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HMS, BM EQ-Bank 430

A student investigated the relationship between perceived exertion and heart rate during a progressive aerobic exercise test. Results showed a strong correlation for most participants, but some individuals consistently reported lower perceived exertion despite similar heart rates to other participants.

Propose TWO further research questions based on these findings and explain how each question could enhance understanding of physiological responses to aerobic exercise.   (5 marks)

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Sample Answer – Any 2 of the following or similar

  • “How does previous exercise experience affect the relationship between perceived exertion and heart rate during aerobic exercise?” directly addresses why some participants might perceive exercise differently despite similar physiological responses.
    • This question would enhance understanding of physiological responses by exploring how familiarity with exercise sensations might influence how individuals interpret and report their body’s responses during aerobic activity.
  • “Would the same disconnect between perceived exertion and heart rate occur during different types of aerobic activities, such as cycling versus running?” examines whether the physiological response patterns are activity-specific.
    • This question would improve understanding of how different exercise modes might produce varying relationships between objective measures (heart rate) and subjective feelings (perceived exertion), helping to explain individual differences in physiological responses.
Show Worked Solution

Sample Answer – Any 2 of the following or similar

  • “How does previous exercise experience affect the relationship between perceived exertion and heart rate during aerobic exercise?” directly addresses why some participants might perceive exercise differently despite similar physiological responses.
    • This question would enhance understanding of physiological responses by exploring how familiarity with exercise sensations might influence how individuals interpret and report their body’s responses during aerobic activity.
  • “Would the same disconnect between perceived exertion and heart rate occur during different types of aerobic activities, such as cycling versus running?” examines whether the physiological response patterns are activity-specific.
    • This question would improve understanding of how different exercise modes might produce varying relationships between objective measures (heart rate) and subjective feelings (perceived exertion), helping to explain individual differences in physiological responses.

PHYSICS, M2 EQ-Bank 4 MC

A 2 kg block is placed on a smooth inclined plane angled at 30\(^{\circ}\) to the horizontal. Assume friction is negligible.

What is the most likely motion of the block?

  1. The block will move down the incline with a constant acceleration.
  2. The block will move down the incline with increasing acceleration.
  3. The block will remain at rest.
  4. The block will move down the incline at a constant speed.
Show Answers Only

\(A\)

Show Worked Solution
  • The block will experience a constant force down the slope which can be calculated using \(F = mg\ \sin \theta\).
  • As the block is subject to a constant force, it will also experience a constant acceleration down the slope (Newton’s 2nd Law).

\(\Rightarrow A\)

HMS, HIC EQ-Bank 002

Explain why people give different meanings to health. In your answer, provide two specific examples of different meanings between groups.   (4 marks)

Show Answers Only
  • People give different meanings to health based on their personal experiences, cultural backgrounds, and socioeconomic circumstances.
  • For example, someone with a chronic illness may define health as being able to manage their condition and maintain independence, while an athlete might define health in terms of optimal physical performance.
  • Cultural values also influence health meanings, with some cultures emphasising spiritual and community aspects of health.
  • Additionally, a person’s age and life stage can affect how they define health, with younger people often focusing on physical capabilities while older individuals may prioritise mental wellbeing and social connections.
Show Worked Solution
  • People give different meanings to health based on their personal experiences, cultural backgrounds, and socioeconomic circumstances.
  • For example, someone with a chronic illness may define health as being able to manage their condition and maintain independence, while an athlete might define health in terms of optimal physical performance.
  • Cultural values also influence health meanings, with some cultures emphasising spiritual and community aspects of health.
  • Additionally, a person’s age and life stage can affect how they define health, with younger people often focusing on physical capabilities while older individuals may prioritise mental wellbeing and social connections.

HMS, HIC EQ-Bank 001

Outline the World Health Organisation's (WHO) definition of health.   (2 marks)

Show Answers Only
  • The World Health Organisation defines health as “a state of complete physical, mental and social well-being and not merely the absence of disease or infirmity”.
  • This definition recognises that health encompasses multiple dimensions beyond just the physical aspect and the absence of illness.
Show Worked Solution
  • The World Health Organisation defines health as “a state of complete physical, mental and social well-being and not merely the absence of disease or infirmity”.
  • This definition recognises that health encompasses multiple dimensions beyond just the physical aspect and the absence of illness.

HMS, HIC EQ-Bank 001 MC

Anne states that health is "the complete absence of disease".

James says health is "feeling good about myself and maintaining a positive outlook".

Samira suggests health is "having meaningful connections with others".

Which dimensions of health are Anne, James, and Samira describing in order?

  1. Physical, emotional, social
  2. Physical, mental, social
  3. Physical, spiritual, social
  4. Physical, emotional, cultural
Show Answers Only

\(B\)

Show Worked Solution
  • B is correct as Anne is describing physical health (absence of disease), James is describing mental health (feeling good, positive outlook), and Samira is describing social health (connections with others).

Other Options:

  • A is incorrect: James is describing mental health rather than just emotional health.
  • C is incorrect: James’s description aligns with mental health, not spiritual health which relates to purpose and meaning.
  • D is incorrect: James is describing mental health, not emotional health, and Samira is describing social health, not cultural health.

Vectors, EXT1 V1 EQ-Bank 3

Given  \(\overrightarrow{OP}=2\underset{\sim}{i}-3\underset{\sim}{j}, \ \overrightarrow{P Q}=-\underset{\sim}{i}+2 \underset{\sim}{j}\), find the expression for \(\overrightarrow{O Q}.\)    (2 marks)

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\(\overrightarrow{O Q}=\underset{\sim}{i}-\underset{\sim}{j}\)

Show Worked Solution

\(\overrightarrow{PQ}\) \(=\overrightarrow{OQ}-\overrightarrow{OP}\)  
\(\displaystyle\binom{-1}{2}\) \(=\displaystyle\binom{x}{y}-\displaystyle\binom{2}{-3}\)  
\(\displaystyle\binom{x}{y}\) \(=\displaystyle\binom{-1}{2}+\displaystyle\binom{2}{-3}=\binom{1}{-1}\)  

 
\(\therefore \overrightarrow{O Q}=\underset{\sim}{i}-\underset{\sim}{j}\)

Vectors, EXT1 V1 EQ-Bank 5 MC

In the diagram, \(OSTU\) is a parallelogram. where  \(\underset{\sim}{s}=\overrightarrow{O S}\)  and  \(\underset{\sim}{u}=\overrightarrow{O U}\).
 

Given \(M\) is the midpoint of \(\overrightarrow{OS}\), determine which expression represents vector \(\overrightarrow{MT}\).

  1. \(\underset{\sim}{s}+\dfrac{1}{2} \underset{\sim}{u}\)
  2. \(\underset{\sim}{s}-\dfrac{1}{2} \underset{\sim}{u}\)
  3. \(\dfrac{1}{2}\underset{\sim}{s}+\underset{\sim}{u}\)
  4. \(\dfrac{1}{2}\underset{\sim}{s}-\underset{\sim}{u}\)
Show Answers Only

\(\Rightarrow C\)

Show Worked Solution
\(\overrightarrow{M T}\) \(=\overrightarrow{O T}-\overrightarrow{O M}\)  
  \(=\underset{\sim}{s}+\underset{\sim}{u}-\left(\dfrac{1}{2} \underset{\sim}{s}\right)\)  
  \(=\dfrac{1}{2}\underset{\sim}{s}+\underset{\sim}{u}\)  

 
\(\Rightarrow C\)

Vectors, EXT1 V1 EQ-Bank 2 MC

Evaluate  \(\abs{\underset{\sim}{u}+\underset{\sim}{w}}\underset{\sim}{v}\)  given  \(\underset{\sim}{u}=\displaystyle\binom{2}{1}, \underset{\sim}{v}=\binom{1}{3}\)  and  \(\underset{\sim}{w}=\displaystyle\binom{-4}{3}\)

  1. \(10\)
  2. \(0\)
  3. \(\displaystyle \binom{\sqrt{20}}{3 \sqrt{20}}\)
  4. \(\displaystyle \binom{-20}{15}\)
Show Answers Only

\(\Rightarrow C\)

Show Worked Solution

\(\displaystyle \underset{\sim}{u}+\underset{\sim}{w}=\binom{2}{1}+\binom{-4}{3}=\binom{-2}{4} \Rightarrow \abs{\underset{\sim}{u}+\underset{\sim}{w}}=\sqrt{4+16}=\sqrt{20}\)

\(\displaystyle \abs{\underset{\sim}{u}+\underset{\sim}{w}} \underset{\sim}{v}=\sqrt{20}\binom{1}{3}=\binom{\sqrt{20}}{3 \sqrt{20}}\)

\(\Rightarrow C\)

HMS, BM EQ-Bank 425

A student investigated how a 6-week aerobic training program affected resting heart rate in Year 11 students. Results showed an average decrease of 7 beats per minute, but some participants showed no change at all.

