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PHYSICS, M7 2021 VCE 20 MC

One of Einstein's postulates for special relativity is that the laws of physics are the same in all inertial frames of reference.

Which one of the following best describes a property of an inertial frame of reference?

  1. It is travelling at a constant speed.
  2. It is travelling at a speed much slower than .
  3. Its movement is consistent with the expansion of the universe.
  4. No observer in the frame can detect any acceleration of the frame.
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Show Worked Solution

→ An inertial frame of reference is one where there is no acceleration acting on the bodies.

→ Therefore, no observer will be able to detect any acceleration.

♦ Mean mark 49%.

PHYSICS, M7 2021 VCE 16 MC

The diagram below shows a circuit that is used to study the photoelectric effect.
 

Which one of the following is essential to the measurement of the maximum kinetic energy of the emitted photoelectrons?

  1. the level of brightness of the light source
  2. the wavelengths that pass through the filter
  3. the reading on the voltmeter when the current is at a minimum value
  4. the reading on the ammeter when the voltage is at a maximum value
Show Answers Only

Show Worked Solution

→ The maximum kinetic energy of the photoelectrons is determined by the stopping voltage. The stopping voltage can be determined by reading the voltmeter when the current is at zero.

→ is incorrect because although knowing the wavelengths of the light passing through will help you determine the total energy of the photons, this is different to the maximum kinetic energy of the photoelectrons (where the work function needs to be taken into account).

♦♦ Mean mark 35%.

PHYSICS, M6 2021 VCE 5 MC

The diagram below shows a small DC electric motor, powered by a battery that is connected via a split-ring commutator. The rectangular coil has sides KJ and LM. The magnetic field between the poles of the magnet is uniform and constant.
 

The switch is now closed, and the coil is stationary and in the position shown in the diagram.

Which one of the following statements best describes the motion of the coil when the switch is closed?

  1. The coil will remain stationary.
  2. The coil will rotate in direction A, as shown in the diagram.
  3. The coil will rotate in direction B, as shown in the diagram.
  4. The coil will oscillate regularly between directions A and B, as shown in the diagram.
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Show Worked Solution

→ When the switch is closed, the current through the side will run from to and the direction magnetic field is to the right.

→ Using the Right hand rule, the force on side JK will be down and the coil will rotate in direction B (anti-clockwise).

♦ Mean mark 51%.

PHYSICS, M8 2022 VCE 17

A materials scientist is studying the diffraction of electrons through a thin metal foil. She uses electrons with an energy of 10.0 keV. The resulting diffraction pattern is shown in Figure 19.
 

  1. Calculate the de Broglie wavelength of the electrons in nanometres.  (4 marks)

  1. The materials scientist then increases the energy of the electrons by a small amount and hence their speed by a small amount.

    Explain what effect this would have on the de Broglie wavelength of the electrons. Justify your answer.  (3 marks)

Show Answers Only

a.    

 

b.    Effects when speed of the electron increases by a small amount:

→ Momentum will increase by a small amount

→ As , if the momentum of the electron increases, its corresponding de Broglie wavelength will decrease.

Show Worked Solution

a.    Convert electron volts to joules:

 
 
   
   

 

 
   
   
   

♦ Mean mark 42%.

b.    Effects when speed of the electron increases by a small amount:

→ Momentum will increase by a small amount

→ As , if the momentum of the electron increases, its corresponding de Broglie wavelength will decrease.

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)
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).

PHYSICS, M7 2022 VCE 11

Explain why muons formed in the outer atmosphere can reach the surface of Earth even though their half-lives indicate that they should decay well before reaching Earth's surface.    (2 marks)

Show Answers Only

Using the special relativity principal of time dilation:

→ The average lifespans of muons are 2.2 μs, but when they are observed from the Earth’s frame of reference this becomes significantly dilated.

→ Thus, they are able to travel from the outer atmosphere and reach the surface of the Earth before they decay.

