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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

Statistics, EXT1 S1 SM-Bank 11

Within a particular population, it is known that the percentage of left-handers is 17%.

A research project randomly selects 200 people from this population.

Assuming this sample proportion is normally distributed, what is the probability that the percentage of people that are left-handed in this sample is

  1. greater than 20%   (2 marks)

  2. less than 10%   (2 marks)

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i.   

 
 
   
 
     

 
   
   

 

 
   
   

 

ii.   
   

 

 
   

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.   

 

Statistics, EXT1 S1 SM-Bank 6

After counting votes in an election, it is known that 40% of people voted for the Katter party.

A sample of 600 voting cards are taken and inspected. Assuming this sample proportion is approximately normally distributed, what is the probability that the percentage of voting cards inspected that chose the Katter party is less than 36%?   (3 marks)

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Statistics, EXT1 S1 SM-Bank 4

In an experiment, a pair of dice are rolled 70 times.

A success is recorded if the sum of the dice roll is 5 or less.

  1. What is the mean of this binomial distribution?

     

    Give your answer to one decimal place.  (3 marks)

  2. What is the standard deviation?

     

    Give your answer to one decimal place.  (1 mark)

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i.   

STRATEGY: A table (or array) can be a very efficient and error minimising strategy in questions like this.

 
 

 

ii.   
 
   
 

Statistics, SPEC2 2019 VCAA 6

A company produces packets of noodles. It is known from past experience that the mass of a packet of noodles produced by one of the company's machines is normally distributed with a mean of 375 grams and a standard deviation of 15 grams.

To check the operation of the machine after some repairs, the company's quality control employees select two independent random samples of 50 packets and calculate the mean mass of the 50 packets for each random sample.

  1. Assume that the machine is working properly. Find the probability that at least one random sample will have a mean mass between 370 grams and 375 grams. Give your answer correct to three decimal places.   (2 marks)

  2. Assume that the machine is working properly. Find the probability that the means of the two random samples differ by less than 2 grams. Give your answer correct to three decimal places.   (3 marks)

To test whether the machine is working properly after the repairs and is still producing packets with a mean mass of 375 grams, the two random samples are combined and the mean mass of the 100 packets is found to be 372 grams. Assume that the standard deviation of the mass of the packets produced is still 15 grams. A two-tailed test at the 5% level of significance is to be carried out.

  1. Write down suitable hypotheses and for this test.   (1 mark)

  2. Find the    value for the test, correct to three decimal places.   (1 mark)

  3. Does the mean mass of the sample of 100 packets suggest that the machine is working properly at the 5% level of significance for a two-tailed test? Justify your answer.   (1 mark)

  4. What is the smallest value of the mean mass of the sample of 100 packets for to be not rejected? Give your answer correct to one decimal place.   (1 mark)

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a.   


 


 

b.   


 

c.   


 

d.   


 

e.   


 

f.   

 
 

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

Calculus, SPEC2 2019 VCAA 3

  1. The growth and decay of a quantity with respect to time is modelled by the differential equation
     

     
    where .

    1. Given that    and  , where is a function of ,

       

      show that  .   (2 marks)

    2. Specify the condition(s) for which  .   (2 marks)

  2. The growth of another quantity with respect to time is modelled by the differential equation
     
       
      
    where    and    when  .

    1. Express this differential equation in the form  .   (1 mark)

    2. Hence, show that  .   (2 marks)

    3. Show that the graph of as a function of does not have a point of inflection.   (2 marks)

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a.i.   
 

 

a.ii.   


 


 

b.i.   
 

 

b.ii.   
 

 

 

 

b.iii.   

 

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, SPEC2 2019 VCAA 18 MC

The masses of a random sample of 36 track athletes have a mean of 65 kg. The standard deviation of the masses of all track athletes is known to be 4 kg.

A 98% confidence interval for the mean of the masses of all track athletes, correct to one decimal place, would be closest to

  1. (51.0, 79.0)
  2. (63.6, 66.4)
  3. (63.3, 66.7)
  4. (63.4, 66.6)
  5. (64.3, 65.7)
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Calculus, SPEC2 2019 VCAA 16 MC

A variable force acts on a particle, causing it to move in a straight line. At time seconds, where  , its velocity metres per second and position metres from the origin are such that  .

The acceleration of the particle, in ms−2, can be expressed as

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Mechanics, SPEC2 2019 VCAA 14 MC

A 4 kg mass is held at rest on a smooth surface. It is connected by a light inextensible string that passes over a smooth pulley to a 2 kg mass, which in turn is connected by the same type of string to a 1 kg mass. This is shown in the diagram below.
 


