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Networks, FUR1 2017 VCE 3 MC

Consider the following graph.
 

 
The matrix for this graph that records the number of direct routes between points, is shown below.
 

 
This adjacency table contains 16 values when complete.

Of the 12 missing elements

  1. eight are ‘1’ and four are ‘2’.
  2. four are ‘1’ and eight are ‘2’.
  3. six are ‘1’ and six are ‘2’.
  4. two are ‘0’, six are ‘1’ and four are ‘2’.
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Networks, FUR1 2013 VCE 6 MC

 
The map above shows the road connections between three towns, .

The graph that could be used to model these road connections is
 

 

vcaa-networks-fur1-2013-6ii

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♦ Mean mark 40%.

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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Show Worked Solution
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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Show Worked Solution

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 34

2005 8a

A cylinder of radius    and height    is to be inscribed in a sphere of radius    centred at    as shown.

  1. Show that the volume of the cylinder is given by
  2.       (1 mark)

  3. Hence, or otherwise, show that the cylinder has a maximum volume when
  4.       (3 marks)

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i.   
   

 

 

 

ii.
   
 
   
 

 

 

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
     
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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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Show Worked Solution

i.  

 

ii.  
   
   

 

iii.   

♦ Mean mark (Vic) 35%.

,

 

iv. 

♦♦♦ Mean mark (Vic) 5%.

 

 vcaa-2011-meth-10ai

 
 
 

Calculus, 2ADV C4 SM-Bank 12

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 .  (2 marks)

     

 

      vcaa-2013-meth-10i

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  2. ²
     
  3. ²
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i.  
   
 

 

ii.   

♦ Mean mark (Vic) 35%.

 

   

²

 

iii.   

♦♦ Mean mark (Vic) 32%.

 

 
 
 

 

vcaa-2013-meth-10ii

 
 
 
  ²

GRAPHS, FUR2 2017 VCAA 3

Lifeguards are required to ensure the safety of swimmers at the beach.

Let be the number of junior lifeguards required.

Let be the number of senior lifeguards required.

The inequality below represents the constraint on the relationship between the number of senior lifeguards required and the number of junior lifeguards required.

 Constraint 1 
 

  1. If eight junior lifeguards are required, what is the minimum number of senior lifeguards required?  (1 mark)

 
There are three other constraints.

 Constraint 2 

 Constraint 3 

 Constraint 4 

  1. Interpret Constraint 4 in terms of the number of junior lifeguards and senior lifeguards required.  (1 mark)

 
The shaded region of the graph below contains the points that satisfy Constraints 1 to 4.

All lifeguards receive a meal allowance per day.

Junior lifeguards receive $15 per day and senior lifeguards receive $25 per day.

The total meal allowance cost per day, , for the lifeguards is given by



  1. Determine the minimum total meal allowance cost per day for the lifeguards.  (2 marks)
  2. On rainy days there will be no set minimum number of junior lifeguards or senior lifeguards required, therefore:

     

    • Constraint 2    and Constraint 3    are removed

     

    • Constraint 1 and Constraint 4 are to remain.
     

     

              Constraint 1 

     

              Constraint 4 

     


    The total meal allowance cost per day,     , for the lifeguards remains as

     

     


    How many junior lifeguards and senior lifeguards work on a rainy day if the total meal allowance cost     is to be a minimum?

     

    Write your answers in the boxes provided below.  (1 mark)


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a.   

b.   

c.   

d.   
Show Worked Solution

a.   

 

b.   

 

c.   

 

 

d.   

♦♦ Mean mark 24%.
MARKER’S COMMENT: A common incorrect answer was 10 and 3.

GRAPHS, FUR2 2017 VCAA 2

A swimming race is being held in the ocean.

From the shore, competitors swim 500 m out to a buoy in the ocean and then return to the shore.

One competitor, Edgar, reaches the buoy after 12.5 minutes and completes the race in 25 minutes.

The graph below shows his distance from shore, in metres, minutes after the race begins.
 

  1. What distance, in metres, has Edgar swum after 15 minutes?  (1 mark)
  2. Let be Edgar’s distance from the shore, in metres, minutes after the race begins.

     

    The linear relation that represents his swim out to the buoy is of the form
     

     

      where 

     

    The slope of the line is the speed at which Edgar is swimming, in metres per minute.

