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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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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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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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a. 
 

 

b.i. 

♦ Mean mark 38%.
 


 


 

b.ii. 

♦♦♦ Mean mark 20%.


 

c. 

♦♦ Mean mark 30%.
 
 

 

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. 

Probability, MET1 2017 VCAA 4

In a large population of fish, the proportion of angel fish is .

Let  be the random variable that represents the sample proportion of angel fish for samples of size drawn from the population.

Find the smallest integer value of such that the standard deviation of is less than or equal to .  (2 marks)

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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.  


 

 

Calculus, MET1 2017 VCAA 2

Let  

  1. Find  .   (2 marks)

  2. Hence, calculate  . Express your answer in the form  , where is a positive integer.   (2 marks)

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a. 

 

 

b. 

MARKER’S COMMENT: Too many students ignored the hence instruction and were not able to form an integral.

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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♦♦♦ 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.  
 

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 7

A dress was on sale with 25% discount.

As a regular customer, Kate received a further 10% on the already discounted price.

What was the overall percentage discount Kate received?  (2 marks)

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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 2

George makes a single deposit of $9000 into an account that pays simple interest.

After 4 years, George's account has a balance of $10 350.

What simple interest rate did George receive on his investment?  (2 marks)

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Show Worked Solution
 

 

 
 

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. 

 
 

Financial Maths, STD2 F1 SM-Bank 1 MC

Rae paid  $40 000  for new office equipment at the start of the 2013 financial year.

At the start of each following financial year, she used flat rate depreciation to revalue her equipment.

At the start of the 2016 financial year she revalued her equipment at  $22 000.

The annual flat rate of depreciation she used, as a percentage of the purchase price, was

  1. 11.25%
  2. 15%
  3. 17.5%
  4. 35%
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♦ Mean mark 50%.

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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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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(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. 

 
 

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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(i) 

 

 

(ii) 

 

 

 

 

 

(iii)  
   

 

♦ Mean mark 47%.

 

 

Measurement, STD2 M1 SM-Bank 4

Steel rods are manufactured in the shape of equilateral triangular prisms.
 


 

  1. Find the volume of the prism (answer correct to 1 decimal place).  (2 marks)

  2. The mass of steel is 7850 kg/m³. Use this information to find the mass of the steel rod correct to the nearest gram.  (2 marks)

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  1. ³
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i.  

 

 
 
  ³

 

ii. 

 

³³

³  
  ³

 

 
 
 

Measurement, STD2 M7 SM-Bank 1

Bikram runs a hot yoga studio.

If it costs 34 cents for 1-kilowatt (1000 watts) for 1 hour, how much does it cost him to run three 3200-watt heaters from 9:00 am to 12:30 pm on a single day? (Give your answer to the nearest cent)  (2 marks)

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Measurement, STD2 M1 SM-Bank 7

Oscar and Nadine select a meal each from the table below.
 

 
Oscar has a hamburger and Nadine has the lamb souvlaki.

After dinner, Oscar goes for a run where he expends energy at 25 kJ/minute.

At the same time, Nadine goes for a brisk walk where she expends 12 kJ/minute.

Who will expend the kilojoule intake from their dinner the quickest, and by how many minutes?  (3 marks)

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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

 

 

 

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)

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i.   

 

ii. 

Volumes, EXT2 2017 HSC 13d

The trapezium whose vertices are    and    forms the base of a solid.

Each cross-section perpendicular to the -axis is an isosceles triangle. The height of the isosceles triangle with base , is 1. The height of the isosceles triangle with base , is 2.  The cross-section through the point is the isosceles triangle , where lies on the line , as shown in the diagram.

Find the volume of the solid.  (4 marks)

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³

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♦ Mean mark 51%.

 

 

 

 
 

 

 
 
  ³

Conics, EXT2 2017 HSC 15c

The ellipse with equation  , where  , has eccentricity .

The hyperbola with equation  , has eccentricity .

The value of is chosen so that the hyperbola and the ellipse meet at  , as shown in the diagram.

