SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

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)

Show Answers Only

  2.  

Show Worked Solution

a.  

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


 

b.   

♦♦♦ Mean mark 16%.

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)

Show Answers Only
  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)

Show Answers Only
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)

Show Answers Only

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 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)

Show Answers Only
  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)

Show Answers Only

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.   
 

NETWORKS, FUR2 2016 VCAA 3

A new skateboard park is to be built in Beachton.

This project involves 13 activities, to .

The directed network below shows these activities and their completion times in days.
 


 

  1. Determine the earliest start time for activity .   (1 mark)

  2. The minimum completion time for the skateboard park is 15 days.

     

    Write down the critical path for this project.   (1 mark)

  3. Which activity has a float time of two days?   (1 mark)

  4. The completion times for activities and can each be reduced by one day.

     

    The cost of reducing the completion time by one day for these activities is shown in the table below.
     

     

       

     

    What is the minimum cost to complete the project in the shortest time possible?   (1 mark)

  5. The original skateboard park project from part (a), before the reduction of time in any activity, will be repeated at another town named Campville, but with the addition of one extra activity.

     

    The new activity, , will take six days to complete and has a float time of one day.

     

    Activity will finish at the same time as the project.

     

     i.  Add activity to the network below.   (1 mark) 


      
    ii.  What is the latest start time for activity ?   (1 mark)

Show Answers Only

Show Worked Solution
a.   
   

  
b.  

♦♦ Mean mark part (c) 37%, part (d) 21%.
MARKER’S COMMENT: In part (d), cannot be crashed, therefore shortest duration is 14 days. Activity is cheapest to reduce.
  

c.  
  

d.  


  

e.i.   

♦♦ Mean mark part (e)(i) 21%, (e)(ii) 27%.
MARKER’S COMMENT: In (e)(ii), activity must have an arrow on it.
  

e.ii.   
   
   

GRAPHS, FUR2 2016 VCAA 3

A company produces two types of hockey stick, the ‘Flick’ and the ‘Jink’.

Let be the number of Flick hockey sticks that are produced each month.

Let be the number of Jink hockey sticks that are produced each month.

Each month, up to 500 hockey sticks in total can be produced.

The inequalities below represent constraints on the number of each hockey stick that can be produced each month.

Constraint 1 Constraint 2
Constraint 3 Constraint 4
  1. Interpret Constraint 4 in terms of the number of Flick hockey sticks and the number of Jink hockey sticks produced each month.  (1 mark)

There is another constraint, Constraint 5, on the number of each hockey stick that can be produced each month.

Constraint 5 is bounded by Line , shown on the graph below.

The shaded region of the graph contains the points that satisfy constraints 1 to 5.

  1. Write down the inequality that represents Constraint 5.  (1 mark)

The profit, , that the company makes from the sale of the hockey sticks is given by

  1. Find the maximum profit that the company can make from the sale of the hockey sticks.  (1 mark)
  2. The company wants to change the selling price of the Flick and Jink hockey sticks in order to increase its maximum profit to $42 000.

    All of the constraints on the numbers of Flick and Jink hockey sticks that can be produced each month remain the same.

    The profit, , that is made from the sale of hockey sticks is now given by

     

    The profit made on the Flick hockey sticks is dollars per hockey stick.

    The profit made on the Jink hockey sticks is dollars per hockey stick.

    The maximum profit of $42 000 is made by selling 400 Flick hockey sticks and 100 Jink hockey sticks.

    What are the values of and ?  (2 marks)

Show Answers Only


Show Worked Solution

a.  

♦♦♦ Mean mark 25%.

 

b.  

 

c.  

 

 

d.  

♦♦♦ Mean mark 7%.
MARKER’S COMMENT: Very few students were able to identify the need for the sliding line concept and execute it correctly in this question.

 

MATRICES, FUR2 2016 VCAA 3

A travel company is studying the choice between air (), land (), sea () or no () travel by some of its customers each year.

Matrix , shown below, contains the percentages of customers who are expected to change their choice of travel from year to year.

Let be the matrix that shows the number of customers who choose each type of travel years after 2014.

Matrix below shows the number of customers who chose each type of travel in 2014.

Matrix below shows the number of customers who chose each type of travel in 2015.

  1. Find the values missing from matrix .   (1 mark)

  2. Write a calculation that shows that 478 customers were expected to choose air travel in 2015.   (1 mark)

  3. Consider the customers who chose sea travel in 2014.
  4. How many of these customers were expected to choose sea travel in 2015?   (1 mark)

  5. Consider the customers who were expected to choose air travel in 2015.
  6. What percentage of these customers had also chosen air travel in 2014?
  7. Round your answer to the nearest whole number.   (1 mark)

In 2016, the number of customers studied was increased to 1360.
Matrix , shown below, contains the number of these customers who chose each type of travel in 2016.