Identify ONE further research question that could be explored based on these findings and explain why this question would be valuable to investigate.   (3 marks)

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Sample Answer – Any ONE of the following or similar with same level of detail

Example 1

  • A valuable follow-up question would be “What factors influenced whether a participant experienced a reduction in resting heart rate?” as this directly addresses the variation in results identified in the original study.
  • This question is valuable because it would help identify which participant characteristics (such as initial fitness level, adherence to the program, or genetic factors) determine training responsiveness.
  • Understanding these factors would allow future training programs to be modified or personalised to increase effectiveness for all participants, improving health outcomes for those who didn’t respond to the standard protocol.

Example 2

  • “Does the frequency of aerobic training sessions per week affect the magnitude of resting heart rate reduction?” would be a valuable follow-up question as it investigates whether training volume was a factor in the varied responses.
  • This question is valuable because it directly addresses program design, potentially identifying whether non-responders simply need more frequent training sessions to achieve similar benefits to those who showed improvements.
  • Results could help establish minimum effective training frequencies for resting heart rate adaptations in adolescents, improving the efficiency of school-based physical education programs.

Example 3

  • “Is there a correlation between baseline fitness levels and the magnitude of resting heart rate reduction following a 6-week aerobic training program?” would explore whether initial conditioning predicts responsiveness.
  • This question is valuable because it would help identify whether the program was more effective for certain fitness profiles, allowing for better participant screening or customised program durations based on starting fitness.
  • Understanding this relationship would enable more personalised and efficient interventions by helping educators predict which students might need modified programs to achieve cardiovascular adaptations.
Show Worked Solution

Sample Answer – Any ONE of the following or similar with same level of detail

Example 1

  • A valuable follow-up question would be “What factors influenced whether a participant experienced a reduction in resting heart rate?” as this directly addresses the variation in results identified in the original study.
  • This question is valuable because it would help identify which participant characteristics (such as initial fitness level, adherence to the program, or genetic factors) determine training responsiveness.
  • Understanding these factors would allow future training programs to be modified or personalised to increase effectiveness for all participants, improving health outcomes for those who didn’t respond to the standard protocol.

Example 2

  • “Does the frequency of aerobic training sessions per week affect the magnitude of resting heart rate reduction?” would be a valuable follow-up question as it investigates whether training volume was a factor in the varied responses.
  • This question is valuable because it directly addresses program design, potentially identifying whether non-responders simply need more frequent training sessions to achieve similar benefits to those who showed improvements.
  • Results could help establish minimum effective training frequencies for resting heart rate adaptations in adolescents, improving the efficiency of school-based physical education programs.

Example 3

  • “Is there a correlation between baseline fitness levels and the magnitude of resting heart rate reduction following a 6-week aerobic training program?” would explore whether initial conditioning predicts responsiveness.
  • This question is valuable because it would help identify whether the program was more effective for certain fitness profiles, allowing for better participant screening or customised program durations based on starting fitness.
  • Understanding this relationship would enable more personalised and efficient interventions by helping educators predict which students might need modified programs to achieve cardiovascular adaptations.

HMS, BM EQ-Bank 423 MC

A student conducted research on the effects of aerobic training on blood pressure in Year 11 students. The results showed a significant decrease in resting blood pressure after 8 weeks of training for male participants, but minimal changes for female participants.

What would be the most appropriate follow-up research question?

  1. Do females require different types of aerobic training to achieve blood pressure reductions?
  2. Is aerobic training an effective intervention for reducing blood pressure?
  3. How long do blood pressure reductions last after the training program ends?
  4. What is the ideal exercise intensity for blood pressure reduction?
Show Answers Only

\(A\)

Show Worked Solution

Consider Option A: Do females require different types of aerobic training to achieve blood pressure reductions?

  • This option directly addresses the gender difference observed in the results, seeking to understand why females responded differently to the same training protocol.
  • This builds directly on the findings and could help develop more effective, targeted interventions.

Other Options:

  • B is incorrect: Asks a question that has already been partly answered by the original study (showing effectiveness for males).
  • C is incorrect: Focuses on the duration of effects rather than addressing the gender difference found in the results.
  • D is incorrect: Asks about optimisation without first addressing why the current protocol was ineffective for females.

\(\Rightarrow A\)

HMS, BM EQ-Bank 421 MC

After completing a 6-week aerobic training program investigating heart rate recovery in teenagers, a student found significant improvements in participants' recovery times but noticed that results varied considerably between individuals.

Which of the following represents the most appropriate further research question based on these findings?

  1. Would a 12-week program show greater improvements in heart rate recovery?
  2. How do different exercise intensities affect heart rate recovery improvements?
  3. Why is heart rate recovery a reliable indicator of cardiovascular fitness?
  4. Does heart rate recovery improve at the same rate for all types of aerobic exercise?
Show Answers Only

\(B\)

Show Worked Solution

Consider Option B: 

  • This option directly addresses the variability in the results.
  • By investigating how different exercise intensities affect heart rate recovery improvements, the student could explore why participants responded differently to the same training program, potentially identifying optimal training zones for different individuals.

Other Options:

  • A is incorrect: Simply extends the duration without addressing the variability in results.
  • C is incorrect: Focuses on validation of the measurement method rather than building on the findings.
  • D is incorrect: Too broad and doesn’t specifically address the individual differences noted in the results.

\(\Rightarrow B\)

HMS, BM EQ-Bank 418

In an aerobic training investigation, a student collected heart rate data immediately after a 12-minute Cooper run test. The student conducted the test on a school oval without marking the exact distance covered.

Identify TWO limitations of this data collection approach and explain how each affects the reliability of the results.   (3 marks)

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Sample Answer – Any 2 of the following

  • Not marking the exact distance covered creates inconsistency in performance measurement. This reduces reliability because participants may run different distances in pre- and post-tests despite similar perceived effort levels.
  • Collecting heart rate only immediately after exercise, rather than at multiple time points during recovery. This limits reliability because single measurements are more susceptible to momentary fluctuations and timing errors.
  • Failure to standardise pre-test conditions such as nutrition, hydration, sleep quality, or prior physical activity. This reduces reliability because physiological variations unrelated to training intervention occur between testing sessions.
  • The absence of consistent environmental controls (temperature, humidity, wind conditions, surface conditions) on the school oval. This creates test-retest variability because environmental factors significantly impact running performance and heart rate responses.
  • Lack of standardised heart rate measurement technique and timing (exact seconds after exercise cessation). This leads to inconsistent recovery phase measurements, as heart rate decreases rapidly after exercise stops.
  • Without precise timing protocols, the duration of the test itself might vary slightly between testing sessions. This affects reliability because different durations change the physiological demands and subsequent heart rate responses.
  • The absence of a standardised warm-up protocol before the test. This results in different starting physiological states, which influences both performance and heart rate responses.
  • Failure to account for or control psychological factors like motivation, competition between participants, or external distractions on the school oval. This affects reliability because these factors influence effort levels and heart rate responses.
  • Without proper equipment calibration (if using heart rate monitors) or consistent manual pulse-taking technique (if done manually). This reduces precision because the heart rate measurements themselves may lack accuracy and consistency.

