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CORE, FUR2 2019 VCAA 4

The relative humidity (%) at 9 am and 3 pm on 14 days in November 2017 is shown in Table 3 below.

A least squares line is to be fitted to the data with the aim of predicting the relative humidity at 3 pm (humidity 3 pm) from the relative humidity at 9 am (humidity 9 am).

  1. Name the explanatory variable.   (1 mark)

  2. Determine the values of the intercept and the slope of this least squares line.
  3. Round both values to three significant figures and write them in the appropriate boxes provided.   (1 mark)

humidity 3 pm = 
 
   +  
 
   × humidity 9 am  (1 mark)
  1. Determine the value of the correlation coefficient for this data set.
  2. Round your answer to three decimal places.   (1 mark)

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a. 
 

b. 


 

c.    

Proof, EXT2 P1 SM-Bank 1 MC

Consider the following statement.

"If you have no treasure, I have no kingdom."

Which of the following is logically equivalent to this statement?

  1.  If I have no kingdom then you have no treasure.
  2.  If you have treasure then I have a kingdom.
  3.  If you have no kingdom then I have no treasure.
  4.  If I have a kingdom then you have treasure.
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CORE, FUR2 2019 VCAA 2

 

The parallel boxplots below show the maximum daily temperature and minimum daily temperature, in degrees Celsius, for 30 days in November 2017.
 

  1. Use the information in the boxplots to complete the following sentences.
  2. For November 2017
  3.    i. the interquartile range for the minimum daily temperature was _____ °C   (1 mark)

  4.  ii. the median value for maximum daily temperature was _____ °C higher than the median value for minimum daily temperature   (1 mark)

  5. iii. the number of days on which the maximum daily temperature was less than the median value for minimum daily temperature was _____    (1 mark)

  1. The temperature difference between the minimum daily temperature and the maximum daily temperature in November 2017 at this location is approximately normally distributed with a mean of 9.4 °C and a standard deviation of 3.2 °C.
  2. Determine the number of days in November 2017 for which this temperature difference is expected to be greater than 9.4 °C.  (1 mark)

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a.i. 
 

a.ii.   
 

 

 

a.iii. 

   
 

b.   
   

CORE, FUR2 2019 VCAA 1

Table 1 shows the day number and the minimum temperature, in degrees Celsius, for 15 consecutive days in May 2017.
 

  1. Which of the two variables in this data set is an ordinal variable?   (1 mark)

The incomplete ordered stem plot below has been constructed using the data values for days 1 to 10.

  1. Complete the stem plot above by adding the data values for days 11 to 15.   (1 mark)

  2. The ordered stem plot below shows the maximum temperature, in degrees Celsius, for the same 15 days.

     

     

    Use this stem plot to determine

  3.  i. the value of the first quartile    (1 mark)

  4. ii. the percentage of days with a maximum temperature higher than 15.3 °C.   (1 mark)

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a.   

b.  

 
c.i.   

 
 

c.ii.   
   

GRAPHS, FUR1 2019 VCAA 2 MC

The cost, , of using kilowatt hours of electricity can be calculated using the equation below.
 


 

From this equation, it can be concluded that there is

  1. no fixed charge and the electricity used is charged at $0.15 per kilowatt hour.
  2. no fixed charge and the electricity used is charged at $52.00 per kilowatt hour.
  3. a fixed charge of $0.15 and the electricity used is charged at $52.00 per kilowatt hour.
  4. a fixed charge of $52.00 and the electricity used is charged at $0.15 per kilowatt hour.
  5. a fixed charge of $52.00 and the electricity used is charged at $15.00 per kilowatt hour.
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GEOMETRY, FUR1 2019 VCAA 3 MC

An ice cream dessert is in the shape of a hemisphere. The dessert has a radius of 5 cm.
 

The top and the base of the dessert are covered in chocolate.

The total surface area, in square centimetres, that is covered in chocolate is closest to

  1.   52
  2. 157
  3. 236
  4. 314
  5. 942
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²

MATRICES, FUR1 2019 VCAA 2 MC

There are two rides called The Big Dipper and The Terror Train at a carnival.

The cost, in dollars, for a child to ride on each ride is shown in the table below.
 

