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HMS, BM EQ-Bank 397

Outline TWO ethical considerations when investigating heart rate responses to aerobic training using student participants.   (3 marks)

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Sample Answer – Any 2 of the following

  • Informed consent must be obtained from all participants (and parents/guardians if under 18) after explaining the purpose of the investigation, the heart rate monitoring procedures, and potential discomforts.
  • Confidentiality of personal health data must be maintained by using participant codes instead of names and securing all heart rate records in password-protected files.
  • Participants must have the right to withdraw from the investigation at any time without penalty or negative consequences.
Show Worked Solution

Sample Answer – Any 2 of the following

  • Informed consent must be obtained from all participants (and parents/guardians if under 18) after explaining the purpose of the investigation, the heart rate monitoring procedures, and potential discomforts.
  • Confidentiality of personal health data must be maintained by using participant codes instead of names and securing all heart rate records in password-protected files.
  • Participants must have the right to withdraw from the investigation at any time without penalty or negative consequences.

HMS, BM EQ-Bank 396 MC

When collecting respiratory rate data during aerobic training investigation, which of the following represents appropriate ethical practice?

  1. Recording participants without their knowledge
  2. Sharing identified personal data with the whole class
  3. Allowing participants to withdraw at any time
  4. Requiring participants to complete the investigation regardless of discomfort
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Show Worked Solution

Consider Option C: Allowing participants to withdraw at any time

  • Respecting participants’ right to withdraw at any time without penalty is a fundamental ethical principle in human research.

Other Options:

  • A is incorrect: Recording without knowledge violates informed consent principles.
  • B is incorrect: Sharing identified data breaches confidentiality.
  • D is incorrect: Requiring completion despite discomfort disrespects participant autonomy and wellbeing.

HMS, BM EQ-Bank 385

Outline TWO methods that could be used to measure heart rate responses during aerobic training.   (3 marks)

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Sample Answer – Any 2 of the following

  • Heart rate monitor with chest strap provides continuous, accurate recordings throughout the aerobic activity without requiring the participant to stop exercising.
  • Manual pulse palpation at the wrist (radial artery) or neck (carotid artery) for 15 seconds multiplied by 4 gives a heart rate reading, though requires momentary pause in activity.
  • Smartphone applications using the phone’s camera and flash to detect pulse through fingertip blood flow changes can provide heart rate measurements between exercise periods.
Show Worked Solution

Sample Answer – Any 2 of the following

  • Heart rate monitor with chest strap provides continuous, accurate recordings throughout the aerobic activity without requiring the participant to stop exercising.
  • Manual pulse palpation at the wrist (radial artery) or neck (carotid artery) for 15 seconds multiplied by 4 gives a heart rate reading, though requires momentary pause in activity.
  • Smartphone applications using the phone’s camera and flash to detect pulse through fingertip blood flow changes can provide heart rate measurements between exercise periods.

HMS, BM EQ-Bank 384 MC

Which of the following is the most practical method for a student to collect data on perceived exertion during aerobic training?

  1. Blood lactate analyser
  2. Electromyography (EMG)
  3. Rating of Perceived Exertion (RPE) scale
  4. Electrocardiogram (ECG)
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Show Worked Solution

Consider Option C: Rating of Perceived Exertion (RPE) scale

  • RPE scale is simple, inexpensive, and doesn’t require specialized equipment, making it practical for student research.

Other Options:

  • A is incorrect: Blood lactate analysers are expensive and invasive, requiring specialised training.
  • B is incorrect: EMG equipment is expensive and complex to operate and interpret.
  • D is incorrect: ECG equipment is expensive and requires technical expertise to operate and interpret.

