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v1 Algebra, STD2 A2 2010 HSC 27c

The graph shows tax payable against taxable income, in thousands of dollars.
  

  1. Use the graph to find the tax payable on a taxable income of .  (1 mark)

  2. Use suitable points from the graph to show that the gradient of the section of the graph marked    is  .    (1 mark)

  3. How much of each dollar earned between    and  is payable in tax? Give your answer correct to the nearest whole number.   (1 mark)

  4. Write an equation that could be used to calculate the tax payable, , in terms of the taxable income, , for taxable incomes between    and  .   (2 marks)

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Show Worked Solution
i.   

  

ii. 

 
 
 

♦♦ Mean mark (ii) 25%.

iii. 

 

♦♦♦ Mean mark (iii) 12%!
MARKER’S COMMENT: Interpreting gradients is an examiner favourite, so make sure you are confident in this area.

iv. 

 
 

♦♦♦ Mean mark (iv) 15%.
STRATEGY: The earlier parts of this question direct students to the most efficient way to solve this question. Make sure earlier parts of a question are front and centre of your mind when devising strategy.

v1 Algebra, STD2 A1 2020 HSC 13 MC

When Stuart stops drinking alcohol at 11:30 pm, he has a blood alcohol content (BAC) of 0.08625.

The number of hours required for a person to reach zero BAC after they stop consuming alcohol is given by the formula:

.

At what time on the next day should Stuart expect his BAC to be 0.05?

  1.  1:33 am
  2.  1:55 am
  3.  2:15 am
  4.  5:15 am
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Show Worked Solution


 

 


 

 

 


♦♦♦ Mean mark 17%.
COMMENT: The rates aspect of this question proved extremely challenging.

EXAMCOPY 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 .
  1. Determine the range of values of such that .   (1 mark)

  2. 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%.

CHEMISTRY, M8 2022 VCE 5*

A chemist uses spectroscopy to identify an unknown organic molecule, Molecule , that contains chlorine.

The spectrum of Molecule is shown below.
 

The infra-red (IR) spectrum of Molecule is shown below.
 

  1. Name the functional group that produces the peak at 168 ppm in the  spectrum on the first image, which is consistent with the IR spectrum shown above. Justify your answer with reference to the IR spectrum.   (2 marks)

The mass spectrum of Molecule is shown below
 

  1. The molecular mass of Molecule is 108.5
  2.  Explain the presence of the peak at 110 m/z.  (1 mark)

The  spectrum of Molecule is shown below.
 

  1. The spectrum consists of two singlet peaks.
  2. What information does this give about the molecule?   (2 marks)

  3. Draw a structural formula for Molecule that is consistent with the information provided in parts a–c.   (2 marks)

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a.    → Absence of a very broad acid peak between 2500–3000. 

→ Molecule must be an ester.
 

b.    The peak at 110 m/z:

→ due to the Chlorine-37 isotope which is slightly heavier than the more abundant Chlorine-35.
 

c.    The two singlet peaks indicate:

→ two different hydrogen environments within the molecule.

→ there are no adjacent hydrogen environments.

→ The relative heights of the peaks show the ratios of the hydrogens in the environments are 2 : 3.
 

d.    Either of the two molecules show below are correct:

Show Worked Solution

a.    → Absence of a very broad acid peak between 2500–3000. 

→ Molecule must be an ester.
 

♦ Mean mark (a) 41%.

b.    The peak at 110 m/z:

→ due to the Chlorine-37 isotope which is slightly heavier than the more abundant Chlorine-35.
 

c.    The two singlet peaks indicate:

→ two different hydrogen environments within the molecule.

→ there are no adjacent hydrogen environments.

→ The relative heights of the peaks show the ratios of the hydrogens in the environments are 2 : 3.

♦♦♦ Mean mark (b) 15%.
COMMENT: Know the masses of common isotopes.

d.    Either of the two molecules show below are correct:
 

♦ Mean mark (d) 40%.

CHEMISTRY, M8 2021 VCE 16 MC

Which one of the following statements about IR spectroscopy is correct?

  1. IR radiation changes the spin state of electrons.
  2. Bond wave number is influenced only by bond strength.
  3. An IR spectrum can be used to determine the purity of a sample.
  4. In an IR spectrum, high transmittance corresponds to high absorption.
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Show Worked Solution

→ Every pure compound has a different fingerprint region on the Infrared spectrum. 

→ Hence the fingerprint region of the sample can be compared against the fingerprint region of the pure substance to determine the purity of the sample.

♦♦♦ Mean mark 15%.

CHEMISTRY, M8 2023 VCE 7-2*

The infrared (IR) spectrum of the molecule 3-methyl-2-butanone is shown below.
 

Explain why different frequencies of infrared radiation can be absorbed by the same molecule as shown in the spectrum above.   (3 marks)

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→ As infrared radiation is passed through the molecules, the different bonds within the molecule vibrate at specific wavelengths leading to the absorption of the infrared radiation.

→ In this way, different frequencies of infrared radiation can be absorbed by a molecule as bonds differ in electronegativity, dipole strengths and in the masses of atoms at the end of bonds.  

→ For example, the double bond between the oxygen atom and carbon atom in the given molecule has a greater dipole than the carbon-hydrogen bonds. This causes a transmittance at 1450 whereas bonds have a transmittance at 3000.

→ An oxygen atom has a higher molecular mass than hydrogen atoms and this also leads to different frequencies of infrared radiation being absorbed in the one molecule.

(Students could have also discussed the strength of bonds, bond length or molecular vibrations)

Show Worked Solution

→ As infrared radiation is passed through the molecules, the different bonds within the molecule vibrate at specific wavelengths leading to the absorption of the infrared radiation.

→ In this way, different frequencies of infrared radiation can be absorbed by a molecule as bonds differ in electronegativity, dipole strengths and in the masses of atoms at the end of bonds.  

→ For example, the double bond between the oxygen atom and carbon atom in the given molecule has a greater dipole than the carbon-hydrogen bonds. This causes a transmittance at 1450 whereas bonds have a transmittance at 3000.

→ An oxygen atom has a higher molecular mass than hydrogen atoms and this also leads to different frequencies of infrared radiation being absorbed in the one molecule.

(Students could have also discussed the strength of bonds, bond length or molecular vibrations)

♦♦ Mean mark 30%.
COMMENT: A deep understanding of the principles behind analytical techniques required here.

CHEMISTRY, M8 2013 VCE 2

The strength of the eggshell of birds is determined by the calcium carbonate, , content of the eggshell.

The percentage of calcium carbonate in the eggshell can be determined by gravimetric analysis.

0.412 g of clean, dry eggshell was completely dissolved in a minimum volume of dilute hydrochloric acid.

An excess of a basic solution of ammonium oxalate, , was then added to form crystals of calcium oxalate monohydrate, .

The suspension was filtered and the crystals were then dried to constant mass.

0.523 g of was collected.

  1. Write a balanced equation for the formation of the calcium oxalate monohydrate precipitate.   (1 mark)

  2. Determine the percentage, by mass, of calcium carbonate in the eggshell.   (3 marks)

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a.    

b.    %

Show Worked Solution

a.    
 

♦♦♦ Mean mark (a) 30%.

b.    

CHEMISTRY, M7 2018 VCE 1a

Organic compounds are numerous and diverse due to the nature of the carbon atom. There are international conventions for the naming and representation of organic compounds.

  1. Draw the structural formula of 2-methyl-propan-2-ol.   (1 mark)

  2. Give the molecular formula of but-2-yne.   (1 mark)

  1. Give the IUPAC name of the compound that has the structural formula shown above.   (1 mark)

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i.    
         

ii.   

→ Structural or semi-structural formulas are not appropriate.
 

iii.  → Longest carbon chain is 6, hence hexane

→ Number carbons to give functional group lowest possible numbers: 2,3-dibromo, 4-methyl

→ Compound name: 2,3-dibromo-4-methylhexane

Show Worked Solution

i.    
         

ii.   

→ Structural or semi-structural formulas are not appropriate.
 

iii.  → Longest carbon chain is 6, hence hexane

→ Number carbons to give functional group lowest possible numbers: 2,3-dibromo, 4-methyl

→ Compound name: 2,3-dibromo-4-methylhexane

♦♦♦ Mean mark (c) 27%.

CHEMISTRY, M8 2016 VCE 6

Brass is an alloy of copper and zinc.