     

    Show that .  (1 mark)

A second competitor, Zlatko, began the race at the same time as Edgar.

Below is the relation that describes Zlatko’s swim, where is his distance from the shore, in metres,  minutes after the race begins and is the time it took Zlatko to finish the race.


  1. The graph below again shows the relation representing Edgar’s swim.

     

    Sketch the relation representing Zlatko’s swim on the graph below.  (2 marks)

  2. How many minutes after the start of the race were Zlatko and Edgar the same distance from the shore?

     

    Round your answer to two decimal places.  (1 mark)

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a.   

b.   

c.   

d.   

Show Worked Solution
a.   
   

 

b.   

♦ Mean mark 49%.
MARKER’S COMMENT: Note that it was not sufficient to type in the equation and write “Solve”.

 

 

c.   

♦ Mean mark 47%

 

d.   

 
 

GEOMETRY, FUR2 2017 VCAA 3

Some hostel buildings are arranged around a grassed area.

The grassed area is shown shaded in the diagram below.

The grassed area is made up of a square overlapping a circle.

The square has side lengths of 65 m.

The circle has a radius of 50 m.

An angle, , is also shown on the diagram.

  1. Use the cosine rule to show that the angle , correct to the nearest degree, is equal to 81°.  (1 mark)
  2. What is the perimeter, in metres, of the entire grassed area?

     

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

  3. The hostel’s management is planning to build a pathway from point A to point B, as shown on the diagram below.
     

     
    Calculate the length, in metres, of the planned pathway.

     

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

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a.   

b.   

c.   

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a.   
   
   
   

 

♦♦ Mean mark part (b) 34%.

b.   
   
   
   

 

♦♦ Mean mark part (c) 34%.

c.   

   
   

GEOMETRY, FUR2 2017 VCAA 2

Miki will travel from Melbourne (38° S, 145° E) to Tokyo (36° N, 140° E) on Wednesday, 20 December.

The flight will leave Melbourne at 11.20 am, and will take 10 hours and 40 minutes to reach Tokyo.

The time difference between Melbourne and Tokyo is two hours at that time of year.

  1. On what day and at what time will Miki arrive in Tokyo?  (1 mark)

Miki will travel by train from Tokyo to Nemuro and she will stay in a hostel when she arrives.

The hostel is located 186 m north and 50 m west of the Nemuro railway station.

    1. What distance will Miki have to walk if she were to walk in a straight line from the Nemuro railway station to the hostel?

       

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

    2. What is the three-figure bearing of the hostel from the Nemuro railway station?

       

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


The city of Nemuro is located 43° N, 145° E.

Assume that the radius of Earth is 6400 km.

  1. The small circle of Earth at latitude 43° N is shown in the diagram below.
     
             
     
    What is the radius of the small circle of Earth at latitude 43° N?

     

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

  2. Find the shortest great circle distance between Melbourne (38° S, 145° E) and Nemuro (43° N, 145° E).

     

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

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  2. i.
  3. ii.
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a.   

♦ Mean mark 40%.
COMMENT: A surprisingly poor result for this standard question.

 
 

 

b.i. 

 
 

 

♦♦ Mean mark part (b)(ii) 27%.
MARKER’S COMMENT: N15°W is not a 3-figure bearing and received no marks.

b.ii.   
 

 

 

c.   

♦ Mean mark part (c) 43%

 
 

 

d.   

♦ Mean mark part (d) 38%

GEOMETRY, FUR2 2017 VCAA 1

Miki is planning a gap year in Japan.

She will store some of her belongings in a small storage box while she is away.

This small storage box is in the shape of a rectangular prism.

The diagram below shows that the dimensions of the small storage box are 40 cm × 19 cm × 32 cm.
 

The lid of the small storage box is labelled on the diagram above.

    1. What is the surface area of the lid, in square centimetres?  (1 mark)
    2. What is the total outside surface area of this storage box, including the lid and base, in square centimetres?  (1 mark)
  2. Miki has a large storage box that is also a rectangular prism.

      

    The large storage box and the small storage box are similar in shape.

      

    The volume of the large storage box is eight times the volume of the small storage box.

      

    The length of the small storage box is 40 cm.

    What is the length of the large storage box, in centimetres?  (1 mark)

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a.i.   ²

a.ii.  ²

b.     

