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Calculus, SPEC2 2022 VCAA 3

A particle moves in a straight line so that its distance, metres, from a fixed origin after time seconds is given by the differential equation , where    when  .

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

  2. ii. Hence, show that  .   (2 marks)

  3. The graph of  has a horizontal asymptote.
    1. Write down the equation of this asymptote.   (1 mark)

    2. Sketch the graph of  and the horizontal asymptote on the axes below. Using coordinates, plot and label the point where  , giving the value of correct to two decimal places.   (2 marks)

  1. Find the speed of the particle when  . Give your answer in metres per second, correct to two decimal places.   (1 mark)

Two seconds after the first particle passed through , a second particle passes through .

Its distance metres from seconds after the first particle passed through , is given by 

  1. Verify that the particles are the same distance from when  .   (1 mark)

  2. Find the ratio of the speed of the first particle to the speed of the second particle when the particles are at the same distance from . Give your answer as in simplest form, where and are positive integers.   (2 marks)

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a.i. 

a.ii. 

b.i.   

b.ii. 
           

c.   

d.   

e.   

Show Worked Solution

a.i. 

 
  

a.ii.

 
b.i.

♦♦♦ Mean mark 23%.

 
b.ii.


 

c.   


 

d.   


 

e.   

CHEMISTRY, M1 EQ-Bank 29

You are given the task to separate the components of two mixtures: a saltwater solution and a mixture of sand and iron filings.

  1.  Suggest a suitable separation technique to extract the salt from the saltwater solution. Explain your reasoning based on the physical property involved.  (2 marks)
  2.  Identify the physical property that allow the mixture of sand and iron fillings to be separated and whether it is a homogeneous or heterogeneous mixture.  (1 marks)
  3. Discuss one safety precaution that should be followed during the separation of the saltwater solution.  (2 marks)
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a.    Separation technique:

→ Evaporation is a suitable technique because water has a much lower boiling point than salt.

→ By heating the solution, the water evaporates, leaving salt crystals behind.

→ Further distillation could be used to collect and condense the evaporated water.
 

b.    Physical property:

→ Iron fillings are magnetic and hence can be separated using a magnet.

→ The mixture is heterogeneous.
 

c.    Safety precaution:

→ For evaporation, wear heat-resistant gloves when handling hot equipment like the tripod or beaker to avoid burns to the hands.

Show Worked Solution

a.    Separation technique:

→ Evaporation is a suitable technique because water has a much lower boiling point than salt.

→ By heating the solution, the water evaporates, leaving salt crystals behind.

→ Further distillation could be used to collect and condense the evaporated water.
 

b.    Physical property:

→ Iron fillings are magnetic and hence can be separated using a magnet.

→ The mixture is heterogeneous.
 

c.    Safety precaution:

→ For evaporation, wear heat-resistant gloves when handling hot equipment like the tripod or beaker to avoid burns to the hands.

Functions, MET2 2022 VCAA 4

Consider the function , where

Part of the graph of is shown below.
 

  1. State the range of .   (1 mark)

  2.  i. Find .   (2 marks)

  3. ii. State the maximal domain over which is strictly increasing.   (1 mark)

  4. Show that .   (1 mark)

  5. Find the domain and the rule of , the inverse of .   (3 marks)

  6. Let be the function , where and .
  7. The inverse function of is defined by .
  8. The area of the regions bound by the functions and can be expressed as a function, .
  9. The graph below shows the relevant area shaded.
     

  1. You are not required to find or define .
  2.  i. Determine the range of values of such that .   (1 mark)
  3. ii. Explain why the domain of does not include all values of .   (1 mark
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a.    
b.i
b.ii
c.
d.
e.i 
e.ii No bounded area for
Show Worked Solution

a.   is the range.

b.i   
 
   
     
   
     

 
b.ii 

c.  

 
 

 
d.   To find the inverse swap and in

 
 
 
 
 
 
 

 
    Domain:
  

e.i   The vertical dilation factor of  is 

For ,

   [Using CAS]


♦♦♦♦ Mean mark (e.i) 10%.
MARKER’S COMMENT: Incorrect responses included and .

e.ii  When for  (or for  ) there is no bounded area.

  There will be no bounded area for .


♦♦♦♦ Mean mark (e.ii) 10%.

Calculus, SPEC2 2022 VCAA 1

Consider the family of functions with rule  , where .

  1. Write down the equations of the two asymptotes of the graph of when .   (2 marks)

  2. Sketch the graph of    for    on the set of axes below. Clearly label any turning points with their coordinates and label any asymptotes with their equations.   (3 marks)
     

  1.  i. Find, in terms of , the equations of the asymptotes of the graph of  .   (1 mark)

  2. ii. Find the distance between the two turning points of the graph of  in terms of .   (2 marks)

  3. Now consider the functions and , where    and  .
  4. The region bounded by the curves of and is rotated about the -axis.
    1. Write down the definite integral that can be used to find the volume of the resulting solid.   (2 marks)

    2. Hence, find the volume of this solid. Give your answer correct to two decimal places.   (1 mark)

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a. 

b.   
       

c.i.   

c.ii. 

d.i. 

d.ii. 

Show Worked Solution

a.   


 

b.    
       

 

c.i.


 

c.ii. 