HMS, BM EQ-Bank 360

Outline the immediate physiological responses to aerobic training.   (3 marks)

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Sample Answer 

  • Heart rate increases to deliver more oxygen to working muscles
  • Ventilation rate increases to take in more oxygen and remove carbon dioxide
  • Cardiac output increases as both heart rate and stroke volume rise to meet increased oxygen demand
Show Worked Solution

Sample Answer 

  • Heart rate increases to deliver more oxygen to working muscles
  • Ventilation rate increases to take in more oxygen and remove carbon dioxide
  • Cardiac output increases as both heart rate and stroke volume rise to meet increased oxygen demand

HMS, BM EQ-Bank 358 MC

A student's ventilation rate increases during a 5km run. This occurs to:

  1. Increase oxygen intake and carbon dioxide removal
  2. Reduce oxygen delivery to the muscles
  3. Increase carbon dioxide in the bloodstream
  4. Decrease cardiac output
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Show Worked Solution

Consider Option A:  Increase oxygen intake and carbon dioxide removal

  • Ventilation rate increases during aerobic exercise to increase oxygen intake and remove the additional carbon dioxide produced.

Other Options:

  • B is incorrect: Exercise increases (not reduces) oxygen delivery to muscles.
  • C is incorrect: Exercise aims to decrease (not increase) carbon dioxide in bloodstream.
  • D is incorrect: Ventilation rate increases are associated with increased (not decreased) cardiac output.

HMS, BM EQ-Bank 341

Outline the immediate response of cardiac output during aerobic training.   (3 marks)

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Sample Answer

  • Cardiac output increases immediately during aerobic training
  • This occurs due to an increase in both heart rate and stroke volume
  • The increase supports the delivery of oxygen and nutrients to working muscles
  • For example, during moderate jogging, cardiac output might increase from 5 L/min at rest to approximately 15 L/min
  • The magnitude of increase depends on the intensity of the training
Show Worked Solution

Sample Answer

  • Cardiac output increases immediately during aerobic training
  • This occurs due to an increase in both heart rate and stroke volume
  • The increase supports the delivery of oxygen and nutrients to working muscles
  • For example, during moderate jogging, cardiac output might increase from 5 L/min at rest to approximately 15 L/min
  • The magnitude of increase depends on the intensity of the training

HMS, BM EQ-Bank 327

Outline how ventilation rate responds to moderate aerobic exercise.   (3 marks)

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Sample Answer

  • Ventilation rate increases from resting rate of 12-15 breaths per minute to 30-40 breaths per minute during moderate aerobic exercise.
  • This immediate physiological response supplies additional oxygen to the working muscles.
  • Carbon dioxide removal is enhanced as a waste product from energy metabolism during exercise.
Show Worked Solution

Sample Answer

  • Ventilation rate increases from resting rate of 12-15 breaths per minute to 30-40 breaths per minute during moderate aerobic exercise.
  • This immediate physiological response supplies additional oxygen to the working muscles.
  • Carbon dioxide removal is enhanced as a waste product from energy metabolism during exercise.

HMS, BM EQ-Bank 320

Outline the immediate heart rate response when an individual begins aerobic exercise.   (3 marks)

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Sample Answer

  • When an individual begins aerobic exercise, there is an immediate increase in heart rate.
  • This occurs due to increased stimulation from the sympathetic nervous system.
  • The heart rate increases in proportion to exercise intensity to deliver more oxygen and nutrients to the working muscles.
Show Worked Solution

Sample Answer

  • When an individual begins aerobic exercise, there is an immediate increase in heart rate.
  • This occurs due to increased stimulation from the sympathetic nervous system.
  • The heart rate increases in proportion to exercise intensity to deliver more oxygen and nutrients to the working muscles.

HMS, BM EQ-Bank 311

Outline TWO immediate cardiac responses to a training session.   (3 marks)

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Sample Answer – Any 2 of the following

  • Heart rate increases immediately when training begins, causing the heart to beat faster to deliver more oxygen to working muscles.
  • Stroke volume increases during training, meaning more blood is pumped out of the heart with each beat.
  • Cardiac output (amount of blood pumped by the heart per minute) increases during training to supply more blood to muscles that are being used.
Show Worked Solution

Sample Answer – Any 2 of the following

  • Heart rate increases immediately when training begins, causing the heart to beat faster to deliver more oxygen to working muscles.
  • Stroke volume increases during training, meaning more blood is pumped out of the heart with each beat.
  • Cardiac output (amount of blood pumped by the heart per minute) increases during training to supply more blood to muscles that are being used.

HMS, BM EQ-Bank 262 MC

A swimmer aims to improve their anaerobic capacity for 100 m sprint events. Which training program based on the FITT principle is most appropriate?