Show Worked Solution

Sample Answer – Any 2 of the following

  • Not marking the exact distance covered creates inconsistency in performance measurement. This reduces reliability because participants may run different distances in pre- and post-tests despite similar perceived effort levels.
  • Collecting heart rate only immediately after exercise, rather than at multiple time points during recovery. This limits reliability because single measurements are more susceptible to momentary fluctuations and timing errors.
  • Failure to standardise pre-test conditions such as nutrition, hydration, sleep quality, or prior physical activity. This reduces reliability because physiological variations unrelated to training intervention occur between testing sessions.
  • The absence of consistent environmental controls (temperature, humidity, wind conditions, surface conditions) on the school oval. This creates test-retest variability because environmental factors significantly impact running performance and heart rate responses.
  • Lack of standardised heart rate measurement technique and timing (exact seconds after exercise cessation). This leads to inconsistent recovery phase measurements, as heart rate decreases rapidly after exercise stops.
  • Without precise timing protocols, the duration of the test itself might vary slightly between testing sessions. This affects reliability because different durations change the physiological demands and subsequent heart rate responses.
  • The absence of a standardised warm-up protocol before the test. This results in different starting physiological states, which influences both performance and heart rate responses.
  • Failure to account for or control psychological factors like motivation, competition between participants, or external distractions on the school oval. This affects reliability because these factors influence effort levels and heart rate responses.
  • Without proper equipment calibration (if using heart rate monitors) or consistent manual pulse-taking technique (if done manually). This reduces precision because the heart rate measurements themselves may lack accuracy and consistency.

HMS, BM EQ-Bank 416

A research team is investigating the relationship between aerobic training and blood pressure responses.

Explain why it is important to use valid and reliable data collection methods when investigating physiological responses to aerobic training.   (5 marks)

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Sample Answer

  • Valid data collection methods are important because they ensure that physiological variables being measured genuinely reflect aerobic training adaptations. This prevents extraneous factors like measurement timing or technique from affecting results. For example, consistent blood pressure measurement protocols eliminate variations from cuff placement or environmental conditions. Therefore, researchers can confidently attribute observed changes to training intervention rather than measurement errors.
  • Reliable methods are essential because they enable accurate quantification of training effects across multiple testing sessions. This consistency eliminates variations from measurement technique, equipment performance, or observer differences. As a result, researchers can identify true physiological adaptations that occur due to aerobic training rather than random variations.
  • Using valid and reliable methods is crucial because it ensures scientific credibility and enables practical applications. This accuracy leads to appropriate exercise prescriptions for health outcomes like blood pressure reduction. In contrast, invalid or unreliable methods could result in incorrect conclusions about training.

Show Worked Solution

Sample Answer

  • Valid data collection methods are important because they ensure that physiological variables being measured genuinely reflect aerobic training adaptations. This prevents extraneous factors like measurement timing or technique from affecting results. For example, consistent blood pressure measurement protocols eliminate variations from cuff placement or environmental conditions. Therefore, researchers can confidently attribute observed changes to training intervention rather than measurement errors.
  • Reliable methods are essential because they enable accurate quantification of training effects across multiple testing sessions. This consistency eliminates variations from measurement technique, equipment performance, or observer differences. As a result, researchers can identify true physiological adaptations that occur due to aerobic training rather than random variations.
  • Using valid and reliable methods is crucial because it ensures scientific credibility and enables practical applications. This accuracy leads to appropriate exercise prescriptions for health outcomes like blood pressure reduction. In contrast, invalid or unreliable methods could result in incorrect conclusions about training.

HMS, BM EQ-Bank 411

When investigating physiological responses to aerobic training, a researcher must consider the validity of their data collection methods.

Outline TWO factors that could impact the validity of heart rate measurements during an aerobic training investigation.   (3 marks)

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Sample Answer – Any 2 of the following

  • Environmental factors such as temperature and humidity can affect heart rate responses, potentially invalidating results if not controlled for, as they may influence the physiological variable being measured rather than the training itself.
  • Psychological factors including anxiety or nervousness around testing equipment can artificially elevate heart rate measurements, reducing validity as the numbers reflect stress responses rather than true training adaptations.
  • Measurement technique or equipment issues, such as incorrect placement of heart rate monitors, use of different monitoring devices between tests, or equipment malfunction/calibration errors, can result in inaccurate readings that don’t reflect true physiological responses.
  • Prior activity and lifestyle factors including recent food intake, caffeine consumption, medication use, or physical activity before testing can artificially alter heart rate responses, reducing the validity of measurements as indicators of training adaptations.
  • Subject-specific physiological variables like hydration status, illness, fatigue levels, or hormonal fluctuations (particularly in female participants) can significantly affect heart rate measurements, compromising validity if not controlled or accounted for.
  • Testing protocol inconsistencies including variations in warm-up procedures, exercise intensity, duration of measurement periods, or recovery intervals between testing components can invalidate comparisons between pre- and post-training measurements.
Show Worked Solution

Sample Answer – Any 2 of the following

  • Environmental factors such as temperature and humidity can affect heart rate responses, potentially invalidating results if not controlled for, as they may influence the physiological variable being measured rather than the training itself.
  • Psychological factors including anxiety or nervousness around testing equipment can artificially elevate heart rate measurements, reducing validity as the numbers reflect stress responses rather than true training adaptations.
  • Measurement technique or equipment issues, such as incorrect placement of heart rate monitors, use of different monitoring devices between tests, or equipment malfunction/calibration errors, can result in inaccurate readings that don’t reflect true physiological responses.
  • Prior activity and lifestyle factors including recent food intake, caffeine consumption, medication use, or physical activity before testing can artificially alter heart rate responses, reducing the validity of measurements as indicators of training adaptations.
  • Subject-specific physiological variables like hydration status, illness, fatigue levels, or hormonal fluctuations (particularly in female participants) can significantly affect heart rate measurements, compromising validity if not controlled or accounted for.
  • Testing protocol inconsistencies including variations in warm-up procedures, exercise intensity, duration of measurement periods, or recovery intervals between testing components can invalidate comparisons between pre- and post-training measurements.

PHYSICS, M1 EQ-Bank 5 MC

A student throws a ball with a speed of 20 m/s at an angle of 40° above the horizontal. What are the horizontal and vertical components of the ball’s velocity?

  1. Horizontal: 15.3 m/s, Vertical: 12.9 m/s
  2. Horizontal: 12.9 m/s, Vertical: 15.3 m/s
  3. Horizontal: 20.0 m/s, Vertical: 0.0 m/s
  4. Horizontal: 17.1 m/s, Vertical: 17.1 m/s
Show Answers Only

\(A\)

Show Worked Solution

\(\sin\,40^{\circ}\) \(=\dfrac{v_v}{20}\)  
\(v_v\) \(=20\times \sin\,40^{\circ} = 12.9\ \text{ms}^{-1}\)  
\(\cos\,40^{\circ}\) \(=\dfrac{v_h}{20}\)  
\(v_h\) \(=20 \times \cos\,40^{\circ} = 15.3\ \text{ms}^{-1}\)  

 
\(\Rightarrow A\)

PHYSICS, M1 EQ-Bank 16

A small plane is flying due south at 80 m/s. It makes a smooth 90° turn over 20 seconds and ends up flying due east at 60 m/s.

  1. Draw a vector diagram showing the change in velocity vector of the small plane.   (2 marks) 
  1. Calculate the average acceleration (magnitude and direction) during the turn.   (2 marks) 
Show Answers Only

a.    
             

b.    \(5\ \text{ms}^{-2}\text{, N}37^{\circ}\text{E}\).

Show Worked Solution

a.    \(\Delta v = v_f-v_i = v_f + (-v_i)\)
 

b.   \(a=\dfrac{\Delta v}{t}=\dfrac{\sqrt{60^2+80^2}}{20}=5\ \text{ms}^{-2}\)

\(\tan \theta\) \(=\dfrac{80}{60}\)  
\(\theta\) \(=\tan^{-1}\left(\dfrac{80}{60}\right)=53^{\circ}\)  

 
\(\therefore\) The average acceleration of the plane is \(5\ \text{ms}^{-2}\text{, N}37^{\circ}\text{E}\).

