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Calculus, MET2 2021 VCAA 5

Part of the graph of    is shown below.
 

  1. State the period of .   (1 mark)

  2. State the minimum value of , correct to three decimal places.   (1 mark)

  3. Find the smallest positive value of for which  .   (1 mark)
Consider the set of functions of the form  , where is a positive integer.
  1. State the value of such that    for all .   (1 mark)

  2. i.  Find an antiderivative of in terms of .   (1 mark)

  3. ii. Use a definite integral to show that the area bounded by and the -axis over the interval    is equal above and below the -axis for all values of .  (3 marks)

  4. Explain why the maximum value of cannot be greater than 2 for all values of and why the minimum value of cannot be less than –2 for all values of .   (1 mark)

  5. Find the greatest possible minimum value of .   (1 mark)

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  5. i. 
    ii.
Show Worked Solution

a.   


 

b.   

♦ Mean mark part (c) 21%.
 

c.     


 

d.   


 

e.i. 

♦ Mean mark part (e)(i) 50%.


 

e.ii.   

♦ Mean mark part (e)(ii) 29%.


 


 

f.   

♦ Mean mark part (f) 13%.


 

g.   

♦ Mean mark part (g) 2%.

Trigonometry, MET2-NHT 2019 VCAA 2

The wind speed at a weather monitoring station varies according to the function

where is the speed of the wind, in kilometres per hour (km/h), and    is the time, in minutes, after 9 am.

  1. What is the amplitude and the period of  ?   (2 marks)

  2. What are the maximum and minimum wind speeds at the weather monitoring station?   (1 mark)

  3. Find  , correct to four decimal places.   (1 mark)

  4. Find the average value of    for the first 60 minutes, correct to two decimal places.   (2 marks)

A sudden wind change occurs at 10 am. From that point in time, the wind speed varies according to the new function

                   

where    is the speed of the wind, in kilometres per hour, is the time, in minutes, after 9 am and  . The wind speed after 9 am is shown in the diagram below.
 

  1. Find the smallest value of , correct to four decimal places, such that    and    are equal and are both increasing at 10 am.   (2 marks)

  2. Another possible value of was found to be 31.4358

     

    Using this value of , the weather monitoring station sends a signal when the wind speed is greater than 38 km/h.

     

    i.  Find the value of at which a signal is first sent, correct to two decimal places.   (1 mark)

     

    ii. Find the proportion of one cycle, to the nearest whole percent, for which  .   (2 marks)

  3. Let    and  .
     
    The transformation    maps the graph of    onto the graph of  .

     

    State the values of  , , and , in terms of where appropriate.   (3 marks)

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  2.  

  6. i.

     

    ii.

Show Worked Solution

a.   

 

b.   

 

c.   
 

 

d.   

 
 

 

e.   

 

f.i. 


 

f.ii. 

 
 

 

g.   

 


 


 

 

Calculus, MET2 2019 VCAA 3

During a telephone call, a phone uses a dual-tone frequency electrical signal to communicate with the telephone exchange.

The strength, , of a simple dual-tone frequency signal is given by the function  , where    is a measure of time and  .

Part of the graph of   is shown below

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

  2. Find the values of    where    for the interval  .   (1 mark)

  3. Find the maximum strength of the dual-tone frequency signal, correct to two decimal places.   (1 mark)

  4. Find the area between the graph of    and the horizontal axis for  .   (2 marks)

Let    be the function obtained by applying the transformation    to the function  , where
 


 

and and are real numbers.

  1. Find the values of and given that    has the same area calculated in part d.   (2 marks)

  2. The rectangle bounded by the line  , the horizontal axis, and the lines    and    has the same area as the area between the graph of    and the horizontal axis for one period of the dual-tone frequency signal.

     

    Find the value of  .   (2 marks)

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  4. ²
Show Worked Solution

a.   
  

b.   
  

c.   

 

d.   
    ²

 

e.   


  

f.   
 
 

Algebra, MET2 2017 VCAA 2

Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point . The height of above the ground, , is modelled by  , where is the time in minutes after Sammy enters the capsule and is measured in metres.

Sammy exits the capsule after one complete rotation of the Ferris wheel.
 


 

  1. State the minimum and maximum heights of above the ground.   (1 mark)

  2. For how much time is Sammy in the capsule?   (1 mark)

  3. Find the rate of change of with respect to and, hence, state the value of at which the rate of change of is at its maximum.   (2 marks)

As the Ferris wheel rotates, a stationary boat at , on a nearby river, first becomes visible at point . is 500 m horizontally from the vertical axis through the centre of the Ferris wheel and angle , as shown below.
 

   
 

  1. Find in degrees, correct to two decimal places.   (1 mark)

Part of the path of is given by  , where and are in metres.

  1. Find .   (1 mark)

As the Ferris wheel continues to rotate, the boat at is no longer visible from the point  onwards. The line through and is tangent to the path of , where angle .
 

   
 

  1. Find the gradient of the line segment in terms of and, hence, find the coordinates of , correct to two decimal places.   (3 marks)

  2. Find in degrees, correct to two decimal places.   (1 mark)

  3. Hence or otherwise, find the length of time, to the nearest minute, during which the boat at is visible.   (2 marks)

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Show Worked Solution
a.   
     

 

b.   

 

c.   

♦ Mean mark 50%.
MARKER’S COMMENT: A number of commons errors here – 2 answers given, calc not in radian mode, etc …

 

   

 

d.   

♦ Mean mark 36%.
MARKER’S COMMENT: Choosing degrees vs radians in the correct context was critical here.

 

 

e.   

 

f.   

 

♦♦♦ Mean mark part (f) 18%.
MARKER’S COMMENT: Many students were unable to use the rise over run information to calculate the second gradient.

 

 

 

 
 

 

 

♦♦♦ Mean mark part (g) 7%.

g.   
   
 

 

h.   

♦♦♦ Mean mark 5%.

 

 
 

Algebra, MET2 2014 VCAA 1

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

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

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

  3. Find  .   (1 mark)

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

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Show Worked Solution

a.   

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


  

b.  


  

c.  
   

d.   

♦ 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. 
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.