To determine the percentage of copper in a particular sample of brass, an analyst prepared a number of standard solutions of copper ions and measured their absorbance using an atomic absorption spectrometer (AAS).

The calibration curve obtained is shown below.
 

  1. A 0.198 g sample of the brass was dissolved in acid and the solution was made up to 100.00 mL in a volumetric flask. The absorbance of this test solution was found to be 0.13
  2. Calculate the percentage by mass of copper in the brass sample.   (3 marks)

  3. If the analyst had made up the solution of the brass sample to 20.00 mL instead of 100.00 mL, would the result of the analysis have been equally reliable? Why?   (2 marks)

  4. Name another analytical technique that could be used to verify the result from part a.   (1 mark)

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a.    55.6%

b.    → No, this would increase the concentration of the copper solution by a factor of 5.

→ The concentration of the solution and absorbance would be too high and outside the range of the calibration curve (it can’t be assumed that the calibration curve remains linear beyond the range of the known data).

c.    Answers could include:

→ UV-vis spectroscopy, colorimetry, volumetric analysis, gravimetric analysis.

Show Worked Solution

a.    Absorbance of 0.13 → concentration of 1.1 gL  (see graph)


 

b.    → No, this would increase the concentration of the copper solution by a factor of 5.

→ The concentration of the solution and absorbance would be too high and outside the range of the calibration curve.

→ It can’t be assumed that the calibration curve remains linear beyond the range of the known data.
 

♦♦♦ Mean mark (b) 25%.

c.    Answers could include:

→ UV-vis spectroscopy, colorimetry, volumetric analysis, gravimetric analysis.

PHYSICS, M8 2019 VCE 17

Students are comparing the diffraction patterns produced by electrons and X-rays, in which the same spacing of bands is observed in the patterns, as shown schematically in Figure 18. Note that both patterns shown are to the same scale.
 

The electron diffraction pattern is produced by 3.0 × 10 eV electrons.

  1. Explain why electrons can produce the same spacing of bands in a diffraction pattern as X-rays.  (3 marks)
  1. Calculate the frequency of X-rays that would produce the same spacing of bands in a diffraction pattern as for the electrons. Show your working.  (4 marks)
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a.    → Electrons with momentum exhibit wave-like properties.

→  In this way, moving electrons produce a de Broglie wavelength.

→  As diffraction is a wave phenomenon and is dependent on wavelengths, if the de Broglie wavelength of an electron matches the wavelength of an X-ray then spacing of the bands will be the same.
 

b.    

Show Worked Solution

a.    → Electrons with momentum exhibit wave-like properties.

→  In this way, moving electrons produce a de Broglie wavelength.

→  As diffraction is a wave phenomenon and is dependent on wavelengths, if the de Broglie wavelength of an electron matches the wavelength of an X-ray then spacing of the bands will be the same.
 

♦ Mean mark (a) 49%.

b.    Find velocity of the electrons using 

 
 
   

 

The de Broglie wavelength of the electron is:

 
   
   

 
Frequency of the X-ray:

♦♦♦ Mean mark (b) 27%.
COMMENT: Multi-step solutions require clear and logical working.

PHYSICS, M5 2019 VCE 8

A 250 g toy car performs a loop in the apparatus shown in Figure 8.
 

The car starts from rest at point and travels along the track without any air resistance or retarding frictional forces. The radius of the car's path in the loop is 0.20 m. When the car reaches point it is travelling at a speed of 3.0 m s.

  1. Calculate the value of . Show your working.   (3 marks)

  2. Calculate the magnitude of the normal reaction force on the car by the track when it is at point . Show your working.   (3 marks)

  3. Explain why the car does not fall from the track at point , when it is upside down.   (2 marks)

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a.    

b.   

c.    Method 1

→ During the car’s circular motion around the loop, the magnitude of the centripetal force exceeds the magnitude of the weight force.

→ This produces a normal reaction force acting downwards which enables the car to travel around the loop at 3 ms.
 

Method 2

→ The minimum speed required for the car so that it stays on the track will be when the centripetal force is equal to the weight force. 

 
 
   
   

 
→ As 3 ms is greater than the minimum speed required, a normal force downwards will be present at point .

→ Therefore the car will not fall from the track at point .

Show Worked Solution

a.    Using the law of the conservation of energy:

 
 
 
 
   

 

b.    
 
     
   

♦ Mean mark (b) 53%.

c.    Method 1

→ During the car’s circular motion around the loop, the magnitude of the centripetal force exceeds the magnitude of the weight force.

→ This produces a normal reaction force acting downwards which enables the car to travel around the loop at 3 ms.
 

Method 2

→ The minimum speed required for the car so that it stays on the track will be when the centripetal force is equal to the weight force. 

 
 
   
   

 
→ As 3 ms is greater than the minimum speed required, a normal force downwards will be present at point .

→ Therefore the car will not fall from the track at point .

♦♦♦ Mean mark (c) 23%.
COMMENT: Normal force is poorly understood here..

PHYSICS, M6 2019 VCE 6

A home owner on a large property creates a backyard entertainment area. The entertainment area has a low-voltage lighting system. To operate correctly, the lighting system requires a voltage of 12 V. The lighting system has a resistance of 12 .

  1. Calculate the power drawn by the lighting system.   (1 mark)

To operate the lighting system, the home owner installs an ideal transformer at the house to reduce the voltage from 240 V to 12 V. The home owner then runs a 200 m long heavy-duty outdoor extension lead, which has a total resistance of 3 , from the transformer to the entertainment area.

  1. The lights are a little dimmer than expected in the entertainment area.
  2. Give one possible reason for this and support your answer with calculations.   (4 marks)

  3. Using the same equipment, what changes could the home owner make to improve the brightness of the lights? Explain your answer.   (2 marks)

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a.   

b.   Potential cause of dimmer lights: 

→ Power loss in the extension lead.

→ The total resistance () in the circuit

→ The new current in the circuit

→ The power loss across the extension lead,

→ Therefore, power supplied to lights .

c.    Changes using the same equipment:

→ The homeowner needs to move the transformer from inside the home (start of the lead) to the outside entertainment area (end of the lead).

→ This would mean a higher voltage/lower current is running through the lead which would decrease the power loss through the lead and improve the brightness of the lights.

Show Worked Solution

a.    


 

b.   Potential cause of dimmer lights: 

→ Power loss in the extension lead.

→ The total resistance () in the circuit

→ The new current in the circuit

→ The power loss across the extension lead,

→ Therefore, power supplied to lights .

♦♦♦ Mean mark (b) 23%.
COMMENT: Many students used 1 Amp in their calculations instead of finding the total current in the system.

c.    Changes using the same equipment:

→ The homeowner needs to move the transformer from inside the home (start of the lead) to the outside entertainment area (end of the lead).

→ This would mean a higher voltage/lower current is running through the lead which would decrease the power loss through the lead and improve the brightness of the lights.

♦♦♦ Mean mark (c) 22%.
COMMENT: Students incorrectly added or changed the equipment used in the question.

PHYSICS, M6 2019 VCE 1

A particle of mass and charge travelling at velocity enters a uniform magnetic field , as shown in Figure 1.
 

  1. Is the charge positive or negative? Give a reason for your answer.   (1 mark)
  1. Explain why the path of the particle is an arc of a circle while the particle is in the magnetic field.   (2 marks)

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a.    Using the right-hand rule:

→ Thumb is in direction of velocity (right) and fingers are in direction of B-field (Into the page).

→ As the force on the charge is down the page (back of the hand), the charge must be negative.
 

b.    For circular motion to occur:

→ The force must be at right-angles to the velocity of the particle.

→ The force must be of constant magnitude .

→ In the given example, the force on the charged particle due to the magnetic field is perpendicular to its velocity and the magnitude of the force remains constant.

→ The particle will therefore follow the arc of a circle while in the magnetic field.

Show Worked Solution

a.    Using the right-hand rule:

→ Thumb is in direction of velocity (right) and fingers are in direction of B-field (Into the page).

→ As the force on the charge is down the page (back of the hand), the charge must be negative.

♦ Mean mark (a) 44%.

b.    For circular motion to occur:

→ The force must be at right-angles to the velocity of the particle.

→ The force must be of constant magnitude .

→ In the given example, the force on the charged particle due to the magnetic field is perpendicular to its velocity and the magnitude of the force remains constant.

