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Probability, MET1 2006 VCAA 5

Let be a normally distributed random variable with a mean of 72 and a standard deviation of 8. Let be the standard normal random variable. Use the result that , correct to two decimal places, to find

  1. the probability that is greater than  (1 mark)

  2. the probability that    (1 mark)

  3. the probability that    given that    (2 marks)

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a.   vcaa-2006-meth-5ai

 

b.  

♦ Mean mark 45%.

 

♦ Mean mark 40%.
MARKER’S COMMENT: Notation was poor, showing a lack of understanding in this area.

c.  

 
 

Probability, MET1 2007 VCAA 11

There is a daily flight from Paradise Island to Melbourne. The probability of the flight departing on time, given that there is fine weather on the island, is 0.8, and the probability of the flight departing on time, given that the weather on the island is not fine, is 0.6.

In March the probability of a day being fine is 0.4.

Find the probability that on a particular day in March

  1. the flight from Paradise Island departs on time  (2 marks)

  2. the weather is fine on Paradise Island, given that the flight departs on time.  (2 marks)

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Show Worked Solution

a.   

 

 

b.  

♦♦ Mean mark 29%.
MARKER’S COMMENT: Students continue to struggle with conditional probability. Attention required here.

 

 

Calculus, MET1 2007 VCAA 9

The graph of  ,    is shown. The normal to the graph of where it crosses the -axis is also shown.
 

MET1 2007 VCAA Q9
 

  1. Find the equation of the normal to the graph of where it crosses the -axis.   (2 marks)

  2. Find the exact area of the shaded region.   (3 marks)

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  2. ²
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a.  

MARKER’S COMMENT: A common error was made in calculating the point of tangency. Be careful!

  

  

 

♦ Mean mark 44%.
b.   
   
   
   
    ²

Functions, MET1 2007 VCAA 8

Let  ,  .

  1. Solve the equation    for  .   (2 marks)

  2. Let  ,  .
  3. Find the smallest positive value of for which is a maximum.   (2 marks)

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a.   

b.   

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a.   

 

  

b.  

♦♦ Mean mark 29%.
STRATEGY: Max/min questions involving trig functions can often use the powerful identity, to solve efficiently and reduce errors.

Functions, MET1 2008 VCAA 10

Let  .

  1. Find the rule and domain of the inverse function  .   (2 marks)

  2. On the axes provided, sketch the graph of    for its maximal domain.   (1 mark)


     

    met1-2008-vcaa-q2
     

  3. Find    in the form    where , and are real constants.   (2 marks)

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  2.  
    met1-2008-vcaa-q10-answer
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a.  

 

 


  

b.  

♦ Mean mark part (b) 19%.
MARKER’S COMMENT: Few students were aware that a function of its own inverse function is the line    over the appropriate domain.

met1-2008-vcaa-q10-answer

 

♦ Mean mark 34%.
c.   
 
   
   
   

Calculus, MET1 2008 VCAA 9

A plastic brick is made in the shape of a right triangular prism. The triangular end is an equilateral triangle with side length cm and the length of the brick is cm.
 

met1-2008-vcaa-q91

The volume of the brick is 1000 cm³.

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

  2. Show that the total surface area, cm², of the brick is given by
  3.       (2 marks)

  4. Find the value of for which the brick has minimum total surface area. (You do not have to find this minimum.)   (3 marks)

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a.  

 

 

 

♦ Mean mark 45%.
b.    
   
   
   

 

♦ Mean mark 42%.

c.  

 

Probability, MET1 2008 VCAA 8

Every Friday Jean-Paul goes to see a movie. He always goes to one of two local cinemas – the Dandy or the Cino.

If he goes to the Dandy one Friday, the probability that he goes to the Cino the next Friday is 0.5. If he goes to the Cino one Friday, then the probability that he goes to the Dandy the next Friday is 0.6.

On any given Friday the cinema he goes to depends only on the cinema he went to on the previous Friday.

If he goes to the Cino one Friday, what is the probability that he goes to the Cino on exactly two of the next three Fridays?  (3 marks)

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Show Worked Solution

MARKER’S COMMENT:
Most efficient strategy here was listing the three possibilities – not a tree diagram.

