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Number and Algebra, NAP-J2-30

Madison uses the number sentence  15 × 12 = 180  to solve a problem.

Which of the following could be the problem?

 
 Madison buys 15 showbags. How much does each showbag cost?
 
 Madison spends $15 on 180 showbags. How much does she spend?
 
 Madison buys 15 showbags that cost $12 each. How many showbags does she buy?
 
 Madison buys 12 showbags that cost $15 each. How much does she spend?
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Geometry, NAP-J2-23

Conrad followed these instructions to create a drawing:

  • Draw a triangle with 2 equal sides.
  • Use one of the sides of this shape to draw another shape with 5 equal sides.
  • Draw a line of symmetry through your completed drawing.

Which of these did he draw?

 
 
 
 
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Polynomials, EXT2 2018 HSC 16c

Let    be the zeros of  , where and are real and  .

  1.  Show that  .  (2 marks)
  2.  By considering the constant term of a cubic equation with roots  and , or otherwise, show that
     
       .  (3marks)
     
  3.  Deduce that if  , then the equation    has 3 distinct real roots.  (2 marks)
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i.   

♦ Mean mark 34%.
COMMENT: Solving from the RHS makes this proof much simpler.

 
 
 
 

 

ii.   
   
   
   

 

 

♦♦♦ Mean mark 6%
COMMENT: This question produced just just the 4th lowest mean mark in a tough 2018 paper.

 

 

 

 

iii.   

♦♦♦ Mean mark 5%.


 


 


 


 

Harder Ext1 Topics, EXT2 2018 HSC 16b

In , point is chosen on side    and point is chosen on side    so that    is parallel to    and  .

The process is repeated two more times. Point is chosen on    and point on    so that    is parallel to    and  .

Point is chosen on side    and on    so that    is parallel to    and  .

The segments    and    intersect at ,   and  intersect at , and    and    intersect at .
 


 

  1. Prove that  .  (3 marks)
  2. Find the exact value of the ratio  .  (2 marks)
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i.    

♦♦♦ Mean mark 5%.

 

 

ii.   
   
   

 

♦♦♦ Mean mark 3%.
COMMENT: As well reported in the papers at the time, this question, in both parts, proved to be a beast.


                   
   
 
 

Proof, EXT2 P1 2018 HSC 15c

Let    be a positive integer and let    be a positive real number.

  1.  Show that  .  (1 mark)

  2.  Hence, show that  .  (2 marks)

  3.  Deduce that for positive real numbers and ,
     
            (2 marks)

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i.   
   
   
   

 

ii.   

♦♦♦ Mean mark 11%.


 


 


 

 

iii.   

♦♦ Mean mark 23%.

Number and Algebra, NAP-K2-20

On Monday, Jeremy went to the doctor and was given 24 tablets.

Jeremy was to take 5 tablets each day starting from Monday.

On which day did Jeremy take the last tablet?

Thursday Friday Saturday Sunday
 
 
 
 
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Statistics, NAP-K2-19

On a school camp, each child chose their meal for dinner.

The table shows how many children chose each meal.
 

 
Select all the statements that are true.

 
 More girls than boys chose spaghetti.
 
 In total 70 children chose pizza.
 
 Less than half the girls chose pizza.
 
 In total, there are the same number of girls on the trip as boys.
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Harder Ext1 Topics, EXT2 2018 HSC 14d

Three people, , and , play a series of n games, where  . In each of the games there is one winner and each of the players is equally likely to win.

  1.  What is the probability that player wins every game?  (1 mark)
  2.  Show that the probability that and win at least one game each but never wins, is
     
         .  (1 mark)
     
  3.  Show that the probability that each player wins at least one game is 
     
         .  (2 marks)
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Show Worked Solution

i.   

 

ii.   

♦ Mean mark 36%.


 

 

iii.   

♦♦♦ Mean mark 19%.


