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Calculus, MET2 2014 VCAA 5

Let  

  1. Express    in the form  .   (2 marks)

  2. Describe the translation that maps the graph of    onto the graph of  .   (1 mark)

  3. Find the values of such that the graph of   has
    1. one positive -axis intercept.   (1 mark)

    2. two positive -axis intercepts.   (1 mark)

  4. Find the value of for which the equation    has one solution.   (1 mark)

  5. At the point  , the gradient of    is and at the point , the gradient is  , where is a positive real number.
    1. Find the value of  .   (2 marks)

    2. Find and if  .   (1 mark)

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

    2. Find the equations of the tangents to the graph of    that pass through the point with coordinates  .   (3 marks)

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  3.  i.
  4. ii.
  6.  i.
  7. ii.
  8. i.
  9. ii.
Show Worked Solution

a. 


 

 
 
 

 

b.   
   
   

 

♦ Mean mark (b) 37%.


 

c.i. 

♦♦♦ Mean mark 7%.

 
met2-2014-vcaa-sec5-answer
 


 

c.ii.  

♦♦♦ Mean mark part (c)(ii) 19%.


 

d.   

♦♦♦ Mean mark 17%.

 

met2-2014-vcaa-sec5-answer1
 

 
 

 

e.i.  

♦♦ Mean mark (e.i.) 28%.

 

e.ii.  

♦♦♦ Mean mark (e.ii.) 10%.


 

f.i.   

 

♦♦ Mean mark (f.i.) 22%.

 

f.ii.  

♦♦♦ Mean mark (f.ii.) 14%.

   
   
   
   

 

Calculus, MET2 2015 VCAA 5

  1. Let  , where  .
    1. Find and .   (1 mark)

    2. The minimum value of occurs when  .
    3. State the value of and the minimum value of .   (2 marks)

    4. On the axes below, sketch the graph of against for  .
    5. Label the end points and the minimum point with their coordinates.   (2 marks)
       
      VCAA 2015 5a

    6. Find the value of the average rate of change of the function over the interval  .   (2 marks)

  2. Let  , where is a real number and   in .

     

    If the minimum value of the function occurs when  , find the value of .   (2 marks)

    1. Find the set of possible values of such that the minimum value of the function occurs when  .   (2 marks)
    2. Find the set of possible values of such that the minimum value of the function occurs when  .   (2 marks)
  4. If the function has a local minimum , where  , it can be shown that  .
  5. Find the value of .   (2 marks)
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    3.  

         vcaa-2015-5a-answer

  3.   
Show Worked Solution
a.i.     
 

 

a.ii.   

 

 

 

a.iii.  vcaa-2015-5a-answer

 

a.iv.   
   
   

 

b.   


 

c.i. 

♦♦ Mean mark part (c)(i) 28%.


 

c.ii. 

♦♦ Mean mark (c.ii.) 27%.


 

♦♦♦ Mean mark (d) 15%.
d.   

Calculus, MET2 2015 VCAA 4

An electronics company is designing a new logo, based initially on the graphs of the functions

These graphs are shown in the diagram below, in which the measurements in the and directions are in metres.

VCAA 2015 4a

The logo is to be painted onto a large sign, with the area enclosed by the graphs of the two functions (shaded in the diagram) to be painted red.

  1. The total area of the shaded regions, in square metres, can be calculated as  .
  2. What is the value of ?   (1 mark)

The electronics company considers changing the circular functions used in the design of the logo.

Its next attempt uses the graphs of the functions  .

  1. On the axes below, the graph of    has been drawn.
  2. On the same axes, draw the graph of  .   (2 marks)

     

          VCAA 2015 4b

  3. State a sequence of two transformations that maps the graph of    to the graph of  .   (2 marks)

The electronics company now considers using the graphs of the functions  , where and are positive integers with   and .

    1. Find the area enclosed by the graphs of    and    in terms of and if is even.
    2. Give your answer in the form  , where and are integers.   (2 marks)

    3. Find the area enclosed by the graphs of    and    in terms of and  if is odd.
    4. Give your answer in the form  , where and are integers.   (2 marks)

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  2.  

    vcaa-2015-4b-answer   

Show Worked Solution
a.  
   

♦ Mean mark part (a) 36%.

 

b.   vcaa-2015-4b-answer

 

c.  

 

d.i.  

