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Statistics, STD2 S1 2017 HSC 27a

Jamal surveyed eight households in his street. He asked them how many kilolitres (kL) of water they used in the last year. Here are the results.

  1. Calculate the mean of this set of data.  (1 mark)

  2. What is the standard deviation of this set of data, correct to one decimal place?  (1 mark)

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i.  
   
♦ Mean mark part (ii) 47%.
IMPORTANT: The population standard deviation is required here.

 

ii.  
   

Measurement, STD2 M1 2017 HSC 25 MC

In the circle, centre , the area of the quadrant is 100 cm².
 


 

What is the arc length , correct to one decimal place?

  1. 8.9 cm
  2. 11.3 cm
  3. 17.7 cm
  4. 25.1 cm
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♦ Mean mark 44%.

 

 
 
 

 

Algebra, STD2 A2 2017 HSC 20 MC

A pentagon is created using matches.

By adding more matches, a row of two pentagons is formed.

Continuing to add matches, a row of three pentagons can be formed.

Continuing this pattern, what is the maximum number of complete pentagons that can be formed if 100 matches in total are available?

A.    

B.    

C.    

D.    

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Probability, STD2 S2 2017 HSC 15 MC

The faces on a twenty-sided die are labelled  $0.05, $0.10, $0.15, … , $1.00.

The die is rolled once.

What is the probability that the amount showing on the upper face is more than 50 cents but less than 80 cents?

A.     

B.     

C.     

D.     

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Show Worked Solution

♦ Mean mark 50%.

 

GEOMETRY, FUR1 SM-Bank 35 MC

Kim lives in Perth (32°S, 115°E). He wants to watch an ice hockey game being played in Toronto  (44°N, 80°W)  starting at 10.00 pm on Wednesday.

What is the time in Perth when the game starts?

A.   9.00 am on Wednesday

B.   7.40 pm on Wednesday 

C.   9.00 pm on Wednesday

D.   12.20 am on Thursday

E.   11.00 am on Thursday

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Show Worked Solution


 

 

 


 

GEOMETRY, FUR2 SM-Bank 15

Osaka is at  34°N, 135°E, and Denver is at  40°N, 105°W. 

  1. Show that there is a 16-hour time difference between the two cities.
    (Ignore time zones.)    (1 mark)
  2. John lives in Denver and wants to ring a friend in Osaka. In Denver it is 9 pm Monday.     

     

    What time and day is it in Osaka then?   (1 mark)

  3. John’s friend in Osaka sent him a text message which happened to take 14 hours to reach him. It was sent at 10 am Thursday, Osaka time.

     

    What was the time and day in Denver when John received the text?    (1 mark)

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a.  
   

 

 

 

b. 

 

 

c. 

GEOMETRY, FUR2 SM-Bank 14

Pontianak has a longitude of 109°E, and Jarvis Island has a longitude of 160°W.

Both places lie on the Equator

  1. Find the shortest great circle distance between these two places, to the nearest kilometre. You may assume that the radius of the Earth is 6400 km.    (2 marks)
  2. The position of Rabaul is 4° to the south and 48° to the west of Jarvis Island. What is the latitude and longitude of Rabaul?     (1 mark)
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a.  
   

 

 

 

 
 
 

 

b.  
 
 
 
   
 
 
 
 

 

 

 

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
Show Worked Solution

a.   

 

 

 

b.   2UA HSC 2006 7b

 

c. 
 
  ­­
 
 
  ²

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

Probability, MET1 2016 VCAA 8*

Let be a continuous random variable with probability density function

Part of the graph of   is shown below. The graph has a turning point at  .

  1. Show by differentiation that    is an antiderivative of  , where is a positive real number.   (2 marks)

  2. Calculate .   (2 marks)

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  2.  

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a.  

♦♦ Mean mark part (a) 28%.
MARKER’S COMMENT: Students who expanded before differentiating tended to score more highly.


 

b.   

♦♦♦ Mean mark 16%.

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.    
Show Worked Solution
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.  

 

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.   
   
   

 

Probability, MET2 2009 VCAA 3

The Bouncy Ball Company (BBC) makes tennis balls whose diameters are normally distributed with mean 67 mm and standard deviation 1 mm. The tennis balls are packed and sold in cylindrical tins that each hold four balls. A tennis ball fits into such a tin if the diameter of the ball is less than 68.5 mm.

  1. What is the probability, correct to four decimal places, that a randomly selected tennis ball produced by BBC fits into a tin?  (2 marks)

BBC management would like each ball produced to have diameter between 65.6 and 68.4 mm.

