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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.   
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a.   

 

b.   

 

c.   

 

 

d.   

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

GRAPHS, FUR2 2017 VCAA 1

The graph below shows the depth of water in a sea bath from 6.00 am to 8.00 pm.
 

 

  1. What was the maximum depth, in metres, of water in the sea bath?   (1 mark)

  2. The sea bath was open to the public when the depth of water was above 1.5 m.
  3. Between which times was the sea bath open?   (1 mark)
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a.  

b.   

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a.  

b.   

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 1

Bus routes connect six towns.

The towns are Northend (), Opera (), Palmer (), Quigley (), Rosebush () and Seatown ().

The graph below gives the cost, in dollars, of bus travel along these routes.

Bai lives in Northend () and he will travel by bus to take a holiday in Seatown ().
 


 

  1. Bai considers travelling by bus along the route Northend () – Opera () – Seatown ().

     

    How much would Bai have to pay?   (1 mark)

  2. If Bai takes the cheapest route from Northend () to Seatown (), which other town(s) will he pass through?   (1 mark)

  3. Euler’s formula, , holds for this graph.

    Complete the formula by writing the appropriate numbers in the boxes provided below.   (1 mark)

     

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a.   

b.   

c. 

       

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a.   
   

 

b.   

 

c.   

MATRICES, FUR2 2017 VCAA 1

A school canteen sells pies (), rolls () and sandwiches ().

The number of each item sold over three school weeks is shown in matrix .

 

  1. In total, how many sandwiches were sold in these three weeks?   (1 mark)

  2. The element in row and column of matrix is .
  3. What does the element indicate?  (1 mark)

  4. Consider the matrix equation



    where = cost of one pie, = cost of one roll and = cost of one sandwich.
  5.  i. What is the cost of one sandwich?   (1 mark)

The matrix equation below shows that the total value of all rolls and sandwiches sold in these three weeks is $915.60

Matrix in this equation is of order .

  1. ii. Write down matrix .   (1 mark)

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a.   
   

 
b.   


 

c.i.   
   

 

 

c.ii.   

CORE, FUR2 2017 VCAA 5

Alex is a mobile mechanic.

He uses a van to travel to his customers to repair their cars.

The value of Alex’s van is depreciated using the flat rate method of depreciation.

The value of the van, in dollars, after years, , can be modelled by the recurrence relation shown below.

  1. Recursion can be used to calculate the value of the van after two years.

     

    Complete the calculations below by writing the appropriate numbers in the boxes provided.   (2 marks)


    1. By how many dollars is the value of the van depreciated each year?   (1 mark)
    2. Calculate the annual flat rate of depreciation in the value of the van.
    3. Write your answer as a percentage.   (1 mark)
  3. The value of Alex’s van could also be depreciated using the reducing balance method of depreciation.
  4. The value of the van, in dollars, after years, , can be modelled by the recurrence relation shown below.

     

           

    At what annual percentage rate is the value of the van depreciated each year?   (1 mark)

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a.

b.i.  

b.ii.

c.  

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a.   

  
b.i.   
  

b.ii.   
   

 

c.   
   

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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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, MET2 2017 VCAA 1

Let  . Part of the graph of is shown below.
 

  1. Find the coordinates of the turning points.   (2 marks)

  2.   and   are two points on the graph of .

     

    1. Find the equation of the straight line through and .   (2 marks)

    2. Find the distance .   (1 mark)

Let  .

  1. Let  and be two points on the graph of .

     

    1. Find the distance in terms of .   (2 marks)

    2. Find the values of such that the distance is equal to  .   (1 mark)

  2. The diagram below shows part of the graphs of and  . These graphs intersect at the points with the coordinates and .
  3.  
       
  4.  
    1. Find the value of in terms of .   (1 mark)

    2. Find the area of the shaded region in terms of .   (2 marks)

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a.   

 


 

b.i.  

 

MARKER’S COMMENT: Students skilled in the use of technology will be much more efficient and minimise errors here.
b.ii.   
   
   

 

c.i.   
   
   

 

c.ii.   


 

d.i.   


 

d.ii.   
    ²

Probability, MET1 2017 VCAA 5

For Jac to log on to a computer successfully, Jac must type the correct password. Unfortunately, Jac has forgotten the password. If Jac types the wrong password, Jac can make another attempt. The probability of success on any attempt is . Assume that the result of each attempt is independent of the result of any other attempt. A maximum of three attempts can be made.

  1. What is the probability that Jac does not log on to the computer successfully?   (1 mark)

  2. Calculate the probability that Jac logs on to the computer successfully. Express your answer in the form , where and are positive integers.   (1 mark)

  3. Calculate the probability that Jac logs on to the computer successfully on the second or on the third attempt. Express your answer in the form , where and are positive integers.   (2 marks)

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a. 

 

b. 

 

c. 

Calculus, MET1 2017 VCAA 3

Let  

  1. Show that  .   (1 mark)

  2. Sketch the graph of   on the axes below. Label the axis intercepts and any stationary points with their coordinates.   (3 marks)

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  2.  
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a. 

 

b.  