♦♦♦ Mean mark 20%.
MARKER’S COMMENT: Note that for even.

 

d.ii.  

♦♦♦ Mean mark 14%.
MARKER’S COMMENT: Note that for odd.

GRAPHS, FUR2 2006 VCAA 3

Harry offers dog washing and dog clipping services.

Let    be the number of dogs washed in one day

   be the number of dogs clipped in one day.

It takes 20 minutes to wash a dog and 25 minutes to clip a dog.

There are 200 minutes available each day to wash and clip dogs.

This information can be written as Inequalities 1 to 3.

Inequality 1:  

Inequality 2:  

Inequality 3:  

  1. Draw the line that represents    on the graph below.  (1 mark)

     

          GRAPHS, FUR2 2006 VCAA 3

In any one day the number of dogs clipped is at least twice the number of dogs washed.

  1. Write Inequality 4 to describe this information in terms of and .  (1 mark)

     

    1. On the graph on page 18 draw and clearly indicate the boundaries of the region represented by Inequalities 1 to 4.  (2 marks)
    2. On a day when exactly five dogs are clipped, what is the maximum number of dogs that could be washed?  (1 mark)

The profit from washing one dog is $40 and the profit from clipping one dog is $30.

Let be the total profit obtained in one day from washing and clipping dogs.

  1. Write an equation for the total profit, , in terms of and .  (1 mark)
    1. Determine the number of dogs that should be washed and the number of dogs that should be clipped in one day in order to maximise the total profit.  (1 mark)
    2. What is the maximum total profit that can be obtained from washing and clipping dogs in one day?  (1 mark) 
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  1.  
    vcaa-graphs-fur2-2006-3ai
    1.   
Show Worked Solution
a.    vcaa-graphs-fur2-2006-3ai

 

b.  

 

c.i.   

 

c.ii. 

♦♦ Mean mark of parts (c)-(e) (combined) was 27%.

 

d.  

 

e.i.   

 

 

e.ii.  

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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  5. ²
Show Worked Solution
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 4

Tessa has a task that involves removing the top 24 cm of the height of her right regular pyramid below.

geo-2007-4 

The shape remaining is shown in Figure 5 below. The top surface, , is parallel to the base, .
 

GEOMETRY, FUR2 2007 VCAA 4
 

  1. What fraction of the height of the pyramid has Tessa removed to produce Figure 5?  (1 mark)
  2. What fraction of the volume of the pyramid remains in Figure 5?  (2 marks)
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Show Worked Solution

a.  

 

b.   

♦♦ Mean mark of both parts (combined) was 18%.

 

MATRICES, FUR2 2008 VCAA 4

By the end of each academic year, students at the university will have either passed, failed or deferred the year. Experience has shown that

  • 88% of students who pass this year will also pass next year
  • 10% of students who pass this year will fail next year
  • 2% of students who pass this year will defer next year
     
  • 52% of students who fail this year will pass next year
  • 44% of students who fail this year will fail next year
  • 4% of students who fail this year will defer next year
     
  • 65% of students who defer this year will pass next year
  • 10% of students who defer this year will fail next year
  • 25% of students who defer this year will defer next year.

Twelve hundred and thirty students began a business degree in 2007.

By the end of the 2007 academic year, 880 students had passed, 230 had failed, while 120 had deferred the year.

No students have dropped out of the business degree permanently.

Use this information to predict the number of business students who will defer the 2009 academic year.   (2 marks)

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Show Worked Solution

♦♦ Exact data unavailable although “few” students were able to write down the correct transition matrix.

 

 

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 2009 VCAA 4

The ferry has two fuel filters, and .

Filter has a hemispherical base with radius 12 cm.

A cylinder of height 30 cm sits on top of this base.
 

GEOMETRY, FUR2 2009 VCAA 41
 

  1. Calculate the volume of filter . Write your answer correct to the nearest cm³.  (2 marks)

Filter  is a right cone with height 50 cm.

 

GEOMETRY, FUR2 2009 VCAA 42

  1. Originally filter was full of oil, but some was removed.

     

    If the height of the oil in the cone is now 20 cm, what percentage of the original volume of oil was removed?  (2 marks)

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  1. ³
Show Worked Solution

a.  

♦ Mean mark of all parts (combined) 36%.

³
 

³

 

³

 

b.  

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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Show Worked Solution

a.i.  