 

d.i 


 

d.ii.

♦♦ Mean mark (d)(ii) 37%.

PHYSICS, M6 2020 VCE 6*

A single loop of wire moves into a uniform magnetic field of strength 3.5 × 10 T over time = 0.20 s from point to point , as shown in the diagram below. The area of the loop is 0.05 m.
 

Determine the magnitude of the average induced EMF in the loop.   (2 marks)

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Show Worked Solution

Functions, EXT1 F1 2023 MET1 7

Consider . Part of the graph of    is shown below.
 

  1. State the range of .   (1 mark)
  2. Sketch the graph of the inverse function  on the axes above. Label any endpoints and axial intercepts with their coordinates.   (2 marks)
  3. Determine the equation of the domain for the inverse function  .   (2 marks)
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a.   

b.   

c.   

Show Worked Solution

a.   

b.   

c.   


 

 


♦ Mean mark (c) 48%.
MARKER’S COMMENT: Common error → writing the function as .

Probability, MET2 2022 VCAA 10 MC

An organisation randomly surveyed 1000 Australian adults and found that 55% of those surveyed were happy with their level of physical activity.

An approximate 95% confidence interval for the percentage of Australian adults who were happy with their level of physical activity is closest to

  1. (4.1, 6.9)
  2. (50.9, 59.1)
  3. (52.4, 57.6)
  4. (51.9, 58.1)
  5. (45.2, 64.8)
Show Answers Only

Show Worked Solution

, , with approximate 95% confidence interval () for

Confidence Interval  
   
   

 

Statistics, SPEC2 2023 VCAA 6

A forest ranger wishes to investigate the mass of adult male koalas in a Victorian forest. A random sample of 20 such koalas has a sample mean of 11.39 kg.

It is known that the mass of adult male koalas in the forest is normally distributed with a standard deviation of 1 kg.

  1. Find a 95% confidence interval for the population mean (the mean mass of all adult male koalas in the forest). Give your values correct to two decimal places.  (1 mark)

  2. Sixty such random samples are taken and their confidence intervals are calculated.
  3. In how many of these confidence intervals would the actual mean mass of all adult male koalas in the forest be expected to lie?  (1 mark)

The ranger wants to decrease the width of the 95% confidence interval by 60% to get a better estimate of the population mean.

  1. How many adult male koalas should be sampled to achieve this?  (1 mark)

It is thought that the mean mass of adult male koalas in the forest is 12 kg. The ranger thinks that the true mean mass is less than this and decides to apply a one-tailed statistical test. A random sample of 40 adult male koalas is taken and the sample mean is found to be 11.6 kg.

  1. Write down the null hypothesis, , and the alternative hypothesis, , for the test.   (1 mark)

The ranger decides to apply the one-tailed test at the 1% level of significance and assumes the mass of adult male koalas in the forest is normally distributed with a mean of 12 kg and a standard deviation of 1 kg.

  1.  i. Find the value for the test correct to four decimal places.  (1 mark)

  2. ii. Draw a conclusion about the null hypothesis in part d. from the value found above, giving a reason for your conclusion.  (1 mark)

  3. What is the critical sample mean (the smallest sample mean for not to be rejected) in this test? Give your answer in kilograms correct to three decimal places.  (1 mark)

Suppose that the true mean mass of adult male koalas in the forest is 11.4 kg, and the standard deviation is 1 kg. The level of significance of the test is still 1%.

  1. What is the probability, correct to three decimal places, of the ranger making a type error in the statistical test?  (1 mark)

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a.   

b.   

c.   

d.   

e.i. 

e.ii.

f.   

g.   

Show Worked Solution

a.   


 

b.   


 

c.   

:


 

♦♦♦ Mean mark (c) 28%.

d.   
 

e.i. 

 

 

 

e.ii.
 

f.     


 

g.   

♦♦ Mean mark (g) 39%.

PHYSICS, M7 2022 VCE 14*

Sam undertakes a photoelectric effect experiment using the apparatus shown in Figure 12. She uses a green filter.
 

   

Sam produces a graph of photocurrent, , in milliamperes, versus voltage, , in volts, as shown in Figure 13.
 

   

  1. Identify what point represents on the graph in Figure 13.  (1 mark)
  1. Sam then significantly increases the intensity of the light.
  2. Sketch the resulting graph on Figure 14. The dashed line in Figure 14 represents the original data.  (2 marks)
     

   

  1. Sam replaces the green filter with a violet filter, keeping the light source at the increased intensity.
  2. Sketch the resulting graph on Figure 15. The dashed line in Figure 15 represents the original data.  (2 marks)
     

Show Answers Only

a.    Stopping voltage

b.
      

c.    
       

Show Worked Solution

a.    Stopping voltage
 

b.    Diagram will exhibit the following points:

→ Increasing the intensity of light will increase the photocurrent due to more photoelectrons being ejected from the metal.

→ However, it will not change the energy of the photoelectrons, hence the stopping voltage remains the same.
 

c.    Diagram will exhibit the following points:

→ Changing the colour from green to violet will increase the frequency of the light.

→ As , the energy of the photoelectrons will increase, hence there needs to be a greater stopping voltage.
 

Calculus, MET2 2023 VCAA 1

Let . Part of the graph of is shown below.

