.

  1. Calculate the new hourly rate (including GST).  (1 mark)
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a.  

 
 

 

 

b.   
   

 

c.   
   
 
 

GRAPHS, FUR2 2015 VCAA 4

The airline that Ben uses to travel to Japan charges for the seat and luggage separately.

The charge for luggage is based on the weight, in kilograms, of the luggage.

If the luggage is paid for at the airport, the graph below can be used to determine the cost, in dollars, of luggage of a certain weight, in kilograms. 

Graphs, FUR2 2015 VCAA 4

  1. Find the cost at the airport for 23 kg of luggage.  (1 mark)

If the luggage is paid for online prior to arriving at the airport, the equation for the relationship between the online cost, in dollars, and the weight, in kilograms, would be

 

  1. Find the online cost for 30 kg of luggage.  (1 mark)
  2. On the graph above, sketch a graph of the online cost of luggage for weight . Include the end points.  (2 marks)

(Answer on the graph above.)

  1. Determine the weight of luggage for which the airport cost and online cost are the same.
    Write your answer correct to one decimal place.  (1 mark)
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  3.  

    Graphs-FUR2-2015-VCAA-4i

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a.   

 

b.   
   

 

c.  
   

Graphs-FUR2-2015-VCAA-4i

 

d.   

 
 

GRAPHS, FUR2 2015 VCAA 3

The graph below shows the relationship between the yen and the dollar on the same day at a different currency exchange agency.

Graphs, FUR2 2015 VCAA 3

The points and are labelled.

The equation for the relationship between the yen and the dollar is

yen = 95 × dollars – k

  1. Use the point to show that the value of is 1425.  (1 mark)
    1. Determine the intercept on the horizontal axis.  (1 mark)
    2. Interpret the intercept on the horizontal axis in the context of converting dollars to yen.  (1 mark)
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    2.  

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a.   
 
 
   

 

b.i.   
 
 
   

 

 

b.ii.  
 

GRAPHS, FUR2 2015 VCAA 2

Ben will use a currency exchange agency to buy some Japanese yen (the Japanese currency unit).

The graph below shows the relationship between Japanese yen and Australian dollars on a particular day.

This graph can be used to calculate a conversion between dollars and yen on that day.

Graphs, FUR2 2015 VCAA 2

  1. Ben converts his dollars into yen using this graph.

     

    How many yen does he receive for $200?  (1 mark)

  2. The slope of this graph is the exchange rate for converting dollars into yen on that particular day.

     

    How many yen will Ben receive for each dollar?  (1 mark)

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a.   

 

b.  

 
 

 

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.

Graphs, FUR2 2015 VCAA 1

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

b.  

Calculus, MET2 2014 VCAA 3

In a controlled experiment, Juan took some medicine at 8 pm. The concentration of medicine in his blood was then measured at regular intervals. The concentration of medicine in Juan’s blood is modelled by the function  , where is the concentration of medicine in his blood, in milligrams per litre, hours after 8 pm. Part of the graph of the function is shown below.
 

VCAA 2014 3a

  1. What was the maximum value of the concentration of medicine in Juan’s blood, in milligrams per litre, correct to two decimal places?   (1 mark)

    1. Find the value of , in hours, correct to two decimal places, when the concentration of medicine in Juan’s blood first reached 0.5 milligrams per litre.   (1 mark)

    2. Find the length of time that the concentration of medicine in Juan’s blood was above 0.5 milligrams per litre. Express the answer in hours, correct to two decimal places.   (2 marks)

    1. What was the value of the average rate of change of the concentration of medicine in Juan’s blood over the interval  ?
    2. Express the answer in milligrams per litre per hour, correct to two decimal places.   (2 marks)
    3. At times and , the instantaneous rate of change of the concentration of medicine in Juan's blood was equal to the average rate of change over the interval  .
    4. Find the values of and , in hours, correct to two decimal places.   (2 marks)

Alicia took part in a similar controlled experiment. However, she used a different medicine. The concentration of this different medicine was modelled by the function    where    and  .

  1. If the maximum concentration of medicine in Alicia’s blood was 0.74 milligrams per litre at    hours, find the value of , correct to the nearest integer.  (3 marks)

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  2.  i.
  3. ii.
  4.  i.
  5. ii.
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a. 


 

b.i.   

 

MARKER’S COMMENT: In part (b)(ii), work to sufficient decimal places to ensure your answer has the required accuracy.

 

b.ii.   
   

 

c.i.   
   
   
   

 

c.ii.  

♦ Mean mark part (c)(ii) 38%.


 

d. 

♦ Mean mark 46%.

 

 

 

 

Calculus, MET2 2014 VCAA 2

On 1 January 2010, Tasmania Jones was walking through an ice-covered region of Greenland when he found a large ice cylinder that was made a thousand years ago by the Vikings.

A statue was inside the ice cylinder. The statue was 1 m tall and its base was at the centre of the base of the cylinder.

VCAA 2014 2a

The cylinder had a height of metres and a diameter of metres. Tasmania Jones found that the volume of the cylinder was 216 m³. At that time, 1 January 2010, the cylinder had not changed in a thousand years. It was exactly as it was when the Vikings made it.

  1. Write an expression for in terms of .   (2 marks)

  2. Show that the surface area of the cylinder excluding the base, square metres, is given by the rule  .   (1 mark)

Tasmania found that the Vikings made the cylinder so that is a minimum.

  1. Find the value of for which is a minimum and find this minimum value of .   (2 marks)

  2. Find the value of when is a minimum.   (1 mark)

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a.   
 
 

 

b.   
   

 

 

 

c.  

²

 

d.  

♦ Mean mark (d) 42%.
MARKER’S COMMENT: An exact answer required here!

   

Algebra, MET2 2014 VCAA 1

The population of wombats in a particular location varies according to the rule  , where is the number of wombats and is the number of months after 1 March 2013.

  1. Find the period and amplitude of the function .   (2 marks)

  2. Find the maximum and minimum populations of wombats in this location.   (2 marks)

  3. Find  .   (1 mark)

  4. Over the 12 months from 1 March 2013, find the fraction of time when the population of wombats in this location was less than  .   (2 marks)

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a.   

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


  

b.  


  

c.  
   

d.   

♦ Mean mark 48%.

 

Probability, MET2 2015 VCAA 3

Mani is a fruit grower. After his oranges have been picked, they are sorted by a machine, according to size. Oranges classified as medium are sold to fruit shops and the remainder are made into orange juice.

The distribution of the diameter, in centimetres, of medium oranges is modelled by a continuous random variable, , with probability density function
 


 

    1. Find the probability that a randomly selected medium orange has a diameter greater than 7 cm.   (2 marks)

    2. Mani randomly selects three medium oranges.
    3. Find the probability that exactly one of the oranges has a diameter greater than 7 cm.
    4. Express the answer in the form , where and are positive integers.   (2 marks)

  2. Find the mean diameter of medium oranges, in centimetres.   (1 mark)

For oranges classified as large, the quantity of juice obtained from each orange is a normally distributed random variable with a mean of 74 mL and a standard deviation of 9 mL.

