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GEOMETRY, FUR2 SM-Bank 13

Melbourne is located at (38°S, 145°E) and Dubai is located at (24°N, 55°E).

  1. Calculate the difference in longitude between Melbourne and Dubai.  (1 mark)
  2. Show that the time difference between Melbourne and Dubai is 6 hours.  (1 mark)
  3. A plane leaves Melbourne on Friday at 11.30 pm. The flight time to Dubai is 15 hours.

     

    What will be the time and the day in Dubai when the plane is due to land?  (1 marks)

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a.  

 

b.  

 

c.   
   

 

Probability, MET2 2007 VCAA 18 MC

The heights of the children in a queue for an amusement park ride are normally distributed with mean 130 cm and standard deviation 2.7 cm. 35% of the children are not allowed to go on the ride because they are too short.

The minimum acceptable height correct to the nearest centimetre is

  1. 126
  2. 127
  3. 128
  4. 129
  5. 130
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vcaa-2007-meth2-18

 

Graphs, MET2 2008 VCAA 22 MC

The graph of the function   with domain    is shown below.

VCAA 2008 22mc

Which one of the following is not true?

  1. The function is not continuous at    and  
  2. The function exists for all values of between and
  3.  for    and  
  4.  The function is positive for  
  5. The gradient of the function is not defined at  
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Probability, MET2 2008 VCAA 13 MC

According to a survey, 30% of employed women have never been married.

If 10 employed women are selected at random, the probability (correct to four decimal places) that at least 7 have never been married is

A.   

B.   

C.   

D.   

E.   

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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. 
 
  ­­
 
 
  ²

Graphs, MET1 SM-Bank 27

The graph shown is  .

  1. Write down the value of  .   (1 mark)

  2. Find the value of  .   (1 mark)

  3. Copy or trace the graph into your writing booklet.

     

    On the same set of axes, draw the graph    for  .   (2 marks)

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a.   

b. 

  

 MARKER’S COMMENT: Graphs are consistently drawn too small by many students. Aim to make your diagrams 1/3 to 1/2 of a page. 
c.

Algebra, MET1 SM-Bank 23

The function    with rule    is drawn below.

Inverse Functions, EXT1 2004 HSC 5b

  1. Copy or trace this diagram into your writing booklet.
  2. On the same set of axes, sketch    where   is the inverse function of  .   (1 mark)

  3. Find the domain of the inverse  .   (1 mark)

  4. Find an expression for    in terms of  .   (2 marks)

  5. The graphs of    and    meet at exactly one point  .Let  α  be the -coordinate of  . Explain why  α  is a root of the equation
  6.      .   (1 mark)

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a.  

Inverse Functions, EXT1 2004 HSC 5b Answer

b.  

 

c. 

 

 

 

d.   

 

α

Calculus, MET1 SM-Bank 21

The rule for function   is  .  The diagram shows the graph  .

 Inverse Functions, EXT1 2010 HSC 3b

The graph has two points of inflection. 

  1. Find the coordinates of these points.   (2 marks)

  2. Explain why the domain of must be restricted if is to have an inverse function.    (1 mark)

  3. Find the rule for the inverse function if the domain of is restricted to     (2 marks)

  4. Find the domain for .    (1 mark)

  5. Sketch the curve  .   (1 mark)

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  5. Inverse Functions, EXT1 2010 HSC 3b Answer

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a.   
 
 
   
   

 

 
   
 
   
COMMENT: It is also valid to show that is an even function and if a P.I. exists at , there must be another P.I. at .
 

 


 

 

 

 

b.  
 
 
 

 

c.  

 

 

 

 

d.  

 

e. 

Inverse Functions, EXT1 2010 HSC 3b Answer

Graphs, MET1 SM-Bank 20

The rule for   is    for  .  This function has an inverse,  .

  1. Sketch the graphs of    and    on the same set of axes. (Use the same scale on both axes.)   (2 marks)

  2. Find the rule for the inverse function  .    (2 marks)

  3. Evaluate  .    (1 mark)

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  1.  
    Inverse Functions, EXT1 2008 HSC 5a Answer
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a. 

Inverse Functions, EXT1 2008 HSC 5a Answer
b.  

 

 

 
 
 

 

 

c.   
   
   
   
   

Calculus, MET1 SM-Bank 5

The diagram shows the graph of the function    where  .

The area under    between   and    is . The area under the curve between   and   is .

The areas and are each equal to 1 square unit.

Find the values of and .  (3 marks)

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Probability, MET1 2016 VCAA 7

A company produces motors for refrigerators. There are two assembly lines, Line A and Line B. 5% of the motors assembled on Line A are faulty and 8% of the motors assembled on Line B are faulty. In one hour, 40 motors are produced from Line A and 50 motors are produced from Line B. At the end of an hour, one motor is selected at random from all the motors that have been produced during that hour.

  1. What is the probability that the selected motor is faulty? Express your answer in the form , where is a positive integer.  (2 marks)

  2. The selected motor is found to be faulty.
  3. What is the probability that it was assembled on Line A? Express your answer in the form , where is a positive integer.  (1 mark)

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a.  


 

 

♦♦ Mean mark 32%.

b.   
   