 

Financial Maths, 2ADV M1 2018 HSC 16c

Kara deposits an amount of $300 000 into an account which pays compound interest of 4% per annum, added to the account at the end of each year. Immediately after the interest is added, Kara makes a withdrawal for expenses for the coming year. The first withdrawal is . Each subsequent withdrawal is 5% greater than the previous one.

Let    be the amount in the account after the th withdrawal.

  1. Show that  .  (1 mark)

  2. Show that  .  (1 mark)

  3. Show that there will be money in the account when
     
      (3 marks)

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Show Worked Solution

♦ Mean mark 47%.

i.  
 
   
   

 

♦ Mean mark part (ii) 30%.

ii.  
   
   

 

iii.  
   
 
   

 

♦♦♦ Mean mark part (iii) 8%.

Probability, 2ADV S1 2018 HSC 16b

A game involves rolling two six-sided dice, followed by rolling a third six-sided die. To win the game, the number rolled on the third die must lie between the two numbers rolled previously. For example, if the first two dice show 1 and 4, the game can only be won by rolling a 2 or 3 with the third die.

  1. What is the probability that a player has no chance of winning before rolling the third die?   (2 marks)

  2. What is the probability that a player wins the game?   (2 marks)

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i.   

♦ Mean mark 40%.
COMMENT: Constructing the full sample space is a critical step here..


 

 
 

 

ii.   

♦♦♦ Mean mark 7%.

 
 
 

Quadratic, 2UA 2018 HSC 8 MC

A radio telescope has a parabolic dish. The width of the opening is 24 m and the distance along the axis from the vertex to the opening is 4 m, as shown in the diagram.
 

 
What is the focal length of the parabola?

(A) 

(B) 

(C) 

(D) 

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♦♦♦ Mean mark 16%.

 

 

 

 

Plane Geometry, EXT1 2018 HSC 14c

In triangle is perpendicular to . Side has length , side has length and side has length . A quadrant of a circle of radius , centered at , is constructed. The arc meets side at . It touches the side at , and meets side at . The interval is perpendicular to .
 


 

  1. Show that and are similar.  (1 mark)
  2. Show that
     
    .  (1 mark)
     

  3. From , a line perpendicular to is drawn to meet at , forming the right-angled triangle . A new quadrant is constructed in triangle touching side at . The process is then repeated indefinitely.
     

                
     

  4. Show that the limiting sum of the areas of all the quadrants is
     
      (4 marks)
     

  5. Hence, or otherwise, show that
     
    .  (1 mark)

 

 

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(i)   


 


 

 

(ii)
   
   
   


(iii) 

♦♦♦ Mean mark part (iii) 19%.

 

 
 
   
 
   
   
 

 

 
 

 

 

(iv) 

♦♦ Mean mark part (iv) 21%.

Networks, STD2 N3 2012 FUR1 8 MC

networks-fur1-2012-vcaa-8-mc 
 

Eight activities,   and  , must be completed for a project.

The network above shows these activities and their usual duration in hours.

The duration of each activity can be reduced by one hour.

To complete this project in 16 hours, the minimum number of activities that must be reduced by one hour each is

A.   

B.   

C.   

D.   

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♦♦ Mean mark 30%.

 

Networks, STD2 N3 2007 FUR2 4

A community centre is to be built on the new housing estate.

Nine activities have been identified for this building project.

The directed network below shows the activities and their completion times in weeks.

 

  1. Determine the minimum time, in weeks, to complete this project.  (1 mark)

  2. Determine the float time, in weeks, for activity .  (2 marks)

The builders of the community centre are able to speed up the project.

Some of the activities can be reduced in time at an additional cost.

The activities that can be reduced in time are , , , and .

  1. Which of these activities, if reduced in time individually, would not result in an earlier completion of the project?  (1 mark)

The owner of the estate is prepared to pay the additional cost to achieve early completion.

The cost of reducing the time of each activity is $5000 per week.

The maximum reduction in time for each one of the five activities, , , , , , is weeks.