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a.i.   
    ²

 

a.ii.   
    ²

 

b.   

♦♦ Mean mark 32%.
MARKER’S COMMENT: A lack of understanding between linear and volume scale factors was again notable.

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.    

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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%.

 

NETWORKS, FUR2 2017 VCAA 2

Bai joins his friends Agatha, Colin and Diane when he arrives for a holiday in Seatown.

Each person will plan one tour that the group will take.

Table 1 shows the time, in minutes, it would take each person to plan each of the four tours.
 

 
The aim is to minimise the total time it takes to plan the four tours.

Agatha applies the Hungarian algorithm to Table 1 to produce Table 2.

Table 2 shows the final result of all her steps of the Hungarian algorithm.
 


 

  1. In Table 2 there is a zero in the column for Colin.
    When all values in the table are considered, what conclusion about minimum total planning time can be made from this zero?   (1 mark)

  2. Determine the minimum total planning time, in minutes, for all four tours.   (1 mark)

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a.   

b.   

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a.   

♦ Mean mark part (a) 37%.
MARKER’S COMMENT: The answer “Colin will plan Tour 2 because he is the fastest” received no marks.

 

b.   

 

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.  
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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%.

MATRICES, FUR2 2017 VCAA 2

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

Students can choose from the areas of performance (), sport () and technology ().

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

 

  1. Of the junior students who choose performance () in one term, what percentage are expected to choose sport () the next term?   (1 mark)

Matrix lists the number of junior students who will be in each elective activity in Term 1.
 


 

  1. 306 junior students are expected to choose sport () in Term 2.
     
    Complete the calculation below to show this.   (1 mark)


  2. In Term 4, how many junior students in total are expected to participate in performance () or sport () or technology ()?   (1 mark)

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  2.  
Show Worked Solution

a.   
 

b. 

♦♦ Mean mark part (c) 30%.

MARKER’S COMMENT: No matrix calculations were required here.


c.   

 
 

CORE, FUR2 2017 VCAA 7

Alex sold his mechanics’ business for $360 000 and invested this amount in a perpetuity.

The perpetuity earns interest at the rate of 5.2% per annum.

Interest is calculated and paid monthly.

  1. What monthly payment will Alex receive from this investment?   (1 mark)

  2. Later, Alex converts the perpetuity to an annuity investment.

     

    This annuity investment earns interest at the rate of 3.8% per annum, compounding monthly.

     

    For the first four years Alex makes a further payment each month of $500 to his investment.

     

    This monthly payment is made immediately after the interest is added.

     

    After four years of these regular monthly payments, Alex increases the monthly payment.

     

    This new monthly payment gives Alex a balance of $500 000 in his annuity after a further two years.

     

    What is the value of Alex’s new monthly payment?

     

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

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Show Worked Solution
a.   
   

♦ Mean mark part (a) 50%.

 

b.   

   

 

♦♦ Mean mark 29%.
MARKER’S COMMENT: A common error was entering the $500 payment as a positive value. Know why this is incorrect!

 

   

 

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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Show Worked Solution
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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Show Worked Solution
i.  
   
   
   
   
   

 

ii.  
 
   
   

 

iii. 

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

 

 

   
 
   

 

 

CORE, FUR2 2017 VCAA 3

The number of male moths caught in a trap set in a forest and the egg density (eggs per square metre) in the forest are shown in the table below.
 

  1. Determine the equation of the least squares line that can be used to predict the egg density in the forest from the number of male moths caught in the trap.
  2. Write the values of the intercept and slope of this least squares line in the appropriate boxes provided below.
  3. Round your answers to one decimal place.  (2 marks)

     

         

  4. The number of female moths caught in a trap set in a forest and the egg density (eggs per square metre) in the forest can also be examined.

     

    A scatterplot of the data is shown below.
     


     
    The equation of the least squares line is

     

                  egg density = 191 + 31.3 × number of female moths

    1. Draw the graph of this least squares line on the scatterplot (provided above).   (1 mark)

    2. Interpret the slope of the regression line in terms of the variables egg density and number of female moths caught in the trap.   (1 mark)

    3. The egg density is 1500 when the number of female moths caught is 55.
    4. Determine the residual value if the least squares line is used to predict the egg density for this number of female moths.   (1 mark)
    5. The correlation coefficient is 
    6. Determine the percentage of the variation in egg density in the forest explained by the variation in the number of female moths caught in the trap.
    7. Round your answer to one decimal place.   (1 mark)
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a.   

b.i.  

b.ii. 

b.iii. 

b.iv. 