  1. State the coordinates of all axial intercepts of .   (1 mark)
  2. Find the coordinates of the stationary points of .   (2 marks)
    1. Let .
    2. Find the values of for which .   (1 mark)
    1. Write down an expression using definite integrals that gives the area of the regions bound by and .  (2 marks)
    2. Hence, find the total area of the regions bound by and , correct to two decimal places.   (1 mark)
  1. Let , where and .
  2. Find the possible values of and .   (4 marks)
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a.   

b.   

c.i. 

c.ii.

 

c.iii.

d.  

Show Worked Solution

a.   
  

b.   


  

ci   

  

cii  


  

ciii 
 

  

d.   

  

 

PHYSICS, M5 2021 VCE 9-10 MC

Lucy is running horizontally at a speed of 6 m s along a diving platform that is 8.0 m vertically above the water.

Lucy runs off the end of the diving platform and reaches the water below after time .

She lands feet first at a horizontal distance from the end of the diving platform.
 

Question 9

Which one of the following expressions correctly gives the distance ?

  1. 0.8
  2. 6
  3. 5
  4. 6 + 5

 
Question 10

Which one of the following is closest to the time taken, , for Lucy to reach the water below?

  1. 0.8 s
  2. 1.1 s
  3. 1.3 s
  4. 1.6 s
Show Answers Only

Show Worked Solution

→ The horizontal displacement formula for projectile motion: 

→ Horizontal distance:


 

 
 
 
   
   

 

Functions, 2ADV F2 2023 MET1 3

  1. Sketch the graph of  on the axes below, labelling all asymptotes with their equation and axial intercepts with their coordinates.   (3 marks)
     


  2. Find the values of for which .   (1 mark)
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a.   

b.   

Show Worked Solution

a.   

b.   


♦♦ Mean mark (b) 38%.
MARKER’S COMMENT: Many students did not use their graph from part (a).

PHYSICS, M5 2022 VCE 8

A Formula 1 racing car is travelling at a constant speed of 144 km h (40 m s) around a horizontal corner of radius 80.0 m. The combined mass of the driver and the car is 800 kg. Figure 8a shows a front view and Figure 8b shows a top view.
 

  1. Calculate the magnitude of the net force acting on the racing car and driver as they go around the corner.  (2 marks)
  1. On Figure 8b, draw the direction of the net force acting on the racing car using an arrow.  (1 mark)
  1. Explain why the racing car needs a net horizontal force to travel around the corner and state what exerts this horizontal force.  (2 marks)
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a.    

b.    
       

c.    → A net horizontal force is required for circular motion.

→ The horizontal force is perpendicular to the direction of travel.

→ This is the force which changes the direction of travel.

→ The force of friction from the road acts on the tires provides the centripetal force in this motion.

Show Worked Solution

a.    

b.   
       

c.    → A net horizontal force is required for circular motion.

→ The horizontal force is perpendicular to the direction of travel.

→ This is the force which changes the direction of travel.

→ The force of friction from the road acts on the tires provides the centripetal force in this motion.

♦ Mean mark 42%.

PHYSICS, M6 2022 VCE 6

Figure 6 shows a simple alternator consisting of a rectangular coil of area 0.060 m and 200 turns, rotating in a uniform magnetic field. The magnetic flux through the coil in the vertical position shown in Figure 6 is 1.2 × 10 Wb.
 

  1. Calculate the strength of the magnetic field in Figure 6 . Show your working.  (2 marks)

  1. The rectangular coil rotates at a frequency of 2.5 Hz.
  2. Calculate the average induced EMF produced in the first quarter of a turn. Begin the quarter with the coil in the vertical position shown in Figure 6.  (3 marks)

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a.   

b.    

Show Worked Solution
a.    
 
   
   

 
b.   

 
 
   
   

PHYSICS, M6 2022 VCE 5*

A wind generator provides power to a factory located 2.00 km away, as shown in Figure 5.

When there is a moderate wind blowing steadily, the generator produces a voltage of 415 V and a current of 100 A.

The total resistance of the transmission wires between the wind generator and the factory is 2.00 .
 

  1. Calculate the power, in kilowatts, produced by the wind generator when there is a moderate wind blowing steadily.  (1 mark)

To operate correctly, the factory's machinery requires a power supply of 40 kW.

  1. Determine whether the energy supply system, as shown, will be able to supply power to the factory when the moderate wind is blowing steadily. Justify your answer with calculations.  (3 marks)

  1. The factory's owner decides to limit transmission energy loss by installing two transformers: a step-up transformer with a turns ratio of 1:10 at the wind generator and a step-down transformer with a turns ratio of 10:1 at the factory. Each transformer can be considered ideal.

    With the installation of the transformers, determine the power, in kilowatts, now supplied to the factory.  (3 marks)

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a.    

b.    

c.    

Show Worked Solution

a.    
 

b.    


 

c.    
 
   
   

 

PHYSICS, M6 2022 VCE 1

Figure 1 shows four positions (1, 2, 3 and 4) of the coil of a single-turn, simple DC motor. The coil is turning in a uniform magnetic field that is parallel to the plane of the coil when the coil is in Position 1, as shown.