  1. Determine the minimum time, in weeks, for the project to be completed now that certain activities can be reduced in time.  (1 mark)

  2. Determine the minimum additional cost of completing the project in this reduced time.  (1 mark)

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a.  
 

 

♦ Mean mark of all parts (combined) 40%.
 

 

b.   
 
 

 

c.  


 

d.  

 

e.   
   

Networks, STD2 N3 2009 FUR2 4

A walkway is to be built across the lake.

Eleven activities must be completed for this building project.

The directed network below shows the activities and their completion times in weeks.
 

NETWORKS, FUR2 2009 VCAA 4
 

  1. What is the earliest start time for activity E?  (1 mark)

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

  3. The project supervisor correctly writes down the float time for each activity that can be delayed and makes a list of these times.

     

    Determine the longest float time, in weeks, on the supervisor’s list.  (1 mark)

A twelfth activity, L, with duration three weeks, is to be added without altering the critical path.

Activity L has an earliest start time of four weeks and a latest start time of five weeks.
 

NETWORKS, FUR2 2009 VCAA 4
 

  1. Draw in activity L on the network diagram above.  (1 mark)
  2. Activity L starts, but then takes four weeks longer than originally planned.

     

    Determine the total overall time, in weeks, for the completion of this building project.  (1 mark)

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  4.  
    NETWORKS, FUR2 2009 VCAA 4 Answer
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a.  

♦ Mean mark of all parts (combined): 44%.

 

b. 
 

 

 

c.   


 

d.    NETWORKS, FUR2 2009 VCAA 4 Answer

 

e.  

Networks, STD2 N3 2006 FUR1 9 MC

The network below shows the activities and their completion times (in hours) that are needed to complete a project.
 


 

The project is to be optimised by reducing the completion time of one activity only.

This will reduce the completion time of the project by a maximum of

A.   1 hour 

B.   3 hours

C.   4 hours

D.   5 hours

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♦♦♦ Mean mark 17%.
MARKER’S COMMENT: When choosing an activity to crash, take care that a new critical path is not created.


 


 


 

Networks, FUR2 2007 VCE 3

As an attraction for young children, a miniature railway runs throughout a new housing estate.

The trains travel through stations that are represented by nodes on the directed network diagram below.

The number of seats available for children, between each pair of stations, is indicated beside the corresponding edge.
 

NETWORKS, FUR2 2007 VCAA 3

 
Cut 1, through the network, is shown in the diagram above.

  1. Determine the capacity of Cut 1.  (1 mark)
  2. Determine the maximum number of seats available for children for a journey that begins at the West Terminal and ends at the East Terminal.  (1 mark)

On one particular train, 10 children set out from the West Terminal.

No new passengers board the train on the journey to the East Terminal.

  1. Determine the maximum number of children who can arrive at the East Terminal on this train.  (1 mark)
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a.  

♦♦ Mean mark for all parts (combined) was 33%.
MARKER’S COMMENT: A common error was counting the edge with “10” in the reverse direction (in part a).

 

b.    networks-fur2-2007-vcaa-3-answer
 
 

 

c.  


  

Networks, FUR2 2013 VCE 3

The rangers at the wildlife park restrict access to the walking tracks through areas where the animals breed.

The edges on the directed network diagram below represent one-way tracks through the breeding areas. The direction of travel on each track is shown by an arrow. The numbers on the edges indicate the maximum number of people who are permitted to walk along each track each day.
 

NETWORKS, FUR2 2013 VCAA 31
 

  1. Starting at , how many people, in total, are permitted to walk to each day?  (1 mark)

One day, all the available walking tracks will be used by students on a school excursion.

The students will start at and walk in four separate groups to .

Students must remain in the same groups throughout the walk.

    1. Group 1 will have 17 students. This is the maximum group size that can walk together from to .

      Write down the path that group 1 will take.  (1 mark)

    2. Groups 2, 3 and 4 will each take different paths from to .

      Complete the six missing entries shaded in the table below.  (2 marks)

       


      NETWORKS, FUR2 2013 VCAA 32

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    2.  
      Networks, FUR2 2013 VCAA 3_2 Answer1
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a.   
   