  1. What is the probability, correct to three decimal places, that a randomly selected large orange produces less than 85 mL of juice, given that it produces more than 74 mL of juice?  (2 marks)

Mani also grows lemons, which are sold to a food factory. When a truckload of lemons arrives at the food factory, the manager randomly selects and weighs four lemons from the load. If one or more of these lemons is underweight, the load is rejected. Otherwise it is accepted.

It is known that 3% of Mani’s lemons are underweight.

    1. Find the probability that a particular load of lemons will be rejected. Express the answer correct to four decimal places.   (2 marks)

    2. Suppose that instead of selecting only four lemons, lemons are selected at random from a particular load.
    3. Find the smallest integer value of such that the probability of at least one lemon being underweight exceeds 0.5  (2 marks)

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a.i.   
   
   

 

a.ii.  

 

 

 

b.  
   
   

 

c.  

 
 
   
 

 

d.i. 

 

 

 
 
 

 

d.ii.  

♦ Mean mark 44%.

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.

GRAPHS, FUR2 2006 VCAA 1

  1. 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.

  1. Draw this information on the graph above.  (1 mark) 
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  2.  
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a.i.   

 

a.ii.   

 

b.   

Calculus, MET2 2015 VCAA 1

Let  . The point    is on the graph of  , as shown below.

The tangent at cuts the y-axis at and the x-axis at

VCAA 2015 1ai

  1. Write down the derivative   of .   (1 mark)

  2.  i. Find the equation of the tangent to the graph of   at the point  .   (1 mark)

    ii. Find the coordinates of points and .   (2 marks)

  3. Find the distance and express it in the form  , where and are positive integers.  (2 marks)

VCAA 2015 1di

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

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  2.  i.
    ii.
  4. ²
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a.   
 
   

 

b.i.   


  

  
b.ii.   


  

 

  

c.  

 
 

 

d.  

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

  ²

GEOMETRY, FUR2 2006 VCAA 1

A farmer owns a flat allotment of land in the shape of triangle shown below.

Boundary is 251 metres.

Boundary is 142 metres.

Angle is 45°.

GEOMETRY, FUR2 2006 VCAA 11 
 

A straight track, , runs perpendicular to the boundary . 

Point is 55 m from along the boundary .

  1. Determine the size of angle .  (1 mark)
  2. Determine the length of .

     

    Write your answer, in metres, correct to one decimal place.  (1 mark)

  3. The bearing of from is 078°.

     

    Determine the bearing of from .  (1 mark)

  4. Determine the shortest distance from to .

     

    Write your answer, in metres, correct to one decimal place.  (2 marks)

  5. Determine the area of triangle correct to the nearest square metre.  (1 mark)

The length of the boundary is 181 metres (correct to the nearest metre).

    1. Use the cosine rule to show how this length can be found.  (1 mark)

    2. Determine the size of angle . 
      Write your answer, in degrees, correct to one decimal place.  (1 mark)

A farmer plans to build a fence, , perpendicular to the boundary . 

The land enclosed by triangle will have an area of 3200 m².

GEOMETRY, FUR2 2006 VCAA 12

  1. Determine the length of the fence .  (2 marks)
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a.   
   

 

b.  

 
 

 

c.   VCAA GEO FUR2 2006 1ci

 

d.   VCAA GEO FUR2 2006 1di

 
 
 

 

e.  

♦ Mean mark of parts (e)-(g) (combined) was 40%.
 
 
  ²²

 

f.i.  

 
 

 

f.ii.  

 
 

 

g.   

MARKER’S COMMENT: Part (g) was “very poorly answered” with many failing to realise the equality of and .

 

 

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

GEOMETRY, FUR2 2007 VCAA 11 

On the front face of Figure 1, , Tessa marks point halfway between and as shown in Figure 2 below. She then draws line segments and as shown.

GEOMETRY, FUR2 2007 VCAA 12

  1. Determine the length, in cm, of .  (1 mark)
  2. Calculate the angle . Write your answer in degrees, correct to one decimal place.  (1 mark)
  3. Calculate the angle correct to one decimal place.  (1 mark)
  4. What fraction of the area of the rectangle does the area of the triangle represent?  (1 mark)

 

Question 2

Tessa carves a triangular prism from her block of wood.

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

GEOMETRY, FUR2 2007 VCAA 2

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

 

Question 3

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

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

GEOMETRY, FUR2 2007 VCAA 3

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

Question 2

  2. ²

Question 3

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a.   
   
   

 

b.   VCAA GEO FUR2 2007 1bi
 
 

 

c.   
   
   

 

d.  
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    ²

 

 

a.   VCAA GEO FUR2 2007 2ai

 
 
 

 

b.  

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a.   
   
    ³

 

b.   VCAA GEO FUR2 2007 3bi

 

 

VCAA GEO FUR2 2007 3bii

 
 
 

 

c.  

 

²

MATRICES, FUR2 2008 VCAA 1

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

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

The matrix lists the proportion of these students expected to be awarded an , , , or grade in each subject.

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

  2. Let the matrix  .
  3.  i. Evaluate the matrix .   (1 mark)

  4. ii. Explain what the matrix element  represents.   (1 mark)

  5. Students enrolled in Biology have to pay a laboratory fee of $110, while students enrolled in Chemistry pay a laboratory fee of $95.
  6.  i. Write down a clearly labelled row matrix, called , that lists these fees.   (1 mark)

  7. ii. Show a matrix calculation that will give the total laboratory fees, , paid in dollars by the students enrolled in Biology and Chemistry. Find this amount.   (1 mark)

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    1.  
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a.   
 

b.i.  
   
   

 
b.ii.

 
 

c.i.  

 
c.ii. 

 

GEOMETRY, FUR2 2008 VCAA 3

A tree, 12 m tall, is growing at point near a shed. 

The distance, , from corner of the shed to the centre base of the tree is 13 m.
 

GEOMETRY, FUR2 2008 VCAA 31
 

  1. Calculate the angle of elevation of the top of the tree from point .

     

    Write your answer, in degrees, correct to one decimal place.  (1 mark)
     

GEOMETRY, FUR2 2008 VCAA 32

 and are two corners at the base of the shed. is due north of .

The angle, , is 65°.

  1. Show that, correct to one decimal place, the distance, , is 12.6 m.  (1 mark)
  2. Calculate the angle, , correct to the nearest degree.  (1 mark)
  3. Determine the bearing of from . Write your answer correct to the nearest degree.  (1 mark)
  4. Is it possible for the tree to hit the shed if it falls?

     

    Explain your answer showing appropriate calculations.  (2 marks)

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Show Worked Solution
a.   VCAA GEO FUR2 2008 3ai
 
 

 

b.   VCAA GEO FUR2 2008 3bi

MARKER’S COMMENT: Show the squareroot step in the calculations as per the solution.
 
 
 

 

c.  

♦♦ Mean mark of parts (c)-(e) (combined) was 23%.
 