  1. What is the probability, correct to four decimal places, that the diameter of a randomly selected tennis ball made by BBC is in this range?  (2 marks)

    1. What is the probability, correct to four decimal places, that the diameter of a tennis ball which fits into a tin is between 65.6 and 68.4 mm?  (1 mark)

    2. A tin of four balls is selected at random. What is the probability, correct to four decimal places, that at least one of these balls has diameter outside the desired range of 65.6 to 68.4 mm?  (2 marks)

BBC management wants engineers to change the manufacturing process so that 99% of all balls produced have diameter between 65.6 and 68.4 mm. The mean is to stay at 67 mm but the standard deviation is to be changed.

  1. What should the new standard deviation be (correct to two decimal places)?  (3 marks)

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a. 

b. 

c.i. 

c.ii. 

d. 

Show Worked Solution

a.  

 

b.  

 

c.i.  

♦ Mean mark 42%.

 

  ii. 

♦♦ Mean mark 30%.

 

d.  

♦♦ Mean mark 35%.

 vcaa-graphs-fur2-2009-3di

 

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. 

Show Worked Solution

 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 2011 VCAA 4

Deep in the South American jungle, Tasmania Jones has been working to help the Quetzacotl tribe to get drinking water from the very salty water of the Parabolic River. The river follows the curve with equation  ,   as shown below. All lengths are measured in kilometres.

Tasmania has his camp site at and the Quetzacotl tribe’s village is at . Tasmania builds a desalination plant, which is connected to the village by a straight pipeline.
 

met2-2011-vcaa-q4

  1. If the desalination plant is at the point  show that the length, kilometres, of the straight pipeline that carries the water from the desalination plant to the village is given by
  2.    .   (3 marks)

  3. If the desalination plant is built at the point on the river that is closest to the village
    1. find and hence find the coordinates of the desalination plant.   (3 marks)

    2. find the length, in kilometres, of the pipeline from the desalination plant to the village.   (2 marks)

The desalination plant is actually built at .

If the desalination plant stops working, Tasmania needs to get to the plant in the minimum time.

Tasmania runs in a straight line from his camp to a point on the river bank where  . He then swims up the river to the desalination plant.

Tasmania runs from his camp to the river at 2 km per hour. The time that he takes to swim to the desalination plant is proportional to the difference between the -coordinates of the desalination plant and the point where he enters the river.

  1. Show that the total time taken to get to the desalination plant is given by

     

    hours where is a positive constant of proportionality.   (3 marks)

The value of varies from day to day depending on the weather conditions.

  1. If  
    1. find    (1 mark)

    2. hence find the coordinates of the point where Tasmania should reach the river if he is to get to the desalination plant in the minimum time.   (2 marks)

  2. On one particular day, the value of is such that Tasmania should run directly from his camp to the point on the river to get to the desalination plant in the minimum time. Find the value of on that particular day.   (2 marks)

  3. Find the values of for which Tasmania should run directly from his camp towards the desalination plant to reach it in the minimum time.   (2 marks)

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  2.  i.
  3. ii.
  5.  i.
  6. ii.
Show Worked Solution

a.   

♦ Mean mark (a) 47%.

 
 

 

b.i.  


 

♦ Mean mark part (b)(ii) 41%.
b.ii.  
   
   

 

c.  

♦♦♦ Mean mark (c) 16%.

 
 

 

d.i.  

 

d.ii.  

♦♦ Mean mark (d.ii.) 33%.

 

e. 

♦♦ Mean mark (e) 39%.

 

f.  

♦♦♦ Mean mark (f) 13%.


 

Calculus, MET2 2011 VCAA 3

  1. Consider the function  .

     

    1. Find     (1 mark)

    2. Explain why   for all .   (1 mark)

  2. The cubic function is defined by  , where , , and are real numbers.

     

    1. If has stationary points, what possible values can have?   (1 mark)

    2. If has an inverse function, what possible values can have?   (1 mark)

  3. The cubic function is defined by  .

     

    1. Write down a expression for  .   (2 marks)

    2. Determine the coordinates of the point(s) of intersection of the graphs of    and  .   (2 marks)

  4. The cubic function is defined by  , where and are real numbers.

     

    1. If has exactly one stationary point, find the value of .   (3 marks)

    2. If this stationary point occurs at a point of intersection of    and  , find the value of .   (3 marks)

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Show Worked Solution

a.i.  
  

a.ii.  

♦ Mean mark 47%.

 

b.i.  

♦♦♦ Mean mark part (b)(i) 9%, and part (b)(ii) 20%.
MARKER’S COMMENT: Good exam strategy should point students to investigate earlier parts for direction. Here, part (a) clearly sheds light on a solution!

 

b.ii.  

 

c.i.  


  

c.ii. 

   

 


  

♦ Mean mark part (d)(i) 44%.
d.i.   
 
 
 
 

 

d.ii.  

♦♦♦ Mean mark part (d)(ii) 14%.

 

Probability, MET2 2011 VCAA 2*

In a chocolate factory the material for making each chocolate is sent to one of two machines, machine A or machine B.