HMS, BM EQ-Bank 407 MC

A student conducted an investigation on heart rate recovery after aerobic exercise by measuring participants' heart rates at 1-minute intervals following a step test. The student found that measurements varied significantly when using different pulse-taking methods (wrist, neck, and chest monitor).

Which statement best explains the issue with the data collection method?

  1. The investigation lacks credibility because heart rate recovery isn't a valid measure of aerobic fitness.
  2. The investigation lacks validity because different pulse-taking methods were not calibrated against each other.
  3. The investigation lacks reliability because inconsistent measuring techniques were used across participants.
  4. The investigation lacks credibility because a step test isn't an appropriate activity for testing aerobic fitness.
Show Answers Only

\(C\)

Show Worked Solution

Consider Option C: 

  • The investigation lacks reliability because using different pulse-taking methods without standardisation creates inconsistency in the measurements.
  • Reliability refers to the consistency of results when repeated measurements are taken, and in this case, the varying methods reduce the reliability of the data.

Other Options:

  • A is incorrect: Heart rate recovery is a valid measure of aerobic fitness; the issue is not with the validity of what’s being measured.
  • B is incorrect: While calibration is important, the primary issue here is the inconsistent application of measuring techniques (a reliability issue), not validity.
  • D is incorrect: Step tests are widely accepted as appropriate for testing aerobic fitness; the credibility of the method itself is not in question.

\(\Rightarrow C\)

PHYSICS, M1 EQ-Bank 3 MC

Two birds take off from the same point at the same time.

    • Bird A flies due north at a speed of 12 m/s.

    • Bird B flies due east at a speed of 16 m/s.

What is the magnitude of the velocity of Bird A relative to Bird B?

  1. 20 m/s
  2. 12.8 m/s
  3. 8 m/s
  4. 28 m/s
Show Answers Only

\(A\)

Show Worked Solution

\(v_{\text{a rel b}} = v_a-v_b\)
 

Magnitude of \(v_{\text{a rel b}} = \sqrt{16^2 + 12^2} = 20\ \text{ms}^{-1}\)

\(\Rightarrow A\)

PHYSICS, M1 EQ-Bank 14

A boat is trying to cross a river that is 200 meters wide. The boat’s speed in still water is 5 m/s. The river flows eastward with a current of 3 m/s.

  1. If the boat heads straight north (perpendicular to the current), using a vector diagram. determine the magnitude and direction of its velocity relative to the ground?   (3 marks)

  1. How far downstream (east) will the boat have drifted by the time it reaches the opposite bank?   (1 mark)

  1. At what angle should the boat be aimed (relative to north) so that it reaches directly across from its starting point?   (2 marks)

Show Answers Only

a.    The velocity of the boat relative to the ground is 5.83 m/s N31\(^{\circ}\)E.

b.    \(120\ \text{m}\)

c.    The boat should head N36.9\(^{\circ}\)W.

Show Worked Solution

a.    Using the vector diagrams below

\(x=\sqrt{3^2+5^2} = 5.83\ \text{ms}^{-1}\)

\(\tan \theta= \dfrac{3}{5}\ \ \Rightarrow\ \ \theta=\tan^{-1}\left(\dfrac{3}{5}\right)=31^{\circ}\)

 \(\therefore\) The velocity of the boat relative to the ground is 5.83 m/s N31\(^{\circ}\)E.
 

b.    
         

\(\tan 31^{\circ}\) \(=\dfrac{d}{200}\)  
\(d\) \(=200 \times \tan\,31^{\circ}=120\ \text{m}\)  

 
c.    
         

\(\sin \theta=\dfrac{3}{5}\ \ \Rightarrow\ \ \theta=\sin^{-1}\left(\dfrac{3}{5}\right)=36.9^{\circ}\)

 \(\therefore\) The boat should head N36.9\(^{\circ}\)W.

HMS, BM EQ-Bank 406

Explain the ethical considerations when using video recording to analyse breathing patterns during an aerobic training investigation.   (4 marks)

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Sample Answer

  • Specific consent for video recording must be obtained separately from general participation consent, clearly explaining how the footage will be used, stored, and eventually destroyed.
  • Participant privacy must be protected by recording in an appropriate setting away from uninvolved observers and ensuring the camera angle focuses only on relevant areas (chest/torso) for breathing analysis.
  • Access to video recordings should be strictly limited to those directly involved in the analysis process to maintain confidentiality.
  • Participants should be given the option to review their own footage and request deletion if they feel uncomfortable with the recording, respecting their ongoing right to control their personal data.
Show Worked Solution

Sample Answer

  • Specific consent for video recording must be obtained separately from general participation consent, clearly explaining how the footage will be used, stored, and eventually destroyed.
  • Participant privacy must be protected by recording in an appropriate setting away from uninvolved observers and ensuring the camera angle focuses only on relevant areas (chest/torso) for breathing analysis.
  • Access to video recordings should be strictly limited to those directly involved in the analysis process to maintain confidentiality.
  • Participants should be given the option to review their own footage and request deletion if they feel uncomfortable with the recording, respecting their ongoing right to control their personal data.

HMS, BM EQ-Bank 405

Outline the ethical responsibilities researchers have when collecting data from participants during an aerobic training investigation.   (3 marks)

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Sample Answer

  • Researchers must obtain voluntary informed consent from all participants (or parents/guardians for minors) before collecting physiological data, clearly explaining procedures and risks.
  • Participant safety must be prioritised through appropriate screening, monitoring during exercise and immediate cessation of activity if adverse responses occur.
  • All data must be handled confidentially and stored securely to protect participant privacy throughout and after the investigation.
  • Participants must be informed of their right to withdraw at any time without penalty.

Show Worked Solution

Sample Answer

  • Researchers must obtain voluntary informed consent from all participants (or parents/guardians for minors) before collecting physiological data, clearly explaining procedures and risks.
  • Participant safety must be prioritised through appropriate screening, monitoring during exercise and immediate cessation of activity if adverse responses occur.
  • All data must be handled confidentially and stored securely to protect participant privacy throughout and after the investigation.
  • Participants must be informed of their right to withdraw at any time without penalty.

HMS, BM EQ-Bank 403

Explain the importance of debriefing participants after they complete an investigation into physiological responses to aerobic training.   (5 marks)

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Sample Answer

  • Debriefing fulfills the ethical obligation to educate participants by explaining the purpose, methods, and findings of the investigation in accessible language.
  • Sharing individual results privately with each participant respects their right to access their own physiological data and understand what it means for their health.
  • Debriefing provides an opportunity to identify and address any negative experiences, discomforts, or misconceptions participants may have had during the investigation.
  • Explaining how participants’ data contributed to the overall findings acknowledges their valuable contribution and demonstrates respect for their involvement.
  • Proper debriefing enhances the educational value of the investigation for participants, turning their participation into a meaningful learning experience about their own physiology.
Show Worked Solution

Sample Answer

  • Debriefing fulfills the ethical obligation to educate participants by explaining the purpose, methods, and findings of the investigation in accessible language.
  • Sharing individual results privately with each participant respects their right to access their own physiological data and understand what it means for their health.
  • Debriefing provides an opportunity to identify and address any negative experiences, discomforts, or misconceptions participants may have had during the investigation.
  • Explaining how participants’ data contributed to the overall findings acknowledges their valuable contribution and demonstrates respect for their involvement.
  • Proper debriefing enhances the educational value of the investigation for participants, turning their participation into a meaningful learning experience about their own physiology.