 

 

d.   VCAA GEO FUR2 2008 3di

 

e.   VCAA GEO FUR2 2008 3ei

MARKER’S COMMENT: Many students had significant difficulty in understanding the 3-dimensional diagram.

 

 

GEOMETRY, FUR2 2014 VCAA 2

The chicken coop has two spaces, one for nesting and one for eating.

The nesting and eating spaces are separated by a wall along the line , as shown in the diagrams below.
 

GEOMETRY, FUR2 2014 VCAA 21
 

  1. Write down a calculation to show that the value of   is 75°.  (1 mark)
  2. The sine rule can be used to calculate the length of the wall  .

     

    Fill in the missing numbers below.  (1 mark)
     

     

          GEOMETRY, FUR2 2014 VCAA 22
     

  3. What is the length of  ?

     

    Write your answer in metres, correct to two decimal places.  (1 mark)

  4. Calculate the area of the floor of the nesting space, .

     

    Write your answer in square metres, correct to one decimal place.  (1 mark)

The height of the chicken coop is 1.8 m.

 

Wire mesh will cover the roof of the eating space.

 

The area of the walls along the lines    will also be covered with wire mesh.

  1. What total area, in square metres, will be covered by wire mesh?

     

    Write your answer, correct to the nearest square metre.  (2 mark)

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a.   
   

 

b.   

 

c.   
     
 
   
   

 

d.   
   
    ²

 

e.   VCAA 2014 fur2 Q2ei
MARKER’S COMMENT: An excellent example where many students do not read the question carefully and give away easy marks.
 
 

 

 

 

 
  ²²

GEOMETRY, FUR2 2014 VCAA 1

The floor of a chicken coop is in the shape of a trapezium.

The floor, , and the chicken coop are shown below.

 GEOMETRY, FUR2 2014 VCAA 1

  1. What is the area of the floor of the chicken coop?

     

    Write your answer in square metres.  (1 mark)

  2. What is the perimeter of the floor of the chicken coop?

     

    Write your answer in metres, correct to one decimal place.  (1 mark)

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a.   
   
    ²

 

b.  

 

 

 

GEOMETRY, FUR2 2008 VCAA 2

A shed has the shape of a prism. Its front face, , is shaded in the diagram below. is a rectangle and is the mid point of .
 

GEOMETRY, FUR2 2008 VCAA 2

  1. Show that the length of is 1.6 m.  (1 mark)
  2. Show that the area of the front face of the shed, , is 18 m².  (1 mark)
  3. Find the volume of the shed in m³.  (1 mark)
  4. All inside surfaces of the shed, including the floor, will be painted.

     

    1. Find the total area that will be painted in m².  (2 marks)

       

      One litre of paint will cover an area of 16 m².

    2. Determine the number of litres of paint that is required.  (1 mark)
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a.  

 
 

 

b.  

MARKER’S COMMENT: “The “Show that” directive requires students to include all mathematical steps, as shown in the solution. Short cuts will not gain full marks!

²

 

c.   
   
    ³

 

d.i.   
   
    ²
 
   
    ²
  ²

 

²

 

  ii.   

GEOMETRY, FUR2 2008 VCAA 1

A shed is built on a concrete slab. The concrete slab is a rectangular prism 6 m wide, 10 m long and 0.2 m deep.

 

Geometry and Trig, FUR2 2008 VCAA 1

  1. Determine the volume of the concrete slab in m³.  (1 mark)
  2. On a plan of the concrete slab, a 3 cm line is used to represent a length of 6 m.

     

    1. What scale factor is used to draw this plan?  (1 mark)

The top surface of the concrete slab shaded in the diagram above has an area of 60 m².

  1. What is the area of this surface on the plan?  (1 mark)

 

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  1. ³
    2. ²
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a.   
   
    ³

 

b.i.   

♦ A poorly answered question, although exact data unavailable.
MARKER’S COMMENT: The most successful answers established the scale factor and then squared it, as shown in the solution.
  ii.   
   
    ²
    ²

 

GEOMETRY, FUR2 2009 VCAA 2

A yacht, , is 7 km from a lighthouse, , on a bearing of 210° as shown in the diagram below.

 

GEOMETRY, FUR2 2009 VCAA 2

  1. A ferry can also be seen from the lighthouse. The ferry is 3 km from on a bearing of 135°. On the diagram above, label the position of the ferry, , and show an angle to indicate its bearing.  (1 mark)
  2. Determine the angle between and .  (1 mark)
  3. Calculate the distance, in km, between the ferry and the yacht correct to two decimal places.  (1 mark)
  4. Determine the bearing of the lighthouse from the ferry.  (1 mark)
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  1.  
Show Worked Solution
a.   

 

b.  

 

c.  

 
 

 

d.  

GEOMETRY, FUR2 2009 VCAA 1

A ferry, , is 400 metres from point at the base of a 50 metre high cliff, .
 

GEOMETRY, FUR2 2009 VCAA 1 
 

  1. Show that the gradient of the line in the diagram is 0.125.  (1 mark)
  2. Calculate the angle of elevation of point from .

     

    Write your answer in degrees, correct to one decimal place.  (1 mark)

  3. Calculate the distance , in metres, correct to one decimal place.  (1 mark)
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a.   
   
   

 

b.   
 
   
   

 

c.   
 
   
   

Calculus, MET2 2013 VCAA 3

Tasmania Jones is in Switzerland. He is working as a construction engineer and he is developing a thrilling train ride in the mountains. He chooses a region of a mountain landscape, the cross-section of which is shown in the diagram below.

VCAA 2013 2a

The cross-section of the mountain and the valley shown in the diagram (including a lake bed) is modelled by the function with rule

Tasmania knows that    is the highest point on the mountain and that and are the points at the edge of the lake, situated in the valley. All distances are measured in kilometres.

  1. Find the coordinates of , the deepest point in the lake.   (3 marks)

Tasmania’s train ride is made by constructing a straight railway line from the top of the mountain, , to the edge of the lake, . The section of the railway line from to passes through a tunnel in the mountain.

  1. Write down the equation of the line that passes through and    (2 marks)

  2.  i. Show that the -coordinate of , the end point of the tunnel, is    (1 mark)

  3. ii. Find the length of the tunnel    (2 marks)

In order to ensure that the section of the railway line from to remains stable, Tasmania constructs vertical columns from the lake bed to the railway line. The column is the longest of all possible columns. (Refer to the diagram above.)

  1.  i. Find the -coordinate of    (2 marks)

  2. ii. Find the length of the column in metres, correct to the nearest metre.   (2 marks)

Tasmania’s train travels down the railway line from to . The speed, in km/h, of the train as it moves down the railway line is described by the function.

where is the -coordinate of a point on the front of the train as it moves down the railway line, and and are positive real constants.

The train begins its journey at . It increases its speed as it travels down the railway line.

The train then slows to a stop at , that is  

  1. Find in terms of    (1 mark)

  2. Find the value of for which the speed, , is a maximum.   (2 marks)

Tasmania is able to change the value of on any particular day. As changes, the relationship between and remains the same.