 

 


 

a.ii.   

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

 

 
  ²

 

  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.
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)

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    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)

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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 2

Michael began his hike at the national park office and followed a track towards a camping ground, 16 km away.

The distance-time graph below shows Michael's distance from the national park office, kilometres, after hours of hiking.

GRAPHS, FUR2 2011 VCAA 2

It took Michael seven hours to complete this hike.

  1. What was Michael's average speed, in kilometres per hour, during this hike?

     

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

The equation of Michael's distance-time graph from    to    is

  1. Determine the value of both and .(2 marks)

Katie hiked along the same track as Michael, but hiked in the opposite direction.

She begun at the camping ground and hiked towards the national park office.

Katie's distance from the national park office, kilometres, after hours of hiking, can be determined from the equation

Katie and Michael both starter hiking at the same time.

  1. After how many hours did Katie pass Michael?  (1 mark)

Katie and Michael both carry radio transmitters that allow them to talk to each other while hiking.

The transmitters will not work if Katie and Michael are more than three kilometres apart.

  1. For how many hours during the hike were Michael and Katie able to use the radio transmitters to talk to each other?

     

    Write your answer in hours correct to two decimal places.  (2 marks)

 

Show Answers Only
Show Worked Solution
a.   
   
   
   

 

b.  

♦ Mean mark for all parts (combined) was 31%.

 

 
 

 

 

c.   

MARKER’S COMMENT: Drawing Katie’s graph was a feature of the best answers here.

 

 

 

d.  

 

♦♦♦ “Few students” answered this part correctly.
MARKER’S COMMENT: Students who had drawn Katie’s graph were the most successful in this part.

 

GEOMETRY, FUR2 2011 VCAA 4

A lighthouse tower, shaded on Figure 5 below, is in the shape of a truncated cone.

It has circular cross-sections that decrease uniformly from a radius of 3.5 metres at ground level to a radius of 2 metres at the walkway.

The height of the lighthouse tower is 18 metres.

The angle marked as is the angle that the outer wall of the lighthouse tower makes with the horizontal at ground level.

GEOMETRY, FUR2 2011 VCAA 41

  1. Determine the size of the angle .

     

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

The lighthouse tower is part of a cone. The height of this cone is metres and the base radius is 3.5 metres, as shown in Figure 6.

GEOMETRY, FUR2 2011 VCAA 42

    1. Determine , the height of this cone, in meters.  (2 marks)
    2. Determine the volume of the lighthouse tower.

       

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

Show Answers Only
    2. ³
Show Worked Solution
a.   VCAA GEO FUR2 2011 4i
 
 
♦♦♦ Mean mark for all parts (combined) 17%.
MARKER’S COMMENT: A common mistake had the base of this triangle equal to 3.5m.

 

b.i.   VCAA GEO FUR2 2011 4ii
 
 
  ii.   VCAA GEO FUR2 2011 4iii

³

 

³

 

³³

MATRICES, FUR2 2014 VCAA 3

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

Mr Choi may need to withdraw from the election at the end of May.

Matrix , shown below, shows the percentage of voters who change their preferred candidate, from month to month, before Mr Choi would withdraw from the election.

Matrix , shown below, shows the percentage of voters who change their preferred candidate, from May to June, after Mr Choi would withdraw from the election.

Consider the voters who preferred Mr Broad in May and who were expected to prefer Mr Choi in June.

  1. What percentage of these voters are now expected to prefer Mr Broad in June?   (1 mark)

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. If Mr Choi withdraws, how many votes is Mr Broad expected to receive in the election in June?
  2. Write your answer, correct to the nearest vote.   (2 marks)
Show Answers Only
Show Worked Solution

a.  

♦♦♦ Mean mark for (a)-(b) was 13% (combined).
MARKER’S COMMENT: In (a), identifying the sub-group expected to change their vote proved very difficult for most students.


  

MARKER’S COMMENT: Few students correctly modified the transition matrix from to in this question.
b.   
   
   
   

 

 
 
 

 

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)
Show Answers Only
  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 2008 VCAA 5

The number of hours spent doing homework each week (homework hours) and the number of hours spent watching television each week (television hours) were recorded for a group of 20 Year 12 students. 

The results are displayed in the table below and a scatterplot constructed as shown.
 

CORE, FUR2 2008 VCAA 5

The relationship between homework hours per week and television hours per week is clearly nonlinear. 