  1. Show that  
  2. .  (2 marks)
  4. If the two conics have the same foci, show that their tangents at are perpendicular.  (3 marks)

 

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(i)   

 

 

(ii)   

♦ Mean mark 51%.

 

 

 
 

 

   
   
 
 

 

 

 

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.  

Volumes, EXT2 2017 HSC 11e

The region bounded by the lines    and the -axis is rotated about the -axis.

Use the method of cylindrical shells to find an integral whose value is the volume of the solid of revolution formed. Do NOT evaluate the integral.  (2 marks)

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Conics, EXT2 2017 HSC 2 MC

Suppose  is a line and is a point NOT on .

The point moves so that the distance from to is half the perpendicular distance from to .

Which conic best describes the locus of ?

  1. A circle
  2. An ellipse
  3. A parabola
  4. A hyperbola
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MATRICES, FUR1 2017 VCAA 2 MC

The matrix below shows how five people, Alan (), Bevan (), Charlie (), Drew () and Esther (), can communicate with each other.
 


 

A 1 in the matrix shows that the person named in that row can send a message directly to the person named in that column.

For example, the 1 in row 3 and column 4 shows that Charlie can send a message directly to Drew.

Esther wants to send a message to Bevan.

Which one of the following shows the order of people through which the message is sent?

  1. Esther – Bevan
  2. Esther – Charlie – Bevan
  3. Esther – Charlie – Alan – Bevan
  4. Esther – Charlie – Drew – Bevan
  5. Esther – Charlie – Drew – Alan – Bevan
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GRAPHS, FUR1 2017 VCAA 4 MC

The annual fee for membership of a car club, in dollars, based on years of membership of the club is shown in the step graph below.

In the Martin family:

• Hayley has been a member of the club for four years
• John has been a member of the club for 20 years
• Sharon has been a member of the club for 25 years.

What is the total fee for membership of the car club for the Martin family?

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NETWORKS, FUR1 2017 VCAA 4-5 MC

The directed graph below shows the sequence of activities required to complete a project.

The time to complete each activity, in hours, is also shown.
 


 

Part 1

The earliest starting time, in hours, for activity is

  1. 3
  2. 10
  3. 11
  4. 12
  5. 13

 

Part 2

To complete the project in minimum time, some activities cannot be delayed.

The number of activities that cannot be delayed is

  1. 2
  2. 3
  3. 4
  4. 5
  5. 6
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GEOMETRY, FUR1 2017 VCAA 3 MC

The locations of four cities are given below.

Adelaide (35° S, 139° E)     Buenos Aires (35° S, 58° W) 
Melilla (35° N, 3° W)      Heraklion (35° N, 25° E)

 

In which order, from first to last, will the sun rise in these cities on New Year’s Day 2018?

  1. Adelaide, Heraklion, Buenos Aires, Melilla
  2. Adelaide, Heraklion, Melilla, Buenos Aires
  3. Heraklion, Adelaide, Melilla, Buenos Aires
  4. Melilla, Adelaide, Heraklion, Buenos Aires
  5. Melilla, Adelaide, Buenos Aires, Heraklion
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Measurement, STD2 M1 2013 HSC 15a*

The diagram shows the front of a tent supported by three vertical poles. The poles are 1.2 m apart. The height of each outer pole is 1.5 m, and the height of the middle pole is 1.8 m. The roof hangs between the poles.

2013 15a

The front of the tent has area ². 

  1. Use the trapezoidal rule to estimate .    (2 marks)

  2. Explain whether the trapezoidal rule give a greater or smaller estimate of  ?  (1 mark)

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  1. ²

  2.  

     

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i.   
   
   
    ²

 

ii. 

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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CORE, FUR1 2017 VCAA 13-15 MC

The wind speed at a city location is measured throughout the day.

The time series plot below shows the daily maximum wind speed, in kilometres per hour, over a three-week period.
  

 

Part 1

The time series is best described as having

  1. seasonality only.
  2. irregular fluctuations only.
  3. seasonality with irregular fluctuations.
  4. a decreasing trend with irregular fluctuations.
  5. an increasing trend with irregular fluctuations.