When the motor is operating, the coil rotates about the axis through the middle of sides and in the direction indicated. The coil is attached to a commutator. Current for the motor is passed to the commutator by brushes that are not shown in Figure 1.
 

  1. When the coil is in Position 1, in which direction is the current flowing in the side from to or from to ? Justify your answer.  (2 marks)

  1. When the coil is in Position 3, in which direction is the current flowing in the side from to or
    from to ?      (1 marks)

  1. The side of the coil has a length of 0.10 m and experiences a magnetic force of 0.15 N due to the magnetic field, which has a magnitude of 0.5 T.

    Calculate the magnitude of the current in the coil.  (2 marks)

Show Answers Only

a.     to

b.     to

c.    

Show Worked Solution

a.    Using the right hand rule:

→ Palm faces up (force), fingers to the right (magnetic field), and the thumb faces out of the page.

→ Therefore the current must run from to .
 

b.    Using the right hand rule:

→ Palm now faces down the page (force), fingers still to the right.

→ The current will run from to (thumb).

 

c.   
 
   
   

PHYSICS, M7 2022 VCE 18 MC

Which one of the following is an example of an inertial frame of reference?

  1. a bus travelling at constant velocity
  2. an express train that is accelerating
  3. a car turning a corner at a constant speed
  4. a roller-coaster speeding up while heading down a slope
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Show Worked Solution

→ is the only frame of reference where no acceleration is imposed on the object.

→ It is therefore an inertial frame of reference.

PHYSICS, M7 2022 VCE 16 MC

Which one of the following phenomena best demonstrates that light waves are transverse?

  1. polarisation
  2. interference
  3. dispersion
  4. diffraction
Show Answers Only

Show Worked Solution

→ Polarisation requires the transfer of energy through a wave to be perpendicular to the direction of oscillation of the wave.

→ As this is a property of a transverse wave, light waves must be traverse.

PHYSICS, M7 2022 VCE 15 MC

Which one of the following best provides evidence of light behaving as a particle?

  1. photoelectric effect
  2. white light passing through a prism
  3. diffraction of light through a single slit
  4. interference of light passing through a double slit
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Show Worked Solution

→ During the photoelectric effect, light transfers energy in small discrete ‘quanta’ which demonstrates light behaving as a particle.

PHYSICS, M8 2022 VCE 3 MC

Particles emitted from a radioactive source travel through a magnetic field, , directed into the page, as shown schematically in the diagram below.

Three particles, and , follow the paths indicated by the arrows.
 

Which of the following correctly identifies the charges on particles and ?

Show Answers Only

Show Worked Solution

→ must have no charge as it travels through the magnetic field with no deflection.

→ Using the right-hand rule, the force on a positive charge would be up the page, hence must be the positive charge.

PHYSICS, M6 2022 VCE 1 MC

A single loop of wire carries a current, , as shown in the diagram below.
 

Which one of the following best describes the direction of the magnetic field at the centre of the circle, , which is produced by the current carrying wire?

  1. to the left
  2. to the right
  3. into the page
  4. out of the page
Show Answers Only

Show Worked Solution

→ Right hand grip rule: fingers curl in a clockwise direction and the thumb points into the page.

Calculus, SPEC2 2023 VCAA 4

A fish farmer releases 200 fish into a pond that originally contained no fish. The fish population, , grows according to the logistic model,  , where is the time in years after the release of the 200 fish.

  1. The above logistic differential equation can be expressed as
  3. Find the values of and .  (1 mark)

One form of the solution for is  ,  where is a real constant.

  1. Find the value of .  (1 mark)

The farmer releases a batch of fish into a second pond, pond 2 , which originally contained no fish. The population, , of fish in pond 2 can be modelled by  ,  where is the time in years after the fish are released.

  1. Find the value of .  (1 mark)

  2. Find the value of when .
  3. Give your answer correct to the nearest integer.  (1 mark)

  4.  i. Given that  ,  express    in terms of .  (1 mark)

  5. ii. Hence or otherwise, find the size of the fish population in pond 2 and the value of when the rate of growth of the population is a maximum. Give your answer for correct to the nearest year.  (2 marks)

  6. Sketch the graph of versus on the set of axes below. Label any axis intercepts and any asymptotes with their equations.  (1 mark)
     

The farmer wishes to take 5.5% of the fish from pond 2 each year. The modified logistic differential equation that would model the fish population, , in pond 2 after years in this situation is

  1. Find the maximum number of fish that could be supported in pond 2 in this situation.  (1 mark)

Show Answers Only

a.   

b.   

c.   

d.   

e.i. 

e.ii.

f.   


g. 

Show Worked Solution

a.   


 

♦ Mean mark (a) 48%.

b.   


 

c.   

:


 

d.   


 

e.i. 

 

♦♦♦ Mean mark (e)(i) 21%.

e.ii.


 

f.   


g. 

♦ Mean mark (g) 40%.

Calculus, SPEC2 2023 VCAA 3

The curve given by  , where  , is rotated about the -axis to form a solid of revolution.

  1.  i. Write down the definite integral, in terms of , for the volume of this solid of revolution.  (1 mark)

  2. ii. Find the volume of the solid of revolution.  (1 mark)

  3.  i. Express the curved surface area of the solid in the form  , where are all positive integers.  (2 marks)

  4. ii. Hence or otherwise, find the curved surface area of the solid correct to three decimal places.  (1 mark)

The total surface area of the solid consists of the curved surface area plus the areas of the two circular discs at each end.