   
♦ Mean mark of all parts (combined) was 41%.

 

 

b.i.  

 

b.ii.   

 

 
  

Networks, FUR2 2013 VCAA 3_2 Answer1

Networks, FUR1 2014 VCE 9 MC

A network of tracks connects two car parks in a festival venue to the exit, as shown in the directed graph below.
 

 
The arrows show the direction that cars can travel along each of the tracks and the numbers show each track’s capacity in cars per minute.

Four cuts are drawn on the diagram.

The maximum number of cars per minute that will reach the exit is given by the capacity of

A.  Cut A

B.  Cut B

C.  Cut C

D.  Cut D

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♦♦♦ Mean mark 24%.
COMMENT: Note that the “4” is not included in Cut D as it is flowing in the opposite direction.

Networks, FUR1 2012 VCE 7 MC

Vehicles from a town can drive onto a freeway along a network of one-way and two-way roads, as shown in the network diagram below.

The numbers indicate the maximum number of vehicles per hour that can travel along each road in this network. The arrows represent the permitted direction of travel.

One of the four dotted lines shown on the diagram is the minimum cut for this network.
 

networks-fur1-2012-vcaa-7-mc-1

 
The maximum number of vehicles per hour that can travel through this network from the town onto the freeway is

A.  

B. 

C. 

D. 

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♦♦♦ Mean mark 23%.


 

Networks, STD2 N2 2009 FUR1 8 MC

An undirected connected graph has five vertices.

Three of these vertices are of even degree and two of these vertices are of odd degree.

One extra edge is added. It joins two of the existing vertices.

In the resulting graph, it is not possible to have five vertices that are

A.   all of even degree.

B.   all of equal degree.

C.   one of even degree and four of odd degree.

D.   four of even degree and one of odd degree. 

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♦♦♦ Mean mark 25%.


 

matrices-fur1-2009-vcaa-8-mc-answer
 

Statistics, NAP-K3-CA06

Clive and Alvin asked their friends how many books they had read in the past month.

Clive draws a picture graph to show the results for his friends.

Alvin draws a column graph to show the results for his friends.
 

 
How many more of Clive's friends read 3-4 books in the last month than Alvin's friends?

 
 
 
 
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Probability, NAP-K3-CA02

There are 20 raffle tickets, numbered 1 to 20, in a box.

Three prizes are given away by choosing three tickets from the box. One ticket can win one prize only.

The first ticket drawn is number 15 and wins the third prize.

Which of the following is not possible?

 
 Second prize is won by number 2.
 
 First prize is won by a prime number.
 
 Second prize is an even number.
 
 First prize is won by number 15.
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Calculus, MET1 SM-Bank 17

The diagram shows a point on the unit circle    at an angle from the positive -axis, where  .

The tangent to the circle at    is perpendicular to  , and intersects the  -axis at  ,  and the line    intersects the  -axis at  

 

  1. Show that the equation of the line is  .   (2 marks)

  2. Find the length of in terms of .   (1 mark)

  3. Show that the area, ,  of the trapezium is given by 
  4.       (2 marks)

  5. Find the angle that gives the minimum area of the trapezium.   (3 marks)

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i.

♦♦ Mean mark 20%

 

ii.  

 

 

 

iii. 

 

               

♦♦ Mean mark (iii) 24%
 
  ²

 

iv.  

 
 
 
 
Mean mark (iv) 19%
IMPORTANT: Look for any opportunity to use the identity as it is an examiner’s favourite and can often be the key to simplifying difficult trig equations.
 

 

 

IMPORTANT: Is the 1st or 2nd derivative test easier here? Students must evaluate and choose. Examiners often make one significantly easier than the other.

.