  1. Frequency: daily; Intensity: 60% MHR; Time: 45 minutes; Type: continuous swimming
  2. Frequency: 3-4 times per week; Intensity: 85-95% MHR; Time: 20-30 minutes; Type: interval training
  3. Frequency: twice weekly; Intensity: 70% MHR; Time: 60 minutes; Type: fartlek training
  4. Frequency: 5 times per week; Intensity: 65-75% MHR; Time: 90 minutes; Type: long slow distance
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Show Worked Solution

Consider Option B: Frequency: 3-4 times per week; Intensity: 85-95% MHR; Time: 20-30 minutes; Type: interval training

  • Anaerobic capacity for sprint events requires high-intensity interval training at 85-95% MHR with adequate recovery between sessions (3-4 times per week).

Other Options:

  • A is incorrect: Daily training with moderate intensity and longer duration targets aerobic capacity, not anaerobic capacity.
  • C is incorrect: Twice weekly is insufficient frequency, and 70% MHR is too low intensity for anaerobic development.
  • D is incorrect: This program describes typical endurance training focusing on aerobic capacity, not anaerobic development for sprinting.

HMS, BM EQ-Bank 258 MC

A volleyball player completes an aerobic continuous training session by riding a stationary bike at a constant pace for 50 minutes at 65% of maximum heart rate.

What is the primary energy system being targeted by this type of training?

  1. ATP-PC system
  2. Lactic acid system
  3. Aerobic energy system
  4. All energy systems equally
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Show Worked Solution

Consider Option C: Aerobic energy system

  • Aerobic continuous training at moderate intensity (65% MHR) for an extended duration (50 minutes) primarily targets the aerobic energy system.

Other Options:

  • A is incorrect: ATP-PC system is anaerobic and only supplies energy for 10-15 seconds of high-intensity activity.
  • B is incorrect: Lactic acid system is anaerobic and provides energy for approximately 30 seconds to 2 minutes of high-intensity activity.
  • D is incorrect: While all energy systems operate simultaneously, this training specifically targets aerobic energy system development.

HMS, BM EQ-Bank 251

Outline the key differences between continuous aerobic training and High Intensity Interval Training (HIIT).   (3 marks)

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Sample Answer

  • Continuous aerobic training involves sustained moderate-intensity exercise (60-75% MHR) for extended periods (typically 20+ minutes) without rest periods, predominantly using the aerobic energy system.
  • HIIT involves alternating between high-intensity exercise periods (85-95% MHR) and recovery periods, with varied work-to-rest ratios depending on fitness goals.
  • Continuous training typically has longer duration (30-60 minutes) compared to HIIT sessions which are shorter in total duration (15-30 minutes) but more intense.
Show Worked Solution

Sample Answer

  • Continuous aerobic training involves sustained moderate-intensity exercise (60-75% MHR) for extended periods (typically 20+ minutes) without rest periods, predominantly using the aerobic energy system.
  • HIIT involves alternating between high-intensity exercise periods (85-95% MHR) and recovery periods, with varied work-to-rest ratios depending on fitness goals.
  • Continuous training typically has longer duration (30-60 minutes) compared to HIIT sessions which are shorter in total duration (15-30 minutes) but more intense.

HMS, BM EQ-Bank 153 MC

Which energy system produces ATP at the fastest rate?

  1. Glycolytic
  2. Aerobic
  3. ATP-PCr
  4. All systems produce ATP at the same rate
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Show Worked Solution

Consider Option C: ATP-PCr

  • ATP-PCr has fastest production rate but limited stores

Other Options:

  • A is incorrect: Slower than ATP-PCr but faster than aerobic
  • B is incorrect: Slowest but most efficient system
  • D is incorrect: Systems have different production rates

HMS, BM EQ-Bank 138

Outline how inefficient jumping technique can affect the skeletal system and require first aid intervention.   (3 marks)

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Sample Answer

  • Incorrect landing can cause immediate joint compression
  • Impact forces travel incorrectly through bones causing stress
  • First aid required includes RICER and joint stabilisation
Show Worked Solution

Sample Answer

  • Incorrect landing can cause immediate joint compression
  • Impact forces travel incorrectly through bones causing stress
  • First aid required includes RICER and joint stabilisation

HMS, BM EQ-Bank 128 MC

During a cross-country run, an athlete experiences severe abdominal cramping. Which first aid response would be most appropriate?