  Ride       Cost ($)      
       The Big Dipper    7
       The Terror Train    8

   
Six children ride once only on The Big Dipper and once only on The Terror Train.

The total cost of the rides, in dollars, for these six children can be determined by which one of the following calculations?

A.   B.  
C.   D.  
E.      
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CORE, FUR1 2019 VCAA 9-10 MC

A least squares line is used to model the relationship between the monthly average temperature and latitude recorded at seven different weather stations. The equation of the least squares line is found to be
 

 
Part 1

When the numbers in this equation are correctly rounded to three significant figures, the equation will be

 
Part 2

The coefficient of determination was calculated to be 0.893743

The value of the correlation coefficient, rounded to three decimal places, is

  1. − 0.945
  2. − 0.898
  3.  0.806
  4.  0.898
  5.  0.945
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CORE, FUR1 2019 VCAA 1-3 MC

The histogram below shows the distribution of the population size of 48 countries in 2018.

Part 1

The number of these countries with a population size between 5 million and 20 million people is

  1.  11
  2.  17
  3.  23
  4.  34
  5.  35

 
Part 2

The shape of this histogram is best described as

  1. positively skewed with no outliers. 
  2. positively skewed with outliers.
  3. approximately symmetric.
  4. negatively skewed with no outliers.
  5. negatively skewed with outliers.

 
Part 3

The histogram below shows the population size for these 48 countries plotted on a scale.

Based on this histogram, the number of countries with a population size that is less than people is

  1. 1
  2. 5
  3. 7
  4. 8
  5. 48
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Part 1


 

Part 2


 

Part 3

Functions, EXT1 F2 SM-Bank 2

The polynomial    has roots  .

  1. Find the value of  .  (1 mark)

  2. Find the value of  .  (1 mark)

  3. It is known that two of the roots are equal in magnitude but opposite in sign.

     

    Find the third root and hence find the value of .  (2 marks)

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i.   

 

ii.   

 

iii.   

 

Functions, 2ADV F1 SM-Bank 43

 Solve     (2 marks)

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 MARKER’S COMMENT: Note that both conditions must be satisfied! Dealing with negative signs and division for inequalities produced many errors.

 
 
 
   

Mechanics, SPEC2 2019 VCAA 5

A mass of    kilograms is initially held at rest near the bottom of a smooth plane inclined at degrees to the horizontal. It is connected to a mass of    kilograms by a light inextensible string parallel to the plane, which passes over a smooth pulley at the end of the plane. The mass    is 2 m above the horizontal floor.

The situation is shown in the diagram below.
 


 

  1. After the mass    is released, the following forces, measured in newtons, act on the system:

     

    • weight forces    and 
    •  the normal reaction force 
    •  the tension in the string 

     

    On the diagram above, show and clearly label the forces acting on each of the masses.  (1 mark) 

  2. If the system remains in equilibrium after the mass    is released, show that  .  (1 mark)
  3. After the mass    is released, the mass    falls to the floor.

     

    1. For what values of    will this occur? Express your answer as an inequality in terms of    and  .  (1 mark)
    2. Find the magnitude of acceleration, in ms−2, of the system after the mass    is released and before the mass    hits the floor. Express your answer in terms of    and  .  (2 marks)
  4. After the mass    is released, it moves up the plane.
    Find the maximum distance, in metres, that the mass    will move up the plane if    and  .  (5 marks)
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  1.   
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a.   

 

b.   

 

c.i.   

 

c.ii.   

 

d.   

 


 

 

 

 

Vectors, SPEC2 2019 VCAA 4

The base of a pyramid is the parallelogram    with vertices at points    and  . The apex (top) of the pyramid is located at  .

  1. Find the values of    and  .  (2 marks)

  2. Find the cosine of the angle between the vectors    and  .  (2 marks)

  3. Find the area of the base of the pyramid.  (2 marks)

  4. Show that    is perpendicular to both    and  , and hence find a unit vector that is perpendicular to the base of the pyramid.  (3 marks)

  5. Find the volume of the pyramid.  (2 marks)

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  5. ³

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a.   
   

 

 

b.   

 
 

 

c.   
 