  1. If, on one particular day, , find the maximum speed of the train, correct to one decimal place.   (2 marks)

  2. If, on another day, the maximum value of is 120, find the value of    (2 marks)

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  3.  i.
  4. ii.
  5.  i.
  6. ii.
Show Worked Solution

a.  


 

b.   
   

 

 
c.i. 


 

 


 

c.ii.  

 

 

 

d.i. 

♦♦♦ Mean mark part (d)(i) 24%.
MARKER’S COMMENT: Many students unfamiliar with this type of question.
 
 

 


 

♦♦♦ Mean mark part (d)(ii) 22%.
d.ii.   
   

 

e.   
 
 

 

f.  


 

g. 

♦ Mean mark 43%.

 

 

h.  

♦ Mean mark 38%.

Graphs, MET2 2013 VCAA 1

Trigg the gardener is working in a temperature-controlled greenhouse. During a particular 24-hour time interval, the temperature   is given by  , where is the time in hours from the beginning of the 24-hour time interval.

  1. State the maximum temperature in the greenhouse and the values of when this occurs.   (2 marks)

  2. State the period of the function    (1 mark)

  3. Find the smallest value of for which     (2 marks)

  4. For how many hours during the 24-hour time interval is     (2 marks)

Trigg is designing a garden that is to be built on flat ground. In his initial plans, he draws the graph of    for    and decides that the garden beds will have the shape of the shaded regions shown in the diagram below. He includes a garden path, which is shown as line segment

  1. The line through points    and    is a tangent to the graph of    at point

    1. Find    when     (1 mark)

    2. Show that the value of is     (1 mark)

In further planning for the garden, Trigg uses a transformation of the plane defined as a dilation of factor from the -axis and a dilation of factor from the -axis, where and are positive real numbers.

  1. Let and be the image, under this transformation, of the points and respectively. 

     

    1. Find the values of and  if    and     (2 marks)

    2. Find the coordinates of the point    (1 mark)

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  5.  i. 
    ii. 
  6.  i. 
    ii. 
Show Worked Solution

a.  

 

b.   
   

 

c.  

 

 

d.  

met2-2013-vcaa-sec1-answer1

 

 

e.i.  

 

e.ii. 

 

 

 

f.i.  

 
♦♦♦ Mean mark part (f)(i) 14%.

 

 
♦♦♦ Mean mark part (f)(ii) 12%.

 

f.ii.   
   

Calculus, MET2 2012 VCAA 5

The shaded region in the diagram below is the plan of a mine site for the Black Possum mining company.

All distances are in kilometres.

Two of the boundaries of the mine site are in the shape of the graphs of the functions

VCAA 2012 5a

    1. Evaluate .   (1 mark)

    2. Hence, or otherwise, find the area of the region bounded by the graph of , the and axes, and the line .   (1 mark)

    3. Find the total area of the shaded region.   (1 mark)

  2. The mining engineer, Victoria, decides that a better site for the mine is the region bounded by the graph of and that of a new function  , where is a positive real number.
    1. Find, in terms of , the -coordinates of the points of intersection of the graphs of and .   (2 marks)

    2. Hence, find the set of values of , for which the graphs of and have two distinct points of intersection.   (1 mark)

  3. For the new mine site, the graphs of and intersect at two distinct points, and . It is proposed to start mining operations along the line segment , which joins the two points of intersection.
  4. Victoria decides that the graph of will be such that the -coordinate of the midpoint of is .
  5. Find the value of in this case.   (2 marks)

Show Answers Only
    2. ²
    3. ²
Show Worked Solution
ai.   
   
   

 

a.ii.  

♦ Mean mark part (a)(ii) 45%.

²

 

♦ Mean mark part (a)(iii) 36%.
a.iii.   
   
    ²

 

b.i.   
 
 
 
 
 

 

 

♦♦♦ Mean mark (b.ii.) 11%.

b.ii.  

 

c.   

♦♦♦ Mean mark (c) 15%.
 

Calculus, MET2 2012 VCAA 4

Tasmania Jones is in the jungle, searching for the Quetzalotl tribe’s valuable emerald that has been stolen and hidden by a neighbouring tribe. Tasmania has heard that the emerald has been hidden in a tank shaped like an inverted cone, with a height of 10 metres and a diameter of 4 metres (as shown below).

The emerald is on a shelf. The tank has a poisonous liquid in it.

VCAA 2012 4a

  1. If the depth of the liquid in the tank is metres.
      
     i.     find the radius, metres, of the surface of the liquid in terms of .   (1 mark)

    ii. show that the volume of the liquid in the tank is  ³.   (1 mark)

The tank has a tap at its base that allows the liquid to run out of it. The tank is initially full. When the tap is turned on, the liquid flows out of the tank at such a rate that the depth, metres, of the liquid in the tank is given by

,

where minutes is the length of time after the tap is turned on until the tank is empty.

  1. Show that the tank is empty when  .   (1 mark)

  2. When   minutes, find
      
    i.     the depth of the liquid in the tank.   (1 mark)

    ii. No longer in syllabus

  1. The shelf on which the emerald is placed is 2 metres above the vertex of the cone.

    From the moment the liquid starts to flow from the tank, find how long, in minutes, it takes until  .

    (Give your answer correct to one decimal place.)   (2 marks)

  2. As soon as the tank is empty, the tap turns itself off and poisonous liquid starts to flow into the tank at a rate of 0.2 m³/minute.
      
    How long, in minutes, after the tank is first empty will the liquid once again reach a depth of 2 metres?   (2 marks)

  3. In order to obtain the emerald, Tasmania Jones enters the tank using a vine to climb down the wall of the tank as soon as the depth of the liquid is first 2 metres. He must leave the tank before the depth is again greater than 2 metres.
      
    Find the length of time, in minutes, correct to one decimal place, that Tasmania Jones has from the time he enters the tank to the time he leaves the tank.   (1 mark)

Show Answers Only
  1.  i.
    ii.
  3.  i.
    ii.
Show Worked Solution

a.i.  

 

a.ii.   
   
 

 

b.   
   
   
   

 

c.i.  

 

 

c.ii.   

 

d.   

 

 

e.  

♦♦♦ Mean mark part (e) 17%.
³
   

 

f.   

♦♦♦ Mean mark part (f) 11%.

Calculus, MET2 2012 VCAA 2

Let  

  1. Sketch the graph of   on the set of axes below. Label the axes intercepts with their coordinates and label each of the asymptotes with its equation.   (3 marks)

           VCAA 2012 2a
     

  2.   i. Find .   (1 mark)

  3.  ii. State the range of  .   (1 mark)

  4. iii. Using the result of part ii. explain why has no stationary points.   (1 mark)

  5. If    is any point on the graph of  , show that the equation of the tangent to    at this point can be written as     (2 marks)

  6. Find the coordinates of the points on the graph of    such that the tangents to the graph at these points intersect at     (4 marks)

  7. A transformation    that maps the graph of   to the graph of the function    has rule
  8.      , where , and are non-zero real numbers.
  9. Find the values of and .   (2 marks)

Show Answers Only
  1. met2-2012-vcaa-sec2-answer
  2.   i.
     ii.
    iii.
Show Worked Solution

a.  

met2-2012-vcaa-sec2-answer

 

b.i.  