HMS, BM EQ-Bank 400

Explain ethical considerations when using a control group in an investigation of how aerobic training affects recovery heart rate.   (4 marks)

Show Answers Only

Sample Answer

  • The control group should not be disadvantaged by participation, which means they should receive an alternative appropriate activity rather than simply sitting idle.
  • All participants (both experimental and control groups) must receive full information about the group they are in and what their participation involves before giving consent.
  • If the aerobic training shows clear benefits during the investigation, there may be an ethical obligation to offer the control group similar training opportunities after the investigation concludes.
  • Random assignment to groups rather than selective placement ensures fairness and scientific validity while reducing potential bias.
Show Worked Solution

Sample Answer

  • The control group should not be disadvantaged by participation, which means they should receive an alternative appropriate activity rather than simply sitting idle.
  • All participants (both experimental and control groups) must receive full information about the group they are in and what their participation involves before giving consent.
  • If the aerobic training shows clear benefits during the investigation, there may be an ethical obligation to offer the control group similar training opportunities after the investigation concludes.
  • Random assignment to groups rather than selective placement ensures fairness and scientific validity while reducing potential bias.

HMS, BM EQ-Bank 399

Discuss the ethical considerations related to screening participants before their involvement in an investigation of physiological responses to moderate-intensity aerobic training.   (5 marks)

Show Answers Only

Sample Answer

  • Pre-participation health screening questionnaires are ethically essential to identify individuals with health conditions that might put them at risk during aerobic exercise.
  • Screening protocols must balance thoroughness with respect for privacy, asking only relevant health questions necessary to determine safe participation.
  • Exclusion of individuals based on screening results must be handled sensitively to avoid stigmatisation while prioritising participant safety.
  • Researchers have an ethical duty of care to ensure participants are physically capable of completing the aerobic training protocol without undue risk of adverse events.
  • Screening results must be kept confidential and stored securely with access limited to only those directly involved in participant safety decisions.
Show Worked Solution

Sample Answer

  • Pre-participation health screening questionnaires are ethically essential to identify individuals with health conditions that might put them at risk during aerobic exercise.
  • Screening protocols must balance thoroughness with respect for privacy, asking only relevant health questions necessary to determine safe participation.
  • Exclusion of individuals based on screening results must be handled sensitively to avoid stigmatisation while prioritising participant safety.
  • Researchers have an ethical duty of care to ensure participants are physically capable of completing the aerobic training protocol without undue risk of adverse events.
  • Screening results must be kept confidential and stored securely with access limited to only those directly involved in participant safety decisions.

HMS, BM EQ-Bank 398

Explain why maintaining confidentiality is an important ethical consideration when collecting physiological data during aerobic training investigations.   (4 marks)

Show Answers Only

Sample Answer

  • Physiological data such as heart rate and fitness levels are personal health information that participants have a right to keep private. Breaches in confidentiality could lead to embarrassment or stigma if individual responses are shared without permission.
  • Maintaining confidentiality builds trust with participants, encouraging honest participation and more reliable data collection. This trust is essential because participants may feel vulnerable when fitness levels are measured.
  • Practical measures to ensure confidentiality include using participant codes instead of names, storing data securely, and presenting only aggregated results. These procedures help protect participants from potential psychological harm or discrimination based on their physiological responses.

Show Worked Solution

Sample Answer

  • Physiological data such as heart rate and fitness levels are personal health information that participants have a right to keep private. Breaches in confidentiality could lead to embarrassment or stigma if individual responses are shared without permission.
  • Maintaining confidentiality builds trust with participants, encouraging honest participation and more reliable data collection. This trust is essential because participants may feel vulnerable when fitness levels are measured.
  • Practical measures to ensure confidentiality include using participant codes instead of names, storing data securely, and presenting only aggregated results. These procedures help protect participants from potential psychological harm or discrimination based on their physiological responses.

HMS, BM EQ-Bank 393 MC

When investigating heart rate responses to aerobic training in classmates, which of the following is the most important ethical consideration?

  1. Getting permission from the school principal
  2. Obtaining informed consent from participants
  3. Using the latest heart rate monitoring technology
  4. Publishing the results in a school newsletter
Show Answers Only

\(B\)

Show Worked Solution
  • B is correct: Obtaining informed consent ensures participants understand the purpose, procedures, potential risks, and their right to withdraw from the investigation.

Other Options:

  • A is incorrect: While administrative approval may be required, participant consent is the primary ethical consideration.
  • C is incorrect: Technology choice is a methodological decision, not a primary ethical consideration.
  • D is incorrect: Publication decisions come after data collection and are secondary to participant consent.

HMS, BM EQ-Bank 391

Compare TWO different methods for collecting data on physiological responses during aerobic training.   (5 marks)

Show Answers Only

Sample Answer 

Similarities:

  • Both methods aim to assess aerobic training responses by measuring physiological indicators during exercise sessions.
  • Both require initial setup and preparation to ensure accurate data collection during aerobic training.
  • Both provide information that can be used to evaluate training intensity and effectiveness.

Differences:

  • A heart rate monitor provides continuous, objective measurement of heart rate throughout aerobic training sessions, giving precise data on cardiovascular responses.
  • Observation method relies on visual assessment of breathing patterns, perceived exertion signs, and recovery indicators, providing subjective evaluation.
  • Heart rate monitors offer real-time numerical data that can be recorded and analysed for patterns, making it easier to track adaptations.
  • Observation method requires trained observers to identify physiological signs like breathing rate changes, skin colour, and fatigue indicators.
  • Heart rate monitors provide consistent, standardised measurements that reduce human error and enable comparison across training sessions.
  • Observation method allows assessment of multiple physiological indicators simultaneously but may lack precision in quantifying responses.

Show Worked Solution

Sample Answer

Similarities:

  • Both methods aim to assess aerobic training responses by measuring physiological indicators during exercise sessions.
  • Both require initial setup and preparation to ensure accurate data collection during aerobic training.
  • Both provide information that can be used to evaluate training intensity and effectiveness.

Differences:

  • A heart rate monitor provides continuous, objective measurement of heart rate throughout aerobic training sessions, giving precise data on cardiovascular responses.
  • Observation method relies on visual assessment of breathing patterns, perceived exertion signs, and recovery indicators, providing subjective evaluation.
  • Heart rate monitors offer real-time numerical data that can be recorded and analysed for patterns, making it easier to track adaptations.
  • Observation method requires trained observers to identify physiological signs like breathing rate changes, skin colour, and fatigue indicators.
  • Heart rate monitors provide consistent, standardised measurements that reduce human error and enable comparison across training sessions.
  • Observation method allows assessment of multiple physiological indicators simultaneously but may lack precision in quantifying responses.

HMS, BM EQ-Bank 388

Explain how a student could collect data on recovery heart rate following a 15-minute aerobic exercise session.   (4 marks)

Show Answers Only

Sample Answer 

  • Record baseline resting heart rate before exercise to establish a comparison point for recovery analysis.
  • Immediately upon completion of the 15-minute session, measure heart rate using a heart rate monitor or manual pulse check. This provides the starting point for recovery measurement.
  • Continue monitoring heart rate at regular intervals (1-minute, 3-minute, 5-minute marks) during recovery. A systematic approach such as this tracks cardiovascular return to baseline.
  • Have participants remain seated in consistent position throughout monitoring to standardise conditions and eliminate variables.
  • Record measurements systematically using a data collection sheet. This enables creation of a recovery profile showing when heart rate returns to resting levels.

Show Worked Solution

Sample Answer

  • Record baseline resting heart rate before exercise to establish a comparison point for recovery analysis.
  • Immediately upon completion of the 15-minute session, measure heart rate using a heart rate monitor or manual pulse check. This provides the starting point for recovery measurement.
  • Continue monitoring heart rate at regular intervals (1-minute, 3-minute, 5-minute marks) during recovery. A systematic approach such as this tracks cardiovascular return to baseline.
  • Have participants remain seated in consistent position throughout monitoring to standardise conditions and eliminate variables.
  • Record measurements systematically using a data collection sheet. This enables creation of a recovery profile showing when heart rate returns to resting levels.