 


 

  ²

 

d.   

 

 

e.   

 

 

 
 

 

 
  ³

Complex Numbers, SPEC2 2019 VCAA 2

  1. Show that the solutions of  , where  , are  .   (1 mark)

  2. Plot the solutions of    on the Argand diagram below.   (1 mark)

     

     

Let  , where  , represent the circle of minimum radius that passes through the solutions of  .

    1. Find    and  .   (2 marks)

    2. Find the cartesian equation of the circle  .   (1 mark)

    3. Sketch the circle on the Argand diagram in part a.ii. Intercepts with the coordinate axes do not need to be calculated or labelled.   (1 mark)

  2. Find all values of  , where  , for which the solutions of    satisfy the relation  .   (2 marks)

  3. All complex solutions of    have non-zero real and imaginary parts.

     

    Let    represent the circle of minimum radius in the complex plane that passes through these solutions, where  .

     

    Find    and    in terms of    and  .   (2 marks)

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  2.   
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a.i.   
   
   
   

 

a.ii.   

 

b.i.   

 

 

b.ii.   
 
 

 

b.iii.   

 

c.   


 

 

d.   

 

Calculus, SPEC2 2019 VCAA 1

A curve is defined parametrically by  , where  .

  1. Show that the curve can be represent in cartesian form by the rule  .   (2 marks)

  2. State the domain and range of the relation given by  .  (2 marks)

  3.  i. Express    in terms of  .   (2 marks)

  4. ii. State the limiting value of    as    approaches  .   (1 mark)

  1. Sketch the curve    on the axes below for  , labelling the endpoints with their coordinates.   (2 marks)

     

     

  2. The portion of the curve given by    for    is rotated about the -axis to form a solid of revolution.
  3. Write down, but do not evaluate, a definite integral in terms of    that gives the volume of the solid formed.   (2 marks)

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  2.  

  4.  
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a.   

 

b.   

 

c.i.   

 

c.ii.   

 

d.   

 

e.   


 

Statistics, MET2 2019 VCAA 4

The Lorenz birdwing is the largest butterfly in Town A.

The probability density function that describes its life span, , in weeks, is given by
 


 

  1. Find the mean life span of the Lorenz birdwing butterfly.  (2 marks)

  2. In a sample of 80 Lorenz birdwing butterflies, how many butterflies are expected to live longer than two weeks, correct to the nearest integer?  (2 marks)

  3. What is the probability that a Lorenz birdwing butterfly lives for at least four weeks, given that it lives for at least two weeks, correct to four decimal places?  (2 marks)

The wingspans of Lorenz birdwing butterflies in Town A are normally distributed with a mean of 14.1 cm and a standard deviation of 2.1 cm.

  1. Find the probability that a randomly selected Lorenz birdwing butterfly in Town A has a wingspan between 16 cm and 18 cm, correct to four decimal places.  (1 mark)

  2. A Lorenz birdwing butterfly is considered to be very small if its wingspan is in the smallest 5% of all the Lorenz birdwing butterflies in Town A.

     

    Find the greatest possible wingspan, in centimetres, for a very small Lorenz birdwing butterfly in Town A, correct to one decimal place.  (1 mark)

Each year, a detailed study is conducted on a random sample of 36 Lorenz birdwing butterflies in Town A.

A Lorenz birdwing butterfly is considered to be very large if its wingspan is greater than 17.5 cm. The probability that the wingspan of any Lorenz birdwing butterfly in Town A is greater than 17.5 cm is 0.0527, correct to four decimal places.

    1. Find the probability that three or more of the butterflies, in a random sample of 36 Lorenz birdwing butterflies from Town A, are very large, correct to four decimal places.  (1 mark)

    2. The probability that or more butterflies, in a random sample of 36 Lorenz birdwing butterflies from Town A, are very large is less than 1%.

