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Calculus, MET1 2018 VCAA 9

Consider a part of the graph of  , as shown below.

  1.  i. Given that  , evaluate    when is a positive even integer or 0.
      
    Give your answer in simplest form.   (2 marks)

    ii. Given that  , evaluate    when is a positive odd integer.
    Give your answer in simplest form.   (1 mark)

  2.  
  3. Find the equation of the tangent to    at the point  .   (2 marks)

  4. The translation maps the graph of    onto the graph of  , where
  6. and is a real constant.
  7. State the value of .   (1 mark)

  8. Let    and  .
    The line is the tangent to the graph of at the point and the line is the tangent to the graph of at , as shown in the diagram below.
     
         
    Find the total area of the shaded regions shown in the diagram above.   (2 marks)

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  1.  i.
    ii.
Show Worked Solution

a.i. 

♦♦ Mean mark 31%.

 

a.ii.

♦♦♦ Mean mark 19%.

 

b.   
 

 

♦ Mean mark part (b) 49%.

 
 

 

 

 

♦♦ Mean mark part (c) 34%.

c.   
   

 

 

d.   

♦♦♦ Mean mark 7%.

 

π

 


 

Calculus, MET2 2016 VCAA 1

Let  .

  1. Find the period and range of .   (2 marks)

  2. State the rule for the derivative function .   (1 mark)

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

  4. Find the equations of the tangents to the graph of    that have a gradient of 1.   (2 marks)

  5. The rule of   can be obtained from the rule of  under a transformation , such that
      

     

     

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

  6. Find the values of  , such that  .   (2 marks)

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Show Worked Solution

a.   

MARKER’S COMMENT: Including round brackets rather than square ones was a common mistake.


  

b.   
 

c.  


 

d.   

♦ Mean mark part (d) 50%.

 

e.  

♦♦ Mean mark part (e) 27%.
   

 
   

 

f.   

♦ Mean mark part (f) 50%.

Calculus, MET1 2015 VCAA 10

The diagram below shows a point, , on a circle. The circle has radius 2 and centre at the point  with coordinates . The angle is , where  .
  

met1-2015-vcaa-q10
  

The diagram also shows the tangent to the circle at . This tangent is perpendicular to and intersects the -axis at point and the -axis at point .

  1. Find the coordinates of in terms of .   (1 mark)

  2. Find the gradient of the tangent to the circle at in terms of .   (1 mark)

  3. The equation of the tangent to the circle at can be expressed as
  5.  i. Point , with coordinates , is on the line segment .
  6.     Find in terms of .   (1 mark)

  7. ii. Point , with coordinates , is on the line segment .
  8.     Find in terms of .   (1 mark)

  9. Consider the trapezium with parallel sides of length and .
  10. Find the value of for which the area of the trapezium is a minimum. Also find the minimum value of the area.   (3 marks)

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  3.  i.
  4. ii.
  6. ²
Show Worked Solution
a.   
   
 
♦♦♦ Part (a) mean mark 20%.
MARKER’S COMMENT: Many students did not include the “+2” in the -coordinate.

 

 

♦♦♦ Part (b) mean mark 16%.
b.   
   
   
     
 

 

c.i.  

 

c.ii.   

♦ Part (c)(ii) mean mark 47%.
 

 

♦♦♦ Part (d) mean mark 19%.
d.   
   
   

 

 

 
 

 

²

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.