→ The particle will therefore follow the arc of a circle while in the magnetic field.

♦♦♦ Mean mark (b) 27%.

PHYSICS, M7 2020 VCE 11

An astronaut has left Earth and is travelling on a spaceship at 0.800 directly towards the star known as Sirius, which is located 8.61 light-years away from Earth, as measured by observers on Earth.

  1. How long will the trip take according to a clock that the astronaut is carrying on his spaceship? Show your working.   (2 marks)
  1. Is the trip time measured by the astronaut in part a. a proper time? Explain your reasoning.   (2 marks)

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a.   

b.   → The trip time of 6.46 years on the spaceship is a proper time.

→ This is due to the Astronaut’s clock being stationary within the astronaut’s frame of reference.

Show Worked Solution

a.    From the earth’s perspective:

.
 

From the astronaut’s perspective:

→  The Earth’s time is going to be dilated compared to his, so the time the astronauts clock will measure is:

 
 
 

♦♦♦ Mean mark (a) 29%.
COMMENT: Light years is a measure of distance, not time!

b.   → The trip time of 6.46 years on the spaceship is a proper time.

→ This is due to the Astronaut’s clock being stationary within the astronaut’s frame of reference.

♦ Mean mark (b) 44%.

PHYSICS, M6 2020 VCE 6

Two Physics students hold a coil of wire in a constant uniform magnetic field, as shown in Figure 5a. The ends of the wire are connected to a sensitive ammeter. The students then change the shape of the coil by pulling each side of the coil in the horizontal direction, as shown in Figure 5b. They notice a current register on the ammeter.
 

  1. Will the magnetic flux through the coil increase, decrease or stay the same as the students change the shape of the coil?  (1 mark)
  1. Explain, using physics principles, why the ammeter registered a current in the coil and determine the direction of the induced current.  (3 marks
  1. The students then push each side of the coil together, as shown in Figure 6a, so that the coil returns to its original circular shape, as shown in Figure 6b, and then changes to the shape shown in Figure 6c. 
     

  1. Describe the direction of any induced currents in the coil during these changes. Give your reasoning.   (2 marks)

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a.    Decrease 

b.    Ammeter registers a current:

→ By Faraday’s law of induction, a change in flux will result in an induced EMF through the coil and induced current. 

→ This induced current acts to oppose the original change in flux that created it. As the magnetic flux is decreasing, the induced current will act to strengthen the magnetic field.

→ Hence by using the right-hand grip rule, the current will run clockwise through the coil.

c.    → There will be an induced current from 6a to 6b.

→ As the magnetic flux is increasing, the induced current will act to decrease the flux and oppose the magnetic field.

→ By the right-hand grip rule, the current will flow anti-clockwise around the coil.

→ There will also be an induced current from 6b to 6c.

→ The magnetic flux this time is decreasing, so the induced current will act to increase the flux and strengthen the magnetic field.

→ Using the right-hand grip rule, the current through the coil will flow clockwise.

Show Worked Solution

a.    Magnetic flux will decrease.

→ The magnetic flux is proportional to how many field lines are passing through a given area.

→ As the number of lines passing through the coil of wire decreases, the magnetic flux will also decrease.
 

b.    Ammeter registers a current:

→ By Faraday’s law of induction, a change in flux will result in an induced EMF through the coil and induced current. 

→ This induced current acts to oppose the original change in flux that created it. As the magnetic flux is decreasing, the induced current will act to strengthen the magnetic field.

→ Hence by using the right-hand grip rule, the current will run clockwise through the coil.
 

♦ Mean mark (b) 42%.

c.    → There will be an induced current from 6a to 6b.

→ As the magnetic flux is increasing, the induced current will act to decrease the flux and oppose the magnetic field.

→ By the right-hand grip rule, the current will flow anti-clockwise around the coil.

→ There will also be an induced current from 6b to 6c.

→ The magnetic flux this time is decreasing, so the induced current will act to increase the flux and strengthen the magnetic field.

→ Using the right-hand grip rule, the current through the coil will flow clockwise.

♦♦♦ Mean mark (c) 28%.
COMMENT: Lenz’s law was poorly understood in this question.

PHYSICS, M5 2020 VCE 4*

The Ionospheric Connection Explorer (ICON) space weather satellite, constructed to study Earth's ionosphere, was launched in October 2019. ICON will study the link between space weather and Earth's weather at its orbital altitude of 600 km above Earth's surface. Assume that ICON's orbit is a circular orbit. 

  1. Calculate the orbital radius of the ICON satellite.   (1 mark)

  2. Calculate the orbital period of the ICON satellite correct to three significant figures. Show your working.   (4 marks)

  3. Explain how the ICON satellite maintains a stable circular orbit without the use of propulsion engines.   (2 marks)

Show Answers Only

a.    

b.    

c.    → The icon satellite is only subject to the gravitational force of the Earth.

→ This force is a centripetal force of constant magnitude and hence the satellite maintains a stable circular orbit. 

Show Worked Solution

a.    Radius of orbit is equal to the altitude plus the radius of the Earth.

 

b.    
 
   
   

 

c.    → The icon satellite is only subject to the gravitational force of the Earth.

→ This force is a centripetal force of constant magnitude and hence the satellite maintains a stable circular orbit. 

♦♦♦ Mean mark (c) 23%.

PHYSICS, M6 2020 VCE 3

Electron microscopes use a high-precision electron velocity selector consisting of an electric field, , perpendicular to a magnetic field, .

Electrons travelling at the required velocity, , exit the aperture at point , while electrons travelling slower or faster than the required velocity, , hit the aperture plate, as shown in Figure 2.
 

  1. Show that the velocity of an electron that travels straight through the aperture to point is given by  = .   (1 mark)

  2. Calculate the magnitude of the velocity, , of an electron that travels straight through the aperture to point if  = 500 kV m  and  = 0.25 T. Show your working.   (2 marks)

  3.  i. At which of the points – , or – in Figure 2 could electrons travelling faster than arrive?   (1 mark)

  4. ii. Explain your answer to part c.i.   (2 marks)

Show Answers Only

a.    Find where forces due to magnetic and electric field are balanced

 
 
 

 
b.    

c.i.    Point

c.ii.  When electrons travel faster than :

→ The force due to the magnetic field will increase but the force due to the electric field will remain unchanged. 

→ Therefore, there will be a net force acting on the electron due to the magnetic force being greater than the electric force.

→ Using the right-hand rule, the force on the electron due to the magnetic field is down the page, hence the electron will arrive at point

Show Worked Solution

a.    Find where forces due to magnetic and electric field are balanced

 
 
 
♦ Mean mark (a) 46%.

b.   
 

c.i.    Point
 

c.ii.  When electrons travel faster than :

→ The force due to the magnetic field will increase but the force due to the electric field will remain unchanged. 

→ Therefore, there will be a net force acting on the electron due to the magnetic force being greater than the electric force.

→ Using the right-hand rule, the force on the electron due to the magnetic field is down the page, hence the electron will arrive at point

♦ Mean mark (c.i.) 40%.
♦♦♦ Mean mark (c.ii.) 15%.
COMMENT: Students needed to include what would happen to the electric force.

CHEMISTRY, M3 EQ-Bank 12

A student stirs 2.80 g of silver nitrate powder into 250.0 mL of 1.00 mol L sodium hydroxide solution until it is fully dissolved. A reaction occurs and a precipitate appears.

  1. Write a balanced chemical equation for the reaction.   (1 mark)

  2. Calculate the theoretical mass of precipitate that will be formed.   (3 marks)

The student weighed a piece of filter paper, filtered out the precipitate and dried it thoroughly in an incubator. The final precipitate mass was higher than predicted in (b).

  1. Identify one scientific reason why the precipitate mass was too high and suggest an improvement to the experimental method which would eliminate this error.   (2 marks)

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a.   

b.   

c.    Possible reasons for the higher mass:

→ Presence of excess solution in the filter paper or incomplete drying.

→ Presence of soluble ions on the filter paper which would crystalise when dried and increase the mass of the precipitate.
 

Experimental adjustment to eliminate error:

→ Ensure the precipitate is thoroughly rinsed with distilled water to remove soluble impurities.

→ Dry precipitate completely in the incubator before weighing.

Show Worked Solution

a.   
 

b.   