→ Other answers could have also referred to length contraction from the perspective of the muons.

Show Worked Solution

Answer 1: Using the special relativity principal of time dilation:

→ The average lifespans of muons are 2.2 μs, but when they are observed from the Earth’s frame of reference this becomes significantly dilated.

→ Thus, they are able to travel from the outer atmosphere and reach the surface of the Earth before they decay.

→ Other answers could have also referred to length contraction from the perspective of the muons.
 

Answer 2: Using the special relativity principal of length contraction:

→ A muon’s frame of reference measures the distance to the Earth as shorter than that measured from the Earth’s frame of reference.

→ This shorter distance allows the muons to reach the Earth before they decay.

♦ Mean mark 41%.

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)
Show Answers Only

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, M8 2022 VCE 9

A star is transforming energy at a rate of 2.90 × 10 W.

Explain the type of transformation involved and what effect, if any, the transformation would have on the mass of the star. No calculations are required.  (2 marks)

Show Answers Only

→ The transformation involved is the transformation of mass to energy using Einstein’s mass-energy equivalence formula,

→ The process undertaken is called nuclear fusion and commonly occurs in the form of the proton-proton chain or CNO cycle.

→ This transformation would decrease the mass of the star as energy is produced and radiates away from the star.

Show Worked Solution

→ The transformation involved is the transformation of mass to energy using Einstein’s mass-energy equivalence formula,

→ The process undertaken is called nuclear fusion and commonly occurs in the form of the proton-proton chain or CNO cycle.

→ This transformation would decrease the mass of the star as energy is produced and radiates away from the star.

♦ Mean mark 47%.

PHYSICS, M6 2022 VCE 4

A square loop of wire connected to a resistor, , is placed close to a long wire carrying a constant current, , in the direction shown in Figure 4.

The square loop is moved three times in the following order:

  • Movement A – Starting at Position 1 in Figure 4, the square loop rotates one full rotation at a steady speed about the -axis. The rotation causes the resistor, , to first move out of the page.
  • Movement B – The square loop is then moved at a constant speed, parallel to the current carrying wire, from Position 1 to Position 2 in Figure 4.
  • Movement C – The square loop is moved at a constant speed, perpendicular to the current carrying wire, from Position 2 to Position 3 in Figure 4.
     

Complete the table below to show the effects of each of the three movements by:

  • sketching any EMF generated in the square loop during the motion on the axes provided (scales and values are not required)
  • stating whether any induced current in the square loop is 'alternating', 'clockwise', 'anticlockwise' or has 'no current'.
Show Answers Only

Show Worked Solution

The magnetic field strength around a current carrying conductor, . Hence, for the movement in C, the change in magnetic field strength and therefore magnetic flux is one of inverse proportionality as depicted in the graph below.

♦ Mean mark 43%.

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)

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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%.

PHYSICS, M7 2023 VCE 13

A group of physics students undertake a Young's double-slit experiment using the apparatus shown in Figure 15. They use a green laser that produces light with a wavelength of 510 nm. The light is incident on two narrow slits, S and S. The distance between the two slits is 100 m.
 

An interference pattern is observed on a screen with points P, P and P being the locations of adjacent bright bands, shown in Figure 15. Point  P is the central bright band.

  1. Calculate the path difference between SP and SP. Give your answer in metres. Show your working.  (2 marks)
  1. The green laser is replaced by a red laser.
  2. Describe the effect of this change on the spacing between adjacent bright bands.  (1 mark)
  1. Explain how Young's double-slit experiment provides evidence for the wave-like nature of light and not the particle-like nature of light.  (3 marks)
Show Answers Only

a.    

b.    The spacing between adjacent bright bands will increase.

c.    Evidence for the wave-like nature of light:

→ Young’s double slit experiment was used to show that light can produce an interference pattern as it diffracts when it passes through the slits.

→ As this is a wave phenomenon, the experiment provided valuable evidence to support the wave-like nature of light.