The 'efficiency ratio' of a body is defined as its total surface area divided by the enclosed volume.

  1. Find the efficiency ratio of the solid of revolution correct to two decimal places.  (2 marks)

  2. Another solid of revolution is formed by rotating the curve given by  about the -axis for  , where . This solid has a volume of .
  3. Find the efficiency ratio for this solid, giving your answer correct to two decimal places.  (3 marks)

Show Answers Only

a.i.
 

a.ii.


 

b.i. 

 
b.ii.


 

c.   


 

d.   

Show Worked Solution

a.i.
 

a.ii.


 

b.i. 

 
b.ii.


 

c.   


 

d.   

Complex Numbers, SPEC2 2023 VCAA 2

Let .

  1. Verify that is a root of  .   (1 marks)
  1. List the other roots of    in polar form.   (1 mark)
  1. On the Argand diagram below, plot and label the points that represent all the roots of  .   (2 marks)

  1.  i. On the Argand diagram below, sketch the ray that originates at the real root of    and passes through the point represented by .   (1 mark)

     

  1. ii. Find the equation of this ray in the form , where , and is measured in radians in terms of .   (1 mark)

  2. Verify that the equation    can be expressed in the form 
  3.     .   (1 mark)
  4.   i. Express  in the form , where .   (1 mark)

  5. ii. Given that    satisfies  ,

use De Moivre's theorem to show that

      1. .   (2 marks)

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a.   


 

b.   

c.   
          

d.i.   
          

d.ii.


 

e.     
     
     

 

f.i.   
 

f.ii. 

Show Worked Solution

a.   


 

b.   

c.   
          

d.i.   
          

d.ii.


 

e.     
     
     

 

f.i.   
 

f.ii. 

Probability, MET2 2023 VCAA 4

A manufacturer produces tennis balls.

The diameter of the tennis balls is a normally distributed random variable , which has a mean of 6.7 cm and a standard deviation of 0.1 cm.

  1. Find  , correct to four decimal places.   (1 mark)
  2. Find the minimum diameter of a tennis ball that is larger than 90% of all tennis balls produced.
  3. Give your answer in centimetres, correct to two decimal places.   (1 mark)

Tennis balls are packed and sold in cylindrical containers. A tennis ball can fit through the opening at the top of the container if its diameter is smaller than 6.95 cm.

  1. Find the probability that a randomly selected tennis ball can fit through the opening at the top of the container.
  2. Give your answer correct to four decimal places.   (1 mark)
  3. In a random selection of 4 tennis balls, find the probability that at least 3 balls can fit through the opening at the top of the container.
  4. Give your answer correct to four decimal places.   (2 marks)

A tennis ball is classed as grade A if its diameter is between 6.54 cm and 6.86 cm, otherwise it is classed as grade B.

  1. Given that a tennis ball can fit through the opening at the top of the container, find the probability that it is classed as grade A.
  2. Give your answer correct to four decimal places.   (2 marks)
  3. The manufacturer would like to improve processes to ensure that more than 99% of all tennis balls produced are classed as grade A.
  4. Assuming that the mean diameter of the tennis balls remains the same, find the required standard deviation of the diameter, in centimetres, correct to two decimal places.   (2 marks)
  5. An inspector takes a random sample of 32 tennis balls from the manufacturer and determines a confidence interval for the population proportion of grade A balls produced.
  6. The confidence interval is (0.7382, 0.9493), correct to four decimal places.
  7. Find the level of confidence that the population proportion of grade A balls is within the interval, as a percentage correct to the nearest integer.   (2 marks)

A tennis coach uses both grade A and grade B balls. The serving speed, in metres per second, of a grade A ball is a continuous random variable, , with the probability density function
 


 

  1. Find the probability that the serving speed of a grade A ball exceeds 50 metres per second.
  2. Give your answer correct to four decimal places.   (1 mark)
  3. Find the exact mean serving speed for grade A balls, in metres per second.   (1 mark)

The serving speed of a grade B ball is given by a continuous random variable, , with the probability density function .

A transformation maps the graph of to the graph of , where .

  1. If the mean serving speed for a grade B ball is metres per second, find the values of and .   (2 marks)
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a.   

b.   

c.   

d.   

e.   

f.   

g.   

h.   

i.   

j.   

Show Worked Solution

a.    


 

b.     

 

c.  


 

d.   

 

e.   
   
 
   
   

  

f.    


♦♦ Mean mark (f) 35%.
MARKER’S COMMENT: Incorrect responses included using . Many answers were accepted in the range as max value of SD was not asked for provided sufficient working was shown.

g.  

    
   
 
 

 

 


 


♦♦ Mean mark (g) 25%.
MARKER’S COMMENT: was often calculated incorrectly with frequently seen.

h.  

i.  

j. 
 


 

 

 

 
 
 
 

♦♦♦ Mean mark (j) 10%.

Calculus, SPEC2 2023 VCAA 1

Viewed from above, a scenic walking track from point to point is shown below. Its shape is given by

The minimum turning point of section occurs at point . Point is a point of inflection and the curves meet at point . Distances are measured in kilometres.
 