 

c.    Possible reasons for the higher mass:

→ Presence of excess solution in the filter paper or incomplete drying.

→ Presence of soluble ions on the filter paper which would crystalise when dried and increase the mass of the precipitate.
 

Experimental adjustment to eliminate error:

→ Ensure the precipitate is thoroughly rinsed with distilled water to remove soluble impurities.

→ Dry precipitate completely in the incubator before weighing.

CHEMISTRY, M3 EQ-Bank 11

A student stirs 2.45 g of copper nitrate powder into 200.0 mL of 1.25 mol L sodium carbonate solution until it is fully dissolved. A reaction occurs and a precipitate appears.

  1. Write a balanced chemical equation for the reaction.   (1 mark)

  2. Calculate the theoretical mass of precipitate that will be formed.   (3 marks)

The student weighed a piece of filter paper, filtered out the precipitate and dried it thoroughly in an incubator. The final precipitate mass was higher than predicted in (b).

  1. Identify one scientific reason why the precipitate mass was too high and suggest an improvement to the experimental method which would eliminate this error.   (2 marks)

Show Answers Only

a.   

b.   

c.    Possible reasons for the higher mass:

→ Presence of excess solution in the filter paper or incomplete drying.

→ Presence of soluble ions on the filter paper which would crystalise when dried and increase the mass of the precipitate.
 

Experimental adjustment to eliminate error:

→ Ensure the precipitate is thoroughly rinsed with distilled water to remove soluble impurities.

→ Dry precipitate completely in the incubator before weighing.

Show Worked Solution

a.   
 

b.   


 

c.    Possible reasons for the higher mass:

→ Presence of excess solution in the filter paper or incomplete drying.

→ Presence of soluble ions on the filter paper which would crystalise when dried and increase the mass of the precipitate.
 

Experimental adjustment to eliminate error:

→ Ensure the precipitate is thoroughly rinsed with distilled water to remove soluble impurities.

→ Dry precipitate completely in the incubator before weighing.

Calculus, MET2 2023 VCE SM-Bank 1

The function is defined as follows.

  1. Sketch the graph of on the axes below. Label the vertical asymptote with its equation, and label any axial intercepts, stationary points and endpoints in coordinate form, correct to three decimal places.   (3 marks)

     

     

  2.  i. Find the equation of the tangent to the graph of at the point where .   (1 mark)

  3. ii. Sketch the graph of the tangent to the graph of at on the axes in part a.   (1 mark)

Newton's method is used to find an approximate -intercept of , with an initial estimate of .

  1. Find the value of .   (1 mark)

  2. Find the horizontal distance between and the closest -intercept of , correct to four decimal places.   (1 mark)

  3.  i. Find the value of , where , such that an initial estimate of    gives the same value of    as found in part . Give your answer correct to three decimal places.   (2 marks)

  4. ii. Using this value of , sketch the tangent to the graph of at the point where    on the axes in part a.   (1 mark)

Show Answers Only
a.    
  
b.i
  
b.ii
  
c.
  
d.
  
e.i
  
e.ii
Show Worked Solution

a.

b.i   
 
 
 

  

b.ii

 
c.   

 
 


  
 

d.    

 

e.i. 

 
e.ii

CHEMISTRY, M2 EQ-Bank 2

During a laboratory experiment, a gas is collected in a sealed syringe. Initially, the gas has a volume of 5.0 litres and a pressure of 1.0 atmosphere. 

  1.  Calculate the new pressure inside the syringe when the volume is decreased to 3.0 litres, assuming no temperature change.   (2 marks)

  2. After reaching the pressure calculated in part a, the volume is further decreased so that the pressure inside the syringe doubles. Calculate the final volume of the gas.   (2 marks)

  3. Discuss two potential experimental errors that could affect the accuracy of the observed results compared to the theoretical predictions of Boyle's Law.   (2 marks)

Show Answers Only

a.   

b.   

c.    Potential experimental errors could include:

→ Temperature Control. Any temperature changes violate the assumptions of Boyle’s Law which holds only at constant temperature.

→ Measurement Accuracy. Errors in measuring volume changes can lead to inaccuracies in pressure calculations.

→ Non-Ideal Behaviour. At high pressures, gases may deviate from ideal behaviour, affecting the accuracy of Boyle’s Law predictions.

Show Worked Solution

a.    Using Boyle’s Law  :


 

b.    To find the new volume when pressure doubles:


 

c.    Potential experimental errors could include:

→ Temperature Control. Any temperature changes violate the assumptions of Boyle’s Law which holds only at constant temperature.

→ Measurement Accuracy. Errors in measuring volume changes can lead to inaccuracies in pressure calculations.

→ Non-Ideal Behaviour. At high pressures, gases may deviate from ideal behaviour, affecting the accuracy of Boyle’s Law predictions.

CHEMISTRY, M1 EQ-Bank 33

Explain why the boiling points of hydrogen halides , and increase from to .

Include in your answer the types of intermolecular forces involved and how molecular mass affects these boiling points.   (3 marks)

Show Answers Only

→ exhibits hydrogen bonding while , and experience dipole-dipole interactions and dispersion forces.

→ As molecular mass increases from to , the number of electrons increases, enhancing the dispersion forces.

→ Despite having hydrogen bonds, the significantly greater molecular mass of results in stronger dispersion forces that lead to a higher boiling point.

Show Worked Solution

→ exhibits hydrogen bonding while , and experience dipole-dipole interactions and dispersion forces.

→ As molecular mass increases from to , the number of electrons increases, enhancing the dispersion forces.

→ Despite having hydrogen bonds, the significantly greater molecular mass of results in stronger dispersion forces that lead to a higher boiling point.

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)

Show Answers Only

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.   

Calculus, MET2 2022 VCAA 5

Consider the composite function , where the function is an unknown but differentiable function for all values of .

Use the following table of values for and .

 

  1. Find the value of .   (1 mark)

The derivative of with respect to is given by .

  1. Show that .   (1 mark)

  2. Find the equation of the tangent to at .   (2 marks)

  3. Find the average value of the derivative function between and .   (2 marks)

  4. Find four solutions to the equation for the interval .   (3 marks)

Show Answers Only

a.   

b.   

c.   

d.   

e.   

Show Worked Solution
 

a. 
 
   
   

 

b.   
 
   

 

c.     and 

 
 
 

♦♦ Mean mark (c) 45%.
MARKER’S COMMENT: Some students did not produce an equation as required.

  
d.   The average value of between and

Average  
   
   
   

♦♦ Mean mark (d) 30%.
MARKER’S COMMENT: Those who used the Average Value formula were generally successful.
Some students substituted , not .

e.  

  or 

(1):         
    
 
 
     
(2):  
 
 

  


♦♦ Mean mark (e) 30%.
MARKER’S COMMENT: Some students were able to find . Some solved or but not both.

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
Show Answers Only
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%.

Probability, MET2 2022 VCAA 3

Mika is flipping a coin. The unbiased coin has a probability of of landing on heads and of landing on tails.

Let be the binomial random variable representing the number of times that the coin lands on heads.

Mika flips the coin five times.

    1. Find .   (1 mark)
    2. Find (1 mark)
    3. Find , correct to three decimal places.   (2 marks)
    4. Find the expected value and the standard deviation for .   (2 marks)

The height reached by each of Mika's coin flips is given by a continuous random variable, , with the probability density function
  


  

where is the vertical height reached by the coin flip, in metres, between the coin and the floor, and and are real constants.

    1. State the value of the definite integral .   (1 mark)
    2. Given that    and  , find the values of and .   (3 marks)

    3. The ceiling of Mika's room is 3 m above the floor. The minimum distance between the coin and the ceiling is a continuous random variable, , with probability density function .
    4. The function is a transformation of the function given by , where is the minimum distance between the coin and the ceiling, and and are real constants.
    5. Find the values of and .   (1 mark)

  1. Mika's sister Bella also has a coin. On each flip, Bella's coin has a probability of of landing on heads and of landing on tails, where is a constant value between 0 and 1 .
  2. Bella flips her coin 25 times in order to estimate .
  3. Let be the random variable representing the proportion of times that Bella's coin lands on heads in her sample.
    1. Is the random variable discrete or continuous? Justify your answer.   (1 mark)
    2. If , find an approximate 95% confidence interval for , correct to three decimal places.   (1 mark)
    3. Bella knows that she can decrease the width of a 95% confidence interval by using a larger sample of coin flips.
    4. If , how many coin flips would be required to halve the width of the confidence interval found in part c.ii.?   (1 mark)

Show Answers Only

a. i.        ii.        iii.    (3 d.p.)

a. iv   

b. i.      

b. ii.   

b. iii.

c. i.      ii.      iii.   

