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GEOMETRY, FUR2 2006 VCAA 2

An allotment of land contains a communications tower, .

Points , and are situated on level ground.

From the angle of elevation of is 20°.

Distance is 125 metres.

Distance is 98 metres. 
 

Geometry and Trig, FUR2 2006 VCAA 2
 

  1. Determine the height, , of the communications tower.

     

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

  2. Determine the angle of depression of from .

     

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

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a.  

♦ Mean mark of both parts (combined) was 50%.
 
 

 

b.     

 

 
 

 

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

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Question 3

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a.   
   
   

 

b.   VCAA GEO FUR2 2007 1bi
 
 

 

c.   
   
   

 

d.  
    ²
 
   
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a.   VCAA GEO FUR2 2007 2ai

 
 
 

 

b.  

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a.   
   
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b.   VCAA GEO FUR2 2007 3bi

 

 

VCAA GEO FUR2 2007 3bii

 
 
 

 

c.  

 

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MATRICES, FUR2 2008 VCAA 3

The bookshop manager at the university has developed a matrix formula for determining the number of Mathematics and Physics textbooks he should order each year.

For 2009, the starting point for the formula is the column matrix . This lists the number of Mathematics and Physics textbooks sold in 2008.

is a column matrix listing the number of Mathematics and Physics textbooks to be ordered for 2009.

is given by the matrix formula

  where     and  

  1. Determine    (1 mark)

The matrix formula above only allows the manager to predict the number of books he should order one year ahead. A new matrix formula enables him to determine the number of books to be ordered two or more years ahead.

The new matrix formula is

where  is a column matrix listing the number of Mathematics and Physics textbooks to be ordered for year .

Here,     and  

The number of books ordered in 2008 was given by

  1. Use the new matrix formula to determine the number of Mathematics textbooks the bookshop manager should order in 2010.   (2 marks)

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a.   
   
   

 

b.  
   
   
 
   

 

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.
 
 

 

 

 

 
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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!

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c.   
   
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d.i.   
   
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  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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    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
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GEOMETRY, FUR2 2015 VCAA 3

Cabins are being built at the camp site.

The dimensions of the front of each cabin are shown in the diagram below.

Geometry and Trig, FUR2 2015 VCAA 31

The walls of each cabin are 2.4 m high.

The sloping edges of the roof of each cabin are 2.4 m long.

The front of each cabin is 4 m wide.

The overall height of each cabin is metres.

  1. Show that the value of is 3.73, correct to two decimal places.  (1 mark)

Each cabin is in the shape of a prism, as shown in the diagram below.

Geometry and Trig, FUR2 2015 VCAA 32

  1. All external surfaces of one cabin are to be painted, excluding the base.
     
    What is the total area of the surface to be painted?

     

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

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a.   
   
   

 

b.  

♦♦ “Few students answered this question correctly” (exact data unavailable).
MARKER’S COMMENT: In extended calculations, clearly label each step and draw supporting diagrams where applicable.

VCAA 2015 fur2 Q3bi

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GEOMETRY, FUR2 2009 VCAA 3

The ferry has a logo painted on its side. The logo is a regular pentagon with centre and side length 30 cm. 

It is shown in the diagram below.

GEOMETRY, FUR2 2009 VCAA 3

  1. Show that angle is equal to 72°.  (1 mark)
  2. Show that, correct to two decimal places, the length is 25.52 cm.  (1 mark)
  3. Find the area of the pentagon. Write your answer correct to the nearest cm².  (2 marks)
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a.   
 
   

 

b.   VCAA GEO FUR2 2009 3i

 

 
 

 

c.  
   
   

 

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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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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.   
 
   
   

GEOMETRY, FUR2 2015 VCAA 2

There are plans to construct a series of straight paths on the flat top of the mountain.

A straight path will connect the cable car station at to a communications tower at , as shown in the diagram below.

Geometry and Trig, FUR2 2015 VCAA 21 

The bearing of the communications tower from the cable car station is 060°.