 

 

Probability, MET1 2008 VCAA 7

Jane drives to work each morning and passes through three intersections with traffic lights. The number of traffic lights that are red when Jane is driving to work is a random variable with probability distribution given b

met1-2008-vcaa-q7

  1. What is the mode of ?   (1 mark)

  2. Jane drives to work on two consecutive days. What is the probability that the number of traffic lights that are red is the same on both days?   (2 marks)

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a.  
 

b.  

Probability, MET1 2008 VCAA 4

The function
 


 

is a probability density function for the continuous random variable .

  1. Show that  .  (2 marks)

  2. Find  .  (3 marks)

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a.   ²

 

b.  

♦ Mean mark 47%.
MARKER’S COMMENT: Few students used the symmetry of the probability density function to calculate the denominator.

Probability, MET1 2009 VCAA 5

Four identical balls are numbered 1, 2, 3 and 4 and put into a box. A ball is randomly drawn from the box, and not returned to the box. A second ball is then randomly drawn from the box.

  1. What is the probability that the first ball drawn is numbered 4 and the second ball drawn is numbered 1?  (1 mark)

  2. What is the probability that the sum of the numbers on the two balls is 5?  (1 mark)

  3. Given that the sum of the numbers on the two balls is 5, what is the probability that the second ball drawn is numbered 1?  (2 marks)

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a.  

 

b.  

 

c.  

♦ Mean mark 46%.

Probability, MET1 2010 VCAA 7

The continuous random variable has a distribution with probability density function given by
 


 

where is a positive constant.

  1. Find the value of .  (3 marks)

  2. Express    as a  definite integral. (Do not evaluate the definite integral.)  (1 mark)

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a.  
 
 
 
 
 

 

b.  

Statistics, MET1 2010 VCAA 5

Let be a normally distributed random variable with mean 5 and variance 9 and let be the random variable with the standard normal distribution.

  1. Find  .  (1 mark)

  2. Find such that  .  (2 marks)

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a.  

 

♦ Part (b) mean mark 43%.

b.  
   
   

 

Calculus, MET1 2011 VCAA 10

The figure shown represents a wire frame where is a convex quadrilateral. The point is on line segment with   and  , where is a positive constant.

Let    where  

 vcaa-2011-meth-10a

  1. Find and in terms of and .   (2 marks)

  2. Find the length, cm, of the wire in the frame, including length , in terms of and .   (1 mark)

  3. Find  , and hence show that    when  .   (2 marks)

  4. Find the maximum value of if  .   (1 mark)

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a.  

 

b.  
   
   

 

c.  

♦ Mean mark 35%.

,

 

d.   

♦♦♦ Mean mark 5%.

vcaa-2011-meth-10ai

 
 
 

Probability, MET1 2011 VCAA 7

A biased coin tossed three times. The probability of a head from a toss of this coin is

  1. Find, in terms of , the probability of obtaining

    1. three heads from the three tosses  (1 mark)

    2. two heads and a tail from the three tosses.  (1 mark)

  2. If the probability of obtaining three heads equals the probability of obtaining two heads and a tail, find .  (2 marks)

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a.i.   
   

 

  ii.  

♦ Part (a)(ii) mean mark 41%.

 

b.  

♦ Mean mark 48%.
MARKER’S COMMENT: Many students incorrectly assumed could not be zero.

Calculus, MET1 2012 VCAA 10

Let  ,  where is a positive rational number.

  1.  i. Find, in terms of , the -coordinate of the stationary point of the graph of     (2 marks)

  2. ii. State the values of such that the -coordinate of this stationary point is a positive number.   (1 mark)

  3. For a particular value of , the tangent to the graph of    at    passes through the origin.
  4. Find this value of .   (3 marks)

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  1.  i.
  2. ii.
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a.i.  

 

 

♦♦♦ Part (a.ii.) mean mark 18%.
  ii.   
 
 
 

 

b.  

♦♦ Mean mark (b) 33%.
MARKER’S COMMENT: Many confused with the more common use of for gradient in  .

 

 

 

Probability, MET1 2012 VCAA 4

On any given day, the number of telephone calls that Daniel receives is a random variable with probability distribution given by

vcaa-2012-meth-4

  1. Find the mean of .   (2 marks)

  2. What is the probability that Daniel receives only one telephone call on each of three consecutive days?   (1 mark)

  3. Daniel receives telephone calls on both Monday and Tuesday.
  4. What is the probability that Daniel receives a total of four calls over these two days?   (3 marks)

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a.   
   