Show Worked Solution

a.i 


 

a.ii  By CAS:        

  
a.iii 

  
By CAS:
 

(3 decimal places)
  

a.iv

  

b.i  


♦♦ Mean mark (b.i) 40%.
MARKER’S COMMENT: Many students did not evaluate the integral or evaluated incorrectly.

b.ii  By CAS:


 


♦♦ Mean mark (b.ii) 47%.
MARKER’S COMMENT: Many students did not give exact answers.

b.iii 

and


♦♦♦♦ Mean mark (b.iii) 10%.
MARKER’S COMMENT: Many students did not attempt this question.

c.i    is discrete.

The number of coin flips must be zero or a positive integer so    is countable and therefore discrete.
 

c.ii 


 

c.iii To halve the width of the confidence interval, the standard deviation needs to be halved.

She would need to flip the coin 100 times


♦♦♦ Mean mark (c.iii) 30%.
MARKER’S COMMENT: Common incorrect answers were 0, 10, 11, 50 and 101.

PHYSICS, M6 2020 VCE 7 MC

An ideal transformer has an input DC voltage of 240 V, 2000 turns in the primary coil and 80 turns in the secondary coil.

The output voltage is closest to

  1. 0 V
  2. 9.6 V
  3. 6.0 × 10 V
  4. 3.8 × 10 V
Show Answers Only

Show Worked Solution

→ Transformers require an AC current to provide the change in flux that generates an EMF in the secondary coil from the primary coil.

→ The question refers to a DC current. There will therefore be no change in flux in the transformer and no output voltage.

♦♦♦ Mean mark 16%.
COMMENT: Attention! DC current will not work on a transformer.

Calculus, MET2 2022 VCAA 2

On a remote island, there are only two species of animals: foxes and rabbits. The foxes are the predators and the rabbits are their prey.

The populations of foxes and rabbits increase and decrease in a periodic pattern, with the period of both populations being the same, as shown in the graph below, for all , where time is measured in weeks.

One point of minimum fox population, (20, 700), and one point of maximum fox population, (100, 2500), are also shown on the graph.

The graph has been drawn to scale.
 

The population of rabbits can be modelled by the rule .

  1.   i. State the initial population of rabbits.   (1 mark)

  2.  ii. State the minimum and maximum population of rabbits.   (1 mark)

  3. iii. State the number of weeks between maximum populations of rabbits.   (1 mark)

The population of foxes can be modelled by the rule .

  1. Show that and .   (2 marks)

  2. Find the maximum combined population of foxes and rabbits. Give your answer correct to the nearest whole number.   (1 mark)

  3. What is the number of weeks between the periods when the combined population of foxes and rabbits is a maximum?   (1 mark)

The population of foxes is better modelled by the transformation of under given by
 

  1. Find the average population during the first 300 weeks for the combined population of foxes and rabbits, where the population of foxes is modelled by the transformation of under the transformation . Give your answer correct to the nearest whole number.   (4 marks)

Over a longer period of time, it is found that the increase and decrease in the population of rabbits gets smaller and smaller.

The population of rabbits over a longer period of time can be modelled by the rule

  1. Find the average rate of change between the first two times when the population of rabbits is at a maximum. Give your answer correct to one decimal place.   (2 marks)

  2. Find the time, where , in weeks, when the rate of change of the rabbit population is at its greatest positive value. Give your answer correct to the nearest whole number.   (2 marks)

  3. Over time, the rabbit population approaches a particular value.
  4. State this value.   (1 mark)

Show Answers Only

ai.   

aii.  Minimum population of rabbits

Maximum population of rabbits

aiii.   weeks

b.    See worked solution.

c.    (nearest whole number)

d.    Weeks between the periods is 160

e.    (nearest whole number)

f.    Average rate of change rabbits/week (1 d.p.)

g.    weeks (nearest whole number)  

h.   

Show Worked Solution

ai.  Initial population of rabbits

From graph when

Using formula when

 
  rabbits  


aii. From graph,

Minimum population of rabbits

Maximum population of rabbits

OR

Using formula

Minimum is when

Maximum is when

 
aiii. Number of weeks between maximum populations of rabbits   weeks

 
b.  Period of foxes = period of rabbits = 160:

which is the same period as the rabbit population.

Using the point

Amplitude when : 

 
 
 
 
 

 

  

c.    Using CAS find :

  
     
  

  
Maximum combined population (nearest whole number)


♦♦ Mean mark (c) 40%.
MARKER’S COMMENT: Many rounding errors with a common error being 5340. Many students incorrectly added the max value of rabbits to the max value of foxes, however, these points occurred at different times.

d.    Using CAS, check by changing domain to 0 to 320.

    
 

 
Therefore, the number of weeks between the periods is 160.
  

e.    Fox population:

   →  

  →  

  
Average combined population  [Using CAS]   
  

  

(nearest whole number)


♦ Mean mark (e) 40%.
MARKER’S COMMENT: Common incorrect answer 1600. Some incorrectly subtracted . Others used average rate of change instead of average value.

f.   Using CAS


 

       
 


 

    
 


 

Av rate of change between the points

  and 


 

Average rate of change rabbits/week (1 d.p.)


♦ Mean mark (f ) 45%.
MARKER’S COMMENT: Some students rounded too early.
was commonly seen.
Some found average rate of change between max and min populations.

g.   Using CAS

,

After testing greatest positive value occurs for  

 
   
   

 
weeks (nearest whole number)


♦♦♦ Mean mark (g) 25%.
MARKER’S COMMENT: Many students solved .
A common answer was 41.8, as .
Another common incorrect answer was 76 weeks.

h.   As ,

Calculus, MET2 2022 VCAA 1

The diagram below shows part of the graph of , where .
 

  1. State the equation of the axis of symmetry of the graph of .   (1 mark)

  2. State the derivative of with respect to .   (1 mark)

The tangent to at point has gradient .

  1. Find the equation of the tangent to at point .   (2 marks)

The diagram below shows part of the graph of , the tangent to at point and the line perpendicular to the tangent at point .
 

 

  1.  i. Find the equation of the line perpendicular to the tangent passing through point .   (1 mark)

  2. ii. The line perpendicular to the tangent at point also cuts at point , as shown in the diagram above.
  3.     Find the area enclosed by this line and the curve .   (2 marks)

  4. Another parabola is defined by the rule , where .
  5. A tangent to and the line perpendicular to the tangent at , where , are shown below.

  1. Find the value of , in terms of , such that the shaded area is a minimum.   (4 marks)

Show Answers Only

a.   

b.   

c.    

di.   

dii.   Area units²

e.   

Show Worked Solution

a.   Axis of symmetry: 
  

b.       
   

  
c.   At gradient

 
 

 
When 
 


 
Equation of tangent at :

 
 
 


d.i   Gradient of tangent

  gradient of normal  

Equation at

 
 
 


d.ii  Points of intersection of and normal are at  and .

So equate and  to find

 
 
 

   
or

Area  
   
   
  units²  

 

e.     

At   

Gradient of tangent

Gradient of normal

Equation of normal at

 
 
 
 

 
Points of intersection of normal and parabola (Using CAS)

solve

  or 

 
Calculate area using CAS


 
Using CAS Solve derivative of   with respect to to find

solve

 
   and 

Given


♦♦ Mean mark (e) 33%.
MARKER’S COMMENT: Common errors: Students found and failed to subtract or had incorrect terminals.

Probability, MET2 2022 VCAA 20 MC

A soccer player kicks a ball with an angle of elevation of °, where is a normally distributed random variable with a mean of 42° and a standard deviation of 8°.

The horizontal distance that the ball travels before landing is given by the function .

The probability that the ball travels more than 40 m horizontally before landing is closest to

  1. 0.969
  2. 0.937
  3. 0.226
  4. 0.149
  5. 0.027
Show Answers Only

Show Worked Solution

Using CAS

Solve:    for    °


 


 


♦♦ Mean mark 30%.
MARKER’S COMMENT: 24% incorrectly chose B and 25% incorrectly chose C.