The length of the straight path between the communications tower and the cable car station is 950 m.

  1. How far north of the cable car station is the communications tower?  (1 mark)

Paths will also connect the cable car station and the communications tower to a camp site at , as shown below.

Geometry and Trig, FUR2 2015 VCAA 22

The length of the straight path between the cable car station and the camp site is 1400 m.

The angle    is 40°.

    1. What will be the length of the straight path between the communications tower and the camp site?

       

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

    2. Use the cosine rule to find the bearing of the camp site from the communications tower.

       

      Write your answer correct to the nearest degree.  (2 marks)

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Show Worked Solution
a.   
 
   

 

b.i   

 
 

 

b.ii.   VCAA 2015 fur2 Q2bii

 

 

 

 

Calculus, MET2 2013 VCAA 4

Part of the graph of a function is shown below.

VCAA 2013 4a

  1. Points and are the positive -intercept and -intercept of the graph , respectively, as shown in the diagram above. The tangent to the graph of at the point is parallel to the line segment
    1. Find the equation of the tangent to the graph of at the point    (2 marks)

    2. The shaded region shown in the diagram above is bounded by the graph of , the tangent at the point , and the -axis and -axis.
    3. Evaluate the area of this shaded region.   (3 marks)
  2. Let be a point on the graph of  .
  3. Find the positive value of the -coordinate of , for which the distance is a minimum and find the minimum distance.   (3 marks)

The tangent to the graph of at a point has a negative gradient and intersects the -axis at point  , where  
 

VCAA 2013 4c
 

  1. Find the gradient of the tangent in terms of    (2 marks)

  2.   i. Find the rule for the function of that gives the area of the shaded region.   (2 marks)

  3.  ii. Find the maximum area of the shaded region and the value of for which this occurs.   (2 marks)

  4. iii. Find the minimum area of the shaded region and the value of for which this occurs.  (2 marks)

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a.i.  

 

 


 

a.ii.   

♦ Mean mark 37%.
MARKER’S COMMENT: A common incorrect answer:
.  Know why it’s wrong!

 

 
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  ²

 

b.  

♦♦♦ Mean mark 20%.
 

 


 

c.  

♦♦♦ Mean mark 8%.


 


 

 


 

d.i.  

♦♦♦ Mean mark 8%.


 

 

 

d.ii.   

♦♦♦ Mean mark 5%.

 

 

 met2-2013-vcaa-sec4-answer


 

d.iii.   

♦♦♦ Mean mark 3%.
 
 

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.
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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%.

Probability, MET2 2013 VCAA 2

FullyFit is an international company that owns and operates many fitness centres (gyms) in several countries. At every one of FullyFit’s gyms, each member agrees to have his or her fitness assessed every month by undertaking a set of exercises called . There is a five-minute time limit on any attempt to complete and if someone completes in less than three minutes, they are considered fit.

  1. At FullyFit’s Melbourne gym, it has been found that the probability that any member will complete in less than three minutes is This is independent of any member.
  2. In a particular week, 20 members of this gym attemp
    1. Find the probability, correct to four decimal places, that at least 10 of these 20 members will complete in less than three minutes (2 marks)

    2. Given that at least 10 of these 20 members complete in less than three minutes, what is the probability, correct to three decimal places, that more than 15 of them complete in less than three minutes?  (3 marks)

  3. When FullyFit surveyed all its gyms throughout the world, it was found that the time taken by members to complete is a continuous random variable , with a probability density function , as defined below.
     

    1. Find , correct to four decimal places.  (2 marks)

    2. In a random sample of 200 FullyFit members, how many members would be expected to take more than four minutes to complete Give your answer to the nearest integer.  (2 marks)

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a.i.   

a.ii.   

b.i.   

b.ii. 

Show Worked Solution

a.i.  

 

 

a.ii.   
   
   

 

b.i.   
   
   
   

 

b.ii.  

♦ Mean mark 48%.

 
 
 
 

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. 
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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)

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ai.   
   
   

 

a.ii.  

♦ Mean mark part (a)(ii) 45%.