   

 

b.  

MARKER’S COMMENT: Many students understood that the calculation of 0.2³ was required but were not able evaluate correctly.

 

c.  

♦ Mean mark 36%.

Calculus, MET1 2013 VCAA 10

Let  

A right-angled triangle has vertex at the origin, vertex on the -axis and vertex on the graph of , as shown. The coordinates of are
 

 vcaa-2013-meth-10

  1. Find the area, , of the triangle in terms of .   (1 mark)

  2. Find the maximum area of triangle and the value of for which the maximum occurs.   (3 marks)

  3. Let be the point on the graph of on the -axis and let be the point on the graph of with the -coordinate .

     

    Find the area of the region bounded by the graph of and the line segment .   (3 marks)

     

    vcaa-2013-meth-10i

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  2. ²
  3. ²
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a.  
   
 

 

b.  

♦ Mean mark 35%.
 

²

 

c.  

♦♦ Mean mark 32%.


 

 
 
 

 

vcaa-2013-meth-10ii

 
 
 
  ²

Probability, MET1 2013 VCAA 7

The probability distribution of a discrete random variable, , is given by the table below

vcaa-2013-meth-7

  1. Show that    or  .   (3 marks)

  2. Let  .

    1. Calculate .   (2 marks)

    2. Find  .   (1 mark)

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a.   

 

b.i.  
   
   
   

 

♦♦ Part (b)(ii) mean mark 32%.

  ii.  
   
   

Probability, MET1 2014 VCAA 9

Sally aims to walk her dog, Mack, most mornings. If the weather is pleasant, the probability that she will walk Mack is , and if the weather is unpleasant, the probability that she will walk Mack is .

Assume that pleasant weather on any morning is independent of pleasant weather on any other morning.

  1. In a particular week, the weather was pleasant on Monday morning and unpleasant on Tuesday morning.
  2. Find the probability that Sally walked Mack on at least one of these two mornings.  (2 marks)

  3. In the month of April, the probability of pleasant weather in the morning was .
  4.  i. Find the probability that on a particular morning in April, Sally walked Mack.  (2 marks)

  5. ii. Using your answer from part b.i., or otherwise, find the probability that on a particular morning in April, the weather was pleasant, given that Sally walked Mack that morning.  (2 marks)

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a.   
   
   

 

b.i.  
 

met1-2014-vcaa-q9-answer1 
 

 

 

♦ Part (b)(ii) mean mark 38%.

b.ii.   
   
   
   

Probability, MET1 2014 VCAA 8

A continuous random variable, , has a probability density function given by
 


 

The median of is  .

  1. Determine the value of  .  (2 marks)
  2. The value of is a number greater than 1.

     

    Find .  (2 marks)

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a.   
 
 
 
 
 

 

 

b.  

♦ Part (b) mean mark 38%.
MARKER’S COMMENT: A common error was assuming obtained in part (a) was equivalent to .
 
 
 
 

Calculus, MET1 2014 VCAA 5

Consider the function  ,  .

  1. Find the coordinates of the stationary points of the function.   (2 marks)

  2. On the axes  below, sketch the graph of .

     

    Label any end points with their coordinates.   (2 marks)

     

     
        met1-2014-vcaa-q5

     

  3. Find the area enclosed by the graph of the function and the horizontal line given by  .   (3 marks)

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  2.  
    met1-2014-vcaa-q5-answer3
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a.   
   
 


 

 

b.    met1-2014-vcaa-q5-answer3

 

♦ Mean mark (c) 48%.
c.    met1-2014-vcaa-q5-answer4

 
 
 
   
²

 

 
 
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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
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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

Calculus, MET2 2010 VCAA 4

Consider the function  

  1. Find the -coordinate of each of the stationary points of   and state the nature of each of these stationary points.   (4 marks)

In the following, is the function   where and are real constants.

  1. Write down, in terms of and , the possible values of for which is a stationary point of .   (3 marks)

  2. For what value of does have no stationary points?   (1 mark)

  3. Find in terms of if has one stationary point.   (2 marks)

  4. What is the maximum number of stationary points that can have?  (1 mark)

  5. Assume that there is a stationary point at and another stationary point  where  .
  6. Find the value of .   (3 marks)

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a.  