Calculus, MET1 SM-Bank 27

A cone is inscribed in a sphere of radius , centred at . The height of the cone is and the radius of the base is , as shown in the diagram.

  1. Show that the volume, , of the cone is given by
  2.    .   (2 marks)

  3. Find the value of for which the volume of the cone is a maximum. You must give reasons why your value of gives the maximum volume.   (3 marks)

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i. 
 

 
 
 

 

ii. 

 

 

 

 

Calculus, MET1 SM-Bank 35

 

The diagram shows two parallel brick walls and joined by a fence from to .  The wall is metres long and  .  The fence is metres long.

A new fence is to be built from to a point somewhere on .  The new fence will cross the original fence at .

Let    metres, where  .

  1. Show that the total area,  square metres, enclosed by and is given by
  2.    .   (3 marks)

  3. Find the value of that makes as small as possible. Justify the fact that this value of gives the minimum value for .   (3 marks)

  4. Hence, find the length of when is as small as possible.   (1 mark)

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i.

♦♦♦ Students found this question extremely challenging (exact results not available).

 

 
 
 
 
 
 
 

 

ii.   

MARKER’S COMMENT: Students who could not complete part (i) are reminded that they can still proceed to part (ii) and attempt to differentiate the result given.
Note that  and are constants when differentiating. 

 

 


 

iii.   

 
 

 

Calculus, MET1 SM-Bank 30

A function is given by  .

  1. Find the coordinates of the stationary points of    and determine their nature.   (3 marks)

  2. Hence, sketch the graph    showing the stationary points.   (2 marks)

  3. For what values of is the function increasing?   (1 mark)

  4. For what values of will    have no solution?   (1 mark)

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  4. 2UA HSC 2012 14ai
     
Show Worked Solution
i.
 
 

 

 

 

 

 
 
 

 

ii.  2UA HSC 2012 14ai

 

♦ Mean mark (HSC) 42%
MARKER’S COMMENT: Be careful to use the correct inequality signs, and not carelessly include  or by mistake.

 

iii.
 

 

iv.   

♦♦♦ Mean mark (HSC) 12%.

Calculus, 2ADV C3 SM-Bank 13

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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i.  

 

ii.  
   
   

 

iii.   

♦ Mean mark (Vic) 35%.

,

 

iv. 

♦♦♦ Mean mark (Vic) 5%.

 

 vcaa-2011-meth-10ai

 
 
 

NETWORKS, FUR2 2017 VCAA 4

The rides at the theme park are set up at the beginning of each holiday season.

This project involves activities A to O.

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

  1. Write down the two immediate predecessors of activity I.   (1 mark)

  2. The minimum completion time for the project is 19 days.

     

     i.  There are two critical paths. One of the critical paths is A–E–J–L–N.
    Write down the other critical path.   (1 mark)

    ii.  Determine the float time, in days, for activity F.   (1 mark)

  3. The project could finish earlier if some activities were crashed.

     

    Six activities, B, D, G, I, J and L, can all be reduced by one day.

     

    The cost of this crashing is $1000 per activity.

     

     i.  What is the minimum number of days in which the project could now be completed?   (1 mark)

    ii.  What is the minimum cost of completing the project in this time?   (1 mark)

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a.     

b.i.   

b.ii.   

c.i.   

c.ii.    

Show Worked Solution

a.   
  

b.i.   

♦♦ Mean mark part (b)(i) 44% and part (b)(ii) 28%.
  

b.ii.   
   

  
c.i.   

♦♦ Mean mark part (c)(i) 35%.

 


  

c.ii.   

♦♦♦ Mean mark part (c)(ii) 15%.

 

MATRICES, FUR2 2017 VCAA 3

Senior students at a school choose one elective activity in each of the four terms in 2018.

Their choices are communication (), investigation (), problem-solving () and service ().

The transition matrix shows the way in which senior students are expected to change their choice of elective activity from term to term.
 


 

Let be the state matrix for the number of senior students expected to choose each elective activity in Term .