 

b.ii.  

MARKER’S COMMENT: Incorrect notation in part (b)(ii) was common, including .

 

b.iii.   
   

 

 

c.  

♦♦ Mean mark 29%.
 

 

 

d.  

♦♦♦ Mean mark 19%.


 

e.  


 

 


 

GRAPHS, FUR2 2011 VCAA 1

Michael is preparing to hike through a national park.

He decides to make some trail mix to eat on the hike.

The trail mix consists of almonds and raisins.

The table below shows some information about the amout of carbohydrate and protein contained in each gram of almonds and raisins.

GRAPHS, FUR2 2011 VCAA 11 

  1. If Michael mixed 180 g of almonds and 250 g of raisins to make some trail mix, calculate the weight, in grams, of carbohydrate in the trail mix.  (1 mark)

Michael wants to make some trail mix that contains 72 g of protein. He already has 320 g of almonds.

  1. How many grams of raisins does he need to add?  (2 marks)

The trail mix Michael takes on his hike must satisfy his dietary requirements.

Let be the weight, in grams, of almonds Michael puts into the trail mix.

Let be the weight, in grams, of raisins Michael puts into the trail mix.

Inequalities 1 to 4 represents Michael's dietary requirements for the weight of carbohydrate and protein in the trail mix.

Inequality 1
Inequality 2
Inequality 3 (carbohydrate)  
Inequality 4 (protein)    

 

Michael also requires a minimum of 16 g of fibre in the trail mix.

Each gram of almonds contains 0.1 g of fibre.

Each gram of raisins contains 0.04 g of fibre.

  1. Write down an inequality, in terms of and , that represents this dietary requirement.

     

     

    Inequality 5 (fibre) _________________________  (1 mark)

The graphs of    and    are shown below.

GRAPHS, FUR2 2011 VCAA 12

  1. On the graph above

     

    1. draw the straight line that relates to Inequality 5  (1 mark)
    2. shade the region that satisfies Inequalities 1 to 5.  (1 mark)
  2. What is the maximum weight, in grams, of trail mix that satisfies Michael's dietary requirements?  (1 mark)

Michael plans to carry at least 500 g of trail mix on his hike.

He would also like this trail mix to cantain the greatest possible weight of almonds.

The trail mix must satisfy all of Michael's dietary requirements.

  1. What is the weight of the almonds, in grams, in this trail mix?  (2 marks) 
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  4. i. & ii.
Show Worked Solution
a.   
   

 

b.   

 

 

c.  

 

d.i. & ii.   

 

e.  

 

f.  

VCAA GRAPHS FUR2 2011 1dii

♦♦ Part f was “Poorly answered” although exact data unavailable.
MARKER’S COMMENT: Ensure you incorporate the new constraint!

 

 

MATRICES, FUR2 2012 VCAA 2

Rosa uses the following six-digit pin number for her bank account: 216342. 

With her knowledge of matrices, she decides to use matrix multiplication to disguise this pin number.

First she writes the six digits in the 2 × 3 matrix .

Next she creates a new matrix by forming the matrix product, ,

where

    1. Determine the matrix .   (1 mark)

    2. From the matrix , Rosa is able to write down a six-digit number that disguises her original pin number. She uses the same pattern that she used to create matrix from the digits 216342.

       

      Write down the new six-digit number that Rosa uses to disguise her pin number.   (1 mark)

  2. Show how the original matrix can be regenerated from matrix .   (1 mark) 
Show Answers Only
Show Worked Solution
a.i.  
   
   

 

a.ii.

 

b.  
 
 

MATRICES, FUR2 2012 VCAA 1

Matrix below shows the flight connections for an airline that serves four cities, Anvil (), Berga (), Cantor () and Dantel ().

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

  1. Complete the following sentence.
  2. On this airline, you can fly directly from Berga to ________ and ________.   (1 mark)

  3. List the route that you must follow to fly from Anvil to Cantor.   (1 mark)

  4. Evaluate the matrix product , where .   (1 mark)

     

  5. In the context of the problem, what information does matrix contain?   (1 mark)

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Show Worked Solution

a.  
 

b.  
 

c.  
   
   

 
d.   

MATRICES, FUR2 2014 VCAA 2

There are three candidates in the election: Ms Aboud (), Mr Broad () and Mr Choi ().

The election campaign will run for six months, from the start of January until the election at the end of June.

A survey of voters found that voting preference can change from month to month leading up to the election.

The transition diagram below shows the percentage of voters who are expected to change their preferred candidate from month to month.
 

MATRICES, FUR2 2014 VCAA 2

    1. Of the voters who prefer Mr Choi this month, what percentage are expected to prefer Ms Aboud next month?   (1 mark)

    2. Of the voters who prefer Ms Aboud this month, what percentage are expected to change their preferred candidate next month?   (1 mark)

In January, 12 000 voters are expected in the city. The number of voters in the city is expected to remain constant until the election is held in June.

The state matrix that indicates the number of voters who are expected to have a preference for each candidate in January, , is given below.

  1. How many voters are expected to change their preference to Mr Broad in February?   (1 mark)

The information in the transition diagram has been used to write the transition matrix, , shown below.

    1. Evaluate the matrix and write it down in the space below.
    2. Write the elements, correct to the nearest whole number.   (1 mark)

    4. What information does matrix contain?   (1 mark)

  2. Using matrix , how many votes would the winner of the election in June be expected to receive?
  3. Write your answer, correct to the nearest whole number.   (1 mark)

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    1.  


Show Worked Solution

a.i.   
 

a.ii.   
 

b.  

 

c.i.   
   

 

c.ii.   

 

 

MARKER’S COMMENT: A common error used . Know why this is incorrect.
d.   
   

 

MATRICES, FUR2 2015 VCAA 2

The ability level of students is assessed regularly and classified as beginner (), intermediate () or advanced ().

After each assessment, students either stay at their current level or progress to a higher level.

Students cannot be assessed at a level that is lower than their current level.

The expected number of students at each level after each assessment can be determined using the transition matrix, , shown below.

  1. The element in the third row and third column of matrix is the number 1.
  2. Explain what this tells you about the advanced-level students.   (1 mark)

Let matrix be a state matrix that lists the number of students at beginner, intermediate and advanced levels after assessments.

The number of students in the school, immediately before the first assessment of the year, is shown in matrix below.

    1. Write down the matrix that contains the expected number of students at each level after one assessment.
    2. Write the elements of this matrix correct to the nearest whole number.   (1 mark)

    3. How many intermediate-level students have become advanced-level students after one assessment?   (1 mark)

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Show Worked Solution

a.   

 

b.i.   
   
   

 

b.ii.  