  1. Continue running at race pace
  2. Increase fluid intake rapidly
  3. Start walking immediately
  4. Stop activity and lie in a comfortable position
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Show Worked Solution

Consider Option D: Stop activity and lie in a comfortable position

  • Stopping activity and finding a comfortable position allows assessment of digestive system stress and prevents further complications.

Other Options:

  • A is incorrect: Would increase digestive system stress
  • B is incorrect: Rapid fluid intake may worsen symptoms
  • C is incorrect: Immediate walking may exacerbate cramping

HMS, BM EQ-Bank 122

Outline how the muscular and skeletal systems work together during movement and identify when first aid intervention is required.   (3 marks)

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Sample Answer

  • Muscles and bones work as a lever system through insertion points and origins to create movement
  • When movement occurs outside normal ranges, strain on attachment points can require first aid intervention
  • Signs requiring intervention include: sudden pain during movement, inability to contract muscle, or loss of joint stability
Show Worked Solution

Sample Answer

  • Muscles and bones work as a lever system through insertion points and origins to create movement
  • When movement occurs outside normal ranges, strain on attachment points can require first aid intervention
  • Signs requiring intervention include: sudden pain during movement, inability to contract muscle, or loss of joint stability

HMS, BM EQ-Bank 109

Outline how the skeletal and muscular systems work together during a squat movement.   (3 marks)

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Sample Answer

Descent:

  • Hip and knee joints bend while thigh muscles lengthen to control movement
  • Skeleton provides stable framework as muscles control lowering

Bottom Position:

  • Skeleton maintains alignment while muscles hold stable position

Rising:

  • Leg muscles contract to push while skeleton provides leverage for upward movement
Show Worked Solution

Sample Answer

Descent:

  • Hip and knee joints bend while thigh muscles lengthen to control movement
  • Skeleton provides stable framework as muscles control lowering

Bottom Position:

  • Skeleton maintains alignment while muscles hold stable position

Rising:

  • Leg muscles contract to push while skeleton provides leverage for upward movement

HMS, BM EQ-Bank 103 MC

During a netball game, a player performs a layup shot. Which body systems are working together to execute this movement?

  1. Skeletal and respiratory only
  2. Muscular and circulatory only
  3. Skeletal, muscular and nervous
  4. Respiratory and circulatory only
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Show Worked Solution

Consider Option C:  Skeletal, muscular and nervous

  • skeletal system provides the framework for movement
  • muscles contract to produce the force
  • nervous system coordinates the timing and precision of the layup shot

Other Options:

  • A is incorrect: Misses muscular control needed
  • B is incorrect: Omits skeletal framework required
  • D is incorrect: Misses skeletal and muscular components

HMS, BM EQ-Bank 26 MC

The muscle group indicated in the image below is primarily responsible for:

  1. Hip flexion
  2. Knee extension
  3. Knee flexion
  4. Hip extension
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Show Worked Solution

Consider Option B:

  • The quadriceps (specifically rectus femoris, vastus lateralis, vastus medialis, and vastus intermedius) are the primary knee extensors

Other Options:

  • A:  Incorrect because hip flexion is primarily performed by the iliopsoas
  • C:  Incorrect because knee flexion is performed by the hamstring muscle group
  • D:  Incorrect because hip extension is primarily performed by the gluteus maximus and hamstrings

Functions, MET1 2024 VCAA 5

The function    models the population of a town after years.

  1. Use the model to predict the population of the town after four years.   (1 mark)

  2. A new function, , models a population where  but decreases at half the rate of at any point in time.
  3. State a sequence of two transformations that maps to this new model .   (2 marks)

  4. In the town, 100 people were randomly selected and surveyed, with 60 people indicating that they were unhappy with the roads.
    1. Determine an approximate 95% confidence interval for the proportion of people in the town who are unhappy with the roads.
    2. Use    for this confidence interval.   (2 marks)
    3. A new sample of people results in the same sample proportion.
    4. Find the smallest value of to achieve a standard deviation of    for the sample proportion.   (1 mark)

Show Answers Only

a.   

b.   

ci.   

cii. 