  1. The company intends to increase the number of customers in the study in 2017 and in 2018.
  2. The matrix that contains the number of customers who are expected to choose each type of travel in 2017 () and 2018 () can be determined using the matrix equations shown below.


 

    1. The element in the fourth row of matrix is – 80.
    2. Explain this number in the context of selecting customers for the studies in 2017 and 2018.   (1 mark)

    3. Determine the number of customers who are expected to choose sea travel in 2018.
    4. Round your answer to the nearest whole number.   (2 marks)

Show Answers Only
  5.  


Show Worked Solution

a.   
 

b.   

♦♦ Mean mark part (b) and (c) was 35% and 45% respectively.

 

c.  

 

 

d.  

♦♦♦ Mean mark 17%.


 

 
 

 

e.i.  

♦ Mean mark 14%.
MARKER’S COMMENT: “80 people chose not to travel” was a common answer that received no marks.

 

e.ii.  
   
   

 

 
 

 

GEOMETRY, FUR2 2016 VCAA 5

A golf course has a sprinkler system that waters the grass in the shape of a sector, as shown in the diagram below. 
 

A sprinkler is positioned at point and can turn through an angle of 100°.

The shaded area on the diagram shows the area of grass that is watered by the sprinkler.

  1. If 147.5 m² of grass is watered, what is the maximum distance, metres, that the water reaches from ?

     

    Round your answer to the nearest metre.  (1 mark)

  2. Another sprinkler can water a larger area of grass.

     

    This sprinkler will water a section of grass as shown in the diagram below.
     

     


    The section of grass that is watered is 4.5 m wide at all points.

     

    Water can reach a maximum of 12 m from the sprinkler at .

     

    What is the area of grass that this sprinkler will water?

     

    Round your answer to the nearest square metre.  (2 marks)

Show Answers Only
  2. ²²
Show Worked Solution
♦ Mean mark 48%.
a.   
 
 
   
 
   
♦♦♦ Mean mark 23%.

 

b.   
   
   
    ²²

GEOMETRY, FUR2 2016 VCAA 4

During a game of golf, Salena hits a ball twice, from  to and then from to .

The path of the ball after each hit is shown in the diagram below.

After Salena’s first hit, the ball travelled 80 m on a bearing of 130° from point to point .

After Salena’s second hit, the ball travelled 100 m on a bearing of 054° from point to point .

  1. Another ball is hit and travels directly from to .

     

    Use the cosine rule to find the distance travelled by this ball.

     

    Round your answer to the nearest metre.  (2 marks)

  2. What is the bearing of from ?

     

    Round your answer to the nearest degree.  (1 mark) 

Show Answers Only
Show Worked Solution
a.   
 
 
 
 

 

b.  

 

 

CORE, FUR2 2016 VCAA 7

 

Ken has borrowed $70 000 to buy a new caravan.

He will be charged interest at the rate of 6.9% per annum, compounding monthly.

  1. For the first year (12 months), Ken will make monthly repayments of $800.

    1. Find the amount that Ken will owe on his loan after he has made 12 repayments.   (1 mark)
    2. What is the total interest that Ken will have paid after 12 repayments?   (1 mark)
  2. After three years, Ken will make a lump sum payment of $L in order to reduce the balance of his loan.
  3. This lump sum payment will ensure that Ken’s loan is fully repaid in a further three years.
  4. Ken’s repayment amount remains at $800 per month and the interest rate remains at 6.9% per annum, compounding monthly.
  5. What is the value of Ken’s lump sum payment, $L?
  6. Round your answer to the nearest dollar.   (2 marks)

Show Answers Only

a.i.

a.ii.

b. 

Show Worked Solution

a.i.  

♦ Mean mark 46%.

 

♦♦ Mean mark 26%.

a.ii.   
   
   

  
b.   

♦♦♦ Mean mark 20%.
MARKER’S COMMENT: Correct input tables allowed for a method mark if answers were calculated incorrectly.

   

  

 

  

 

CORE*, FUR2 2016 VCAA 6

Ken’s first caravan had a purchase price of $38 000.

After eight years, the value of the caravan was $16 000.

  1. Show that the average depreciation in the value of the caravan per year was $2750.   (1 mark)

  2. Let be the value of the caravan years after it was purchased.

     

    Assume that the value of the caravan has been depreciated using the flat rate method of depreciation.