A reciprocal transformation applied to the variable, television hours, can be used to linearise the scatterplot.

  1. Apply this reciprocal transformation to the data and determine the equation of the least squares regression line that allows homework hours to be predicted from the reciprocal of television hours.
  2. Write the coefficients correct to two decimal places.  (2 marks)

  3. If a student spends 12 hours per week watching television, use the least squares regression line to predict the number of hours that the student spends doing homework. Give your answer correct to one decimal place.  (1 mark)

Show Answers Only
Show Worked Solution

a.  

♦♦♦ Avg mean mark for both parts 16%.
MARKER’S COMMENT: A majority of students didn’t understand the term “reciprocal”.

 

b.   
   
   

CORE, FUR2 2011 VCAA 4

The average age of women at first marriage in years (average age) and average yearly income in dollars per person (income) were recorded for a group of 17 countries.

The results are displayed in Table 2. A scatterplot of the data is also shown.
 

2011 4-1

The relationship between average age and income is nonlinear.

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

  1. Apply this log transformation to the data and determine the equation of the least squares regression line that allows average age to be predicted from log (income).
  2. Write the coefficients for this equation, correct to two decimal places, in the spaces provided.   (2 marks)

    2011 4-2

  1. Use this equation to predict the average age of women at first marriage in a country with an average yearly income of $20 000 per person.
  2. Write your answer correct to one decimal place.   (1 mark) 
Show Answers Only
Show Worked Solution

a.  

♦♦ Parts (i) and (ii) had a combined mean mark 0f 34%.
MARKER’S COMMENT: Students struggled with the log transformation, incorrect data entry and choosing the correct dependent variable.

 

b.  

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)

Show Answers Only



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 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)
Show Answers Only
  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 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)

Show Answers Only
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

 

GRAPHS, FUR1 2008 VCAA 9 MC

The shaded region in the graph below represents the feasible region for a linear programming problem.

GRAPHS, FUR1 2008 VCAA 9 MC

Which objective function, , has its maximum value at the point ?

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 17%.
MARKER’S COMMENT: Even though (option E) has a larger value at than option A, its maximum value occurs at the vertex (0, 60).

 

GRAPHS, FUR1 2012 VCAA 9 MC

The cost of posting a parcel that weighs 500 grams or less depends on its weight, as shown on the graph below.

graphs 2012 VCAA 9mc

The total cost of posting three separate parcels weighing respectively 100 grams, 150 grams and 210 grams is $11.

The total cost of posting three separate parcels weighing respectively 60 grams, 110 grams and 350 grams is $8.

The total cost of posting two separate parcels weighing respectively 170 grams and 500 grams is

A.     

B.     

C.     

D.   

E.    

Show Answers Only

Show Worked Solution

 

GRAPHS, FUR1 2013 VCAA 8 MC

The shaded region in the graph below represents the feasible region for a linear programming problem.

An objective function    has its value maximised at both vertex and vertex .

The values of and could be

A.  

B.  

C.  

D.  

E.  

Show Answers Only

Show Worked Solution
♦♦ Mean mark 35%.

 

 

GEOMETRY, FUR1 2015 VCAA 8 MC

A composite shape is made up of a parallelogram and a triangle, as shown below.

GEO & TRIG, FUR1 2015 VCAA 8 MC

Which one of the following is always true?

A.  

B.  

C.  

D.  

E.  

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Show Worked Solution

GEO & TRIG, FUR1 2015 VCAA 8 MC Answer1 

♦♦ Mean mark 25%.

 

CORE*, FUR1 2015 VCAA 8 MC

Cindy took out a reducing balance loan of $8400 to finance an overseas holiday.

Interest was charged at a rate of 9% per annum, compounding quarterly.

Her loan is to be fully repaid in six years, with equal quarterly payments.

After three years, Cindy will have reduced the balance of her loan by approximately

A.     9%

B.   35%

C.   43%

D.   50%

E.   57%

Show Answers Only

Show Worked Solution

♦♦ Mean mark 33%.

 

 

 

 
 

CORE, FUR1 2010 VCAA 11 MC

A student uses the following data to construct the scatterplot shown below.
 

CORE, FUR1 2015 VCAA 11 MC
 

A reciprocal transformation is applied to the -axis and is used to linearise the scatterplot.