Show Worked Solution

a. 

b.    
 
 
 

♦♦♦ Mean mark (a) 17%.


 


 

ci.   

 
 

  

cii.  
 
 
 
 
   
♦♦ Mean mark (c.ii.) 33%.

BIOLOGY, M8 2024 HSC 31

A study monitored the changes in the body temperature of a kookaburra (an Australian bird) and a human over a 24-hour period. The results of the study are shown in the graph.
 

  1. At what time was the kookaburra's body temperature the lowest?   (1 mark)

  1. Some endothermic organisms can display torpor (a significant decrease in physiological activity).

    With reference to the graph, explain whether the human or the kookaburra was displaying torpor and if so, state the time this occurred.   (3 marks)

  1. Outline an adaptation that may lead to an increase in the kookaburra's body temperature during the inactive period.   (2 marks)

Show Answers Only

a.   4 am

b.   Signs of torpor:

→ The human maintained a steady body temperature throughout the observed period, showing no signs of torpor or reduced physiological activity

→ In contrast, the kookaburra exhibited classic torpor behaviour.

→ Its body temperature dropped significantly between 5 pm and 4 am, demonstrating the characteristic reduction in physiological functions during this period.
 

c.   Kookaburra adaptation:

→ Kookaburras have an effective insulation mechanism where they puff out their feathers, creating space between them.

→ This fluffing action traps a layer of warm air between the feathers and the bird’s body, forming an insulating barrier

→ The trapped air pocket acts like natural insulation, minimising heat loss and helping the kookaburra maintain its body temperature efficiently.

Show Worked Solution

a.   4 am
 

b.   Signs of torpor:

→ The human maintained a steady body temperature throughout the observed period, showing no signs of torpor or reduced physiological activity

→ In contrast, the kookaburra exhibited classic torpor behaviour.

→ Its body temperature dropped significantly between 5 pm and 4 am, demonstrating the characteristic reduction in physiological functions during this period.

Mean mark (b) 56%.

c.   Kookaburra adaptation:

→ Kookaburras have an effective insulation mechanism where they puff out their feathers, creating space between them.

→ This fluffing action traps a layer of warm air between the feathers and the bird’s body, forming an insulating barrier

→ The trapped air pocket acts like natural insulation, minimising heat loss and helping the kookaburra maintain its body temperature efficiently.

♦ Mean mark (c) 47%.

Data Analysis, GEN2 2024 VCAA 1

Table 1 lists the Olympic year, , and the gold medal-winning height for the men's high jump, , in metres, for each Olympic Games held from 1928 to 2020. No Olympic Games were held in 1940 or 1944, and the 2020 Olympic Games were held in 2021.

Table 1

  1. For the data in Table 1, determine:
  2.  i. the maximum in metres   (1 mark)

  3. ii. the percentage of values greater than 2.25 m.   (1 mark)

  4. The mean of these values is 2.23 m, and the standard deviation is 0.15 m.
  5. Calculate the standardised -score for the 2000 of 2.35 m.   (1 mark)

  6. Construct a boxplot for the data in Table 1 on the grid below.   (2 marks)

     

  1. A least squares line can also be used to model the association between and .
  2. Using the data from Table 1, determine the equation of the least squares line for this data set.
  3. Use the template below to write your answer.
  4. Round the values of the intercept and slope to three significant figures.   (2 marks)

     

  1. The coefficient of determination is 0.857
  2. Interpret the coefficient of determination in terms of and .   (1 mark)

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a.i.   

a.ii. 

b.   

c.     

d.   
    

e.   

Show Worked Solution

a.i.   

a.ii. 

b.    
   
   

  
c.   


  

d.  


  
 

Mean mark (d) 52%.
Mean mark (e) 52%.

e.   

Proof, EXT2 P1 2024 HSC 3 MC

Consider the statement:

'If a polygon is a square, then it is a rectangle.'

Which of the following is the converse of the statement above?