Show Worked Solution

a.   

 

b.i.   

♦♦ Mean mark 26%.
MARKER’S COMMENT: Not well answered! Many students did not realise the graph started at 10 on the -axis and many did not use a ruler!

 

♦ Mean mark 39%.
MARKER’S COMMENT: Students must clearly refer to the increase in egg density for every one-unit increase in female moths.

b.ii.   

 

 

 

b.iii.   

♦ Mean mark 48%.

 

 

b.iv.   

♦ Mean mark 47%.

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)

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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)


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  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)

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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)

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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%.

   
 
   

 

 

 

 

Probability, MET1 2017 VCAA 8

For events and from a sample space, and .  Let  .

  1. Find   in terms of .  (1 mark)

  2. Find   in terms of .  (2 marks)

  3. Given that  , state the largest possible interval for .  (2 marks)

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Show Worked Solution

a.   
   
   

 

b. 

♦ Mean mark 40%.
MARKER’S COMMENT: The most successful answers used a Venn diagram or table.

 

c. 

♦ Mean mark 37%.

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)

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Show Worked Solution

a. 
 

 

b.i. 

♦ Mean mark 38%.
 


 


 

b.ii. 

♦♦♦ Mean mark 20%.


 

c. 

♦♦ Mean mark 30%.
 
 

 

Algebra, MET1 2017 VCAA 6

Let  .

  1. State all possible values of  .   (1 mark)

  2. Hence, find all possible solutions for  , where  .   (2 marks)

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Show Worked Solution

a. 

♦ Mean mark 42%.

 

 

 

b. 

♦ Mean mark 42%.

 

 

Financial Maths, STD2 F1 SM-Bank 5

Alex is buying a used car which has a sale price of  $13 380. In addition to the sale price there are the following costs:

2014 27a1

  1. Stamp Duty for this car is calculated at $3 for every $100, or part thereof, of the sale price.  
    Calculate the Stamp Duty payable.   (1 mark)

  2. Alex wishes to take out comprehensive insurance for the car for 12 months.

     

    The cost of comprehensive insurance is calculated using the following:
     
          2014 27a2
    Find the total amount that Alex will need to pay for comprehensive insurance.   (3 marks)

  3. Alex has decided he will take out the comprehensive car insurance rather than the less expensive non-compulsory third-party car insurance.

  4.  

    What extra cover is provided by the comprehensive car insurance?   (1 mark)

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Show Worked Solution
♦♦♦ Mean mark 12%
IMPORTANT: “or part thereof ..” in the question requires students to round up to 134 to get the right multiple of $3 for their calculation.
i.   
 

 

ii.  

 

 

 
 

 

 
 

 

 

 

♦ Mean mark 34%.
iii.  
 

Calculus, EXT2 C1 2017 HSC 15a

Let  , for  

  1. Find the value of .  (1 mark)

  2. Using integration by parts, or otherwise, show that for  
     
         .  (3 marks)

  3. Find the value of  .  (1 mark)

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Show Worked Solution
i.  
   
   
   
   

 

ii. 

♦ Mean mark 48%.

   
 
 
 

 

 

iii.  
   
   
   

Mechanics, EXT2 2017 HSC 14c

A smooth double cone with semi-vertical angle    is rotating about its axis with constant angular velocity .

Two particles, each of mass , are sitting on the cone as it rotates, as shown in the diagram.

Particle 1 is inside the cone at vertical distance above the apex, , and moves in a horizontal circle of radius .

Particle 2 is attached to the apex by a light inextensible string so that it sits on the cone at vertical distance below the apex. Particle 2 also moves in a horizontal circle of radius .

The acceleration due to gravity is .

  1. The normal reaction force on Particle 1 is .
    By resolving into vertical and horizontal components, or otherwise, show that  .  (2 marks)
  2. The normal reaction force on Particle 2 is and the tension in the string is  .

  3. By considering horizontal and vertical forces, or otherwise, show that

  4. .  (2 marks)
  6. Show that  .  (2 marks)

 

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Show Worked Solution
(i)  

 

(ii)  

 

 

 

 

(iii) 

♦♦ Mean mark 34%.