For the given matrix , a matrix rule that can be used to predict the number of senior students in each elective activity in Terms 2, 3 and 4 is
 


 

  1. How many senior students will not change their elective activity from Term 1 to Term 2?   (1 mark)

  2. Complete , the state matrix for Term 2, below.   (1 mark)


             

  3. Of the senior students expected to choose investigation () in Term 3, what percentage chose service () in Term 2?   (2 marks)

  4. What is the maximum number of senior students expected in investigation () at any time during 2018?   (1 mark)

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  2.  
Show Worked Solution

a.   

♦ Mean mark 47%.

 

b.   
   

 

c.   
   

♦♦♦ Mean mark 13%.
MARKER’S COMMENT: A poorly understood and answered question worthy of careful attention.

 
 

 

d.   
   

♦ Mean mark 43%.

CORE, FUR2 2017 VCAA 6

Alex sends a bill to his customers after repairs are completed.

If a customer does not pay the bill by the due date, interest is charged.

Alex charges interest after the due date at the rate of 1.5% per month on the amount of an unpaid bill.

The interest on this amount will compound monthly.

  1. Alex sent Marcus a bill of $200 for repairs to his car.

     

    Marcus paid the full amount one month after the due date.

     

    How much did Marcus pay?   (1 mark)

Alex sent Lily a bill of $428 for repairs to her car.

Lily did not pay the bill by the due date.

Let be the amount of this bill months after the due date.

  1. Write down a recurrence relation, in terms of , and , that models the amount of the bill.   (2 marks)

  2. Lily paid the full amount of her bill four months after the due date.

     

    How much interest was Lily charged?

     

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

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a.   
   
   

♦ Mean mark part (b) 47%.
MARKER’S COMMENT: A recurrence relation has the initial value written first. Know why    is incorrect.

 

b.   

 

c.   
   

♦♦ Mean mark part (c) 29%.

 

CORE, FUR2 2017 VCAA 4

The eggs laid by the female moths hatch and become caterpillars.

The following time series plot shows the total area, in hectares, of forest eaten by the caterpillars in a rural area during the period 1900 to 1980.

The data used to generate this plot is also given.
 

The association between area of forest eaten by the caterpillars and year is non-linear.

A log10 transformation can be applied to the variable area to linearise the data.

  1. When the equation of the least squares line that can be used to predict log10 (area) from year is determined, the slope of this line is approximately 0.0085385
  2. Round this value to three significant figures.   (1 mark)
  3. Perform the log10 transformation to the variable area and determine the equation of the least squares line that can be used to predict log10 (area) from year.
  4. Write the values of the intercept and slope of this least squares line in the appropriate boxes provided below.
  5. Round your answers to three significant figures.  (2 marks)

The least squares line predicts that the log10 (area) of forest eaten by the caterpillars by the year 2020 will be approximately 2.85

  1. Using this value of 2.85, calculate the expected area of forest that will be eaten by the caterpillars by the year 2020.
  2.  i. Round your answer to the nearest hectare.   (1 mark)

  3. ii. Give a reason why this prediction may have limited reliability.   (1 mark)

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a. 

b. 

c.i. 

c.ii.
       

Show Worked Solution

a.   

♦ Mean marks of part (a) and (b) 44%.

 

b.   

 

♦♦ Mean mark part (c)(i) 29%.
COMMENT: When the question specifies using the value 2.85, use it!

c.i.   
 
   
   

 

c.ii.   

 

Calculus, 2ADV C3 SM-Bank 7

The graph of    for    is shown below.
 


 

  1. Calculate the area between the graph of   and the -axis.  (2 marks)

  2. For in the interval , show that the gradient of the tangent to the graph of    is  .  (1 mark)

The edges of the right-angled triangle are the line segments and , which are tangent to the graph of  , and the line segment , which is part of the horizontal axis, as shown below.

Let be the angle that makes with the positive direction of the horizontal axis.
 