 


 

 

vcaa-graphs-fur2-2010-4ai


 

b.  

♦ Mean mark (b) 46%.
MARKER’S COMMENT: Issues with use of CAS caused significant difficulties in this question.

 


 

c.  

♦ Mean mark part (c) 47%.


 

d.  

♦♦♦ Mean mark (d) 16%.

 

♦ Mean mark (e) 37%.

e.   

 

f.    


 

♦♦♦ Mean mark (f) 9%.

 

   


 

 

Calculus, MET2 2010 VCAA 3

An ancient civilisation buried its kings and queens in tombs in the shape of a square-based pyramid,

The kings and queens were each buried in a pyramid with  

Each of the isosceles triangle faces is congruent to each of the other triangular faces.

The base angle of each of these triangles is , where  

Pyramid and a face of the pyramid, , are shown here.
 

VCAA 2010 3a

is the midpoint of

  1. i. Find in terms of .   (1 mark)

  2. ii. Find in terms of .   (1 mark)

  3. Show that the total surface area (including the base), , of the pyramid, , is given by 
  4.      .   (2 marks)

  5. Find , the height of the pyramid , in terms of .    (2 marks)

  6. The volume of any pyramid is given by the formula  .
  7. Show that the volume, ³, of the pyramid  is  .   (1 mark)

Queen Hepzabah’s pyramid was designed so that it had the maximum possible volume.

  1. Find    and hence find the exact volume of Queen Hepzabah’s pyramid and the corresponding value of .   (4 marks)

Queen Hepzabah’s daughter, Queen Jepzibah, was also buried in a pyramid. It also had

The volume of Jepzibah’s pyramid is exactly one half of the volume of Queen Hepzabah’s pyramid. The volume of Queen Jepzibah’s pyramid is also given by the formula for obtained in part d.

  1. Find the possible values of , for Jepzibah’s pyramid, correct to two decimal places.   (2 marks)

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a.i.

a.ii.

b.   

c.   

d.   

e.   

f.   

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a.i.  
 

 

  ii.  
 

 

b.  
   
 
   
♦ Mean mark 47%.

 

 

 

c.  

 vcaa-graphs-fur2-2010-ci

 
♦♦♦ Mean mark part (d) 22%.

 

d.  
   
   

 

 

 

e.  

♦ Mean mark part (e) 45%.

 

  ³

♦♦♦ Mean mark part (f) 16%.

 

f.  

Probability, MET1 2015 VCAA 9

An egg marketing company buys its eggs from farm A and farm B. Let be the proportion of eggs that the company buys from farm A. The rest of the company’s eggs come from farm B. Each day, the eggs from both farms are taken to the company’s warehouse.

Assume that  of all eggs from farm A have white eggshells and of all eggs from farm B have white eggshells.

  1. An egg is selected at random from the set of all eggs at the warehouse.
  2. Find, in terms of , the probability that the egg has a white eggshell.  (1 mark)

  3. Another egg is selected at random from the set of all eggs at the warehouse.

     

    1. Given that the egg has a white eggshell, find, in terms of , the probability that it came from farm B.  (2 marks)

    2. If the probability that this egg came from farm B is 0.3, find the value of .  (1 mark)

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a.  

met1-2015-vcaa-q9-answer1

 

 
 

 

♦ Part (b) mean mark 41%.
MARKER’S COMMENT: Algebraic fractions “were not handled well”!

b.i.   
   
   
   

 

♦♦♦ Part (c) mean mark 19%.
STRATEGY: Previous parts of a question are gold dust for directing your strategy in many harder questions.

b.ii.   
 
 
 
 

Probability, MET1 2015 VCAA 8

For events  and  from a sample space,  and  .

  1. Calculate  .   (1 mark)

  2. Calculate  , where denotes the complement of .   (1 mark)

  3. If events and are independent, calculate  .   (1 mark)

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a.  

 

b.    met1-2015-vcaa-q8-answer
 
 

 

c.  

♦♦ Mean mark 28%.
MARKER’S COMMENT: A lack of understanding of independent events was clearly evident.

 

 

Functions, MET1 2015 VCAA 5

On any given day, the depth of water in a river is modelled by the function

where is the depth of water, in metres, and    is the time, in hours, after 6 am. 