Calculus, MET2 2023 VCAA 3

Consider the function .

  1. State the value of .   (1 mark)
  2. The derivative, , can be expressed in the form .
  3. Find the real number .   (1 mark)
  4.  i. Let be a real number. Find, in terms of , the equation of the tangent to at the point .   (1 mark)

    ii. Hence, or otherwise, find the equation of the tangent to that passes through the origin, correct to three decimal places.   (2 marks)

  1.  

Let .

  1. Find the coordinates of the point of inflection for , correct to two decimal places.   (1 mark)
  2. Find the largest interval of values for which is strictly decreasing.
  3. Give your answer correct to two decimal places.   (1 mark)
  4. Apply Newton's method, with an initial estimate of , to find an approximate -intercept of .
  5. Write the estimates and in the table below, correct to three decimal places.   (2 marks)
      


  6. For the function , explain why a solution to the equation should not be used as an initial estimate in Newton's method.   (1 mark)
  7. There is a positive real number for which the function has a local minimum on the -axis.
  8. Find this value of .   (2 marks)
Show Answers Only

a.   

b.   

ci. 

cii.

d.

e.

f. 

g.

h.

Show Worked Solution

a.   


  

b.    
   
   
 
     
 
   
ci. 


♦♦ Mean mark (c)(i) 50%.

cii. 

  

♦♦♦ Mean mark (c)(ii) 30%.
MARKER’S COMMENT: Some students did not substitute (0, 0) into the correct equation or did not find the value of and used .
d.    
 
 

e.   


♦♦ Mean mark (e) 40%.
MARKER’S COMMENT: Round brackets were often used which were incorrect as endpoints were included. Some responses showed the interval where the function was strictly increasing.

f.     

g.   

♦♦♦ Mean mark (g) 20%.

h.   

 


♦♦♦ Mean mark (h) 10%.
MARKER’S COMMENT: Many students showed that but failed to couple it with . Ensure exact values are given where indicated not approximations.

PHYSICS, M6 2021 VCE 5

Figure 5 shows a stationary electron (e) in a uniform magnetic field between two parallel plates. The plates are separated by a distance of 6.0 × 10 m, and they are connected to a 200 V power supply and a switch. Initially, the plates are uncharged. Assume that gravitational effects on the electron are negligible.
 

  1. Explain why the magnetic field does not exert a force on the electron. Justify your answer with an appropriate formula.  (2 marks)

The switch is now closed.

  1. Determine the magnitude and the direction of any electric force now acting on the electron. Show your working.  (3 marks)

  1. Ravi and Mia discuss what they think will happen regarding the size and the direction of the magnetic force on the electron after the switch is closed.

    Ravi says that there will be a magnetic force of constant magnitude, but it will be continually changing direction.

    Mia says that there will be a constantly increasing magnetic force, but it will always be acting in the same direction.

    Evaluate these two statements, giving clear reasons for your answer.  (4 marks)

Show Answers Only

a.    → Force of a magnetic field on an electron  .

→ Electron is stationary ()

→ The force on the electron = .

b.    

→ Since the bottom plate is positively charged, the direction of the electron will be down the page (towards the bottom plate).

c.    Both statements contain a correct assertion and incorrect assertion.

→ Ravi was incorrect to say that the magnitude of the magnetic force on the electron will be constant.

→ Mia, however, was correct in saying that the magnetic force on the electron will increase in magnitude.

→ This is due to the velocity of the electron increasing under the acceleration of the electric field. As the velocity of the electron increases, so will the magnitude of the force of the magnetic field on the electron (). 

→ Mia was incorrect to say that the direction of the magnetic force on the electron would not change.

→ Instead, Ravi was correct in saying that the magnetic force would be continuously changing direction.

→ This is because the force of the magnetic field on the electron will act perpendicular to the velocity of the electron and will cause the electron to move in circular motion. As the force is always perpendicular to the velocity of the electron, it will continually change to act towards the centre of the electrons circular motion.

Show Worked Solution

a.    → Force of a magnetic field on an electron  .

→ Electron is stationary ()

→ The force on the electron = .
 

♦♦ Mean mark (a) 35%. 
COMMENT: Many students confused the forces of electric (not applicable here) and magnetic fields.

b.    

→ Since the bottom plate is positively charged, the direction of the electron will be down the page (towards the bottom plate).
 

c.    Both statements contain a correct assertion and incorrect assertion.

→ Ravi was incorrect to say that the magnitude of the magnetic force on the electron will be constant.

→ Mia, however, was correct in saying that the magnetic force on the electron will increase in magnitude.

→ This is due to the velocity of the electron increasing under the acceleration of the electric field. As the velocity of the electron increases, so will the magnitude of the force of the magnetic field on the electron (). 

→ Mia was incorrect to say that the direction of the magnetic force on the electron would not change.

→ Instead, Ravi was correct in saying that the magnetic force would be continuously changing direction.

→ This is because the force of the magnetic field on the electron will act perpendicular to the velocity of the electron and will cause the electron to move in circular motion. As the force is always perpendicular to the velocity of the electron, it will continually change to act towards the centre of the electrons circular motion.

♦♦♦ Mean mark (c) 18%.
COMMENT: Students need well planned responses to properly explain the physics principles required for 4 marks.

PHYSICS, M6 2021 VCE 2

A schematic side view of one design of an audio loudspeaker is shown in Figure 2. It uses a current carrying coil that interacts with permanent magnets to create sound by moving a cone in and out.
 

Figure 3 shows a schematic view of the loudspeaker from the position of the eye shown in Figure 2. The direction of the current is clockwise, as shown.
 

   

  1. Draw four magnetic field lines on Figure 3, showing the direction of each field line using an arrow.  (1 mark)

  2. Which one of the following gives the direction of the force acting on the current carrying coil shown in Figure 3?  (1 mark)
A. left B. right
C. up the page D. down the page
E. into the page F. out of the page
  1. The current carrying coil has a radius of 5.0 cm and 20 turns of wire, and it carries a clockwise current of 2.0 A. Its magnetic field strength is 200 mT.
  2. Calculate the magnitude of the force, , acting on the current carrying coil. Show your working.  (2 marks)

Show Answers Only

a.   
         

b.    

c.    

Show Worked Solution

a.    Magnetic fields run from the north pole to the south pole.
 

   

 

b.    Apply the right-hand rule:

→ Thumb to the right and fingers down, the force on the current carrying coil must be into the page.


 

c.   

♦♦♦ Mean mark 24%.

Functions, MET2 2023 VCAA 2

The following diagram represents an observation wheel, with its centre at point . Passengers are seated in pods, which are carried around as the wheel turns. The wheel moves anticlockwise with constant speed and completes one full rotation every 30 minutes.When a pod is at the lowest point of the wheel (point ), it is 15 metres above the ground. The wheel has a radius of 60 metres.
 

Consider the function for some , which models the height above the ground of a pod originally situated at point , after time minutes.

  1. Show that and .   (2 marks)
  2. Find the average height of a pod on the wheel as it travels from point to point .
  3. Give your answer in metres, correct to two decimal places.   (2 marks)
  4. Find the average rate of change, in metres per minute, of the height of a pod on the wheel as it travels from point to point .   (1 mark)

After 15 minutes, the wheel stops moving and remains stationary for 5 minutes. After this, it continues moving at double its previous speed for another 7.5 minutes.

The height above the ground of a pod that was initially at point , after minutes, can be modelled by the piecewise function :
 

 
where and .

  1.   i.State the values of and .   (1 mark)
       
     ii. Find all possible values of .   (2 marks)

      

    iii. Sketch the graph of the piecewise function on the axes below, showing the coordinates of the endpoints.   (3 marks)


     
  1.  
Show Answers Only

a.   

b.   

c.   

d.i. 

d.ii.

d.iii. 

Show Worked Solution
a.   
   

  

 

 

b.    
   
   
   
   
 
♦♦ Mean mark (b) 45%.
MARKER’S COMMENT: was a common error.
Others incorrectly found average rate of change instead of average value.

c.   

 
 
 
 
♦ Mean mark (c) 50%.
MARKER’S COMMENT: Common incorrect answer was – 8.
di.   
 

  

 
 
 
   
 
♦♦ Mean mark (d)(i) 40%.
MARKER’S COMMENT: Many students could find but not with a common error.
dii.  
  