     

    Write down a recurrence relation, in terms of and , that models the value of the caravan.   (1 mark)

  3. The caravan has travelled an average of 5000 km in each of the eight years since it was purchased.

     

    Assume that the value of the caravan has been depreciated using the unit cost method of depreciation.

     

    By how much is the value of the caravan reduced per kilometre travelled?   (1 mark)

Show Answers Only
Show Worked Solution
a.    


♦♦ Mean mark (a) 39%.
MARKER’S COMMENT: A “show that” question should include an equation
  

b.   
 

♦♦ Mean mark (b) 33%.
MARKER’S COMMENT: A lack of attention to detail and careless errors were common!
  

c.   
   

 


♦♦ Mean mark (c) 29%.
  

NETWORKS, FUR1 2016 VCAA 8 MC

Five children, Alan, Brianna, Chamath, Deidre and Ewen, are each to be assigned a different job by their teacher. The table below shows the time, in minutes, that each child would take to complete each of the five jobs.
 

     
 

The teacher wants to allocate the jobs so as to minimise the total time taken to complete the five jobs.

In doing so, she finds that two allocations are possible.

If each child starts their allocated job at the same time, then the first child to finish could be either

  1. Alan or Brianna.
  2. Brianna or Deidre.
  3. Chamath or Deidre.
  4. Chamath or Ewen.
  5. Deidre or Ewen. 
Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 30%.

 

GRAPHS, FUR1 2016 VCAA 8 MC

Megan walks from her house to a shop that is 800 m away.

The equation for the relationship between the distance, in metres, that Megan is from her house minutes after leaving is

If Megan reaches the shop 10 minutes after leaving her house, the value of is

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 26%.

 

 

GRAPHS, FUR1 2016 VCAA 6 MC

The point  (2, 12)  lies on the graph of  , as shown below.

Another graph that represents this relationship between and could be

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 17%.
STRATEGY: Use the gradient and -axis variable to form an equation for each linear graph.

GRAPHS, FUR1 2016 VCAA 5 MC

The feasible region for a linear programming problem is shaded in the diagram below.

The equation of the objective function for this problem is of the form

,              where and

The dotted line in the diagram has the same slope as the objective function for this problem.

The maximum value of the objective function can be determined by calculating its value at

  1. point A.
  2. point B.
  3. point C.
  4. point D.
  5. any point along line segment BC. 
Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 29%.
MARKER’S COMMENT: The majority of students had no idea of the sliding rule technique or applied it inaccurately and answered E.

 

GEOMETRY, FUR1 2016 VCAA 8 MC

A string of seven flags consisting of equilateral triangles in two sizes is hanging at the end of a racetrack, as shown in the diagram below.
 


 

The edge length of each black flag is twice the edge length of each white flag.

For this string of seven flags, the total area of the black flags would be

  1. two times the total area of the white flags.
  2. four times the total area of the white flags.
  3. times the total area of the white flags.
  4. times the total area of the white flags.
  5. times the total area of the white flags. 
Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 30%.

 

GEOMETRY, FUR1 2016 VCAA 4 MC

All towns in the state of Victoria are in the same time zone.

Mallacoota (38°S, 150°E) and Portland (38°S, 142°E) are two coastal towns in the state of Victoria.

On one day in January, the sun rose in Mallacoota at 6.03 am.

Assuming that 15° of longitude equates to a one-hour time difference, the time that the sun was expected to rise in Portland is

  1. 5.31 am.
  2. 5.55 am.
  3. 6.03 am.
  4. 6.11 am.
  5. 6.35 am.
Show Answers Only

Show Worked Solution
 

 

 
♦♦♦ Mean mark 26%.

 


 

 

 

CORE, FUR1 2016 VCAA 24 MC

Mai invests in an annuity that earns interest at the rate of 5.2% per annum compounding monthly.

Monthly payments are received from the annuity.

The balance of the annuity will be $130 784.93 after five years.

The balance of the annuity will be $66 992.27 after 10 years.

The monthly payment that Mai receives from the annuity is closest to

  1. $1270
  2. $1400
  3. $1500
  4. $2480
  5. $3460
Show Answers Only

Show Worked Solution

♦♦ Mean mark 30%.
MARKER’S COMMENT: Many students incorrectly chose , by allocating the same sign to both the PV and FV.
   

 

Number and Algebra, NAP-B2-20

A group of soccer fans all go to a soccer game.

Entry to the game costs $3.

Altogether the group pay $156.

How many soccer fans are there in the group?

 
 
 
 
Show Answers Only

Show Worked Solution