  1. Find the minimum depth of the water in the river.   (1 mark)

  2. Find the values of    for which  .   (2 marks)

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a.   

MARKER’S COMMENT: Students who used calculus to find the minimum were less successful.
 

 

b.   
 

 

 

 

Calculus, MET1 2015 VCAA 4

Consider the function  .

  1. Find the coordinates of the stationary points of the function.   (2 marks)

The rule for   can also be expressed as  .

  1. On the axes below, sketch the graph of  , clearly indicating axis intercepts and turning points.

     

    Label the end points with their coordinates.   (2 marks)


      

    met1-2015-vcaa-q4
     

  2. Find the average value of   over the interval    (2 marks)

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  2.  

     

    met1-2015-vcaa-q4-answer

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a.  

 

 

 

b.    met1-2015-vcaa-q4-answer
♦ Part (c) mean mark 50%.
MARKER’S COMMENT: Most students recalled the average value definition but then did not integrate correctly.

 

c.   
   
   
   
   

CORE*, FUR2 2006 VCAA 3

The company prepares for this expenditure by establishing three different investments.

  1. $7000 is invested at a simple interest rate of 6.25% per annum for eight years.
  2. Determine the total value of this investment at the end of eight years.   (2 marks)
  3. $10 000 is invested at an interest rate of 6% per annum compounding quarterly for eight years.
  4. Determine the total value of this investment at the end of eight years.
  5. Write your answer correct to the nearest dollar.   (1 mark)
  6. $500 is deposited into an account with an interest rate of 6.5% per annum compounding monthly.
  7. Deposits of $200 are made to this account on the last day of each month after interest has been paid.
  8. Determine the total value of this investment at the end of eight years.
  9. Write your answer correct to the nearest dollar.   (1 mark) 

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a.   
   
   

  


    

b.  

 

  


  

c.   

   

CORE*, FUR2 2006 VCAA 2

It is estimated that inflation will average 2% per annum over the next eight years.

If a new machine costs $60 000 now, calculate the cost of a similar new machine in eight years time, adjusted for inflation. Assume no other cost change.

Write your answer correct to the nearest dollar.   (1 mark)

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CORE*, FUR2 2006 VCAA 1

A company purchased a machine for $60 000.

For taxation purposes the machine is depreciated over time.

Two methods of depreciation are considered.

  1. Flat rate depreciation

    The machine is depreciated at a flat rate of 10% of the purchase price each year.

    i.      By how many dollars will the machine depreciate annually?   (1 mark)

    ii.  Calculate the value of the machine after three years.   (1 mark)

  2. iii. After how many years will the machine be $12 000 in value?   (1 mark)

  3. Reducing balance depreciation

    The value, , of the machine after years is given by the formula .

    i.      By what percentage will the machine depreciate annually?   (1 mark)

    ii.  Calculate the value of the machine after three years.   (1 mark)

  4. iii. At the end of which year will the machine's value first fall below $12 000?   (1 mark)

  1. At the end of which year will the value of the machine first be less using flat rate depreciation than it will be using reducing balance depreciation?  (2 marks)

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  1. i.   
    ii. 
    iii.
  2. i.    
    ii. 
    iii.
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a.i.   
   

a.ii.  

 

a.iii.  

 

 

b.i.   
   


  

b.ii.  

 

 

b.iii.  

  


  

c.   

BUSINESS, FUR2 2006 VCAA 1 Answer

CORE*, FUR2 2007 VCAA 3

Khan paid $900 for a fax machine.

This price includes 10% GST (goods and services tax).

  1. Determine the price of the fax machine before GST was added. Write your answer correct to the nearest cent.  (1 mark)

  2. Khan will depreciate his $900 fax machine for taxation purposes.
  3. He considers two methods of depreciation.
  4. Flat rate depreciation
  5. Under flat rate depreciation the fax machine will be valued at $300 after five years.
    1. Calculate the annual depreciation in dollars.   (1 mark)
    2. ii. Determine the value of the fax machine after five years.   (1 mark)

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  2. i.
    ii.  

Show Worked Solution

a.  

MARKER’S COMMENT: Reverse GST questions continue to cause problems for many students.

 
 

  
b.i.  

 

b.ii.