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Calculus, MET1-NHT 2018 VCAA 7

Let    and  .

  1. Sketch the graph of and the graph of on the axes provided below.   (2 marks)

     
     

     

  2. Let be such that  ,  where 

     
    Find the value of    and the value of  .   (3 marks)

     

  3.  Let be the region enclosed by the horizontal axis, the graph of and the graph of .
    1. Shade the region on the axes provided in part a. and also label the position of on the horizontal axis.   (1 mark)

    2. Calculate the area of the region .   (3 marks)

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  1.  

     

  2.  

  3. i.

     
    ii.
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a.   

 

b.   

 

c. i.

 

   ii.       
 
 
 
 
 
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Graphs, MET1 2019 VCAA 4

  1. Solve    for  .   (2 marks) 

  2. The function    is shown on the axes below.
     

     

     

    Let  .

     

     

    Sketch the graph of    on the axes above. Label all points of intersection of the graphs of    and  , and the endpoints of  , with their coordinates.   (2 marks)

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a.   
 

 

 

b.   

Graphs, MET1 2018 VCAA 3

Let  .

  1. Solve the equation    for  .   (2 marks)

  2. Sketch the graph of the function  on the axes below. Label the endpoints and local minimum point with their coordinates.   (3 marks)

 

 

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  2.  
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a.
 

 

 

b.   

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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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%.

 

 
 

Calculus, MET1 SM-Bank 28

The function    has the rule  .

  1. Show that the graph of    cuts the -axis at  .   (1 mark)

  2. Sketch the graph    for    showing where the graph cuts each of the axes.   (2 marks)

  3. Find the area under the curve    between    and  .   (3 marks)

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  1.  
  2.  
       
  3. ²
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a.   

 

 

 

b.   2UA HSC 2006 7b

 

c. 
 
  ­­
 
 
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Functions, MET1 2006 VCAA 4

For the function  

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

  2. sketch the graph of the function on the set of axes below. Label axes intercepts with their coordinates.

     

    Label endpoints of the graph with their coordinates.   (3 marks)


VCAA 2006 meth 4b

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  2.  
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a.  

 

b.  

Functions, MET1 2012 VCAA 6

The graphs of  ,  where is a real constant, have a point of intersection at  

  1. Find the value of .  (2 marks)

  2. If  , find the -coordinate of the other point of intersection of the two graphs.  (1 mark)

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a.  

 

b.  
 
 

Algebra, MET2 2014 VCAA 1

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

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

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

  3. Find  .   (1 mark)

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

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a.   

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


  

b.  


  

c.  
   

d.   

♦ Mean mark 48%.

 

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.