  1. Show that    and  .  (1 mark)

  2. Verify that the two curves meet smoothly at point .  (2 marks)

  3.  i. Find the coordinates of point .  (1 mark)

  4. ii. Find the coordinates of point .  (1 mark)

The return track from point to point follows an elliptical path given by

, where .

  1. Find the Cartesian equation of the elliptical path.  (2 marks)

  2. Sketch the elliptical path from to on the diagram above.  (1 mark)

  3.  i. Write down a definite integral in terms of that gives the length of the elliptical path from to .  (1 mark)

  4. ii. Find the length of the elliptical path from to .
  5.     Give your answer in kilometres correct to three decimal places.   (1 mark)

Show Answers Only

a.   

   


 

b.   


 

c.i. 
 

c.ii. 
 

d.   
 

e.    
          
 

f.i.   
 

f.ii. 

Show Worked Solution

a.   

   


 

b.   


 

c.i. 


 

c.ii. 


 

d.   


 

e.    
          
 

f.i.   


 

f.ii.   

PHYSICS, M6 2023 VCE 7*

Two high-voltage transmission lines span a distance of 260 km between Power Plant A and Town B, as shown in Figure 7. Power Plant A provides 350 MW of power. The potential difference at Power Plant A is 500 kV. The current in the transmission lines has a value of 700 A and the power loss in the transmission lines is 20 MW.
 

   

  1. Show, using calculations, that the total resistance of the two transmission lines is 41 .  (2 marks)
  1. Town B needs a minimum of 480 kV.
  2. Determine whether 480 kV will be available to Town B. Show your working.  (3 marks)
  1. Explain what would happen if the electricity between Power Plant A and Town B were to be transmitted at 50 kV instead of 500 kV. Assume that the resistance of the transmission lines is still 41 .  (2 marks)
Show Answers Only

a.    

b.    The power generated in the power plant is 500 kV.

480V will not be available to Town B.
 

c.    If the voltage was decreased by a factor of 10 to 50 kV:

→ The current through the power lines would increase by a factor of 10.

→ Power remains constant

→ As , this would dramatically increase the power loss and significantly less power would be transmitted to the town.

Show Worked Solution
a.    
 
   
   

 

b.    The power generated in the power plant is 500 kV.

480V will not be available to Town B.

 

c.    If the voltage was decreased by a factor of 10 to 50 kV:

→ Power remains constant

→ The current through the power lines would increase by a factor of 10.

→ As , this would dramatically increase the power loss and significantly less power would be transmitted to the town.

PHYSICS, M6 2023 VCE 6

Kim and Charlie are attempting to create a DC generator and have arranged the magnets along the axis of rotation of the wire loop, J, K, L and M, as shown in Figure 6. They are having some trouble getting it to work. They rotate the loop in the direction of the arrow, as shown in Figure 6.
 

  1. Using physics concepts, explain why this orientation of the magnets will not generate an EMF.  (2 marks)
  1. Kim and Charlie decide to move the magnets so that an EMF is generated. On Figure 6, draw the positions of the magnets to ensure that an EMF is generated.  (1 mark)

Show Answers Only

a.    → The generator has the magnetic field parallel to the plane of the loop. 

→ Therefore, there is no flux cutting through the loop of wire.

→ Further, as the loop rotates there is no change in flux and no current or EMF produced.

b.   → The magnets should be rotated by 90°.
 

Show Worked Solution

a.    → The generator has the magnetic field parallel to the plane of the loop. 

→ Therefore, there is no flux cutting through the loop of wire.

→ Further, as the loop rotates there is no change in flux and no current or EMF produced.

♦ Mean mark (a) 46%.

b.   → The magnets should be rotated by 90°.
 

PHYSICS, M7 2023 VCE 14 MC

Polarisation of visible light provides evidence that electromagnetic radiation can be explained using a

  1. standing wave model for light.
  2. transverse wave model for light.
  3. mechanical wave model for light.
  4. longitudinal wave model for light.
Show Answers Only

Show Worked Solution

→ For polarisation to occur, the plane of vibration of the wave must be perpendicular to direction of motion. 

→ This is a property of the transverse wave model of light.

Calculus, MET1 2023 VCAA 9

The shapes of two walking tracks are shown below.
 

 

Track 1 is described by the function  .

Track 2 is defined by the function  .

The unit of length is kilometres.

  1. Given that   and  , verify that   and  .   (1 mark)
  2. Verify that and both have a turning point at .
  3. Give the co-ordinates of .  (2 marks)
  4. A theme park is planned whose boundaries will form the triangle where is the origin, is at and is at , as shown below, where .
  5. Find the maximum possible area of the theme park, in km².   (3 marks)
     

Show Answers Only

a.   

b.   

c.   

Show Worked Solution
a.   
 

 

 

b.   
   
 

 

 

 

 

 

 

 


♦ Mean mark (b) 48%.

c.  

 
 

 

 

 
 
 
 

♦♦♦ Mean mark (c) 27%.
MARKER’S COMMENT: Many students made arithmetic errors substituting the fractional values of into to find max area.

Calculus, MET1 2023 VCAA 7

Consider . Part of the graph of    is shown below.
 