CORE, FUR2 2006 VCAA 1

Table 1 shows the heights (in cm) of three groups of randomly chosen boys aged 18 months, 27 months and 36 months respectively.

Core, FUR2 2006 VCAA 11

  1. Complete Table 2 by calculating the standard deviation of the heights of the 18-month-old boys.

     

    Write your answer correct to one decimal place.   (1 mark)

     

    Core, FUR2 2006 VCAA 12

A 27-month-old boy has a height of 83.1 cm.

  1. Calculate his standardised height ( score) relative to this sample of 27-month-old boys.
  2. Write your answer correct to one decimal place.   (1 mark)

The heights of the 36-month-old boys are normally distributed.

A 36-month-old boy has a standardised height of 2.

  1. Approximately what percentage of 36-month-old boys will be shorter than this child?   (1 mark)

Using the data from Table 1, boxplots have been constructed to display the distributions of heights of 36-month-old and 27-month-old boys as shown below. 

     Core, FUR2 2006 VCAA 13

  1. Complete the display by constructing and drawing a boxplot that shows the distribution of heights for the 18-month-old boys.   (2 marks)

  2. Use the appropriate boxplot to determine the median height (in centimetres) of the 27-month-old boys.   (1 mark)

The three parallel boxplots suggest that height and age (18 months, 27 months, 36 months) are positively related.

  1. Explain why, giving reference to an appropriate statistic.   (1 mark)

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  3.  
  4.  
    Core, FUR2 2006 VCAA 13 Asnwer
Show Worked Solution

a.   

 

b.   
   
   
   

 

♦♦ MARKER’S COMMENT: Attention required here as this standard question was “very poorly answered”.

c.  

CORE, FUR2 2006 VCAA Answer 111

 

d.   

Core, FUR2 2006 VCAA 13 Asnwer

e.   

MARKER’S COMMENT: A boxplot statistic was required, so mean values were not relevant.

 

f.  

CORE, FUR2 2007 VCAA 3

The table below displays the mean surface temperature (in °C) and the mean duration of warm spell (in days) in Australia for 13 years selected at random from the period 1960 to 2005.
 

CORE, FUR2 2007 VCAA 31
 

This data set has been used to construct the scatterplot below. The scatterplot is incomplete.

  1. Complete the scatterplot below by plotting the bold data values given in the table above. Mark the point with a cross (×).  (1 mark)

          

     

  2. Mean surface temperature is the explanatory variable.

     

    1. Determine the equation of the least squares regression line for this set of data. Write the equation in terms of the variables mean duration of warm spell and mean surface temperature. Write the value of the coefficients correct to one decimal place.  (2 marks)

    2. Plot the least squares regression line on Scatterplot 1.  (1 mark)

The residual plot below was constructed to test the assumption of linearity for the relationship between the variables mean duration of warm spell and the mean surface temperature.

CORE, FUR2 2007 VCAA 33

  1. Explain why this residual plot supports the assumption of linearity for this relationship.  (1 mark)

  2. Write down the percentage of variation in the mean duration of a warm spell that is explained by the variation in mean surface temperature. Write your answer correct to the nearest per cent.  (1 mark)

  3. Describe the relationship between the mean duration of a warm spell and the mean surface temperature in terms of strength, direction and form.  (2 marks)

Show Answers Only
  1.  
    CORE, FUR2 2007 VCAA 3 Answer



Show Worked Solution
a.    CORE, FUR2 2007 VCAA 3 Answer

 

b.i.   

 

MARKER’S COMMENT: A consistent error in these type of questions is taking 2 points that are too close together!

b.ii.   

 

 

 

c.   


 

d.  


 

e.  

CORE, FUR2 2007 VCAA 2

The mean surface temperature (in °C) of Australia for the period 1960 to 2005 is displayed in the time series plot below.

2007 2-1

  1. In what year was the lowest mean surface temperature recorded?   (1 mark)

The least squares method is used to fit a trend line to the time series plot.

  1.   i. The equation of this trend line is found to be
  2.          mean surface temperature = – 12.361 + 0.013 × year
  3.      Use the trend line to predict the mean surface temperature (in °C) for 2010.
  4.      Write your answer correct to two decimal places.   (1 mark)

  5.  ii. The actual mean surface temperature in the year 2000 was 13.55°C.
  6.      Determine the residual value (in °C) when the trend line is used to predict the mean surface temperature for this year.
  7.      Write your answer correct to two decimal places.   (1 mark)

  8. iii. By how many degrees does the trend line predict Australia's mean surface temperature will rise each year?
  9.      Write your answer correct to three decimal places.   (1 mark)

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Show Worked Solution

a.   
 

b.i.   


 

b.ii.   

MARKER’S COMMENT: A common error was to omit the negative sign.

 

 
 

 

b.iii.   

 

 

CORE, FUR2 2007 VCAA 1

The histogram below shows the distribution of mean yearly rainfall (in mm) for Australia over 103 years.
 

CORE, FUR2 2007 VCAA 1

  1. Describe the shape of the histogram.   (1 mark)

  2. Use the histogram to determine
  3.  i. the number of years in which the mean yearly rainfall was 500 mm or more.   (1 mark)

  4. ii. the percentage of years in which the mean yearly rainfall was between 500 mm and 600 mm.
  5.     Write your answer correct to one decimal place.  (1 mark)

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Show Worked Solution

a.  
 

b.i.   


 

b.ii.   

CORE, FUR2 2008 VCAA 3

The arm spans (in cm) were also recorded for each of the Years 6, 8 and 10 girls in the larger survey. The results are summarised in the three parallel box plots displayed below.
 

CORE, FUR2 2008 VCAA 3

  1. Complete the following sentence.
  2. The middle 50% of Year 6 students have an arm span between _______ and _______ cm.   (1 mark)
  3. The three parallel box plots suggest that arm span and year level are associated.
  4. Explain why.   (1 mark)

  5. The arm span of 110 cm of a Year 10 girl is shown as an outlier on the box plot. This value is an error. Her real arm span is 140 cm. If the error is corrected, would this girl’s arm span still show as an outlier on the box plot? Give reasons for your answer showing an appropriate calculation.   (2 marks)

Show Answers Only


Show Worked Solution

a.  
 

b.   

♦ Sub 50% mean.
MARKER’S COMMENT: Use specific metrics! Stating “arm span increases” did not receive a mark.

 

c.   

MARKER’S COMMENT: The final comparison here, “Since 140 < 145” is worth a full mark.

CORE, FUR2 2008 VCAA 2

In a larger survey, Years 6, 8 and 10 girls were asked what they did (walked, sat, stood, ran) for most of the time during a typical school lunch time. The results are displayed in the percentage segmented bar chart below.
 

CORE, FUR2 2008 VCAA 2 

Does the percentage segmented bar chart support the opinion that, for these girls, the lunch time activity (walked, sat or stood, ran) undertaken is associated with year level? Justify your answer by quoting appropriate percentages.   (2 marks)

Show Answers Only

Show Worked Solution

CORE, FUR2 2008 VCAA 2 Answer

MARKER’S COMMENT: An association does not have to be a consistent increase or decrease over the years.