²

 

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

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  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%.

Probability, MET2 2012 VCAA 3

Steve, Katerina and Jess are three students who have agreed to take part in a psychology experiment. Each student is to answer several sets of multiple-choice questions. Each set has the same number of questions, , where is a number greater than 20. For each question there are four possible options A, B, C or D, of which only one is correct.

  1. Steve decides to guess the answer to every question, so that for each question he chooses A, B, C or D at random.
  2. Let the random variable be the number of questions that Steve answers correctly in a particular set.
    1. What is the probability that Steve will answer the first three questions of this set correctly?   (1 mark)

    2. Find, to four decimal places, the probability that Steve will answer at least 10 of the first 20 questions of this set correctly.   (2 marks)

    3. Use the fact that the variance of is to show that the value of is 25.   (1 mark)

  1. The probability that Jess will answer any question correctly, independently of her answer to any other question, is  . Let the random variable be the number of questions that Jess answers correctly in any set of 25.
  2. If  , show that the value of   is .   (2 marks)

  3. From these sets of 25 questions being completed by many students, it has been found that the time, in minutes, that any student takes to answer each set of 25 questions is another random variable, , which is normally distributed with mean and standard deviation .
  4. It turns out that, for Jess, and also .
  5. Calculate the values of and , correct to three decimal places.   (4 marks)

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Show Worked Solution

a.i.   
   

 

a.ii.  

 

 

a.iii.   
 
 
 

 

b.i. & b.ii. 

 

c.  

♦♦♦ Mean mark part (c) 19%.

 

 

d.  

♦♦♦ Mean mark part (d) 19%.

   

MARKER’S COMMENT: Many students didn’t use binomial calculations for  here.

 

   

 

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)

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  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!

 

 

GEOMETRY, FUR2 2011 VCAA 3

A lighthouse has a lightroom, shown shaded in Figure 2 below.

The floor of the lightroom is in the shape of a regular octagon.

The longest distance across the floor is 4 metres.

The lightroom floor and are shown in Figure 3 below.

GEOMETRY, FUR2 2011 VCAA 31

  1. Show that the size of the angle is 45°.  (1 mark)
  2. Determine the area of triangle .

     

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

The lightroom is surrounded by a walkway of diameter 6.4 metres.

An outer circular wall surrounds the walkway.

The walkway is shown in Figure 4 below.

GEOMETRY, FUR2 2011 VCAA 32

  1. Determine the minimum distance between the lightroom wall and the outer circular wall.  (1 mark)
  2. The walkway is the shaded area in Figure 4. Determine its area correct to the nearest square metre.  (2 marks)
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  2. ²
  4. ²
Show Worked Solution
a.   
 
   

 

b.  

VCAA GEO FUR2 2011 3i

 
 
  ²

 

c.   
   

 

d.   
    ²

 

²

 

²²

GEOMETRY, FUR2 2011 VCAA 2

Ship A and Ship B can both be seen from a lighthouse.

Ship A is 5 kilometres from the lighthouse, on a bearing of 028°.

Ship B is 5 kilometres from Ship A, on a bearing of 130°.

This information is shown in Figure 1 below.

GEOMETRY, FUR2 2011 VCAA 2

  1. Two angles, and , are shown in Figure 1 above.

    1. Determine the size of the angle in degrees.  (1 mark)
    2. Determine the size of the angle in degrees.  (1 mark)
  2. Determine the bearing of the lighthouse from Ship A.  (1 mark)
  3. Determine the bearing of Ship B from the lighthouse.  (1 mark)

 

Show Answers Only
Show Worked Solution

a.i. 

VCAA GEO FUR2 2011 2i

 

 

   ii.   
   

 

b.  

 

c.  

MARKER’S COMMENT: True bearings are required and an answer of 79° is incorrect.

 

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) 
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Show Worked Solution
a.i.  
   
   

 

a.ii.

 

b.  
 
 

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 2006 VCAA 3

The heights (in cm) and ages (in months) of the 15 boys are shown in the scatterplot below.
 