  1. If a polygon is a rectangle, then it is a square.
  2. If a polygon is a rectangle, then it is not a square.
  3. If a polygon is not a rectangle, then it is not a square.
  4. If a polygon is not a square, then it is not a rectangle.
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Show Worked Solution

Proof, EXT2 P1 2024 HSC 2 MC

Consider the following statement written in the formal language of proof

.

Which of the following best represents this statement?

  1. There exists a in the second quadrant such that for all in the third quadrant  .
  2. There exists a in the third quadrant such that for all in the second quadrant  .
  3. For all in the second quadrant there exists a in the third quadrant such that  .
  4. For all in the third quadrant there exists a in the second quadrant such that  .
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Show Worked Solution

Trigonometry, 2ADV T1 2024 HSC 20

A vertical tower is 40 metres high. The point is due east of the base of the tower . The angle of elevation to the top of the tower from is 35°. A second point is on a different bearing from the tower as shown. The angle of elevation to the top of the tower from is 30°. The points and are 100 metres apart.
 

  1. Show that distance is 57.13 metres, correct to 2 decimal places.  (1 mark)
  1. Find the bearing of from to the nearest degree.  (3 marks)
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Show Worked Solution

a.   

 
 
   
   

 
b.   

 
 
   

  

 
  …  
 

Networks, STD2 N2 2024 HSC 16

A network of towns and the distances between them in kilometres is shown.
 

  1. What is the shortest path from to ?   (2 marks)

  2. A truck driver needs to travel from to but knows that the road from to is closed.
  3. What is the length of the shortest path the truck driver can take from to after the road closure?  (2 marks)

Show Answers Only

a.   

b. 

Show Worked Solution

a.   


 

b.   

BIOLOGY, M1 EQ-Bank 4 MC

Which of the following structures is present in both prokaryotic and eukaryotic cells?

  1. Golgi apparatus
  2. Mitochondria
  3. Ribosomes
  4. Endoplasmic reticulum
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Show Worked Solution

→ Both prokaryotic and eukaryotic cells contain ribosomes, which are responsible for protein synthesis.

→ Prokaryotes lack membrane-bound organelles like mitochondria or the Golgi apparatus.

BIOLOGY, M6 2019 VCE 27 MC

Farmers and supermarkets agree that green beans are bought more frequently than yellow beans. A supermarket has asked a farmer to produce only green beans.

One way this could be achieved is by

  1. condensation polymerisation.
  2. DNA hybridisation.
  3. selective breeding.
  4. adaptive radiation.
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Show Worked Solution

→ Selective breeding involves choosing parents with desirable traits and breeding them to produce offspring with those traits.

→ In this case, the farmer would selectively breed green bean plants to produce more green beans, as that is the preferred variety by consumers and supermarkets.

→ Selective breeding does not involve genetic engineering techniques like DNA hybridization (option B) and is a common agricultural practice used to enhance desirable traits in crops, unlike options A and D which are not relevant to this scenario.

BIOLOGY, M5 2019 VCE 5 MC

Which one of the following statements about proteins is correct?

  1. The activity of a protein may be affected by the temperature and pH of its environment.
  2. The primary structure of a protein refers to its three-dimensional protein shape.
  3. Proteins are not involved in the human immune response.
  4. A protein with a quaternary structure will be an enzyme.
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Show Worked Solution

Consider option A:

→ Changes in temperature or pH can disrupt the interactions that stabilise a protein’s higher-order structure.

→ This causes it to lose its three-dimensional shape and become denatured which usually results in a non-functional protein (Option A is Correct)

BIOLOGY, M7 2021 VCE 36 MC

A study assessed the effectiveness and safety of a drug called doxycycline. One hundred and fifty adults hospitalised with malaria were involved. These adults were randomly placed into two groups of equal size. One group received doxycycline in addition to standard care. The other group received standard care only.

The group receiving standard care only was the

  1. control group.
  2. variable group.
  3. unsupported group.
  4. experimental group.
Show Answers Only

Show Worked Solution

→ A control group does not receive the experimental treatment or any intervention being tested.

v1 Algebra, STD2 A2 2022 HSC 16

Nicole is 38 years old, and likes to keep fit by doing cross-fit classes.