 

  1. Find the equation of the line through and in the form  , for  .  (3 marks)

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i.  
   
   
   
   
   

 

ii.  
 
   
   

 

iii. 

♦♦♦ Mean mark (Vic) part (iii) 20%.
MARKER’S COMMENT: Most successful answers introduced a pronumeral such as    to solve.

 

 

   
 
   

 

 

Algebra, MET2 2017 VCAA 4

Let  . Part of the graph of  is shown below.
 

  1. The transformation    maps the graph of    onto the graph of  .

     

    State the values of and .   (2 marks)

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

  3. Find the area bounded by the graphs of  and  .   (3 marks)

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

         
     
    Find the gradient of  and the gradient of    at  .   (2 marks)

The functions of  , where  , are defined with domain such that  .

  1. Find the value of such that  .  (1 mark)

  2. Find the rule for the inverse functions  of  , where  .   (1 mark)

  3. i. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

    ii. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

  4. The lines and are the tangents at the origin to the graphs of    and    respectively.
  5. Find the value(s) of for which the angle between and is 30°.   (2 marks)

  6. Let be the value of for which    has only one solution.
  7.  i. Find .   (2 marks)

  8. ii. Let    be the area bounded by the graphs of    and    for all  .
  9.     State the smallest value of such that  .   (1 mark)

Show Answers Only

a. 

b. 

c.  ²

d.

e. 

f. 

g.i. 

g.ii. 

h. 

i.i. 

i.ii. 

Show Worked Solution

a.   

   

 

 

b.   

 

 

c.   

MARKER’S COMMENT: Specifically recommends using    in this type of integral to avoid errors in stating the integral.

 

  ²

 

d.   
 

 

e.   

 

f.   

 

♦♦ Mean mark part (g)(i) 31%.

g.i.   
 

 

 

♦♦ Mean mark part (g)(ii) 30%.

g.ii.   
 

 

 

h.   

♦♦♦ Mean mark part (h) 13%.

 

 

 

i.i   

♦♦♦ Mean mark part (i)(i) 4%.

 

 

i.ii 

♦♦♦ Mean mark part (i)(ii) 2%.

 

Probability, MET2 2017 VCAA 3

The time Jennifer spends on her homework each day varies, but she does some homework every day.

The continuous random variable , which models the time, in minutes, that Jennifer spends each day on her homework, has a probability density function , where

 

 

  1. Sketch the graph of on the axes provided below.  (3 marks)


     

       
     

  2. Find  .  (2 marks)

  3. Find  .  (2 marks)

  4. Find such that  , correct to four decimal places.  (2 marks)

  5. The probability that Jennifer spends more than 50 minutes on her homework on any given day is . Assume that the amount of time spent on her homework on any day is independent of the time spent on her homework on any other day.

     

    1. Find the probability that Jennifer spends more than 50 minutes on her homework on more than three of seven randomly chosen days, correct to four decimal places.  (2 marks)

    2. Find the probability that Jennifer spends more than 50 minutes on her homework on at least two of seven randomly chosen days, given that she spends more than 50 minutes on her homework on at least one of those days, correct to four decimal places.  (2 marks)

Let be the probability that on any given day Jennifer spends more than minutes on her homework.

Let be the probability that on two or three days out of seven randomly chosen days she spends more than minutes on her homework.

  1. Express as a polynomial in terms of .  (2 marks)

    1. Find the maximum value of , correct to four decimal places, and the value of for which this maximum occurs, correct to four decimal places.  (2 marks)

    2. Find the value of for which the maximum found in part g.i. occurs, correct to the nearest minute.  (2 marks)


Show Answers Only

  1.  

Show Worked Solution

a.   

MARKER’S COMMENT: Many did not draw graph along -axis between 0 and 20 and for  .

 

b.   

 

c.   

 

d.   

♦ Mean mark part (d) 36%.

 

 

e.i.   

 

e.ii.   
   
   

 

f.   

♦ Mean mark part (f) 36%.