With as the dependent variable, the slope of the least squares regression line that is fitted to the linearised plot is closest to

A.   

B.     

C.         

D.        

E.      

Show Answers Only

Show Worked Solution

q11 2010

♦♦ Mean mark 35%.

CORE*, FUR1 2013 VCAA 9 MC

The following information relates to the repayment of a home loan of $300 000.
 
• The loan is to be repaid fully with monthly payments of $2500.
• Interest compounds monthly.
• After the first monthly payment has been made, the amount owing on the loan is $299 000.

Which one of the following statements is true?

  1. After two months, $297 995 is still owing on the loan.
  2. $1000 of interest has been paid in the first month.
  3. The loan will be fully repaid in less than 15 years.
  4. Halfway through the term of the loan, the amount still owing will be $150 000.
  5. Payments of $2750 rather than $2500 per month will reduce the time to repay the loan fully by more than three years. 
Show Answers Only

Show Worked Solution

♦♦ Mean mark 29%.

 
 

 

 

PATTERNS, FUR 1 2006 VCAA 9 MC

A healthy eating and gym program is designed to help football recruits build body weight over an extended period of time.

Roh, a new recruit who initially weighs 73.4 kg, decides to follow the program.

In the first week he gains 400 g in body weight.

In the second week he gains 380 g in body weight.

In the third week he gains 361 g in body weight.

 If Roh continues to follow this program indefinitely, and this pattern of weight gain remains the same, his eventual body weight will be closest to

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

♦♦ Mean mark 33%.

 

 
 

 

CORE, FUR1 2006 VCAA 11-13 MC

The following information relates to Parts 1, 2 and 3.

The table shows the seasonal indices for the monthly unemployment numbers for workers in a regional town.

Part 1

The seasonal index for October is missing from the table.

The value of the missing seasonal index for October is

A.   

B.   

C.   

D.   

E.   

 

Part 2

The actual number of unemployed in the regional town in September is 330.

The deseasonalised number of unemployed in September is closest to

A.   

B.   

C.   

D.   

E.   

 

Part 3

A trend line that can be used to forecast the deseasonalised number of unemployed workers in the regional town for the first nine months of the year is given by

deseasonalised number of unemployed = 373.3 – 3.38 × month number

where month 1 is January, month 2 is February, and so on.

The actual number of unemployed for June is predicted to be closest to

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

 
 

 

 
 

 

♦♦ Mean mark 29%.
MARKERS’ COMMENT: 59% of students correctly found the deseasonalised number but failed to convert it to the actual.

 
 

PATTERNS, FUR1 2013 VCAA 9 MC

Eleven speed bumps are placed on a road.

The speed bumps are placed so that the distance between consecutive speed bumps decreases according to an arithmetic sequence.

The distance between the first and last speed bumps is exactly 100 m.

The smallest distance between consecutive speed bumps is 2 m.

The largest distance, in m, between two consecutive speed bumps is

A.  

B.  

C.  

D. 

E.  

Show Answers Only

Show Worked Solution

♦♦ Mean mark 33%.
MARKER’S COMMENT: The not often used formula provides the most efficient route to this solution, although can also be used after finding the value of .

 

 

Graphs, MET2 2015 VCAA 11 MC

The transformation that maps the graph of    onto the graph of    is a

  1. dilation by a factor of from the -axis.
  2. dilation by a factor of from the -axis.
  3. dilation by a factor of from the -axis.
  4. dilation by a factor of from the -axis.
  5. dilation by a factor of from the -axis.
Show Answers Only

Show Worked Solution
♦♦ Mean mark 24%.
 

 

Mechanics, EXT2* M1 2008 HSC 7

EXT1 2008 HSC 7
 

A projectile is fired from    with velocity    at an angle of inclination    across level ground. The projectile passes through the points    and  , which are both    metres above the ground, at times    and    respectively. The projectile returns to the ground at  .

The equations of motion of the projectile are

  and   . (Do NOT prove this.)

  1. Show that    AND   .  (2 marks)

Let  α  and  β. It can be shown that
 
        and   .  (Do NOT prove this.)

  1. Show that  .  (2 marks)

  2. Show that  .  (1 mark)

Let   and  .

  1. Show that    and  .  (2 marks)

Let the gradient of the parabola at    be .

  1. Show that  .  (3 marks)

  2. Show that  .  (2 marks)

Show Answers Only
Show Worked Solution

i.   