Harder Ext1 Topics, EXT2 2017 HSC 14b

Two circles, and , intersect at the points and . Point is chosen on the arc of  as shown in the diagram.

The line segment produced meets  at .

The line segment produced meets  at .

The line segment produced meets  at .

The line segment produced meets the line segment at .

Copy or trace the diagram into your writing booklet.

  1. State why .  (1 mark)
  2. Show that .  (1 mark)
  3. Hence, or otherwise, show that and are concyclic points.  (3 marks)
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Show Worked Solution

(i) 

 

 

(ii) 

 

 

 

 

 

(iii)  
   

 

♦ Mean mark 47%.

 

 

Algebra, STD2 A4 SM-Bank 4

Penny is a baker and makes meat pies every day.

The cost of making pies, ,  can be calculated using the equation

Penny sells the pies for $5.75 each, and her income is calculated using the equation

  1. On the graph, draw the graphs of    and  .  (2 marks)

  2. On the graph, label the breakeven point and the loss zone.  (2 marks)

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(i) and (ii)

Show Worked Solution
i.   

 

ii. 

Mechanics, EXT2 2017 HSC 9 MC

A particle is travelling on the circle with equation  .

It is given that  .

Which statement about the motion of the particle is true?

  1.  and the particle travels clockwise
  2.  and the particle travels anticlockwise
  3.  and the particle travels clockwise
  4.  and the particle travels anticlockwise
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Show Worked Solution
♦ Mean mark 46%.
 
 

 

Conics, EXT2 2017 HSC 16b

The hyperbola with equation

has eccentricity 2.

The distance from one of the foci to one of the vertices is 1.

What are the possible values of ?  (2 marks)

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Show Worked Solution

♦ Mean mark 45%.

 

 

Harder Ext1 Topics, EXT2 2017 HSC 16a

  Let  , where  .

  1. Show that  , for any integer .  (1 mark)
  3. Let  , where is a positive integer.
  4. By summing the series, prove that  

    .  (3 marks)

  6. Deduce, from parts (i) and (ii), that.  (2 marks)

  7. Show that  
     is independent of .  (1 mark)

 

 

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Show Worked Solution

(i)   

 
 

 

(ii)   

♦♦ Mean mark 34%.

 
 
 
 
 
♦ Mean mark part (iii) 48%.

 

(iii)   
   
   
   

 

 
 
 
 

♦♦♦ Mean mark part (iv) 28%.

 

(iv)   

 

CORE, FUR2 2017 VCAA 1

The number of eggs counted in a sample of 12 clusters of moth eggs is recorded in the table below.
     

  1. From the information given, determine
  2.  i. the range   (1 mark)

  3. ii. the percentage of clusters in this sample that contain more than 170 eggs.   (1 mark)

In a large population of moths, the number of eggs per cluster is approximately normally distributed with a mean of 165 eggs and a standard deviation of 25 eggs.

  1. Using the 68–95–99.7% rule, determine
  2.  i. the percentage of clusters expected to contain more than 140 eggs.   (1 mark)

  3. ii. the number of clusters expected to have less than 215 eggs in a sample of 1000 clusters.   (1 mark)

  4. The standardised number of eggs in one cluster is given by  
  5. Determine the actual number of eggs in this cluster.   (1 mark)

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a.   

b.i.  

b.ii. 

c. 

Show Worked Solution

a.i. 

a.ii. 

 

 

b.i.   
   

 

b.ii.   
   

 

 

c.   
 
 
   

MATRICES, FUR1 2017 VCAA 5 MC

Four teams, , , and , competed in a round-robin competition where each team played each of the other teams once. There were no draws.

The results are shown in the matrix below.
 


 

A 1 in the matrix shows that the team named in that row defeated the team named in that column.

For example, the 1 in row 2 shows that team defeated team .

In this matrix, the values of  , and are

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Show Worked Solution

GRAPHS, FUR1 2017 VCAA 7 MC

Connor makes 200 meat pies to sell at his local market.

The cost, , of producing pies can be determined from the rule below.

Connor sells the first 150 pies at full price and sells the remaining 50 pies at half-price.

To break even, the full price of each pie must be closest to

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