HMS, BM EQ-Bank 387

Explain the advantages and limitations of using a Rating of Perceived Exertion (RPE) scale when collecting data on aerobic training responses.   (5 marks)

Show Answers Only

Sample Answer 

  • The RPE scale provides subjective data on exercise intensity without requiring specialised equipment, making it accessible for all research settings. This accessibility is beneficial because it reduces costs and enables widespread use.
  • Participants can provide RPE ratings without interrupting their exercise, allowing continuous data collection throughout an aerobic session. This enables researchers to track intensity changes.
  • The scale’s portability enables data collection in natural training environments where laboratory equipment is impractical. This creates more realistic training conditions.
  • However, the subjective nature of RPE introduces potential reliability issues as perception varies between individuals based on fitness levels, pain tolerance and motivation. This variability can affect consistency of results.
  • Prior training of participants in using the RPE scale correctly is essential to ensure consistency in how ratings are assigned. This training requirement adds time and complexity.
  • RPE provides valuable complementary data when used alongside objective physiological measures, creating a complete picture.
Show Worked Solution

Sample Answer

  • The RPE scale provides subjective data on exercise intensity without requiring specialised equipment, making it accessible for all research settings. This accessibility is beneficial because it reduces costs and enables widespread use.
  • Participants can provide RPE ratings without interrupting their exercise, allowing continuous data collection throughout an aerobic session. This enables researchers to track intensity changes.
  • The scale’s portability enables data collection in natural training environments where laboratory equipment is impractical. This creates more realistic training conditions.
  • However, the subjective nature of RPE introduces potential reliability issues as perception varies between individuals based on fitness levels, pain tolerance and motivation. This variability can affect consistency of results.
  • Prior training of participants in using the RPE scale correctly is essential to ensure consistency in how ratings are assigned. This training requirement adds time and complexity.
  • RPE provides valuable complementary data when used alongside objective physiological measures, creating a complete picture.

HMS, BM EQ-Bank 386

A student has access to a respiratory rate monitor with a chest strap sensor to investigate ventilation responses during aerobic training. Describe how this equipment should be used to collect ventilation rate data during a 20-minute aerobic exercise session.   (4 marks)

Show Answers Only

Sample Answer 

  • The chest strap sensor should be calibrated and securely fastened around the participant’s chest before beginning the exercise session to ensure accurate measurement.
  • The device should be set to record breath frequency continuously throughout the 20-minute session with data stored automatically at preset intervals (every 30 seconds).
  • The participant’s baseline ventilation rate should be recorded during a 2-minute rest period before commencing exercise to establish individual reference values.
  • After completing the aerobic session, ventilation rate should continue to be monitored for a 5-minute recovery period to observe how quickly breathing returns to baseline levels.
  • All collected data should be transferred to a spreadsheet for analysis, allowing calculation of average rates, maximum rates, and recovery patterns.
  • Consistent placement of the chest strap sensor between participants and testing sessions is essential to ensure reliable and comparable data.
Show Worked Solution

Sample Answer

  • The chest strap sensor should be calibrated and securely fastened around the participant’s chest before beginning the exercise session to ensure accurate measurement.
  • The device should be set to record breath frequency continuously throughout the 20-minute session with data stored automatically at preset intervals (every 30 seconds).
  • The participant’s baseline ventilation rate should be recorded during a 2-minute rest period before commencing exercise to establish individual reference values.
  • After completing the aerobic session, ventilation rate should continue to be monitored for a 5-minute recovery period to observe how quickly breathing returns to baseline levels.
  • All collected data should be transferred to a spreadsheet for analysis, allowing calculation of average rates, maximum rates, and recovery patterns.
  • Consistent placement of the chest strap sensor between participants and testing sessions is essential to ensure reliable and comparable data.

HMS, BM EQ-Bank 381 MC

A student wants to collect data on the immediate cardiovascular response to a 10-minute jog. Which method would be most appropriate?

  1. Blood sample analysis
  2. Heart rate monitoring
  3. Muscle biopsy
  4. Bone density scan
Show Answers Only

\(B\)

Show Worked Solution
  • B is correct: Heart rate monitoring provides immediate, non-invasive data about cardiovascular responses during aerobic exercise.

Other Options:

  • A is incorrect: Blood sample analysis is invasive and better suited for measuring metabolic markers, not immediate cardiovascular responses.
  • C is incorrect: Muscle biopsy is highly invasive and measures muscle tissue changes, not immediate cardiovascular responses.
  • D is incorrect: Bone density scanning measures skeletal characteristics, not cardiovascular responses to exercise.

HMS, BM EQ-Bank 379

A student is studying physiological responses to aerobic training. Outline the steps they should take to develop a focused research question on this topic.   (4 marks)

Show Answers Only

Sample Answer 

  • Identify a specific physiological response of interest (e.g., heart rate, ventilation rate, body temperature) that can be reliably measured.
  • Determine the exact aerobic training activity or protocol to be studied, including details of duration and intensity.
  • Consider the particular aspect of the response to investigate (immediate changes, recovery patterns, or adaptations over time).
  • Formulate a question that establishes a clear relationship between the training and the physiological response, using precise language and avoiding ambiguity.
Show Worked Solution

Sample Answer 

  • Identify a specific physiological response of interest (e.g., heart rate, ventilation rate, body temperature) that can be reliably measured.
  • Determine the exact aerobic training activity or protocol to be studied, including details of duration and intensity.
  • Consider the particular aspect of the response to investigate (immediate changes, recovery patterns, or adaptations over time).
  • Formulate a question that establishes a clear relationship between the training and the physiological response, using precise language and avoiding ambiguity.

HMS, BM EQ-Bank 376

A student wants to investigate how heart rate changes during a 5km run. Explain how they could refine this initial idea into a well-structured research question for investigating physiological responses to aerobic training.   (5 marks)

Show Answers Only

Sample Answer 

  • Specify the exact physiological response by focusing on particular aspects of heart rate (resting, maximum, recovery rate). Refinement such as this is necessary because it provides measurable and specific variables for investigation.
  • Define the exercise conditions clearly, including the pace of the run (time-based or percentage of maximum effort). Thus ensuring standardised conditions and enables reliable data collection.
  • Consider participant characteristics that might be relevant (age, fitness level, training experience). This helps control variables that could influence heart rate responses and improves validity.
  • Identify whether the question will examine immediate responses or adaptations over multiple training sessions. Making distinctions like this is important because it determines study design.
  • Include measurement timing and control variables such as environmental conditions and warm-up protocols. This creates a comprehensive framework.
  • A refined research question might be: “How does heart rate respond and recover in 16-18 year old students when completing a 5km run at 70% estimated maximum?”

Show Worked Solution

Sample Answer 

  • Specify the exact physiological response by focusing on particular aspects of heart rate (resting, maximum, recovery rate). Refinement such as this is necessary because it provides measurable and specific variables for investigation.
  • Define the exercise conditions clearly, including the pace of the run (time-based or percentage of maximum effort). Thus ensuring standardised conditions and enables reliable data collection.
  • Consider participant characteristics that might be relevant (age, fitness level, training experience). This helps control variables that could influence heart rate responses and improves validity.
  • Identify whether the question will examine immediate responses or adaptations over multiple training sessions. Making distinctions like this is important because it determines study design.
  • Include measurement timing and control variables such as environmental conditions and warm-up protocols. This creates a comprehensive framework.
  • A refined research question might be: “How does heart rate respond and recover in 16-18 year old students when completing a 5km run at 70% estimated maximum?”