→ Using the particle-like nature of light as a model, two bright bands would have been expected behind the two slits as the particles would have passed through the slits in a straight line which was not observed.

Show Worked Solution

a.   

.

♦ Mean mark (a) 45%.

b.    The spacing will increase.

→ , where is the length between the screen and the slits and is the distance between adjacent bright bands.

→ Since , as the wavelength of light increases from green to red, so will the spacing between adjacent bright bands.
 

c.    Evidence for the wave-like nature of light:

→ Young’s double slit experiment was used to show that light can produce an interference pattern as it diffracts when it passes through the slits.

→ As this is a wave phenomenon, the experiment provided valuable evidence to support the wave-like nature of light.

→ Using the particle-like nature of light as a model, two bright bands would have been expected behind the two slits as the particles would have passed through the slits in a straight line which was not observed.

PHYSICS, M5 2023 VCE 9

Giorgos is practising his tennis serve using a tennis ball of mass 56 g.

  1. Giorgos practises throwing the ball vertically upwards from point A to point B, as shown in Figure 10 . His daughter Eka, a physics student, models the throw, assuming that the ball is at the level of Giorgos's shoulder, point A, both when it leaves his hand and also when he catches it again. Point A is 1.8 m from the ground. The ball reaches a maximum height, point B, 1.8 m above Giorgos's shoulder.

  1. Show that the ball is in the air for 1.2 s from the time it leaves Giorgos's hand, which is level with his shoulder, until he catches it again at the same height.  (2 marks)
  1. Giorgos swings his racquet from point D through point C, which is horizontally behind him at shoulder height, as shown in Figure 11, to point B. Eka models this swing as circular motion of the racquet head. The centre of the racquet head moves with constant speed in a circular arc of radius 1.8 m from point C to point B.

  1. The racquet passes point C at the same time that the ball is released at point A and then the racquet hits the ball at point B.
  2. Calculate the speed of the racquet at point C.  (2 marks)
  1. The ball leaves Giorgos's racquet with an initial speed of 24 m s in a horizontal direction, as shown in Figure 12. A tennis net is located 12 m in front of Giorgos and has a height of 0.90 m.
     

  1. How far above the net will the ball be when it passes above the net? Assume that there is no air resistance. Show your working.  (3 marks)

Show Answers Only

a.    

b.    

c.    

Show Worked Solution

a.    Time to for the ball to fall from max height back to shoulder level:

 
 
 
   

 


 

b.    The racket travels a quarter of the circumference of the circle in 0.606 seconds.


 

♦♦ Mean mark (b) 27%.

c.    Time for the ball to reach the net:

Vertical displacement from max height in 0.5 seconds:

.

.

♦ Mean mark (c) 47%.

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.   

Calculus, MET2 2023 VCAA 17 MC

A cylinder of height and radius is formed from a thin rectangular sheet of metal of length and , by cutting along the dashed lines shown below.

The volume of the cylinder, in terms of and , is given by

Show Answers Only

Show Worked Solution

 
 

 

 
 
 

♦♦ Mean mark 28%.
MARKER’S COMMENT: 26% incorrectly chose both C and D.

 

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%.

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, M6 2023 VCE 5

Figure 4a shows a single square loop of conducting wire placed just outside a constant uniform magnetic field, . The length of each side of the loop is 0.040 m. The magnetic field has a magnitude of 0.30 T and is directed out of the page.

Over a time period of 0.50 s, the loop is moved at a constant speed, , from completely outside the magnetic field, Figure 4a, to completely inside the magnetic field, Figure 4b.
 

  1. Calculate the average EMF produced in the loop as it moves from the position just outside the region of the field, Figure , to the position completely within the area of the magnetic field, Figure 4b.
  2. Show your working.  (2 marks)
  1. On the small square loop in Figure 5, show the direction of the induced current as the loop moves into the area of the magnetic field.  (1 mark)
     

Show Answers Only

a.    

b.    
       