 

Part 2

The seven-median smoothed maximum wind speed, in kilometres per hour, for day 4 is closest to

 

Part 3

The table below shows the daily maximum wind speed, in kilometres per hour, for the days in week 2.

A four-point moving mean with centring is used to smooth the time series data above.

The smoothed maximum wind speed, in kilometres per hour, for day 11 is closest to

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`

CORE, FUR1 2017 VCAA 12 MC

Data collected over a period of 10 years indicated a strong, positive association between the number of stray cats and the number of stray dogs reported each year () in a large, regional city.

A positive association was also found between the population of the city and both the number of stray cats () and the number of stray dogs ().

During the time that the data was collected, the population of the city grew from 34 564 to 51 055.

From this information, we can conclude that

  1. if cat owners paid more attention to keeping dogs off their property, the number of stray cats reported would decrease.
  2. the association between the number of stray cats and stray dogs reported cannot be causal because only a correlation of +1 or –1 shows causal relationships.
  3. there is no logical explanation for the association between the number of stray cats and stray dogs reported in the city so it must be a chance occurrence.
  4. because larger populations tend to have both a larger number of stray cats and stray dogs, the association between the number of stray cats and the number of stray dogs can be explained by a common response to a third variable, which is the increasing population size of the city.
  5. more stray cats were reported because people are no longer as careful about keeping their cats properly contained on their property as they were in the past.
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CORE, FUR1 2017 VCAA 11 MC

Which one of the following statistics can never be negative?

  1. the maximum value in a data set
  2. the value of a Pearson correlation coefficient
  3. the value of a moving mean in a smoothed time series
  4. the value of a seasonal index
  5. the value of the slope of a least squares line fitted to a scatterplot
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CORE, FUR1 2017 VCAA 8-10 MC

The scatterplot below shows the wrist circumference and ankle circumference, both in centimetres, of 13 people. A least squares line has been fitted to the scatterplot with ankle circumference as the explanatory variable.
 

Part 1

The equation of the least squares line is closest to

  1. ankle = 10.2 + 0.342 × wrist
  2. wrist = 10.2 + 0.342 × ankle
  3. ankle = 17.4 + 0.342 × wrist
  4. wrist = 17.4 + 0.342 × ankle
  5. wrist = 17.4 + 0.731 × ankle

 

Part 2

When the least squares line on the scatterplot is used to predict the wrist circumference of the person with an ankle circumference of 24 cm, the residual will be closest to

  4.    
  5.    

 

Part 3

The residuals for this least squares line have a mean of 0.02 cm and a standard deviation of 0.4 cm.

The value of the residual for one of the data points is found to be  – 0.3 cm.

The standardised value of this residual is

  4.    
  5.    
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CORE, FUR1 2017 VCAA 5-7 MC

A study was conducted to investigate the association between the number of moths caught in a moth trap (less than 250, 250–500, more than 500) and the trap type (sugar, scent, light). The results are summarised in the percentaged segmented bar chart below.
 

Part 1

There were 300 sugar traps.

The number of sugar traps that caught less than 250 moths is closest to

  1. 30
  2. 90
  3. 250
  4. 300
  5. 500

 
Part 2

The data displayed in the percentaged segmented bar chart supports the contention that there is an association between the number of moths caught in a moth trap and the trap type because

  1. most of the light traps contained less than 250 moths.
  2. 15% of the scent traps contained 500 or more moths.
  3. the percentage of sugar traps containing more than 500 moths is greater than the percentage of scent traps containing less than 500 moths.
  4. 20% of sugar traps contained more than 500 moths while 50% of light traps contained less than 250 moths.
  5. 20% of sugar traps contained more than 500 moths while 10% of light traps contained more than 500 moths.

 
Part 3

The variables number of moths (less than 250, 250–500, more than 500) and trap type (sugar, scent, light) are

  1. both nominal variables.
  2. both ordinal variables.
  3. a numerical variable and a categorical variable respectively.
  4. a nominal variable and an ordinal variable respectively.
  5. an ordinal variable and a nominal variable respectively.
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