 

 

Harder Ext1 Topics, EXT2 2017 HSC 15b

Consider the curve  , for    and  , where is a positive constant.

  1. Show that the equation of the tangent to the curve at the point    is given by  .  (2 marks)
  2. The tangent to the curve at the point meets the and axes at   and  respectively. Show that  , where is the origin.  (3 marks)

 

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(i)   

 

 

 

(ii) 

 

 

 

 
 
 
 
 
 

Financial Maths, STD2 F1 SM-Bank 12

A golf shop is having a Boxing Day sale.

  1. What is the percentage discount on a putter which is reduced from $120.00 to $102.00?  (1 mark)

  2. The same discount applies storewide. What discount amount is applicable to a box of golf balls whose original price was $25.00?  (1 mark)

  3. What is the sale price of a golf bag which originally cost $160.00  (1 mark)

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i.    
   
   

 

ii.   
   

 

iii.   
   
   

Financial Maths, STD2 F1 SM-Bank 6

Michelle intends to keep a car purchased for $17 000 for 15 years. At the end of this time its value will be $3500.

  1. By what amount, in dollars, would the car’s value depreciate annually if Michelle used the flat rate method of depreciation?  (1 mark)

  2. Determine the annual flat rate of depreciation correct to one decimal place.  (1 mark) 

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i.   
   

 

 

ii.   

Financial Maths, STD2 F1 SM-Bank 5

Khan paid $900 for a printer.

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

  1. Determine the price of the printer before GST was added.

     

    Write your answer correct to the nearest cent.  (2 marks)

  2. Khan is able to depreciate the full $900 purchase price of his printer for taxation purposes.

     

    Under flat rate depreciation the printer will be valued at $300 after five years.

     

    Calculate the annual depreciation in dollars.  (1 mark)

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i.   

COMMENT: Reverse GST questions regularly cause problems for many students.
 
 

 

ii.   

Financial Maths, STD2 F1 SM-Bank 3

A company purchased a machine for $60 000.

For taxation purposes the machine is depreciated over time using the straight line depreciation method.

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

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

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

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

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i.   
   

 

ii.   

 
 

 

iii.   

 

Financial Maths, STD2 F1 SM-Bank 10

Hugo is a professional bike rider.

The value of his bike will be depreciated over time using the flat rate method of depreciation.

The graph below shows his bike’s initial purchase price and its value at the end of each year for a period of three years.
 

  1. What was the initial purchase price of the bike?  (1 mark)

  2. Use calculations to show that the bike depreciates in value by $1500 each year.  (1 mark)

  3. Assume that the bike’s value continues to depreciate by $1500 each year. Determine its value five years after it was purchased.  (1 mark)

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i.   
 

ii.   

 

 

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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(i)  

 

(ii)  

 

 

 

 

(iii) 

♦♦ Mean mark 34%.

Calculus, EXT2 C1 2017 HSC 14a

It is given that  .

  1. Find and so that  .  (1 mark)

  2. Hence, or otherwise, show that for any real number
     
         .  (2 marks)

  3. Find the limiting value as    of
     
         .  (1 mark)

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i. 

 

 


 

ii. 

 

iii. 

 
 

Measurement, STD2 M1 SM-Bank 8

A cannon ball is made out of steel and has a diameter of 23 cm.

  1. Find the volume of the sphere in cubic centimetres (correct to 1 decimal place).  (2 marks)

  2. It is known that the mass of the steel used is 8.2 tonnes/m³. Use this information to find the mass of the cannon ball to the nearest gram.  (2 marks)

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  1. ³
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i.   

 
 
  ³

 

ii.   ³³

³
  ³

 

 

 

Measurement, STD2 M1 SM-Bank 3

A 250-watt television is turned on for an average of 4 hours per day during off-peak periods for a week.

If the television is not running at any other time and electricity is charged at $0.36/kWh during off-peak, how much does it cost to run the television for a week?  (2 marks)

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Mechanics, EXT2 M1 2017 HSC 13c

A particle is projected upwards from ground level with initial velocity  , where is the acceleration due to gravity and is a positive constant. The particle moves through the air with speed    and experiences a resistive force.

The acceleration of the particle is given by  . Do NOT prove this.

The particle reaches a maximum height, , before returning to the ground.

Using  , or otherwise, show that    metres.  (4 marks)

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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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(i)   

 
 

 

(ii)   

♦♦ Mean mark 34%.

 
 
 
 
 
♦ Mean mark part (iii) 48%.

 

(iii)   
   
   
   

 

 
 
 
 

♦♦♦ Mean mark part (iv) 28%.

 

(iv)   

 

Functions, EXT1′ F2 2017 HSC 12d

Let be a polynomial.

  1. Given that    is a factor of , show that
     
    .  (2 marks)

  2. Given that the polynomial    has a factor  , find the value of .  (2 marks)

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i.  
 
   

 

 

 

ii.  
 
   
   

 

 

CORE, FUR2 2017 VCAA 2

The back-to-back stem plot below displays the wingspan, in millimetres, of 32 moths and their place of capture (forest or grassland).