       

      Find the smallest value of , where is an integer.  (2 marks)

    3. For random samples of 36 Lorenz birdwing butterflies in Town A, is the random variable that represents the proportion of butterflies that are very large.
    4. Find the expected value and the standard deviation of , correct to four decimal places.  (2 marks)

    5. What is the probability that a sample proportion of butterflies that are very large lies within one standard deviation of 0.0527, correct to four decimal places? Do not use a normal approximation.  (2 marks)

  2. The Lorenz birdwing butterfly also lives in Town B.

     

    In a particular sample of Lorenz birdwing butterflies from Town B, an approximate 95% confidence interval for the proportion of butterflies that are very large was calculated to be (0.0234, 0.0866), correct to four decimal places.

     

    Determine the sample size used in the calculation of this confidence interval.  (2 marks)

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a.   
   
   

 

b.   
   
   

 

 
 

 

c.   
   
   

 

d.   

 

e.   

 

 

f.i.   

 

f.ii.   
 

 

f.iii.   
 

 

f.iv.       

 

g.   

Calculus, MET2 2019 VCAA 3

During a telephone call, a phone uses a dual-tone frequency electrical signal to communicate with the telephone exchange.

The strength, , of a simple dual-tone frequency signal is given by the function  , where    is a measure of time and  .

Part of the graph of   is shown below

  1. State the period of the function.   (1 mark)

  2. Find the values of    where    for the interval  .   (1 mark)

  3. Find the maximum strength of the dual-tone frequency signal, correct to two decimal places.   (1 mark)

  4. Find the area between the graph of    and the horizontal axis for  .   (2 marks)

Let    be the function obtained by applying the transformation    to the function  , where
 


 

and and are real numbers.

  1. Find the values of and given that    has the same area calculated in part d.   (2 marks)

  2. The rectangle bounded by the line  , the horizontal axis, and the lines    and    has the same area as the area between the graph of    and the horizontal axis for one period of the dual-tone frequency signal.

     

    Find the value of  .   (2 marks)

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  4. ²
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a.   
  

b.   
  

c.   

 

d.   
    ²

 

e.   


  

f.   
 
 

Complex Numbers, SPEC1 2019 VCAA 7

  1. Show that  .   (1 mark)

  2. Find  , expressing your answer in the form  , where  ,  .   (2 marks)

  3. Find the integer values of    for which    is real.   (1 mark)

  4. Find the integer values of    for which  , where    is a real number.   (1 mark)

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a. 

   

b.   
   
   
   

 

c.   

 

d. 

Calculus, SPEC1 2019 VCAA 5

The graph of    over the domain    is shown below.

  1.  i.  Find .  (1 mark)

  2. ii. Hence, find the coordinates of the turning points of the graph in the interval  .  (2 marks)

  3. Sketch the graph of    on the set of axes above. Clearly label the turning points and endpoints of this graph with their coordinates.  (3 marks)
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  1. i. 
  2. ii.
  3.  

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a.i.   
  `f^{′}(x)`
   

 

a.ii.   

 

 

b.   

Calculus, MET2 2019 VCAA 1

Let  .

  1. Find .  (1 mark)

  2. i.  State the nature of the stationary point on the graph of    at the origin.  (1 mark)

  3. ii.  Find the maximum value of the function    and the values of    for which the maximum occurs.  (2 marks)

  4. iii. Find the values of    for which    is always negative.  (1 mark)

  5. i.  Find the equation of the tangent to the graph of    at  .  (1 mark)

  6. ii. Find the area enclosed by the graph of    and the tangent to the graph of    at  , correct to four decimal places.  (2 marks)

  7. Let    be a point on the graph of  , where  .
  8. Find the minimum distance between    and the point  , and the value of    for which this occurs, correct to three decimal places.  (3 marks)

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  2. i. 
  3. ii.
  4. iii.
  5. i.   
  6. ii.

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a.   
   

​

b.i.   
 

 

b.ii.   
 

 

b.iii.   
 

 

c.i.   
 

 

c.ii.   
   

 

d.   

Statistics, SPEC1 2019 VCAA 3

A machine produces chocolate in the form of a continuous cylinder of radius 0.5 cm. Smaller cylindrical pieces are cut parallel to its end, as shown in the diagram below.

The lengths of the pieces vary with a mean of 3 cm and a standard deviation of 0.1 cm.
 


 

  1. Find the expected volume of a piece of chocolate in cm³.   (1 mark)

  2. Find the variance of the volume of a piece of chocolate in cm6.   (1 mark)

  3. Find the expected surface area of a piece of chocolate in cm².   (1 mark)

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  1. ³
  3. ²
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a.   
   