 

CORE, FUR2 2009 VCAA 4

  1. Table 2 shows the seasonal indices for rainfall in summer, autumn and winter. Complete the table by calculating the seasonal index for spring.   (1 mark)

     

    CORE, FUR2 2009 VCAA 4

  2. In 2008, a total of 188 mm of rain fell during summer.

     

    Using the appropriate seasonal index in Table 2, determine the deseasonalised value for the summer rainfall in 2008. Write your answer correct to the nearest millimetre.   (1 mark)

  3. What does a seasonal index of 1.05 tell us about the rainfall in autumn?   (1 mark)

Show Answers Only

Show Worked Solution

a.  

 

b.  

♦ Part (c) was “poorly answered” (no exact data).
MARKER’S COMMENT: A common error was to say rainfall was above average monthly rainfall.

 

c.   

 

CORE, FUR2 2009 VCAA 2

The time series plot below shows the rainfall (in mm) for each month during 2008.
 

CORE, FUR2 2009 VCAA 2
 

  1. Which month had the highest rainfall?   (1 mark)

  2. Use three-median smoothing to smooth the time series. Plot the smoothed time series on the plot above.
  3. Mark each smoothed data point with a cross (×).   (2 marks)

  4. Describe the general pattern in rainfall that is revealed by the smoothed time series plot.   (1 mark)

Show Answers Only
  2.  
    CORE, FUR2 2009 VCAA 2 Answer
     


  4.  
Show Worked Solution

a.  

 

b.    CORE, FUR2 2009 VCAA 2 Answer
MARKER’S COMMENT: Locate medians graphically by inspection. Explaining a general pattern with more than one trend proved challenging.

c.   

 

MATRICES, FUR2 2015 VCAA 1

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

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

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

The music school has four teachers, David (), Edith (), Flavio () and Geoff ().

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

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

  1.  i. Complete matrix , shown below, by writing the missing elements in the shaded boxes.   (1 mark) 

 Matrices, FUR2 2015 VCAA 1

  1. ii. 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 below.

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

  1.  i. Write down a matrix calculation in terms of and that results in a matrix that lists the amount paid to each teacher each week.   (1 mark)

  2. ii. How much is paid to Geoff each week?   (1 mark)

Show Answers Only
Show Worked Solution

a.   
 

b.i.  
   
   

 
b.ii.   

 
c.i.   

 
c.ii.   

CORE, FUR2 2009 VCAA 1

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

CORE, FUR2 2009 VCAA 11

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

CORE, FUR2 2009 VCAA 12

  1. Circle the dot on the dot plot that represents the number of rainy days in April 2008.   (1 mark)

  2. For the year 2008, determine

     

  3.  i. the median number of rainy days per month.   (1 mark)
  4. ii. 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.  
    CORE, FUR2 2009 VCAA 12 Answer
Show Worked Solution
a.    CORE, FUR2 2009 VCAA 12 Answer

 

b.i.   
   
   

 

b.ii.   

CORE, FUR2 2010 VCAA 3

Table 2 shows the Australian gross domestic product (GDP) per person, in dollars, at five yearly intervals for the period 1980 to 2005.
 

CORE, FUR2 2010 VCAA 31 

  1. Complete the time series plot above by plotting the GDP for the years 2000 and 2005.   (1 mark)

  2. Briefly describe the general trend in the data.   (1 mark)

In Table 3, the variable year has been rescaled using 1980 = 0, 1985 = 5 and so on. The new variable is time.

CORE, FUR2 2010 VCAA 32

  1. Use the variables time and GDP to write down the equation of the least squares regression line that can be used to predict GDP from time. Take time as the independent variable.   (2 marks)

  2. In the year 2007, the GDP was $34 900. Find the error in the prediction if the least squares regression line calculated in part c. is used to predict GDP in 2007.   (2 marks)

Show Answers Only
  1.  
    CORE, FUR2 2010 VCAA 31 Answer
Show Worked Solution
a.    CORE, FUR2 2010 VCAA 31 Answer

 

b.  

 

c.   

MARKER’S COMMENT: Students are expected to use the variables in the diagram rather than just and .


 

d.  

CORE, FUR2 2010 VCAA 2

In the scatterplot below, average annual female income, in dollars, is plotted against average annual male income, in dollars, for 16 countries. A least squares regression line is fitted to the data.
 


 

The equation of the least squares regression line for predicting female income from male income is

female income = 13 000 + 0.35 × male income

  1. What is the explanatory variable?  (1 mark)

  2. Complete the following statement by filling in the missing information.

     

    From the least squares regression line equation it can be concluded that, for these countries, on average, female income increases by for each $1000 increase in male income.  (1 mark)

    1. Use the least squares regression line equation to predict the average annual female income (in dollars) in a country where the average annual male income is $15 000.  (1 mark)

    2. The prediction made in part c.i. is not likely to be reliable.

       

      Explain why.  (1 mark)


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  4.  

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a.  
 

b.   


 

c.i.  

♦♦ This part was poorly answered (exact data unavailable).
MARKER’S COMMENT: Many students offered “real world” explanations which did not gain a mark here.

 
c.ii.   

   

   

CORE, FUR2 2010 VCAA 1

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

CORE, FUR2 2010 VCAA 11
 

  1. What proportion of these 22 countries have a higher percentage of women ministers in their parliament than Australia?  (1 mark)

  2. 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.
 

    CORE, FUR2 2010 VCAA 12
 

  1. Complete the stemplot above by adding the value for Canada.  (1 mark)

  2. Both the median and the mean appropriate measures of centre for this distribution.
  3. Explain why.  (1 mark)

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  4.  

     

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a.  

b.  

 
 

 

 

 

 

 

c.   

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

 

d.   

CORE, FUR2 2011 VCAA 2

Table 1 shows information about a particular country. It shows the percentage of women by age at first marriage, for the years 1986, 1996 and 2006.

CORE, FUR2 2011 VCAA 2

  1. Of the women who first married in 1986, what percentage were aged 20 to 29 years inclusive?   (1 mark)

  2. Does the information in Table 1 support the opinion that, for the years 1986, 1996 and 2006, the age of women at first marriage was associated with years of marriage?
  3. Justify your answer by quoting appropriate percentages. It is sufficient to consider one age group only when justifying your answer.   (2 marks) 
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a.  


 

b.  

CORE, FUR2 2011 VCAA 1

The stemplot in Figure 1 shows the distribution of the average age, in years, at which women first marry in 17 countries.
 

CORE, FUR2 2011 VCAA 11
 

  1. For these countries, determine
    1. the lowest average age of women at first marriage  (1 mark)

    2. the median average age of women at first marriage  (1 mark)

The stemplot in Figure 2 shows the distribution of the average age, in years, at which men first marry in 17 countries.
 