Core, FUR2 2006 VCAA 3
 

  1. Fit a three median line to the scatterplot. Circle the three points you used to determine this three median line.  (2 marks)
  2. Determine the equation of the three median line. Write the equation in terms of the variables height and age and give the slope and intercept correct to one decimal place.  (2 marks)
  3. Explain why the three median line might model the relationship between height and age better than the least squares regression line.  (1 mark)
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  1.   
    Core, FUR2 2006 VCAA 3 Answer

Show Worked Solution
a.    Core, FUR2 2006 VCAA 3 Answer
♦ Average mean mark for all parts 36%.

b.  

 
 

 

 

 

 
 

 

 

c.   

CORE, FUR2 2006 VCAA 2

The heights (in cm) and ages (in months) of a random sample of 15 boys have been plotted in the scatterplot below. The least squares regression line has been fitted to the data.
 


The equation of the least squares regression line is 

The correlation coefficient is 

  1. Complete the following sentence.

     

    On average, the height of a boy increases by _______ cm for each one-month increase in age.   (1 mark)

  2.  i. Evaluate the coefficient of determination.
  3.     Write your answer, as a percentage, correct to one decimal place.   (1 mark)

  4. ii. Interpret the coefficient of determination in terms of the variables height and age.   (1 mark)

Show Answers Only

a.   

b.i.   

b.ii.

Show Worked Solution

a.   

♦ Average mean mark for all parts 44%.
MARKER’S COMMENT: b.i. errors included not converting to % and rounding as a decimal before converting and answering 60%.

 

b.i.   
   
   
MARKER’S COMMENT: Any reference to causation in b.ii. was marked incorrect.

 

b.ii.  

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)

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  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 2008 VCAA 4

The arm spans (in cm) and heights (in cm) for a group of 13 boys have been measured. The results are displayed in the table below.
 

CORE, FUR2 2008 VCAA 4 

The aim is to find a linear equation that allows arm span to be predicted from height.

  1. What will be the explanatory variable in the equation?   (1 mark)

  2. Assuming a linear association, determine the equation of the least squares regression line that enables arm span to be predicted from height. Write this equation in terms of the variables arm span and height. Give the coefficients correct to two decimal places.   (2 marks)

  3. Using the equation that you have determined in part b., interpret the slope of the least squares regression line in terms of the variables height and arm span.   (1 mark)

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  3.  

Show Worked Solution

a.  

♦ Mean mark sub 50% (exact data not available).
MARKER’S COMMENT: Many students did not understand the term co-efficients as it applies to the regression equation.

 

b.   

 

c.   

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)

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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 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)

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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 3

The scatterplot below shows the rainfall (in mm) and the percentage of clear days for each month of 2008. 
 

An equation of the least squares regression line for this data set is

rainfall = 131 – 2.68 × percentage of clear days

  1. Draw this line on the scatterplot.   (1 mark)

  2. Use the equation of the least squares regression line to predict the rainfall for a month with 35% of clear days. Write your answer in mm correct to one decimal place.   (1 mark)

  3. The coefficient of determination for this data set is 0.8081.
  4.  i. Interpret the coefficient of determination in terms of the variables rainfall and percentage of clear days.   (1 mark)

  5. ii. Determine the value of Pearson’s product moment correlation coefficient. Write your answer correct to three decimal places.   (2 marks)

Show Answers Only
  1.  
    CORE, FUR2 2009 VCAA 3 Answer


Show Worked Solution
a.    CORE, FUR2 2009 VCAA 3 Answer

 

b.   
   
MARKER’S COMMENT: Any answers that suggested causation with terms like “is due to” etc.., did not receive a mark.

 

c.i.   

 

 

 

MARKER’S COMMENT: Many students failed to include the negative sign as indicated by the negative slope of the graph.
c.ii.   
   
   

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)

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Show Worked Solution

a.   
 

b.i.  
   
   

 
b.ii.   

 
c.i.   

 
c.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)

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  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.  

Show Worked Solution

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.  