   

 
♦♦♦ Mean mark (d)(ii) 20%.
MARKER’S COMMENT: Many students set up the correct equations to solve but did not provide the general solution. Some incorrectly stated the variable as an element of R.

diii.   

 
♦♦♦ Mean mark (d)(iii) 40%.
MARKER’S COMMENT: Students are reminded to include all endpoints and be particular about the curvature of graph.

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)
Show Answers Only

a.   

b.   

c.i. 

c.ii.

 

c.iii.

d.  

Show Worked Solution

a.   
  

b.   


  

ci   

  

cii  


  

ciii 
 

  

d.   

  

 

PHYSICS, M5 2022 VCE 2

There are over 400 geostationary satellites above Earth in circular orbits. The period of orbit is one day (86 400 seconds). Each geostationary satellite remains stationary in relation to a fixed point on the equator. Figure 2 shows an example of a geostationary satellite that is in orbit relative to a fixed point, , on the equator.
 

  1. Explain why geostationary satellites must be vertically above the equator to remain stationary relative to Earth's surface.  (2 marks)
  1.  Using    and  , show that the altitude of a geostationary satellite must be equal to .  (4 marks)
  1. Calculate the speed of an orbiting geostationary satellite.  (3 marks)

Show Answers Only

a.   Reasons the satellite must orbit relative to a fixed point on the equator:

→ So that it is in the same rotational plane/axis of the Earth. 

→ The gravitational force on the satellite must be directly towards the centre of the Earth and it must orbit at a set altitude to the Earth where the speed of the satellite matches the rotational speed of the Earth.

→ As the speed and direction of the satellite matches that of the Earth, it will remain stationary relative to the motion of the Earth.

b.    See worked solutions

c.    3076 ms

Show Worked Solution

a.   Reasons the satellite must orbit relative to a fixed point on the equator:

→ So that it is in the same rotational plane/axis of the Earth. 

→ The gravitational force on the satellite must be directly towards the centre of the Earth and it must orbit at a set altitude to the Earth where the speed of the satellite matches the rotational speed of the Earth.

→ As the speed and direction of the satellite matches that of the Earth, it will remain stationary relative to the motion of the Earth.
 

♦♦♦ Mean mark (a) 19%.
b.    
 
   
   

 

 

c.    
   
   

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)

Show Answers Only

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)
Show Answers Only

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, MET2 2023 VCAA 5

Let and .

  1. Complete a possible sequence of transformations to map to .   (2 marks)
    •    Dilation of factor from the axis.

Two functions and are created, both with the same rule as but with distinct domains, such that is strictly increasing and is strictly decreasing.

  1. Give the domain and range for the inverse of .   (2 marks)

Shown below is the graph of , the inverse of and , and the line .
 

The intersection points between the graphs of and the inverses of and , are labelled and .

    1. Find the coordinates of and , correct to two decimal places.   (1 mark)
    1. Find the area of the region bound by the graphs of , the inverse of and the inverse of .
    2. Give your answer correct to two decimal places.   (2 marks)

Let , where .

  1. The turning point of always lies on the graph of the function , where is an integer.
  2. Find the value of .  (1 mark)

Let .

The rule for the inverse of is

  1. What is the smallest value of such that will intersect with the inverse of ?\
  2. Give your answer correct to two decimal places.   (1 mark)

It is possible for the graphs of and the inverse of to intersect twice. This occurs when .

  1. Find the area of the region bound by the graphs of and the inverse of , where .
  2. Give your answer correct to two decimal places.   (2 marks)

Show Answers Only

a.   

b.   

c.i.  , Q(4.09, 4.09)\)

c.ii.

d.   

e.

f.

Show Worked Solution

a.   

b.   

• 

• 
 

•
  
 

• 
  
  


♦ Mean mark 50%.
MARKER’S COMMENT: Note: maximal domain not requested so there were many correct answers. Many students seemed to be confused by notation. Often domain and range reversed.

c.i. 

c.ii.   
   
   

♦♦ Mean mark (c)(ii) 30%.
MARKER’S COMMENT: Some students set up the integral but did not provide an answer.
Others did not double the integral. Some incorrectly used .
Others had correct answer but incorrect integral.

d.   


♦♦♦ Mean mark (d) 10%.
MARKER’S COMMENT: Question not well done. a common error.

e.   

 

 

 


♦♦♦♦ Mean mark (e) 0%.
MARKER’S COMMENT: Question poorly done with many students not attempting.

f.   

 
   

♦♦♦ Mean mark (f) 15%.
MARKER’S COMMENT: Errors were made with terminals. Common error using , which was the value of the pt of intersection of and .

Calculus, MET1 2022 VCAA 8

Part of the graph of is shown below. The rule gives the area bounded by the graph of , the horizontal axis and the line .
 

  1. State the value of .   (1 mark)

  2. Evaluate .   (2 marks)

  3. Consider the average value of the function over the interval , where .
  4. Find the value of that results in the maximum average value.   (2 marks)

Show Answers Only

a.   

b.   

c.   

Show Worked Solution
a.   
   
   
b.    
 
   
 
   
   
   

♦♦♦ Mean mark (b) 25%.
MARKER’S COMMENT: Common error was giving derivative of as .
c.   Average value  
   
   
   

 
Average value has a maximum value of 1 when


♦♦♦ Mean mark (c) 25%.
MARKER’S COMMENT: Students often chose unnecessarily complicated methods or inconsistently applied nomenclature when attempting this question.

Calculus, MET1 2022 VCAA 7

A tilemaker wants to make square tiles of size 20 cm × 20 cm.

The front surface of the tiles is to be painted with two different colours that meet the following conditions:

  • Condition 1 - Each colour covers half the front surface of a tile.
  • Condition 2 - The tiles can be lined up in a single horizontal row so that the colours form a continuous pattern.

An example is shown below.
 

There are two types of tiles: Type A and Type B.

For Type A, the colours on the tiles are divided using the rule , where .

The corners of each tile have the coordinates (0,0), (20,0), (20,20) and (0,20), as shown below.
 

  1.  i. Find the area of the front surface of each tile.   (1 mark)

    ii. Find the value of so that a Type A tile meets Condition 1.   (1 mark)


Type B tiles, an example of which is shown below, are divided using the rule .
 

  1. Show that a Type B tile meets Condition 1.   (3 marks)

  2. Determine the endpoints of and on each tile. Hence, use these values to confirm that Type A and Type B tiles can be placed in any order to produce a continuous pattern in order to meet Condition 2.   (2 marks)

Show Answers Only

a.i.    cm²

a.ii.  

b.     cm²

c.     See worked solution

Show Worked Solution
a.i   Area  
  cm²  

  
a.ii 
 

b.   Area  
   
   
   
  cm²  

 
The area of the coloured section of the Type B tile is 200 cm² which is half the 400 cm² area of the tile.
 

c.   
 
   
   
   

 
→
 

   The endpoints for are and and for are also and .

So the tiles can be placed in any order to make the continuous pattern.


♦♦ Mean mark (c) 30%.
MARKER’S COMMENT: Students often only found and , however, and also needed to be found to verify the pattern match.

Graphs, MET1 2022 VCAA 6

The graph of , where , is shown below.
 

  1. On the axes above, draw the graph of , where is the reflection of in the horizontal axis.   (2 marks)
  2. Find all values of such that and .   (3 marks)
  3. Let , where has the same rule as with a different domain.
  4. The graph of is translated units in the positive horizontal direction and units in the positive vertical direction so that it is mapped onto the graph of , where .
    1. Find the value for .   (1 mark)

    2. Find the smallest positive value for .   (1 mark)

    3. Hence, or otherwise, state the domain, , of .   (1 mark)

Show Answers Only

a.    Graph

b.   

c.i.   

cii.   

ciii.  

Show Worked Solution

a.

b.          
 
 
 

 

c.i    
 
 

 

c.ii   
 
 
 

♦ Mean mark (c.ii) 50%.
MARKER’S COMMENT: Students confused vertical and horizontal translations. Common error

c.iii The domain for is

is

 
♦♦♦ Mean mark (c.iii) 10%.
MARKER’S COMMENT: Common error was translating in the wrong direction. A common incorrect answer was

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.

Statistics, MET1 2023 VCAA 6

Let be the random variable that represents the sample proportion of households in a given suburb that have solar panels installed.

From a sample of randomly selected households in a given suburb, an approximate 95% confidence interval for the proportion of households having solar panels installed was determined to be (0.04, 0.16).