   
   

Graphs, MET1 2016 VCAA 5

Let  , where    and  , where  .

  1.   i. Find the rule for , where  .   (1 mark)

    ii. State the domain and range of .   (2 marks)

  2. iii. Show that  .   (2 marks)

  3. iv. Find the coordinates of the stationary point of and state its nature.   (2 marks)

     

  4. Let    where  .
  5.  i. Find the rule for  .   (2 marks)

  6. ii. State the domain and range of  .   (2 marks)

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  1.   i.
  2.  ii.
  3. iii.
  4. iv.
  5.  i.
  6. ii.
  7.    
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a.i.   
   

 

a.ii.  

♦♦ Mean mark part (a)(ii) 30%.
 
 

 

MARKER’S COMMENT: Many students were unsure of how to present their working in this question. Note the layout in the solution.
a.iii.  
   
   
   

 

 

 

 

a.iv.  

♦ Mean mark part (a)(iv) 41%.
MARKER’S COMMENT: Solving a fraction is zero

   

 

 

b.i.  

♦ Mean mark (b)(i) 49%.
MARKER’s COMMENT: Many students failed to consider the restrictions on the domain in and only select the negative root.

 

 

♦ Mean mark part (b)(ii) 44%.

 

b.ii.  

 

Probability, MET1 2016 VCAA 4

A paddock contains 10 tagged sheep and 20 untagged sheep. Four times each day, one sheep is selected at random from the paddock, placed in an observation area and studied, and then returned to the paddock.

  1. What is the probability that the number of tagged sheep selected on a given day is zero?  (1 mark)

  2. What is the probability that at least one tagged sheep is selected on a given day?  (1 mark)

  3. What is the probability that no tagged sheep are selected on each of six consecutive days?
  4. Express your answer in the form , where , and are positive integers.  (1 mark)

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a.  

 

 

b.   
   
   

 

c.  

 

Calculus, MET1 2016 VCAA 3

Let    where  .

  1. Sketch the graph of  . Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation.   (3 marks)
     

     

  2. Find the area enclosed by the graph of  , the lines    and  , and the -axis.   (2 marks)

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  1.  
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a.   

 

b.   
   
   
    ²

Calculus, MET1 2016 VCAA 2

Let  , where  .

  1. Find  .   (1 mark)

  2. Find the angle from the positive direction of the -axis to the tangent to the graph of   at  , measured in the anticlockwise direction.   (2 marks)

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a.   
 
   

 

♦ Mean mark 40%.
b.   
   
   

 

Calculus, MET2 2009 VCAA 2

VCAA 2009 2a

A train is travelling at a constant speed of km/h along a straight level track from towards

The train will travel along a section of track

Section passes along a bridge over a valley.

Section passes through a tunnel in a mountain.

Section is 6.2 km long.

From to , the curve of the valley and the mountain, directly below and above the train track, is modelled by the graph of
 

where and are real numbers.
 

All measurements are in kilometres.

  1. The curve defined from to passes through . The gradient of the curve at is – 0.06 and the curve has a turning point at  .
  2.  i. From this information write down three simultaneous equations in , and .   (3 marks)

  3. ii. Hence show that  , and .   (2 marks)

  4. Find, giving exact values
  5.   i. the coordinates of .   (2 marks)

  6.  ii. the length of the tunnel.   (1 mark)

  7. iii. the maximum depth of the valley below the train track.   (1 mark)

The driver sees a large rock on the track at a point , 6.2 km from . The driver puts on the brakes at the instant that the front of the train comes out of the tunnel at .

From its initial speed of km/h, the train slows down from point so that its speed km/h is given by

,

where km is the distance of the front of the train from and is a real constant.

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

  2. Find the exact distance from the front of the train to the large rock when the train finally stops.   (2 marks)

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a.i.   

a.ii.   

a.iii. 

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 a.i.  


 


 


 

  ii.   

♦ Mean mark part (a)(ii) 41%.

 


 


 

b.i.  

   

 

 

  ii.  

♦ Mean mark part (a) 41%.
 
 

 

   iii. 

♦ Mean mark part (b) 50%.


 

c. 

♦ Mean mark part (c) 35%.

 

 

 

d.  

♦ Mean mark part (d) 40%.

 

e.  

♦♦ Mean mark part (e) 32%.

 

Calculus, MET2 2009 VCAA 1

Let  

The graph of    is shown below.

VCAA 2009 1a

  1. State the interval for which the graph of is strictly decreasing.   (2 marks)

  2. Points and are the points of intersection of    with the -axis. Point has coordinates  and point has coordinates .

     

    Find the length of such that the area of rectangle is equal to the area of the shaded region.   (2 marks)
     
    VCAA 2009 1c

  3. The points and are labelled on the diagram.

      
     

          VCAA 2009 1d

     

    1. Find , the gradient of the chord .   (Exact value to be given.)   (1 mark)

    2. Find    such that  .  (Exact value to be given.)   (2 marks)

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a. 

b. 

c.i.   

c.ii. 

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a.   vcaa-2009-1ai

 

 

b.   vcaa-2009-1ci
 
 

 

c.i.  
 

 

 c.ii.