     

Show Worked Solution

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 3

The following time series plot shows the average age of women at first marriage in a particular country during the period 1915 to 1970.
 

CORE, FUR2 2011 VCAA 31

  1. Use this plot to describe, in general terms, the way in which the average age of women at first marriage in this country has changed during the period 1915 to 1970.   (1 mark)

During the period 1986 to 2006, the average age of men at first marriage in a particular country indicated an increasing linear trend, as shown in the time plot below.

CORE, FUR2 2011 VCAA 32

A three-median line could be used to model this trend.

  1. On the graph above
  2.  i. clearly mark with a cross (×) the three points that would be used to fit a three-median line to this time series plot.   (2 marks)

  3. ii. draw in the three-median line.   (1 mark)

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  2.  
    CORE, FUR2 2011 VCAA 3 Answer
Show Worked Solution

a.  

 

b.i. & ii.

CORE, FUR2 2011 VCAA 3 Answer

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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Show Worked Solution

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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Show Worked Solution

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 4

The wind speeds (in km/h) that were recorded at the weather station at 9.00 am and 3.00 pm respectively on 18 days in November are given in the table below. A scatterplot has been constructed from this data set.
 

CORE, FUR2 2012 VCAA 41
 

Let the wind speed at 9.00 am be represented by the variable ws9.00am and the wind speed at 3.00 pm be represented by the variable ws3.00pm. 

The relationship between ws9.00am and ws3.00pm shown in the scatterplot above is nonlinear. 

A squared transformation can be applied to the variable ws3.00pm to linearise the data in the scatterplot.

  1. Apply the squared transformation to the variable ws3.00pm and determine the equation of the least squares regression line that allows (ws3.00pm)² to be predicted from ws9.00am.

     

    In the boxes provided, write the coefficients for this equation, correct to one decimal place.   (2 marks)

    2012 4-1

  2. Use this equation to predict the wind speed at 3.00 pm on a day when the wind speed at 9.00 am is 24 km/h.

     

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

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Show Worked Solution

a.  

MARKER’S COMMENT: Many students were unable to deal with the squared transformation.


 

b.  

CORE, FUR2 2012 VCAA 3

A weather station records the wind speed and the wind direction each day at 9.00 am. 

The wind speed is recorded, correct to the nearest whole number. 

The parallel boxplots below have been constructed from data that was collected on the 214 days from June to December in 2011.
 

CORE, FUR2 2012 VCAA 3

  1. Complete the following statements.
  2. The wind direction with the lowest recorded wind speed was ________.
  3. The wind direction with the largest range of recorded wind speeds was _______.   (1 mark)

  4. The wind blew from the south on eight days.
  5. Reading from the parallel boxplots above we know that, for these eight wind speeds, the
first quartile
median
third quartile
  1. Given that the eight wind speeds were recorded to the nearest whole number, write down the eight wind speeds.   (1 mark) 
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Show Worked Solution

a.  
 

b.  

♦♦ MARKER’S COMMENT: Although specific data isn’t available, “few” students answered this part correctly.


  

CORE, FUR2 2012 VCAA 2

The maximum temperature and the minimum temperature at this weather station on each of the 30 days in November 2011 are displayed in the scatterplot below.

CORE, FUR2 2012 VCAA 2

The correlation coefficient for this data set is  . 

The equation of the least squares regression line for this data set is

maximum temperature = × minimum temperature

  1. Draw this least squares regression line on the scatterplot above.   (1 mark)

  2. Interpret the vertical intercept of the least squares regression line in terms of maximum temperature and minimum temperature.   (1 mark)

  3. Describe the relationship between the maximum temperature and the minimum temperature in terms of strength and direction.   (1 mark)

  4. Interpret the slope of the least squares regression line in terms of maximum temperature and minimum temperature.   (1 mark)

  5. Determine the percentage of variation in the maximum temperature that may be explained by the variation in the minimum temperature.
  6. Write your answer, correct to the nearest percentage.   (1 mark)

On the day that the minimum temperature was 11.1 °C, the actual maximum temperature was 12.2 °C.