 

 
 

 

ii.   

 
 
 
 

 

iii. 

 
 
 

 

iv.   

 

 

 

 
 
 

 

v.   
 
 
 
 
 
 

 

vi.   

 
 
 

 

Binomial, EXT1 2008 HSC 6c

Let    and    be positive integers with  .

  1. Use the binomial theorem to expand  , and hence write down the term of
     
  2.  which is independent of  .  (2 marks)
     

  3. Given that
     
  4. ,
     
    apply the binomial theorem and the result of part(i) to find a simpler expression for
     
  5. .  (3 marks)

 

 

Show Answers Only
Show Worked Solution

(i)  

 

(ii)  


 


 


 

Calculus, 2ADV C3 2004 HSC 10b

2004 10b

 
The diagram shows a triangular piece of land    with dimensions   metres,  metres and    metres, where  .

The owner of the land wants to build a straight fence to divide the land into two pieces of equal area. Let    and    be points on    and    respectively so that    divides the land into two pieces of equal area.

Let   metres, metres and   metres.

  1. Show that  .  (1 mark)

  2. Use the cosine rule in triangle    to show that
     
           (2 marks)

  3. Show that the value of    in the equation in part (ii) is a minimum when
     
         .  (4 marks)

  4. Show that the minimum length of the fence is   metres, where  . 

     

    (You may assume that the value of    given in part (iii) is feasible.)  (2 marks)

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Show Worked Solution
i.    
 

 

 

ii. 

   

 

iii.  
 
   

 

 

 

iv. 

 
 
 
 
 
 
 
 
 

 

Calculus, 2ADV C3 2004 HSC 9c

Consider the function  , for  .

  1. Show that the graph of    has a stationary point at  .  (2 marks)

  2. By considering the gradient on either side of  , or otherwise, show that the stationary point is a maximum.  (1 mark)

  3. Use the fact that the maximum value of    occurs at    to deduce that    for all  .  (2 marks)

Show Answers Only
Show Worked Solution

i.     

 
 

 

 

 

ii.   

Geometry and Calculus, 2UA 2004 HSC 9c Answer

 

 

 

iii.  
   

MARKER’S COMMENT: Very challenging for most students. Successful students recognised the link to part (ii).

 

Calculus, EXT1* C1 2004 HSC 9b

A particle moves along the -axis. Initially it is at rest at the origin. The graph shows the acceleration, , of the particle as a function of time    for  .
 

2004 9b
 

  1. Write down the time at which the velocity of the particle is a maximum.  (1 marks)

  2. At what time during the interval    is the particle furthest from the origin? Give brief reasons for your answer.  (2 marks)

Show Answers Only

Show Worked Solution

i.   

 

 

ii. 

Calculus, 2ADV C4 2004 HSC 8b

The diagram shows the graph of the parabola  . The points    and    are on the parabola, and is the point where the tangent to the parabola at intersects the directrix.
 

2004 8s
 

  1. Write down the equation of the directrix of the parabola  .  (1 mark)
  2. Find the equation of the tangent to the parabola at the point .  (2 marks)
  3. Show that is the point  .  (1 mark)
  4. Given that the equation of the line is  , find the area bounded by the line and the parabola.  (2 marks)
  5. Hence, or otherwise, find the shaded area bounded by the parabola, the tangent at and the line .  (3 marks)
Show Answers Only

  4. ²
  5. ²
Show Worked Solution
(i)   
   
 
 

 

 

(ii)  
 
 
 
   

 

 

(iii)  

 

 

(iv)  
 

 

²

 

(v)  

   
 

 

 
 
 
 
 
 
 

 

 

²

  ²

Polynomials, EXT2 2015 HSC 16b

Let    be a positive integer.

  1. By considering  , show that
  3.  
  4.  
  5. Let  ,  for    (2 marks)

  6.  
  7. Show that
    1.   (2 marks)

  9.  
  10. By considering the roots of  , find the value of
    1.   (3 marks)

    2.  
  11. Prove that
    1.   (2 marks)

 

 

Show Answers Only
  4.  
Show Worked Solution

(i)   

 

 

(ii)  

 

♦♦♦ Mean mark 8%!
STRATEGY: Note that finding the correct roots gained 2 full marks in this part.

 

(iii) 

 

 

 

♦♦ Mean mark 14%.