Show Worked Solution
a.    
   
   
   

 

b.  
         

♦ Mean mark (b) 45%.

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 .

PHYSICS, M6 2023 VCE 4*

A transformer is used to provide a low-voltage supply for six outdoor garden globes. The circuit is shown in Figure 3. Assume there is no power loss in the connecting wires. 
 

The input of the transformer is connected to a power supply that provides an AC voltage of 240 V. The globes in the circuit are designed to operate with an AC voltage of 12 V. Each globe is designed to operate with a power of 20 W.

  1. Assuming that the transformer is ideal, calculate the ratio of primary turns to secondary turns of the transformer.  (1 mark)

The globes are turned on.

  1. Calculate the current in the primary coil of the transformer.  (2 marks)
  1. Explain why the input current to the primary coil of the transformer must be AC rather than constant DC for the globes to shine.  (2 marks)

Show Answers Only

a.    

b.    

c.    → A change in flux from the primary coil induces an EMF in the secondary coil.

→ Only an AC current will provide the change in flux required for the transformer to work. 

→ DC currents provide a constant flux and therefore will not work on a transformer.

Show Worked Solution

a.    
 

b.    The total power drawn on the secondary side

Total power on the primary side is .

♦ Mean mark (b) 43%.

c.    → A change in flux from the primary coil induces an EMF in the secondary coil.

→ Only an AC current will provide the change in flux required for the transformer to work. 

→ DC currents provide a constant flux and therefore will not work on a transformer.

♦ Mean mark (c) 42%.

PHYSICS, M5 2023 VCE 2

Phobos is a small moon in a circular orbit around Mars at an altitude of 6000 km above the surface of Mars.

The gravitational field strength of Mars at its surface is 3.72 N kg. The radius of Mars is 3390 km.

  1. Show that the gravitational field strength 6000 km above the surface of Mars is 0.48 N kg.   (2 marks)
  1. Calculate the orbital period of Phobos. Give your answer in seconds.  (3 marks)
  1. Phobos is very slowly getting closer to Mars as it orbits.
  2. Will the orbital period of Phobos become shorter, stay the same or become longer as it orbits closer to Mars? Explain your reasoning.  (2 marks)

Show Answers Only

a.    \(0.48\ \text{N kg}^{-1}\)

b.   

c.    The orbital period will become shorter.

Show Worked Solution

a.   

        
 
   
   

 
 

        
   
   

 

b.    
 
   
   
   

♦ Mean mark (b) 53%. 
COMMENT: Rounded calculations, such as using 6.41 × 10^23 were marked as wrong.

c.    

→

PHYSICS, M5 2023 VCE 3 MC

Space scientists want to place a satellite into a circular orbit where the gravitational field strength of Earth is half of its value at Earth's surface.

Which one of the following expressions best represents the altitude of this orbit above Earth's surface, where is the radius of Earth?

Show Answers Only

Show Worked Solution

→ For gravitational field strength,

→ Therefore if → , the radius will become .

→ Thus the altitude above the surface of the earth is .

♦♦ Mean mark 39%.

Vectors, SPEC2 2023 VCAA 16 MC

A student throws a ball for his dog to retrieve. The position vector of the ball, relative to an origin at ground level seconds after release, is given by  . Displacement components are measured in metres, where is a unit vector to the east, is a unit vector to the north and is a unit vector vertically up.

The total distance, in metres, travelled by the ball before it hits the ground is closest to

  1. 1.5
  2. 11.5
  3. 13.0
  4. 24.5
  5. 26.0
Show Answers Only

Show Worked Solution


 

Calculus, SPEC2 2023 VCAA 12 MC

The acceleration, ms, of a particle that starts from rest and moves in a straight line is described by  , where ms is its velocity after seconds.