  1. Determine the residual value for this day if the least squares regression line is used to predict the maximum temperature.
  2. Write your answer, correct to the nearest degree.   (2 marks)

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Show Worked Solution

a.  

 CORE, FUR2 2012 VCAA 2 Answer 

b.  

 

♦ Parts (i) to (vi) have an average mean mark of 41%.

c.   

 

d.  

 

 

 

e.   
   
   

 

f.  

MARKER’S COMMENT: Students had particular difficulty with this part, with many using the incorrect calculation of  12.2 – 11.1 = 1.1.
 
 
 

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 4

The time series plot below shows the average pay rate, in dollars per hour, for workers in a particular country for the years 1991 to 2005.
 

CORE, FUR2 2013 VCAA 4

A three median line will be used to model the increasing trend in the average pay rate shown in this time series.

The explanatory variable to be used is year.

  1. Three medians will be used to draw the three median line.

     

    1. On the time series plot above, mark the location of each of the three medians with a cross ().  (2 marks)
    2. Draw the three median line on the time series plot.  (1 mark)
  2. Calculate the slope of the three median line.

     

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

  3. Interpret the slope of the three median line in terms of the relationship between the variables average pay rate and year.  (1 mark)
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  1. i. & ii.

  3.  

Show Worked Solution

a.i. & ii.   

♦ Mean mark 45%.
MARKER’S COMMENT: Few students found the correct middle point.

 

CORE, FUR2 2013 VCAA 4 Answer

♦♦ Parts (ii) and (iii) produced an overall mean mark of 32%.

 

b.   
   
   
   

 

MARKER’S COMMENT: Answering “an increase of 0.8 each year” received no marks (refer to actul units).

 

c.   

 

CORE, FUR2 2013 VCAA 3

The development index and the average pay rate for workers, in dollars per hour, for a selection of 25 countries are displayed in the scatterplot below.

CORE, FUR2 2013 VCAA 31   

The table below contains the values of some statistics that have been calculated for this data.

     CORE, FUR2 2013 VCAA 32

  1. Determine the standardised value of the development index ( score) for a country with a development index of 91.
  2. Write your answer, correct to one decimal place.   (1 mark)

  3. Use the information in the table to show that the equation of the least squares regression line for a country’s development index, , in terms of its average pay rate, , is given by
  4.           (2 marks)

  5. The country with an average pay rate of $14.30 per hour has a development index of 83.
  6. Determine the residual value when the least squares regression line given in part (b) is used to predict this country’s development index.
  7. Write your answer, correct to one decimal place.   (2 marks)

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Show Worked Solution
MARKER’S COMMENT: Many students made a careless error by using the Average pay rate (-variable).
a.   
   
   
   

 

♦ Parts (b) and (c) had a combined mean mark of 48%.
b.   
   
   

 

 
 

 

 

c.   
   

 

 
 

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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Show Worked Solution

a.   

 

b.  

 

 

 

CORE, FUR2 2014 VCAA 4

The scatterplot below shows the population density, in people per square kilometre, and the area, in square kilometres, of 38 inner suburbs of the same city.

Core, FUR2 2015 VCAA 41

For this scatterplot,

  1. Describe the association between the variables population density and area for these suburbs in terms of strength, direction and form.   (1 mark)

  2. The mean and standard deviation of the variables population density and area for these 38 inner suburbs are shown in the table below.
     
    Core, FUR2 2015 VCAA 42
  3.   i. One of these suburbs has a population density of 3082 people per square kilometre.
  4.     Determine the standard -score of this suburb’s population density.
  5.     Write your answer, correct to one decimal place.   (1 mark)

Assume the areas of these inner suburbs are approximately normally distributed.

  1.  ii. How many of these 38 suburbs are expected to have an area that is two standard deviations or more above the mean?
  2.     Write your answer, correct to the nearest whole number.   (1 mark)

  3. iii. How many of these 38 inner suburbs actually have an area that is two standard deviations or more above the mean?   (1 mark)

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Show Worked Solution
♦♦ Mean mark 30%.
MARKER’S COMMENT: The most common error was to describe the association as positive.
a.   
   