 

(iv) 

 
 
 

Harder Ext 1 Topics, EXT2 2015 HSC 16a

  1. A table has rows and columns, creating cells as shown.
  2. Counters are to be placed randomly on the table so that there is one counter in each cell. There are identical black counters and identical white counters.
  4. Show that the probability that there is exactly one black counter in each column is   (2 marks)

  5. The table is extended to have rows and columns. There are counters, where are identical black counters and the remainder are identical white counters. The counters are placed randomly on the table with one counter in each cell.

  6. Let be the probability that each column contains exactly one black counter.

  7. Show that   (2 marks)
  10. Find   (2 marks)
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Show Worked Solution

(i) 

 

 

 

(ii) 

♦♦ Mean mark 16%.

 

 

♦♦♦ Mean mark 5%.
COMMENT: Lowest State mean mark on the 2015 exam.
(iii)  
   
   
   
   

Proof, EXT2 P1 2015 HSC 15b

Suppose that    and    is a positive integer.

  1. Show that    (2 marks) 

  2. Hence, or otherwise, show that 
     
            (2 marks)

  3. Hence, explain why
     
          (1 mark)

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Show Worked Solution
i.   
 
 

 

♦♦ Mean mark 21%.
STRATEGY: The conversion of the middle term from a fraction into a logarithm should flag the need for integration of each term.
ii.   
 
 
 

 

 

iii.   
 
♦♦ Mean mark 29%.
 

Harder Ext1 Topics, EXT2 2006 HSC 8b

For  , let  , where    is an integer and  

  1. The two points of inflection of    occur at    and  , where  . Find    and    in terms of    (4 marks)

  2. Show that
    1.   (2 marks)

  3. Using the following
  4. If    then   (DO NOT prove this)
    2. show that    (2 marks)

  5. What can be said about the ratio    as    (1 mark)

 

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Show Worked Solution
(i)
 
 
   
   

 

 

 

 

(ii)   ­
­
­

 

(iii)  

 

 

(iv)   
 

Calculus, EXT2 C1 2006 HSC 7b

  1. Let  ,  where  .  
     
    Show that     (3 marks)

  2. Hence find the exact value of
     
           (2 marks)

Show Answers Only
Show Worked Solution
STRATEGY: The choice of and was extremely important in being able to answer this question. Take note!
i.   
 

­
 
 
 
 

 

 

ii.   ­
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Harder Ext1 Topics, EXT2 2006 HSC 4d

In the acute-angled triangle    is the midpoint of    is the midpoint of    and    is the midpoint of  . The circle through    and    also cuts    at    as shown in the diagram.

Copy or trace the diagram into your writing booklet.

  1. Prove that    is a parallelogram.  (1 mark)
  2. Prove that    (1 mark)
  3. Prove that    (2 marks)
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(i)  


   

 

(ii)  

 

(iii)  
 

 

 

 

Harder Ext1 Topics, EXT2 2007 HSC 8c

The diagram shows a regular  -sided polygon with vertices  . Each side has unit length. The length    of the ‘diagonal’    where    is given by

    (Do NOT prove this.)

  1. Show, using the identity,
  2.     (Do NOT prove this.)
    1. that     (2 marks)

  5. Let    be the perimeter of the polygon and  
  6. . 
  7. Show that     (2 marks)

  8. Hence calculate the limiting value of    as    (1 mark)

 

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(i)  

 

(ii)   
   
   
MARKER’S COMMENT: Applying the result    was poorly done.

 

(iii)  
   
   
   
   

Complex Numbers, EXT2 N2 2007 HSC 8b

  1. Let    be a positive integer. Show that if    then
     
        .  (2 marks)

  2. By substituting    where  , into part (i), show that
     
       
     
              (3 marks)

  3. Suppose  .  Using part (ii), show that
     
          (2 marks)

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i.  

 
 
 

 

ii.   
   
 
   

 

 
 
 
   

 
 
 

 

iii. 

Calculus, EXT2 C1 2007 HSC 8a

  1. Using a suitable substitution, show that
     
           (1 mark)

  2. A function    has the property that  

     

    Using part (i), or otherwise, show that
          
           (2 marks)

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i. 


 

 
 
MARKER’S COMMENT: Integrating was poorly done in part (ii). Pay careful attention to this in the Worked Solution.

 

ii.