 

When the 4 kg mass is released, the tension in the string connecting the 1 kg and 2 kg masses is newtons. The value of is

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Calculus, MET2 2019 VCAA 5

Let  . The tangent to the graph of at  , where  , intersects the graph of again at and intersects the horizontal axis at . The shaded regions shown in the diagram below are bounded by the graph of , its tangent at    and the horizontal axis.
 

  1. Find the equation of the tangent to the graph of at  , in terms of .   (1 mark)

  2. Find the -coordinate of , in terms of .   (1 mark)

  3. Find the -coordinate of , in terms of .   (2 marks)

Let    be the function that determines the total area of the shaded regions.

  1. Find the rule of , in terms of .   (3 marks)

  2. Find the value of for which is a minimum.   (2 marks)

Consider the regions bounded by the graph of , the tangent to the graph of at  , where  , and the vertical axis.

  1. Find the value of for which the total area of these regions is a minimum.   (2 marks)

  2. Find the value of the acute angle between the tangent to the graph of and the tangent to the graph of at  .   (1 mark)

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a.   


 

b.   


 

c.   


 

d.   

 
 

 

e.   


 

f.   


 

g.   

 

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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a.   
  

b.   
  

c.   

 

d.   
    ²

 

e.   


  

f.   
 
 

Calculus, MET2 2019 VCAA 2

An amusement park is planning to build a zip-line above a hill on its property.

The hill is modelled by  , where    is the horizontal distance, in metres, from an origin and    is the height, in metres, above this origin, as shown in the graph below.
  

  1. Find  .   (1 mark)

  2. State the set of values for which the gradient of the hill is strictly decreasing.   (1 mark)

The cable for the zip-line is connected to a pole at the origin at a height of 10 m and is straight for  , where  . The straight section joins the curved section at  . The cable is then exactly 3 m vertically above the hill from  , as shown in the graph below.

  1. State the rule, in terms of , for the height of the cable above the horizontal axis for  .   (1 mark)

  2. Find the values of for which the gradient of the cable is equal to the average gradient of the hill for  .   (3 marks)

The gradients of the straight and curved sections of the cable approach the same value at  , so there is a continuous and smooth join at .

    1. State the gradient of the cable at , in terms of .   (1 mark)

    2. Find the coordinates of , with each value correct to two decimal places.   (3 marks)

    3. Find the value of the gradient at , correct to one decimal place.   (1 mark)

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a.   
 

b.   


 

c.   


 

d.   


 

e.i.   


 

e.ii.   


 

 

 

e.iii.   
   

Mechanics, SPEC1 2019 VCAA 9

  1. A light inextensible string is connected at each end to a horizontal ceiling. A mass of kilograms hangs in equilibrium from a smooth ring on the string, as shown in the diagram below. The string makes an angle with the ceiling.
      

     
    Express the tension, newtons, in the string in terms of , and .  (1 mark)
  2. A different light inextensible sting is connected at each end to a horizontal ceiling. A mass of kilograms hangs from a smooth ring on the string. A horizontal force of newtons is applied to the ring. The tension in the sting has a constant magnitude and the system is in equilibrium. At one end the string makes an angle with the ceiling and at the other end the string makes an angle with the ceiling, as shown in the diagram below.
     

     
    Show that  .  (3 marks)
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a.   

 

b.   

 

 

 
 
 
 
 
 
 

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.   
   

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b.i.   
 

 

b.ii.   
 

 

b.iii.   
 

 

c.i.   
 

 

c.ii.   
   

 

d.   

Vectors, SPEC1 2019 VCAA 4

The position vectors of two particles    and    at time  seconds after they have started moving are given by    and    respectively, where    is a real constant and  .

Find the value of    if the particles collide after they have started moving.   (3 marks)

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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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  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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  2.   
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i.   
 

 

ii.   

 

Statistics, MET2 2019 VCAA 14 MC

The weights of packets of lollies are normally distributed with a mean of 200 g.

If 97% of these packets of lollies have a weight of more than 190 g, then the standard deviation of the distribution, correct to one decimal place, is

  1.  3.3 g
  2.  5.3 g
  3.  6.1 g
  4.  9.4 g
  5. 12.1 g
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Probability, MET2 2019 VCAA 8 MC

An archer can successfully hit a target with a probability of 0.9. The archer attempts to hit the target 80 times. The outcome of each attempt is independent of any other attempt.

Given that the archer successfully hits the target at least 70 times, the probability that the archer successfully hits the target exactly 74 times, correct to four decimal places, is

A.   0.3635

B.   0.8266

C.   0.1494

D.   0.3005

E.   0.1701

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