  1. Find the value of that was used to obtain this approximate 95% confidence interval.   (1 mark)

Use to approximate the 95% confidence interval.

  1. Find the size of the sample from which this 95% confidence interval was obtained.   (2 marks)
  2. A larger sample of households is selected, with a sample size four times the original sample.
  3. The sample proportion of households having solar panels installed is found to be the same.
  4. By what factor will the increased sample size affect the width of the confidence interval?   (1 mark)
Show Answers Only

a.   

b.   

c.   

Show Worked Solution

a.   


♦ Mean mark (a) 52%.
b.   
 
 
 
 
 

♦♦ Mean mark (b) 38%.

c.   

→

→


♦♦♦ Mean mark (c) 23%.

Probability, MET1 2023 VCAA 8

Suppose that the queuing time, (in minutes), at a customer service desk has a probability density function given by
 

 
for some  .

  1. Show that  .   (1 mark)
  2. Find  .   (2 marks)
  3. What is the probability that a person has to queue for more than two minutes, given that they have already queued for one minute?   (3 marks)
Show Answers Only
a.   
 
 
 
 

b.   

c.   

Show Worked Solution
a.   
 
 
 
 

♦ Mean mark (a) 43%.
b.   
   
   
   
   

♦♦ Mean mark (b) 38%.
c.   
   
   
   
   

♦♦♦ Mean mark (c) 24%.
MARKER’S COMMENT: Simplifying fractions caused problems. Cancelling factors will assist with calculations.

Networks, GEN2 2023 VCAA 14

One of the landmarks in state requires a renovation project.

This project involves 12 activities, to . The directed network below shows these activities and their completion times, in days.
 

The table below shows the 12 activities that need to be completed for the renovation project.

It also shows the earliest start time (EST), the duration, and the immediate predecessors for the activities.

The immediate predecessor(s) for activity and the EST for activity are missing.

  1. Write down the immediate predecessor(s) for activity .  (1 mark)

  2. What is the earliest start time, in days, for activity ?  (1 mark)

  3. How many activities have a float time of zero?  (1 mark)

The managers of the project are able to reduce the time, in days, of six activities.

These reductions will result in an increase in the cost of completing the activity.

The maximum decrease in time of any activity is two days.

  1. If activities and have their completion time reduced by two days each, the overall completion time of the project will be reduced.
  2. What will be the maximum reduction time, in days?  (1 mark)
  3. The managers of the project have a maximum budget of $15 000 to reduce the time for several activities to produce the maximum reduction in the project's overall completion time.
  4. Complete the table below, showing the reductions in individual activity completion times that would achieve the earliest completion time within the $ 15 000 budget.  (1 mark)

Show Answers Only

a.   

b.   

c.   

d.   

e.    

Show Worked Solution

a.   


 

b.   


 

c.   


 

d.   

♦♦ Mean mark (c) 35%.
♦♦ Mean mark (d) 33%.

e.    

♦♦♦ Mean mark (e) 9%.

Networks, GEN2 2023 VCAA 13

The state has nine landmarks, and .

The edges on the graph represent the roads between the landmarks.

The numbers on each edge represent the length, in kilometres, along each road.
 

 

Three friends, Eden, Reynold and Shyla, meet at landmark .

  1. Eden would like to visit landmark .
  2. What is the minimum distance Eden could travel from to ?  (1 mark)

  3. Reynold would like to visit all the landmarks and return to .
  4. Write down a route that Reynold could follow to minimise the total distance travelled.  (1 mark)

  5. Shyla would like to travel along all the roads.
  6. To complete this journey in the minimum distance, she will travel along two roads twice.
  7. Shyla will leave from landmark but end at a different landmark.
  8. Complete the following by filling in the boxes provided.  
  9. The two roads that will be travelled along twice are the roads between:  (1 mark)

Show Answers Only

a.   


 

b.   


 

c.   

Show Worked Solution

a.   


 

b.   

♦ Mean mark (b) 44%.

c.   

♦♦♦ Mean mark (c) 12%.

Recursion and Finance, GEN2 2023 VCAA 7

Arthur takes out a new loan of $60 000 to pay for an overseas holiday.

Interest on this loan compounds weekly.

The balance of the loan, in dollars, after weeks, , can be determined using a recurrence relation of the form

  1. Show that the interest rate for this loan is 7.8% per annum.   (1 mark)

  2. Determine the value of in the recurrence relation if
    1. Arthur makes interest-only repayments   (1 mark)
    2. Arthur fully repays the loan in five years. Round your answer to the nearest cent.   (1 mark)
  3. Arthur decides that the value of will be 300 for the first year of repayments.  
  4. If Arthur fully repays the loan with exactly three more years of repayments, what new value of will apply for these three years? Round your answer to the nearest cent.   (1 mark)
  5. For what value of does the recurrence relation generate a geometric sequence?   (1 mark)

Show Answers Only

a.   

b.i. 

b.ii.

c.   

d.   

Show Worked Solution


a.   



♦♦ Mean mark (a) 32%.



 
b.i. 
 

b.ii.

 
 
 
 
 
 

 

 



♦ Mean mark (b)(i) 47%.
♦ Mean mark (b)(ii) 40%.



c.   

 
 
 
 
 
 

 

 

 
 
 
 
 
 

 

 



♦♦ Mean mark (c) 32%.



d.    



♦♦♦ Mean mark (d) 11%.



Matrices, GEN1 2022 VCAA 8 MC

Two types of computers - laptops and desktops - can be serviced by Henry , Irvine or Jean .

Matrix shows the time, in minutes, it takes each person to service a laptop and a desktop.

Matrix shows the number of laptops and desktops in four different departments: marketing , advertising , publishing and editing .

A calculation that determines the total time that it would take each of Henry, Irvine or Jean, working alone, to service all the laptops and desktops in all four departments is

Show Answers Only

Show Worked Solution

CHEMISTRY, M7 2020 VCE 16 MC

The following table provides information about three organic compounds, and .

Which one of the following is the best estimate for the boiling point of Compound ?

  1. 31 °C
  2. 101 °C
  3. 114 °C
  4. 156 °C
Show Answers Only

Show Worked Solution

→ Boiling points of molecular compounds are related to the strength of their intermolecular bonding.

→ All given molecules are polar molecules. Each intermolecular bonding will have contributions from both dispersion forces and dipole-dipole bonding.

→ Since the molar mass of each compound is the same, the contribution from dispersion forces will be similar for all.

→ Compounds and will both have hydrogen bonding, due to the presence of the  functional group.

→ Compound does not have intermolecular hydrogen bonding and therefore would have the lowest boiling point.

♦♦♦ Mean mark 20%.

Complex Numbers, EXT2 N2 2023 HSC 16c

The complex numbers and both have modulus 1, and  , where denotes the principal argument.

For real numbers and , consider the complex number .

On an -plane, clearly sketch the region that contains all points for which .

Show Answers Only

Show Worked Solution

 

♦♦♦ Mean mark 3%.
COMMENT: On the podium for the toughest questions in Ext2 in the last decade.

Vectors, EXT2 V1 2023 HSC 10 MC

Consider any three-dimensional vectors    and    that satisfy these three conditions

.

Which of the following statements about the vectors is true?

  1. Two of and could be unit vectors.
  2. The points and could lie on a sphere centred at .
  3. For any three-dimensional vector , vectors and can be found so that and satisfy these three conditions.
  4. and satisfying the conditions, and such that and are positive real numbers and  .
Show Answers Only

Show Worked Solution


 

♦♦♦ Mean mark 15%.

Complex Numbers, EXT2 N2 2023 HSC 16a

Let be the complex number  .

  1. Show that  .   (2 marks)

The vertices of a triangle can be labelled and in anticlockwise or clockwise direction, as shown.
 

Three complex numbers and are represented in the complex plane by points and respectively.

  1. Show that if triangle is anticlockwise and equilateral, then  .   (2 marks)

  2. It can be shown that if triangle is clockwise and equilateral, then . (Do NOT prove this.)
  3. Show that if is an equilateral triangle, then

  (2 marks)

Show Answers Only
Show Worked Solution

i.   

 
   
   
   

 
ii.   

 
 
♦♦♦ Mean mark (ii) 18%.

iii.   


 

 
 
 

 

♦♦♦ Mean mark (iii) 11%.