 

  1. Which variable, wingspan or place of capture, is a categorical variable?  (1 mark)

  2. Write down the modal wingspan, in millimetres, of the moths captured in the forest.  (1 mark)

  3. Use the information in the back-to-back stem plot to complete the table below.  (2 marks)

     

     

  4. Show that the moth captured in the forest that had a wingspan of 52 mm is an outlier.  (2 marks)

  5. The back-to-back stem plot suggests that wingspan is associated with place of capture.
  6. Explain why, quoting the values of an appropriate statistic.  (2 marks)

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  3.    

     

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a.   

 

b.   

 

c.   

 

 

d.   
 

 

 

 

 

e.   

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. 

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a.i. 

a.ii. 

 

 

b.i.   
   

 

b.ii.   
   

 

 

c.   
 
 
   

Functions, EXT1′ F1 2017 HSC 12a

Consider the function  .

  1.  Show that    is increasing for all .  (1 mark)

  2.  Show that    is an odd function.  (1 mark)

  3.  Describe the behaviour of    for large positive values of .  (1 mark)

  4.  Hence sketch the graph of  .  (1 mark)

  5.  Hence, or otherwise, sketch the graph of  .  (1 mark)

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  4.  
  5.  

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i. 


 

 
 

 


 

 
 

 

 

ii.  
     
   
   

 

 

iii. 

 

iv.  

 

v.  

Conics, EXT2 2017 HSC 11b

An asymptote to the hyperbola    makes an angle with the positive -axis, as shown.

Find the value of .  (2 marks)

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GRAPHS, FUR1 2017 VCAA 2 MC

The graph below shows the volume of water in a water tank between 7 am and 5 pm on one day.

Which one of the following statements is true?

  1. The volume of water in the tank decreases between 8 am and 11 am.
  2. The volume of water in the tank increases at the greatest rate between 4 pm and 5 pm.
  3. The volume of water in the tank is constant between 12 noon and 2 pm.
  4. The tank is filled with water at a constant rate of 100 L per hour.
  5. More water enters the tank during the first five hours than during the last five hours.
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NETWORKS, FUR1 2017 VCAA 2 MC

Two graphs, labelled Graph 1 and Graph 2, are shown below.
 

 
The sum of the degrees of the vertices of Graph 1 is

  1. two less than the sum of the degrees of the vertices of Graph 2.
  2. one less than the sum of the degrees of the vertices of Graph 2.
  3. equal to the sum of the degrees of the vertices of Graph 2.
  4. one more than the sum of the degrees of the vertices of Graph 2.
  5. two more than the sum of the degrees of the vertices of Graph 2.
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Algebra, STD2 A1 SM-Bank 5

Fried's formula is used to calculate the medicine dosages for children aged 1-2 years.
 


 

Ben is 1.5 years old and receives a daily dosage of 450 mg of a medicine.

According to Fried's formula, what would the appropriate adult daily dosage of the medicine be?  (2 marks)

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GEOMETRY, FUR1 2017 VCAA 1 MC

A wheel has five spokes equally spaced around a central hub, as shown in the diagram below.
 


 

The angle between two spokes is labelled on the diagram.

What is the angle ?

  1.  
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CORE, FUR1 2017 VCAA 19-20 MC

Shirley would like to purchase a new home. She will establish a loan for $225 000 with interest charged at the rate of 3.6% per annum, compounding monthly.

Each month, Shirley will pay only the interest charged for that month.

Part 1

After three years, the amount that Shirley will owe is

  1.    $73 362
  2.  $170 752
  3.  $225 000
  4.  $239 605
  5.  $245 865

 
Part 2

Let be the value of Shirley’s loan, in dollars, after months.

A recurrence relation that models the value of is

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Algebra, STD2 A4 SM-Bank 3

Temperature can be measured in degrees Celsius () or degrees Fahrenheit ().

The two temperature scales are related by the equation  .

  1. Calculate the temperature in degrees Fahrenheit when it is  −20 degrees Celsius.  (1 mark)

  2. The following two graphs are drawn on the axes below:
     
             and  
     

         

    Explain what happens at the point where the two graphs intersect.  (1 mark)

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  2.  

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i.  
   

 

ii.   

CORE, FUR1 2017 VCAA 1-3 MC

The boxplot below shows the distribution of the forearm circumference, in centimetres, of 252 people.
 

Part 1

The percentage of these 252 people with a forearm circumference of less than 30 cm is closest to

 

Part 2

The five-number summary for the forearm circumference of these 252 people is closest to

 

Part 3

The table below shows the forearm circumference, in centimetres, of a sample of 10 people selected from this group of 252 people.
 

 
The mean, , and the standard deviation, , of the forearm circumference for this sample of people are closest to

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Measurement, STD2 M2 SM-Bank 01

Island A and island B are both on the equator. Island B is west of island A. The longitude of island A is 5°E and the angle at the centre of Earth (O), between A and B, is 30°.
 


 

  1. What is the longitude of island B?  (1 mark)
  2. What time is it on island B when it is 10 am on island A?  (1 mark)
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(i)   
   
   

 

 

(ii)