 

 

b.i   
   
   
   

 

b.ii.   

 

b.iii.   

♦♦ Very few students answered this part correctly.

²

 

CORE, FUR2 2014 VCAA 2

The scatterplot below shows the population and area (in square kilometres) of a sample of inner suburbs of a large city.
 

Core, FUR2 2015 VCAA 2

The equation of the least squares regression line for the data in the scatterplot is

population = 5330 + 2680 × area

  1. Write down the response variable.   (1 mark)

  2. Draw the least squares regression line on the scatterplot above.   (1 mark)

  3. Interpret the slope of this least squares regression line in terms of the variables area and population.  (2 marks)

  4. Wiston is an inner suburb. It has an area of 4 km² and a population of 6690.
  5. The correlation coefficient, , is equal to 0.668
  6.  i. Calculate the residual when the least squares regression line is used to predict the population of Wiston from its area.  (1 mark)

  7. ii. What percentage of the variation in the population of the suburbs is explained by the variaton in area.
  8.     Write your answer, correct to one decimal place.  (1 mark)

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  2.  
          Core, FUR2 2015 VCAA 2 Answer

  3. ²
Show Worked Solution

a.  
 

♦ Mean mark 36%.
MARKER’S COMMENT: Use the equation to draw the line and use points at the extremities.
b.   

Core, FUR2 2015 VCAA 2 Answer

 

c.   

♦ Mean mark 41% (part (iii)).

²
 

d.i. 


 

♦ Part (iv) in total had a mean mark 42%.
d.ii.   
 
   

 

CORE, FUR2 2014 VCAA 1

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

Core, FUR2 2015 VCAA 1

  1. Write down the percentage of people in Australia who were aged 0 – 14 years in 2010.
  2. Write your answer, correct to the nearest percentage.  (1 mark)

  3. In 2010, the population of Japan was 128 000 000.
  4. How many people in Japan were aged 65 years and over in 2010?  (1 mark)

  5. From the graph above, it appears that there is no association between the percentage of people in the 15 – 64 age group and the country in which they live.
  6. Explain why, quoting appropriate percentages to support your explanation.  (1 mark)

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  3.  
     
     

Show Worked Solution

a.   

 

b.  

 

c.   

♦ Mean mark 41%.

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.  

       

Show Worked Solution

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 4

The table below shows male life expectancy (male) and female life expectancy (female) for a number of countries in 2013. The scatterplot has been constructed from this data.
 

Core, FUR2 2015 VCAA 4

  1. Use the scatterplot to describe the association between male life expectancy and female life expectancy in terms of strength, direction and form.   (1 mark)

  2. Determine the equation of a least squares regression line that can be used to predict male life expectancy from female life expectancy for the year 2013.

     

    Complete the equation for the least squares regression line below by writing the intercept and slope in the space provided.

     

    Write these values correct to two decimal places.  (1 mark)

male = ______________ + ______________ ×  female

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Show Worked Solution

a.   

♦ Mean mark 49%.
MARKER’S COMMENT: Common errors included using the 1st column as the independent () variable and poor rounding.

 

b.  

Probability, 2UG SM-Bank 01 MC

A circle has 5 equally spaced points on its circumference, as shown.

How many different triangles using 3 of these points as vertices can be drawn?  (3 marks)

2UG SM-bank 01 mc

A.  

B.  

C.  

D.  

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Show Worked Solution

STRATEGY: The number of triangles is divided by because the selections of any 3 vertices are unordered.

Financial Maths, 2ADV M1 SM-Bank 10 MC

The graph above shows the first six terms of a sequence.

This sequence could be

  1. an arithmetic sequence that sums to one.
  2. an arithmetic sequence with a common difference of one.
  3. a geometric sequence with an infinite sum of one.
  4. a geometric sequence with a common ratio of one. 
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