  1. State the range of .   (1 mark)
  2. Sketch the graph of the inverse function  on the axes above. Label any endpoints and axial intercepts with their coordinates.   (2 marks)
  3. Determine the equation of the domain for the inverse function  .   (2 marks)
  4. Calculate the area of the regions enclosed by the curves of   and  .   (2 marks)
Show Answers Only

a.   

b.   

c.   

d.   

Show Worked Solution

a.   

b.   

c.   


 

 


♦ Mean mark (c) 48%.
MARKER’S COMMENT: Common error → writing the function as .

d.     

 
   
   
   
   

♦♦♦ Mean mark (d) 24%.
MARKER’S COMMENT: Using the symmetry properties of the graph and its inverse helped answer this question efficiently.

Graphs, MET1 2023 VCAA 3

  1. Sketch the graph of  on the axes below, labelling all asymptotes with their equation and axial intercepts with their coordinates.   (3 marks)
     


  2. Find the values of for which .   (1 mark)
Show Answers Only

a.   

b.   

Show Worked Solution

a.   

b.   


♦♦ Mean mark (b) 38%.
MARKER’S COMMENT: Many students did not use their graph from part (a).

Functions, SPEC1 2023 VCAA 1

Consider the function with rule  .

  1. Show that the rule for the function can be written as  .   (1 mark)
  1. Sketch the graph of on the axes below, labelling any asymptotes with their equations.   (3 marks)

   

Show Answers Only
a.    
   
   

 
b.   

 

Show Worked Solution
a.    
   
   

 
b.   

 

Statistics, SPEC1 2023 VCAA 6

Josie travels from home to work in the city. She drives a car to a train station, waits, and then rides on a train to the city. The time, minutes, taken to drive to the station is normally distributed with a mean of 20 minutes and standard deviation of 6 minutes . The waiting time, minutes, for a train is normally distributed with a mean of 8 minutes and standard deviation of minutes . The time, minutes, taken to ride on a train to the city is also normally distributed with a mean of 12 minutes and standard deviation of 5 minutes . The three times are independent of each other.

  1. Find the mean and standard deviation of the total time, in minutes, it takes for Josie to travel from home to the city.   (2 marks)
  1. Josie's waiting time for a train on each work day is independent of her waiting time for a train on any other work day. The probability that, for 12 randomly chosen work days, Josie's average waiting time is between 7 minutes 45 seconds and 8 minutes 30 seconds is equivalent to , where and and are real numbers.
  2. Find the values of and .   (2 marks)
Show Answers Only

a.   

b.   

Show Worked Solution

a.   

 
   
   

 

 
   
   
 

 
b.   

 
   
   

 

Networks, GEN2 2023 VCAA 12

A country has five states, and .

A graph can be drawn with vertices to represent each of the states.

Edges represent a border shared between two states.

  1. What is the sum of the degrees of the vertices of the graph above?   (1 mark)

  2. Euler's formula, , holds for this graph.
  3. i. Complete the formula by writing the appropriate numbers in the boxes provided below.   (1 mark)


  4. ii. Complete the sentence by writing the appropriate word in the space provided below.   (1 mark)


    Euler’s formula holds for this graph because the graph is connected and ______________.
  5. The diagram below shows the position of state on a map of this country.
  6. The four other states are indicated on the diagram as 1, 2, 3 and 4.
     

  1. Use the information in the graph above to complete the table below. Match the state and with the corresponding state number and 4 given in the map above.   (1 mark)

Show Answers Only

a.   
 

b.i.   


 

b.ii.     
 

c.   

Show Worked Solution

a.   
 

b.i.   


 

b.ii.     
 

c.   

Recursion and Finance, GEN2 2023 VCAA 5

Arthur borrowed $30 000 to buy a new motorcycle.

Interest on this loan is charged at the rate of 6.4% per annum, compounding quarterly.

Arthur will repay the loan in full with quarterly repayments over six years.

  1. How many repayments, in total, will Arthur make?  (1 mark)

The balance of the loan, in dollars, after quarters, , can be modelled by the recurrence relation

  1. Showing recursive calculations, determine the balance of the loan after two quarters.
  2. Round your answer to the nearest cent.   (1 mark)

  3. The final repayment required will differ slightly from all the earlier repayments of $1515.18
  4. Determine the value of the final repayment.
  5. Round your answer to the nearest cent.   (1 mark)

Show Answers Only

a.   

b.   

c.   

Show Worked Solution

a.   

  
b.   

Mean mark (b) 51%.
♦♦ Mean mark (c) 25%

c.    

 
 
 
 
 
 

 

Data Analysis, GEN2 2023 VCAA 3

The scatterplot below plots the average monthly ice cream consumption, in litres/person, against average monthly temperature, in °C. The data for the graph was recorded in the Northern Hemisphere.
 

When a least squares line is fitted to the scatterplot, the equation is found to be:

consumption = 0.1404 + 0.0024 × temperature

The coefficient of determination is 0.7212

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

  2. Determine the value of the correlation coefficient .
  3. Round your answer to three decimal places.  (1 mark)

  4. Describe the association between average monthly ice cream consumption and average monthly temperature in terms of strength, direction and form.  (1 mark)

  5. Referring to the equation of the least squares line, interpret the value of the intercept in terms of the variables consumption and temperature.  (1 mark)

  6. Use the equation of the least squares line to predict the average monthly ice cream consumption, in litres per person, when the monthly average temperature is –6°C.  (1 mark)

  7. Write down whether this prediction is an interpolation or an extrapolation.  (1 mark)

Show Answers Only

a.    
         

b.   

c.   

 
d.   

 
e.   

 
f.    