HMS, BM EQ-Bank 373

Explain how you would develop an appropriate research question to investigate the relationship between aerobic training and recovery heart rate.   (5 marks)

Show Answers Only

Sample Answer 

  • Begin by identifying the specific aspect of recovery heart rate to be examined (rate of decline, time to return to baseline, or percentage recovery at set time points).
  • Define the aerobic training stimulus precisely, including intensity (e.g., percentage of maximum heart rate) and duration of the exercise.
  • Consider the timeframe for measuring recovery (immediate post-exercise and at what intervals thereafter).
  • Ensure the question is measurable using available equipment such as heart rate monitors or pulse palpation techniques.
  • Refine the question to be specific, clear, and focused on a single relationship rather than multiple variables, such as “How does a 15-minute run at 70% maximum heart rate affect the time taken for heart rate to return to within 10% of resting values?”
Show Worked Solution

Sample Answer 

  • Begin by identifying the specific aspect of recovery heart rate to be examined (rate of decline, time to return to baseline, or percentage recovery at set time points).
  • Define the aerobic training stimulus precisely, including intensity (e.g., percentage of maximum heart rate) and duration of the exercise.
  • Consider the timeframe for measuring recovery (immediate post-exercise and at what intervals thereafter).
  • Ensure the question is measurable using available equipment such as heart rate monitors or pulse palpation techniques.
  • Refine the question to be specific, clear, and focused on a single relationship rather than multiple variables, such as “How does a 15-minute run at 70% maximum heart rate affect the time taken for heart rate to return to within 10% of resting values?”

HMS, BM EQ-Bank 372

Describe the key components that should be included when creating a research question about physiological responses to aerobic training.   (4 marks)

Show Answers Only

Sample Answer 

  • Specific physiological response being measured must be clearly identified (e.g., resting heart rate, ventilation rate, blood lactate levels).
  • Type of aerobic training should be defined with specific characteristics including duration, intensity, frequency, and mode of exercise.
  • Target population characteristics should be specified including age range, fitness level, and health status.
  • Timeframe and measurement points must be established, indicating when data will be collected (e.g., pre-training, post-exercise).
  • Control variables that could affect results should be identified, such as environmental conditions and prior exercise history.
  • Measurability considerations should ensure the investigation can be practically conducted.

Show Worked Solution

Sample Answer 

  • Specific physiological response being measured must be clearly identified (e.g., resting heart rate, ventilation rate, blood lactate levels).
  • Type of aerobic training should be defined with specific characteristics including duration, intensity, frequency, and mode of exercise.
  • Target population characteristics should be specified including age range, fitness level, and health status.
  • Timeframe and measurement points must be established, indicating when data will be collected (e.g., pre-training, post-exercise).
  • Control variables that could affect results should be identified, such as environmental conditions and prior exercise history.
  • Measurability considerations should ensure the investigation can be practically conducted.

HMS, BM EQ-Bank 370 MC

Which of the following research questions would best allow investigation of a physiological response to aerobic training?

  1. "How many students participate in school aerobic activities?"
  2. "What is the most popular aerobic activity among teenagers?"
  3. "Why do some people prefer swimming to running?"
  4. "How does a 20-minute cycling session affect cardiac output?"
Show Answers Only

\(D\)

Show Worked Solution
  • D is correct: Specifically identifies both the aerobic training stimulus (20-minute cycling) and the physiological response being measured (cardiac output).

Other Options:

  • A is incorrect: Measures participation rates, not physiological responses.
  • B is incorrect: Examines popularity of activities, not physiological responses.
  • C is incorrect: Investigates psychological preferences, not physiological responses.

HMS, BM EQ-Bank 367 MC

When creating a research question about physiological responses to aerobic training, which of the following would be most appropriate?

  1. "Does aerobic training make you tired?"
  2. "How does heart rate respond to a 12-minute aerobic fitness test?"
  3. "Is aerobic exercise better than weight training?"
  4. "Why do people enjoy aerobic training?"
Show Answers Only

\(B\)

Show Worked Solution
  • B is correct: This option is specific, measurable, and directly investigates a physiological response (heart rate) to a defined aerobic training activity.

Other Options:

  • A is incorrect: Too vague and subjective without specifying a physiological response.
  • C is incorrect: Compares different exercise types rather than investigating a physiological response.
  • D is incorrect: Focuses on psychological aspects rather than physiological responses

HMS, BM EQ-Bank 366

A student has been asked to design a research question investigating the physiological responses to aerobic training in school students. Evaluate what ethical considerations should be addressed and how the student might ensure validity and reliability in their data collection.   (7 marks)

Show Answers Only

Sample Answer 

  • Obtain informed consent from participants and parents/guardians for minors, clearly explaining procedures and potential discomforts
  • Screen participants for pre-existing conditions that might affect results or pose health risks during testing
  • Ensure privacy and confidentiality of physiological data collected from participants
  • Control variables such as time of day, prior food intake, and recent activity to ensure reliable results
  • Use standardised protocols for measuring heart rate, ventilation rate and other physiological responses
  • Employ calibrated equipment such as heart rate monitors and respiratory measurement devices to ensure accuracy
  • Include appropriate sample size and demographic representation to allow for meaningful conclusions
Show Worked Solution

Sample Answer 

  • Obtain informed consent from participants and parents/guardians for minors, clearly explaining procedures and potential discomforts
  • Screen participants for pre-existing conditions that might affect results or pose health risks during testing
  • Ensure privacy and confidentiality of physiological data collected from participants
  • Control variables such as time of day, prior food intake, and recent activity to ensure reliable results
  • Use standardised protocols for measuring heart rate, ventilation rate and other physiological responses
  • Employ calibrated equipment such as heart rate monitors and respiratory measurement devices to ensure accuracy
  • Include appropriate sample size and demographic representation to allow for meaningful conclusions

HMS, BM EQ-Bank 364

A student is conducting research on their physiological responses to a 30-minute jog. Outline what data they should collect and how they could analyse the results to understand their aerobic training responses.   (5 marks)

Show Answers Only

Sample Answer 

  • Collect baseline resting heart rate and ventilation rate before the jog to establish comparison points for aerobic training responses.
  • Monitor heart rate continuously during the 30-minute jog using a heart rate monitor to track cardiovascular response patterns and training zones.
  • Measure ventilation rate by counting breaths per minute at 5-minute intervals during exercise to assess respiratory system responses.
  • Record ratings of perceived exertion (RPE) on a scale of 6-20 every 10 minutes to correlate physiological responses with subjective feelings.
  • Track recovery time by measuring how long it takes for heart rate to return to within 10% of resting levels after exercise.
  • Note environmental conditions (temperature, humidity) that might influence physiological responses and training adaptations.
  • Analyse data by creating line graphs showing heart rate and ventilation rate changes over time, calculating average values for each phase (rest, exercise, recovery).
  • Compare results to established aerobic training norms and previous personal results to identify improvements in cardiovascular fitness.

Show Worked Solution

Sample Answer 

  • Collect baseline resting heart rate and ventilation rate before the jog to establish comparison points for aerobic training responses.
  • Monitor heart rate continuously during the 30-minute jog using a heart rate monitor to track cardiovascular response patterns and training zones.
  • Measure ventilation rate by counting breaths per minute at 5-minute intervals during exercise to assess respiratory system responses.
  • Record ratings of perceived exertion (RPE) on a scale of 6-20 every 10 minutes to correlate physiological responses with subjective feelings.
  • Track recovery time by measuring how long it takes for heart rate to return to within 10% of resting levels after exercise.
  • Note environmental conditions (temperature, humidity) that might influence physiological responses and training adaptations.
  • Analyse data by creating line graphs showing heart rate and ventilation rate changes over time, calculating average values for each phase (rest, exercise, recovery).
  • Compare results to established aerobic training norms and previous personal results to identify improvements in cardiovascular fitness.