The velocity of the particle after seconds is

Show Answers Only

Show Worked Solution

 
 
 

 

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

Functions, MET1 2022 VCAA 5b

Find the maximal domain of , where .   (3 marks)

Show Answers Only

∪

Show Worked Solution

exists where

From the graph and

∪


♦ Mean mark (b) 53%
MARKER’S COMMENT: Students often used incorrect notation. Drawing the graph of the parabola assisted with recognition of correct domain.

Probability, MET1 2022 VCAA 4

A card is drawn from a deck of red and blue cards. After verifying the colour, the card is replaced in the deck. This is performed four times.

Each card has a probability of of being red and a probability of of being blue.

The colour of any drawn card is independent of the colour of any other drawn card.

Let be a random variable describing the number of blue cards drawn from the deck, in any order.

  1. Complete the table below by giving the probability of each outcome.   (2 marks)
     

  1. Given that the first card drawn is blue, find the probability that exactly two of the next three cards drawn will be red.  (1 mark)

  2. The deck is changed so that the probability of a card being red is and the probability of a card being blue is .
  3. Given that the first card drawn is blue, find the probability that exactly two of the next three cards drawn will be red.   (2 marks)

Show Answers Only

a.   Complete table

b.   

c.   

Show Worked Solution

a.   Using binomial distribution where and 

b.   Trials are independent of first trial.

Binomial where


♦♦ Mean mark (b) 40%.

c.   Trials are independent.

Binomial where


♦ Mean mark 55%.
MARKER’S COMMENT: Many students wrongly treated this question as conditional probability.

Algebra, MET1 2022 VCAA 3

Consider the system of equations

Determine the value of for which the system of equations above has an infinite number of solutions.   (3 marks)

Show Answers Only

for infinite solutions

Show Worked Solution

Making the subject of each equation:

   
  (1)
     
 
  (2)

 
From equations (1) and (2)

and
 

An infinite number of solutions occur when

 
 
 
 

 

 

Let and be the -intercepts of equations (1) and (2)

then   and 
 

 
 
 
   

 
for infinite solutions and only value to satisfy both equations.


♦ Mean mark 50%.
MARKER’S COMMENT: Students are reminded to always use the correct variables.

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.

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.

Calculus, MET1 2022 VCAA 2a

Let

Find the rule for an antiderivative of .   (1 mark)

Show Answers Only

Show Worked Solution
 
   

Notes:

→ As the domain is   the natural log to the base is used

→ A constant is not required in the solution as the question only asks for AN antiderivative not a family of solutions.


♦ Mean mark 50%.

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).

Vectors, SPEC1 2023 VCAA 10

The position vector of a particle at time seconds is given by

, where .

  1. Write in the form , where .  (1 mark)
  1. Show that the Cartesian equation of the path of the particle is   (2 marks)
  1. The particle is at point when and at point when , where is a positive real constant.
  2. If the distance travelled along the curve from to is , find .   (1 mark)
  1. Find all values of for which the position vector of the particle, , is perpendicular to its velocity vector, .   (2 marks)
Show Answers Only

a.   

b.   

c.   

d.   

Show Worked Solution

a. 


 

b.   

 
c. 


 

d.   

Vectors, SPEC1 2023 VCAA 9

A plane contains the points and , where is a real constant. The plane has a -axis intercept of 2 at the point .

  1. Write down the coordinates of point .   (1 mark)
  1. Show that and are   and  , respectively.   (1 mark)
  1. Hence find the equation of the plane in Cartesian form.  (2 marks)
  1.  Find .   (1 mark)
  1.   and are adjacent sides of a parallelogram. Find the area of this parallelogram.   (1 mark)
Show Answers Only

a.   

b.   

c.   

d.   

e.   

Show Worked Solution

a.   

 
b.   

 
c.   

 
d.   

e.    
   
   
   

Calculus, SPEC1 2023 VCAA 7

The curve defined by the parametric equations

, where 

is rotated about the -axis to form an open hollow surface of revolution.

Find the surface area of the surface of revolution.