  1. Use this formula to find her maximum heart rate (bpm).
      
  2.      Maximum heart rate = 220 – age in years
      
  3. Nicole's maximum heart rate is ........................... bpm.   (1 mark)
  4. Nicole will get the most benefit from this exercise if her heart rate is between 50% and 85% of her maximum heart rate.
  5. Between what two heart rates should Nicole be aiming for to get the most benefit from her exercise?  (2 marks)

Show Answers Only

a.   

b.   

Show Worked Solution
a.    
   

 

b.   

v1 Algebra, STD2 A4 2022 HSC 22

The formula    is used to calculate the cost of producing desktop computers, where is the cost in dollars, is the number of desktop computers produced and is the fixed cost in dollars.

  1. Find the cost when 2458 desktop computers are produced and the fixed cost is .  (1 mark)

  2. Some desktop computers have extra features added. The formula to calculate the production cost for these desktop computers is
  3.    
  4. where is the additional cost in dollars per desktop computer produced.
  5. Find the number of desktop computers produced if the additional cost is $35 per desktop computer and the total production cost is .  (2 marks)

Show Answers Only
Show Worked Solution

a.   

 
   

 

b.   

 

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.

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.

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.

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.   

  

 

Data Analysis, GEN2 2023 VCAA 2a

The following data shows the sizes of a sample of 20 oysters rated as small, medium or large.

  1. Use the data above to complete the following frequency table.   (1 mark)

  1. Use the percentages in the table to construct a percentage segmented bar chart below. A key has been provided.   (1 mark)

     

Show Answers Only

i.    

ii.    
     

Show Worked Solution

i.    

 
ii.    
         

Data Analysis, GEN1 2023 VCAA 1-2 MC

The dot plot below shows the times, in seconds, of 40 runners in the qualifying heats of their 800 m club championship.
 

Question 1

The median time, in seconds, of these runners is

  1. 135.5
  2. 136
  3. 136.5
  4. 137
  5. 137

 
Question 2

The shape of this distribution is best described as

  1. positively skewed with one or more possible outliers.
  2. positively skewed with no outliers.
  3. approximately symmetric with one or more possible outliers.
  4. approximately symmetric with no outliers.
  5. negatively skewed with one or more possible outliers.
Show Answers Only

Show Worked Solution


 

ENGINEERING, TE 2023 HSC 3 MC

Why is pure copper preferred over a copper alloy in telecommunications applications?

  1. It has higher stiffness.
  2. It has better conductivity.
  3. It can be precipitation hardened.
  4. It has a better strength to weight ratio.
Show Answers Only

Show Worked Solution

→ Pure copper offers lower stiffness than copper alloys (eliminate A), is work hardened (eliminate C), and generally has lower strength to weight ratios (eliminate D).

→ In telecommunications, where high conductivity is crucial for transmitting electrical signals, pure copper is the preferred choice to ensure efficient signal transmission.

ENGINEERING, PPT 2023 HSC 24a

Roller coaster support structures can be made from either timber or steel.

Compare the properties of the two materials in roller coaster support structures.  (2 marks)

Show Answers Only

→ Unlike steel, timber has the ability to flex and bend, which can absorb some of the forces exerted by the roller coaster.

→ However, timber has less mechanical strength than steel and is more susceptible to rot and insect damage over time.
 

Answers could include:

→ Steel frames are much more easily fabricated and assembled than timber frames, which can save time and costs in the construction process.

→ Steel is much more resistant to fire than timber, which makes it a safer material to use in roller coasters.

→ Steel has greater mechanical strength and is a more durable material than timber. It is able to withstand the higher stresses and forces exerted on a roller coaster.

Show Worked Solution

→ Unlike steel, timber has the ability to flex and bend, which can absorb some of the forces exerted by the roller coaster.

→ However, timber has less mechanical strength than steel and is more susceptible to rot and insect damage over time.
 

Answers could include:

→ Steel frames are much more easily fabricated and assembled than timber frames, which can save time and costs in the construction process.

→ Steel is much more resistant to fire than timber, which makes it a safer material to use in roller coasters.

→ Steel has greater mechanical strength and is a more durable material than timber. It is able to withstand the higher stresses and forces exerted on a roller coaster.