Show Worked Solution

a.    
         

b.   

c.   

 
d.   

 
e.   

 
f.    

Proof, EXT2 P1 2023 SPEC2 1 MC

Consider the following statement.

'If my football team plays badly, then they are not training enough.'

Which one of the following statements is the contrapositive of the statement above?

  1. If my football team plays badly, then they need more training.
  2. If they are training enough, then my football team does not play badly.
  3. If my football team doesn't play badly, then they are training enough.
  4. If they are training enough, then my football team will most likely win.
Show Answers Only

Show Worked Solution

Proof, SPEC2 2023 VCAA 1 MC

Consider the following statement.

'If my football team plays badly, then they are not training enough.'

Which one of the following statements is the contrapositive of the statement above?

  1. If they are not training enough, then my football team plays badly.
  2. If my football team plays badly, then they need more training.
  3. If they are training enough, then my football team does not play badly.
  4. If my football team doesn't play badly, then they are training enough.
  5. If they are training enough, then my football team will most likely win.
Show Answers Only

Show Worked Solution

Networks, GEN1 2022 VCAA 1 MC

The network below shows the distances, in kilometres, along a series of roads.

The vertices and represent the intersections of these roads.
 

Prim's algorithm can be used to find the

  1. critical path.
  2. shortest path.
  3. minimum cut.
  4. minimum allocation.
  5. minimum spanning tree.
Show Answers Only

Show Worked Solution

Prim’s algorithm is used to find the minimum spanning tree.

Matrices, GEN1 2022 VCAA 4 MC

The communication matrix below shows the communication links between five people: Steph , Tran ,Ursula , Vinh and Wanda .

In this matrix:

  • the '1' in row , column indicates that Steph can communicate directly with Tran
  • the '0' in row , column indicates that Vinh cannot communicate directly with Wanda.

Ursula needs to communicate with Steph.

The sequence of communication links that will successfully allow Ursula to communicate with Steph is

Show Answers Only

Show Worked Solution

By elimination:

Option A:  (eliminate A)

Option B:  (eliminate B)

Option C:  (correct)

Option D:  (eliminate D)

Option E:  (eliminate E)

Matrices, GEN1 2022 VCAA 1-2 MC

A bike rental business rents road bikes and mountain bikes in three sizes: child , junior and adult .

Matrix shows the daily rental cost, in dollars, for each type of bike.

 
The element in row and column in matrix is .

 
Question 1

The daily cost of renting an adult mountain bike is shown in element

 
Question 2

On Sundays, the business increases the daily rental price for each type of bike by 10%.

To determine the rental cost for each type of bike on a Sunday, which one of the following matrix calculations needs to be completed?

Show Answers Only

Show Worked Solution


 

Recursion and Finance, GEN1 2022 VCAA 18-19 MC

The balance of a loan, , in dollars, after months is modelled by the recurrence relation


 

Question 18

The balance of the loan first falls below $398 000 after how many months?

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

 
Question 19

With a small change to the final payment, the loan is expected to be repaid in full in

  1. 25 years.
  2. 26 years.
  3. 28 years.
  4. 29 years.
  5. 30 years.
Show Answers Only

Show Worked Solution

 
 
 

  

 

  

 
 
 
 
 
 

 


♦ Mean mark (Q19) 47%.

CHEMISTRY, M7 2016 VCE 7a

Butanoic acid is the simplest carboxylic acid that is also classified as a fatty acid. Butanoic acid may be synthesised as outlined in the following reaction flow chart.
 

  1. Draw the structural formula of but-1-ene in the box provided.  (1 mark)

  2. State the reagent(s) needed to convert but-1-ene to Compound Y in the box provided.  (1 mark)

  3. Write the systematic name of Compound Y in the box provided.  (1 mark)

  4. Write the semi-structural formula of butanoic acid in the box provided.  (1 mark)

  5. Write a balanced half-equation for the conversion of .  (2 marks)

Show Answers Only

i.    
       

ii.    and (catalyst)

iii.   butan-1-ol or 1-butanol

iv.   

v.   

Show Worked Solution

i.    
       

ii.    and (catalyst)

iii.   butan-1-ol or 1-butanol

iv.   

v.   

♦♦ Mean mark (ii) 29%.

CHEMISTRY, M7 2015 VCE 5b

Draw the full structural formula of an isomer of butan-2-ol.   (1 mark)

Show Answers Only

Possible images include:

                
 

Show Worked Solution

Possible images include:

                
 

CHEMISTRY, M7 2015 VCE 5a

A reaction pathway is designed for the synthesis of the compound that has the structural formula shown below.
 

The table below gives a list of available organic reactants and reagents.
 


Complete the reaction pathway design flow chart below. Write the corresponding letter for the structural formula of all organic reactants in each of the boxes provided. The corresponding letter for the formula of other necessary reagents should be shown in the boxes next to the arrows.   (5 marks)

Show Answers Only

Show Worked Solution