    ³

 

b.   
   
   
   

 

c.   
   
   
    ²

Functions, 2ADV F1 SM-Bank 35

  1. Sketch the function    where    on a number plane, labelling all intercepts.  (1 mark)

  2. On the same graph, sketch  . Label all intercepts.  (2 marks)

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  1.   
  2.   
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i.   
 

 

ii.   

 

Statistics, 2ADV S3 EQ-Bank 17

The diastolic measurement for blood pressure in 35-year-old people is normally distributed, with a mean of 75 and a standard deviation of 12.

  1. A person is considered to have low blood pressure if their diastolic measurement is 63 or less.What percentage of 35-year-olds have low blood pressure?  (1 mark)

  2. Calculate the -score for a diastolic measurement of 57.  (1 mark)

  3. The probability that a 35-year-old person has a diastolic measurement for blood pressure between 57 and 63 can be found by evaluating
     

     
    where and are constants and where
     

     
    is the normal probability density function with mean 0 and standard deviation 1.

     

    By first finding the values and , calculate an approximate value for this probability by using the trapezoidal rule with 3 function values.  (3 marks)

  4. Hence, find the approximate probability that a 35-year-old person chosen at random has a diastolic measurement of 57 or less.  (1 mark)

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i.    σ
   
   

 

 

ii.   
   

 

iii. 

 
 
 
 

 

iv.   

 

 

Calculus, MET1 2019 VCAA 7

The graph of the relation    is shown on the axes below. is a point on the graph of this relation, is the point and is the point .
 

  1. Find an expression for the length in terms of only.   (1 mark) 

  2. Find the maximum area of the triangle .  (3 marks)

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a.   

 

b.   
   
 
   
   

 

 
 

Probability, MET1 2019 VCAA 6

Fred owns a company that produces thousands of pegs each day. He randomly selects 41 pegs that are produced on one day and finds eight faulty pegs.

  1. What is the proportion of faulty pegs in this sample?  (1 mark) 

  2. Pegs are packed each day in boxes. Each box holds 12 pegs. Let    be the random variable that represents the proportion of faulty pegs in a box.
  3. The actual proportion of faulty pegs produced by the company each day is .
  4. Find  . Express your answer in the form  , where    and    are positive rational numbers and    is a positive integer.  (2 marks)

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a.   
 

b.   

 
 
 
 

Calculus, MET1 2019 VCAA 5

Let  .

    1. Evaluate  .   (1 mark)

    2. Sketch the graph of on the axes below, labelling all asymptotes with their equations.   (2 marks)

        

  2. Find the area bounded by the graph of , the -axis, the line    and the line  .   (2 marks)

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a.i.   
   

​

a.ii.     

 

b.   
   
   
   
   

L&E, 2ADV E1 SM-Bank 14

The spread of a highly contagious virus can be modelled by the function

Where is the number of days after the first case of sickness due to the virus is diagnosed and    is the total number of people who are infected by the virus in the first    days.

  1. Calculate  .  (1 mark)

  2. Find the value of    and interpret it result.  (2 marks)

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i.  
   
   

 

ii.   
   
   

 

Vectors, EXT1 V1 SM-Bank 20

Consider the vector  , where    and    are unit vectors in the positive direction of the and axes respectively.

  1. Find the unit vector in the direction of  .    (1 mark)

  2. Find the acute angle that    makes with the positive direction of the -axis.   (1 mark)

  3. The vector  .

     

    Given that    is perpendicular to  , find the value of  .   (1 mark)

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i.   

 

ii.   

 
 
     

 

iii.   

Vectors, EXT1 V1 SM-Bank 19

Consider the following vectors

  1. Find  .  (1 mark)

  2. The points , and are vertices of a triangle. Prove that the triangle has a right angle at .  (2 marks)

  3. Find the length of the hypotenuse of the triangle.  (1 mark)

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i. 

COMMENT: Many teachers recommend column vector notation to simplify calculations and minimise errors – we agree!

 
 
 

 

ii.   
   
   
   

 

 

iii.