The time, seconds, taken to produce a chocolate by machine A, is normally distributed with mean 3 and standard deviation 0.8.

The time, seconds, taken to produce a chocolate by machine B, has the following probability density function
 


 

  1. Find correct to four decimal places

    1.    (1 mark)

    2.    (3 marks)

  2. Find the mean of , correct to three decimal places.   (3 marks)

  3. It can be shown that  . A random sample of 10 chocolates produced by machine B is chosen. Find the probability, correct to four decimal places, that exactly 4 of these 10 chocolate took 3 or less seconds to produce.   (2 marks)

All of the chocolates produced by machine A and machine B are stored in a large bin. There is an equal number of chocolates from each machine in the bin.

It is found that if a chocolate, produced by either machine, takes longer than 3 seconds to produce then it can easily be identified by its darker colour.

  1. A chocolate is selected at random from the bin. It is found to have taken longer than 3 seconds to produce.
  2. Find, correct to four decimal places, the probability that it was produced by machine A.   (3 marks)

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a.i. 

a.ii.

b.   

c.   

d.   

Show Worked Solution

a.i.  

 

a.ii.   
   
   

 

b.   
   

 

c.  


 

   
 

♦♦♦ Mean mark part (e) 19%.
MARKER’S COMMENT: Students who used tree diagrams were the most successful.

 

d.   
 
 

Calculus, MET2 2011 VCAA 1

Two ships, the Elsa and the Violet, have collided. Fuel immediately starts leaking from the Elsa into the sea.

The captain of the Elsa estimates that at the time of the collision his ship has 6075 litres of fuel on board and he also forecasts that it will leak into the sea at a rate of liters per minute, where  is the number of minutes that have elapsed since the collision.

  1. At this rate how long, in minutes, will it take for all the fuel from the Elsa to leak into the sea?   (3 marks)

*Parts (b) - (d) are no longer in the syllabus. 

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a.  

♦♦♦ Mean mark 23%.


  

 

 

 

*Parts (b) – (d) are no longer in the syllabus.

CORE, FUR2 SM-Bank VCE 4

Damon runs a swim school.

The value of his pool pump is depreciated over time using flat rate depreciation.

Damon purchased the pool pump for $28 000 and its value in dollars after years, , is modelled by the recursion equation below:

  1. Write down calculations, using the recurrence relation, to find the pool pump's value after 3 years.   (1 mark)

  2. After how many years will the pump's depreciated value reduce to $7000?   (1 mark)

The reducing balance depreciation method can also be used by Damon.

Using this method, the value of the pump is depreciated by 15% each year.

A recursion relation that models its value in dollars after years, , is:

  1. After how many years does the reducing balance method first give the pump a higher valuation than the flat rate method in part (a)?   (2 marks)

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a.   
 
 

  

  

b.  

 

  
c.  

  


  

Calculus, MET2 2016 VCAA 4

  1. Express    in the form  ,  where and are non-zero integers.   (2 marks)

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

    2. Part of the graphs of and    are shown in the diagram below.
       

    3. Find the area of the shaded region.   (1 mark)

    4. Part of the graphs of and are shown in the diagram below.
       

    5. Find the area of the shaded region.   (1 mark) 
  1. Part of the graph of  is shown in the diagram below.
     
       
     

     

    The point  is on the graph of .

     

    Find the exact values of and such that the distance of this point to the origin is a minimum, and find this minimum distance.   (3 marks)

Let  , where .

  1. Show that    implies that   where  .   (2 marks)

  2. Let be the point of intersection of the graphs of  .
    1. Find the coordinates of in terms of .   (2 marks)

    2. Find the value of for which the coordinates of are  .   (2 marks)

    3. Let    and be the vertices of the triangle . Let    be the square of the area of triangle .
       

       

         
       

       

      Find the values of such that  .   (2 marks)

  3. The graph of and the line    enclose a region of the plane. The region is shown shaded in the diagram below.
     

     

     

    Let    be the rule of the function that gives the area of this enclosed region. The domain of is  .

    1. Give the rule for .   (2 marks)

    2. Show that    for all  .   (2 marks)

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  2.   i.
  3.  ii. ²
  4. iii. ²
  8.   i.
  9.  ii.
  10. iii.
  11.  i.
  12. ii.
Show Worked Solution

a.  

 

 

 

 

b.i.  

 

 

b.ii.   

 
  ²

 

b.iii.   
   
    ²
♦♦♦ Mean mark part (c) 25%.

 

c.   
 
 

 

d.   

♦♦♦ Mean mark part (d) 8%.

 

 

 

♦♦ Mean mark part (e)(i) 32%.
e.i.   
 
 
 
 
   

 

 

e.ii.   

♦ Mean mark part (e)(ii) 44%.