CORE, FUR2 2011 VCAA 12

  1. For these countries, determine the interquartile range (IQR) for the average age of men at first marriage.  (1 mark)

  2. If the data values displayed in Figure 2 were used to construct a boxplot with outliers, then the country for which the average age of men at first marriage is 26.0 years would be shown as an outlier.
  3. Explain why is this so. Show an appropriate calculation to support your explanation.  (2 marks)

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a.i.  

 

a.ii.  

   

 

b.  

 
 

 

c.  

MARKER’S COMMENT: Many students correctly calculated 28.25 but then failed to discuss how 26.0 related to it.

 

 

CORE, FUR2 2012 VCAA 1

The dot plot below displays the maximum daily temperature (in °C) recorded at a weather station on each of the 30 days in November 2011. 
  

CORE, FUR2 2012 VCAA 1

  1. From this dot plot, determine

     

  2.  i. the median maximum daily temperature, correct to the nearest degree  (1 mark)
  3. ii. the percentage of days on which the maximum temperature was less than 16 °C.
  4.     Write your answer, correct to one decimal place.  (1 mark)

Records show that the minimum daily temperature for November at this weather station is approximately normally distributed with a mean of 9.5 °C and a standard deviation of 2.25 °C.

  1. Determine the percentage of days in November that are expected to have a minimum daily temperature less than 14 °C at this weather station.
  2. Write your answer, correct to one decimal place.  (1 mark)

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Show Worked Solution

a.i.  

.


 

a.ii.   


 

b.   
   
   

 

CORE, FUR2 2012 VCAA 1 Answer 
 

CORE, FUR2 2013 VCAA 2

The development index for each country is a whole number between 0 and 100.

The dot plot below displays the values of the development index for each of the 28 countries that has a high development index.
 

CORE, FUR2 2013 VCAA 21 
 

  1. Using the information in the dot plot, determine each of the following.  (1 mark)

     

     

    Core, FUR2 2013 VCAA 2_2   
     

  2. Write down an appropriate calculation and use it to explain why the country with a development index of 70 is an outlier for this group of countries.  (2 marks)

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a.   

 

b.  

 

 

 

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.
 

CORE, FUR2 2013 VCAA 1

  1. How many of these countries have a very high development index?   (1 mark)

  2. What percentage of the 153 countries has either a low or medium development index?
  3. Write your answer, correct to the nearest percentage.   (1 mark)

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a.  

 

b.  

 
 

CORE, FUR2 2015 VCAA 5

The time series plot below displays the life expectancy, in years, of people living in Australia and the United Kingdom (UK) for each year from 1920 to 2010.
 

Core, FUR2 2015 VCAA 51

  1. By how much did life expectancy in Australia increase during the period 1920 to 2010?
  2. Write your answer correct to the nearest year.   (1 mark)

  3. In 1975, the life expectancies in Australia and the UK were very similar.
  4. From 1975, the gap between the life expectancies in the two countries increased, with people in Australia having a longer life expectancy than people in the UK.
  5. To investigate the difference in life expectancies, least squares regression lines were fitted to the data for both Australia and the UK for the period 1975 to 2010.
  6. The results are shown below. 

  Core, FUR2 2015 VCAA 52

  1. The equations of the least squares regression lines are as follows.
 

 

  1. Use these equations to predict the difference between the life expectancies of Australia and the UK in 2030.
  2. Give your answer correct to the nearest year.   (2 marks)

  3. Explain why this prediction may be of limited reliability.   (1 mark)

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    2.  

       

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a.   

 

b.i.   
   
     
 
   

 

 
♦ Mean mark 45%.
MARKER’S COMMENT: Relate answers directly to the limitations of the given statistical data rather than future events in (b)(ii).

 

b.ii.   

 

 

CORE, FUR2 2015 VCAA 3

The scatterplot below plots male life expectancy (male) against female life expectancy (female) in 1950 for a number of countries. A least squares regression line has been fitted to the scatterplot as shown.
 


 

The slope of this least squares regression line is 0.88

  1. Interpret the slope in terms of the variables male life expectancy and female life expectancy.  (1 mark)

The equation of this least squares regression line is

male = 3.6 + 0.88 × female

  1. In a particular country in 1950, female life expectancy was 35 years.

     

    Use the equation to predict male life expectancy for that country.  (1 mark)

  2. The coefficient of determination is 0.95

     

    Interpret the coefficient of determination in terms of male life expectancy and female life expectancy.  (1 mark)

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a.   

♦ Mean mark 40%.
MARKER’S COMMENT: Many students did not describe the slope, despite being specifically asked about it!

 

b.   

 

c.   

MARKER’S COMMENT: A common error: use of as the basis of the interpretation.

 

CORE, FUR2 2015 VCAA 2

The parallel boxplots below compare the distribution of life expectancy for 183 countries for the years 1953, 1973 and 1993.
 

Core, FUR2 2015 VCAA 2

  1. Describe the shape of the distribution of life expectancy for 1973.   (1 mark)

  2. Explain why life expectancy for these countries is associated with the year. Refer to specific statistical values in your answer.   (2 marks)

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Show Worked Solution

a.   


 

b.   

♦ Mean mark 45%.
MARKER’S COMMENT: Means are typically not discernible from a box plot (unless the data is perfectly symmetrical) and shouldn’t be referred to.

 

CORE, FUR2 2015 VCAA 1

The histogram below shows the distribution of life expectancy of people for 183 countries.
 

CORE, FUR2 2015 VCAA 1 

  1. For this distribution, the modal interval is  ___________  (1 mark)

  2. In how many of these countries is life expectancy less than 55 years?  (1 mark)

  3. In what percentage of these 183 countries is life expectancy between 75 and 80 years?
  4. Write your answer correct to one decimal place.  (1 mark)

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a.   

MARKER’S COMMENT: 40% of students answered this basic question incorrectly.

 

b.   

 

c.   

MARKER’S COMMENT: Answers of 16% or an unrounded figure received no marks.

NETWORKS, FUR1 2015 VCAA 3 MC

The plan shows the layout of a section of pipes, drawn in bold lines, that supplies water to nine houses in a new estate.
 

NETWORKS, FUR1 2015 VCAA 3 MC
 

Which one of the following types of graph could be used to represent the layout of water pipe connections to the water supply and these houses?

A.   a bipartite graph

B.   a complete graph

C.   a loop

D.   a Hamiltonian path

E.   a tree

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MATRICES, FUR1 2015 VCAA 1 MC

Matrix below shows the number of photography (), art () and cooking () books owned by Steven (), Trevor (), Ursula (), Veronica () and William ().

 

The element in row and column of matrix is .

The element is the number of

A.   art books owned by Trevor.

B.   art books owned by Ursula.

C.   art books owned by Veronica.

D.   cooking books owned by Ursula.

E.   cooking books owned by Trevor.

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GRAPHS, FUR1 2015 VCAA 3 MC

A local supermarket delivers groceries that are ordered online by its customers.

The graph below is used to determine the delivery fee from the amount spent by the customer.

GRAPHS, FUR1 2015 VCAA 3 MC

The delivery fee for an online customer who spends $250 on groceries is

A.   

B.   

C.   

D.   

E.   

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