Give your answer in the form , where and .   (4 marks)

Show Answers Only

Show Worked Solution

 
   

 

 
   
   
   

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%.

Matrices, GEN2 2023 VCAA 11

The circus requires 180 workers to put on each show.

From one show to the next, workers can either continue working or they can leave the circus .

Once workers leave the circus, they do not return.

It is known that 95% of the workers continue working at the circus.

This situation can be modelled by the matrix recurrence relation

  1. Write down matrix , the transition matrix, for this recurrence relation.   (1 mark)

  1. Write down matrix for this recurrence relation to ensure that the circus always has 180 workers.   (1 mark)

Show Answers Only

a.   

b.   

Show Worked Solution

a.   


 

b.   

♦♦♦ Mean mark (a) 25%.
♦♦ Mean mark (b) 39%.

Matrices, GEN2 2023 VCAA 10

Within the circus, there are different types of employees: directors , managers , performers and sales staff Customers attend the circus.

Communication between the five groups depends on whether they are customers or employees, and on what type of employee they are.

Matrix below shows the communication links between the five groups.

In this matrix:

    • The ' 1 ' in row , column indicates that the directors can communicate directly with the managers.
    • The ' 0 ' in row , column indicates that the performers cannot communicate directly with the directors.
  1. A customer wants to make a complaint to a director.
  2. What is the shortest communication sequence that will successfully get this complaint to a director?  (1 mark)

  3. Matrix below shows the number of two-step communication links between each group. Sixteen elements in this matrix are missing.

  2.  i. Complete matrix above by filling in the missing elements.  (1 mark)
  4. ii. What information do elements and provide about the communication between the circus employees?  (1 mark)

Show Answers Only

a.   
 

b.i. 

b.ii.

Show Worked Solution

a.   
 

b.i. 

b.ii.

♦♦♦ Mean mark (b)(ii) 25%.

Matrices, GEN2 2023 VCAA 9

The circus is held at five different locations, and .

The table below shows the total revenue for the ticket sales, rounded to the nearest hundred dollars, for the last 20 performances held at each of the five locations.

The ticket sales information is presented in matrix below.

  1. Complete the matrix equation below that calculates the average ticket sales per performance at each of the five locations.  (1 mark)

The circus would like to increase its total revenue from the ticket sales from all five locations.

The circus will use the following matrix calculation to target the next 20 performances.

  1. Determine the value of if the circus would like to increase its revenue from ticket sales by 25%.  (1 mark)

The circus moves from one location to the next each month. It rotates through each of the five locations, before starting the cycle again.

The following matrix displays the movement between the five locations.

  1. The circus started in town .
  2. What is the order in which the circus will visit the five towns?  (1 mark)

Show Answers Only

a.   

b.   

c.   

Show Worked Solution

a.   

 
b.   

 
c.   

♦♦♦ Mean mark (a) 25%
♦ Mean mark (b) 47%.

Matrices, GEN2 2023 VCAA 8

A circus sells three different types of tickets: family , adult and child .

The cost of admission, in dollars, for each ticket type is presented in matrix below.

The element in row and column of matrix is .

  1. Which element shows the cost for one child ticket?  (1 mark)

  2. A family ticket will allow admission for two adults and two children.
  3. Complete the matrix equation below to show that purchasing a family ticket could give families a saving of $10.  (1 mark)

     

  1. On the opening night, the circus sold 204 family tickets, 162 adult tickets and 176 child tickets.
  2. The owners of the circus want a 3 × 1 product matrix that displays the revenue for each ticket type: family, adult and child.
  3. This product matrix can be achieved by completing the following matrix multiplication.

  1. Write down matrix .  (1 mark)

Show Answers Only

a.    

b.   

c.   

Show Worked Solution

a.    

♦ Mean mark (a) 48%.

 
b.   

♦ Mean mark (b) 44%.
c.   
♦♦ Mean mark 34%.