 
 
   
 

 

g.i.   

♦♦ Mean mark part (g)(i) 30%.

 

 

g.ii.   

♦♦♦ Mean mark part (g)(ii) 8%.

 

 

 

Algebra, MET2 2017 VCAA 2

Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point . The height of above the ground, , is modelled by  , where is the time in minutes after Sammy enters the capsule and is measured in metres.

Sammy exits the capsule after one complete rotation of the Ferris wheel.
 


 

  1. State the minimum and maximum heights of above the ground.   (1 mark)

  2. For how much time is Sammy in the capsule?   (1 mark)

  3. Find the rate of change of with respect to and, hence, state the value of at which the rate of change of is at its maximum.   (2 marks)

As the Ferris wheel rotates, a stationary boat at , on a nearby river, first becomes visible at point . is 500 m horizontally from the vertical axis through the centre of the Ferris wheel and angle , as shown below.
 

   
 

  1. Find in degrees, correct to two decimal places.   (1 mark)

Part of the path of is given by  , where and are in metres.

  1. Find .   (1 mark)

As the Ferris wheel continues to rotate, the boat at is no longer visible from the point  onwards. The line through and is tangent to the path of , where angle .
 

   
 

  1. Find the gradient of the line segment in terms of and, hence, find the coordinates of , correct to two decimal places.   (3 marks)

  2. Find in degrees, correct to two decimal places.   (1 mark)

  3. Hence or otherwise, find the length of time, to the nearest minute, during which the boat at is visible.   (2 marks)

Show Answers Only
Show Worked Solution
a.   
     

 

b.   

 

c.   

♦ Mean mark 50%.
MARKER’S COMMENT: A number of commons errors here – 2 answers given, calc not in radian mode, etc …

 

   

 

d.   

♦ Mean mark 36%.
MARKER’S COMMENT: Choosing degrees vs radians in the correct context was critical here.

 

 

e.   

 

f.   

 

♦♦♦ Mean mark part (f) 18%.
MARKER’S COMMENT: Many students were unable to use the rise over run information to calculate the second gradient.

 

 

 

 
 

 

 

♦♦♦ Mean mark part (g) 7%.

g.   
   
 

 

h.   

♦♦♦ Mean mark 5%.

 

 
 

Calculus, MET1 2017 VCAA 9

The graph of    is shown below.
 

  1. Calculate the area between the graph of and the -axis.   (2 marks)

  2. For in the interval , show that the gradient of the tangent to the graph of is  .   (1 mark)

The edges of the right-angled triangle are the line segments and , which are tangent to the graph of , and the line segment , which is part of the horizontal axis, as shown below.

Let be the angle that makes with the positive direction of the horizontal axis, where  .

  1. Find the equation of the line through and in the form  , for  .   (2 marks)

  2. Find the coordinates of when  .   (4 marks)

Show Answers Only
Show Worked Solution
a.  
   
   
   
   
   

 

♦♦ Mean mark part (b) 35%.
MARKER’S COMMENT: Establishing the common denominator in the working was required!
b.  
 
   
   

 

c. 

♦♦♦ Mean mark part (c) 20%.
MARKER’S COMMENT: Most successful answers introduced a pronumeral such as to solve.

 

 

   
 
   

 

 

 

d. 

♦♦♦ Mean mark part (d) 17%.

   
 
   

 

 

 

 

Functions, MET1 2017 VCAA 7

Let  .

  1.  State the range of .   (1 mark)

  2.  Let  .
    1. Find the largest possible value of such that the range of is a subset of the domain of .   (2 marks)

    2. For the value of found in part b.i., state the range of .   (1 mark) 

  3. Let  .
  4. State the range of .   (1 mark)

Show Answers Only
Show Worked Solution

a. 
 

 

b.i. 

♦ Mean mark 38%.
 


 


 

b.ii. 

♦♦♦ Mean mark 20%.


 

c. 

♦♦ Mean mark 30%.