 

e.iii.  
   

 

♦♦♦ Mean mark (e)(iii) 6%.

 
   

 

 

♦♦ Mean mark (f)(i) 28%.
f.i.   
   
   
   
   
   
   

 

♦♦♦ Mean mark part (f)(ii) 4%.
f.ii.  
 
 
 
 
 

Probability, MET2 2016 VCAA 3*

A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges.

On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student’s plugging-in is independent of any other student’s correctness.

  1. Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to four decimal places.   (2 marks)

  2. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley.
  3. Given this, find the probability that fewer than five laptops are not correctly plugged in. Give your answer correct to four decimal places.   (2 marks)

The time for which a laptop will work without recharging (the battery life) is normally distributed, with a mean of three hours and 10 minutes and standard deviation of six minutes. Suppose that the laptops remain out of the recharging trolley for three hours.

  1. For any one laptop, find the probability that it will stop working by the end of these three hours. Give your answer correct to four decimal places.   (2 marks)

A supplier of laptops decides to take a sample of 100 new laptops from a number of different schools. For samples of size 100 from the population of laptops with a mean battery life of three hours and 10 minutes and standard deviation of six minutes,  is the random variable of the distribution of sample proportions of laptops with a battery life of less than three hours.

  1. Find the probability that . Give your answer correct to three decimal places. Do not use a normal approximation.   (3 marks)

It is known that when laptops have been used regularly in a school for six months, their battery life is still normally distributed but the mean battery life drops to three hours. It is also known that only 12% of such laptops work for more than three hours and 10 minutes.

  1. Find the standard deviation for the normal distribution that applies to the battery life of laptops that have been used regularly in a school for six months, correct to four decimal places.   (2 marks)

The laptop supplier collects a sample of 100 laptops that have been used for six months from a number of different schools and tests their battery life. The laptop supplier wishes to estimate the proportion of such laptops with a battery life of less than three hours.

  1. Suppose the supplier tests the battery life of the laptops one at a time.
  2. Find the probability that the first laptop found to have a battery life of less than three hours is the third one.   (1 mark)

The laptop supplier finds that, in a particular sample of 100 laptops, six of them have a battery life of less than three hours.

  1. Determine the 95% confidence interval for the supplier’s estimate of the proportion of interest. Give values correct to two decimal places.   (1 mark)

  2. The supplier also provides laptops to businesses. The probability density function for battery life, (in minutes), of a laptop after six months of use in a business is
     

     


     

  3. Find the mean battery life, in minutes, of a laptop with six months of business use, correct to two decimal places.   (1 mark)

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Show Worked Solution

a.  

 

 
 

 

 b.   

MARKER’S COMMENT: Early rounding was a common mistake, producing 0.9312.

 

c.   

MARKER’S COMMENT: Some working must be shown for full marks in questions worth more than 1 mark.

 

d.   

♦ Mean mark part (d) 33%.

 
 
 

 

e.   

 

f.   
   

 

g.   

 

h.   
 

Calculus, MET2 2016 VCAA 2

Consider the function  

  1.  i. Given that  ,
  2.      show that  .   (1 mark)

  3. ii. Find the values of for which the graph of    has a stationary point.   (1 mark)

The diagram below shows part of the graph of  , the tangent to the graph at    and a straight line drawn perpendicular to the tangent to the graph at  . The equation of the tangent at the point with coordinates    is  .

The tangent cuts the -axis at . The line perpendicular to the tangent cuts the -axis at .
 


 

  1.   i. Find the coordinates of .   (1 mark)

  2.  ii. Find the equation of the line that passes through and and, hence, find the coordinates of .   (2 marks)

  3. iii. Find the area of triangle .   (2 marks)

  4. The tangent at is parallel to the tangent at . It intersects the line passing through and at .

     


     
     i. Find the coordinates of .   (2 marks)

  5. ii. Find the length of .   (3 marks)

Show Answers Only
  1.  i.
  2. ii.
  3.   i.
  4. ii.
  5.      
  6. iii. ²
Show Worked Solution
a.i.   
   
   
 

 

  

 

a.ii.   


 

b.i.   


 

b.ii.   

   

   


 

b.iii.  
   
    ²

 

♦ Mean mark part (c)(i) 43%.
MARKER’S COMMENT: Many students gave an incorrect -value here. Be careful!
c.i.   
 
 
   

  

 

c.ii.  

 

♦♦ Mean mark 32%.

 

 

 

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%.

Probability, MET2 2010 VCAA 12 MC

A soccer player is practising her goal kicking. She has a probability of    of scoring a goal with each attempt. She has 15 attempts.

The probability that the number of goals she scores is less than 7 is closest to

A.  

B.  

C.  

D.  

E.  

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Show Worked Solution

♦ Mean mark 44%.