HMS, BM EQ-Bank 363

Explain what happens to blood lactate levels during aerobic training of increasing intensity.   (5 marks)

Show Answers Only

Sample Answer 

  • At rest, blood lactate levels remain low at approximately 1-2 mmol/L. This occurs because the aerobic system meets all energy demands without producing excess lactate.
  • During light to moderate aerobic exercise, lactate production increases slightly as a result of increased energy demands. However, it remains in equilibrium with clearance due to sufficient oxygen availability.
  • When exercise intensity increases to the aerobic threshold, lactate levels begin to rise gradually. The reason for this is that energy demands are approaching the capacity of the aerobic system, though the body can still manage lactate clearance effectively.
  • At the lactate threshold/inflection point, lactate production exceeds the body’s ability to clear it. This happens because insufficient oxygen delivery forces a shift toward anaerobic metabolism.
  • Beyond this point, lactate accumulates rapidly in the bloodstream. Consequently, increased muscle acidity develops, which leads to fatigue and limits performance duration.
Show Worked Solution

Sample Answer 

  • At rest, blood lactate levels remain low at approximately 1-2 mmol/L. This occurs because the aerobic system meets all energy demands without producing excess lactate.
  • During light to moderate aerobic exercise, lactate production increases slightly as a result of increased energy demands. However, it remains in equilibrium with clearance due to sufficient oxygen availability.
  • When exercise intensity increases to the aerobic threshold, lactate levels begin to rise gradually. The reason for this is that energy demands are approaching the capacity of the aerobic system, though the body can still manage lactate clearance effectively.
  • At the lactate threshold/inflection point, lactate production exceeds the body’s ability to clear it. This happens because insufficient oxygen delivery forces a shift toward anaerobic metabolism.
  • Beyond this point, lactate accumulates rapidly in the bloodstream. Consequently, increased muscle acidity develops, which leads to fatigue and limits performance duration.

HMS, BM EQ-Bank 361

Explain how the body adjusts ventilation during aerobic exercise.   (4 marks)

Show Answers Only

Sample Answer 

  • Ventilation rate increases immediately when aerobic exercise begins. This occurs because working muscles demand more oxygen for energy production.
  • Breathing depth increases to maximise lung capacity. As a result, more oxygen enters the lungs with each breath, improving oxygen availability.
  • Breathing frequency rises from rest to meet metabolic demands. This happens because the body needs to maintain adequate oxygen supply and carbon dioxide removal.
  • Therefore, increased ventilation prevents acid-base imbalance in the blood. Higher breathing rates enable efficient removal of carbon dioxide produced during metabolism.
  • Respiratory muscles work harder to support these changes. Consequently, the diaphragm and intercostal muscles contract more forcefully to facilitate air movement.
Show Worked Solution

Sample Answer 

  • Ventilation rate increases immediately when aerobic exercise begins. This occurs because working muscles demand more oxygen for energy production.
  • Breathing depth increases to maximise lung capacity. As a result, more oxygen enters the lungs with each breath, improving oxygen availability.
  • Breathing frequency rises from rest to meet metabolic demands. This happens because the body needs to maintain adequate oxygen supply and carbon dioxide removal.
  • Therefore, increased ventilation prevents acid-base imbalance in the blood. Higher breathing rates enable efficient removal of carbon dioxide produced during metabolism.
  • Respiratory muscles work harder to support these changes. Consequently, the diaphragm and intercostal muscles contract more forcefully to facilitate air movement.

HMS, BM EQ-Bank 357 MC

During a 30 minute jog, a student recorded an increase in their cardiac output. This change is primarily due to:

  1. Decreased blood pressure
  2. Increased respiratory rate
  3. Increased heart rate and stroke volume
  4. Decreased peripheral resistance
Show Answers Only

\(C\)

Show Worked Solution

Consider Option C: Increased heart rate and stroke volume

  • Cardiac output increases during aerobic exercise due to both increased heart rate and stroke volume.

Other Options:

  • A is incorrect: Blood pressure typically increases (not decreases) during exercise.
  • B is incorrect: Respiratory rate increases but doesn’t directly cause cardiac output increase.
  • D is incorrect: Peripheral resistance decreases but isn’t the primary cause of increased cardiac output.

\(\Rightarrow C\)

Advanced Trigonometry, 2ADV T2 SM-Bank 31

Given  \(\tan\,\theta = -\dfrac{7}{24}\)  for  \(-90^{\circ} \lt \theta \lt 90^{\circ}\), find the exact value of

  1.  \(\dfrac{1}{\cos\,\theta}\)   (2 marks)

  2.  \(\sin\,\theta\)   (1 mark)

Show Answers Only

a.   \(\dfrac{25}{24}\)

b.   \(-\dfrac{7}{25}\)

Show Worked Solution

a.   \(\tan\,\theta = -\dfrac{7}{24}\)

\(\text{Graphically, given}\ \ -90^{\circ} \lt \theta \lt 90^{\circ}:\)

\(x= \sqrt{24^2 + 7^2}=25\)

\(\dfrac{1}{\cos\,\theta}=\dfrac{1}{\frac{24}{25}}=\dfrac{25}{24}\)
  

b.   \(\sin\,\theta = -\dfrac{7}{25}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 43v4

Find the exact value of

\(\sin(-300^{\circ})\).   (2 marks)

Show Answers Only

\(\dfrac{\sqrt{3}}{2}\)

Show Worked Solution

\(\sin(-300^{\circ})= \sin\,60^{\circ}\)

\(\text{Reference angle:}\ 60^{\circ}\)

\(\text{Since sin is positive in 1st quadrant:}\)

\(\sin(-300^{\circ})= \sin\,60^{\circ}=\dfrac{\sqrt{3}}{2}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 43v3

Find the exact value of

\(\sin(-210^{\circ})\).   (2 marks)

Show Answers Only

\(\dfrac{1}{2}\)

Show Worked Solution

\(\sin(-210^{\circ})= \sin\,150^{\circ}\)

\(\text{Reference angle:}\ 180-150=30^{\circ}\)

\(\text{Since sin is positive in 2nd quadrant:}\)

\(\sin(-210^{\circ})= \sin\,30^{\circ}=\dfrac{1}{2}\)

Advanced Trigonometry, 2ADV T2 SM-Bank 43v2

Find the exact value of

\(\cos(-240^{\circ})\).   (2 marks)

Show Answers Only

\(-\dfrac{1}{2}\)

Show Worked Solution

\(\cos(-240^{\circ})= \cos\,120^{\circ}\)

\(\text{Reference angle:}\ 180-120=60^{\circ}\)

\(\text{Since cos is negative in 2nd quadrant:}\)

\(\cos(-240^{\circ})= -\cos\,60^{\circ}=-\dfrac{1}{2}\)

Advanced Trigonometry, 2ADV T2 2015 HSC 12a

Find the solutions of  \(2\sin\,\theta = 1\)  for  \(0^{\circ} \leq \theta \leq 360^{\circ}\).   (2 marks)

Show Answers Only

\(30^{\circ}, 150^{\circ}\)

Show Worked Solution

\(2\sin\,\theta=1\ \ \Rightarrow\ \ \sin\,\theta=\dfrac{1}{2}\)

\(\text{Reference angle:}\ \sin\,30^{\circ}=\dfrac{1}{2}\)

\(\text{Since sin is positive in the 1st/2nd quadrants:}\)

\(\therefore\ \theta\) \(= 30^{\circ}, 180-30^{\circ}\)
  \(= 30^{\circ}, 150^{\circ}\)

Advanced Trigonometry, 2ADV T2 2009 HSC 1e

Find the exact value of \(\theta\) such that  \(2\cos\,\theta = 1\), where  \(0^{\circ} \leq \theta \leq 90^{\circ}\).   (2 marks)

Show Answers Only

 \(\theta = 60^{\circ}\)

Show Worked Solution
\(2 \cos\,\theta\) \(= 1\)
\(\cos\,\theta\) \(= \dfrac{1}{2}\)
\(\therefore \theta\) \(= 60^{\circ},\ \ \ \ 0^{\circ} \leq \theta \leq 90^{\circ}\)

Advanced Trigonometry, 2ADV T2 2007 HSC 4a

Solve  \(\sqrt{2}\,\sin\,x = 1\)  for  \(0^{\circ} \leq x \leq 360^{\circ}.\)   (2 marks)

Show Answers Only

\(45^{\circ}, 135^{\circ}\)

Show Worked Solution

\(\sqrt{2}\,\sin\,x = 1\ \ \Rightarrow \ \ \sin\,x= \dfrac{1}{\sqrt{2}} \)

\(\text{Reference angle:}\ \sin\,45^{\circ} = \dfrac{1}{\sqrt{2}}\)

\(\text{Since sin is positive in 1st/2nd quadrants:}\)

\(\therefore x\) \(= 45^{\circ}, (180-45)^{\circ}\)
  \(= 45^{\circ}, 135^{\circ}\)

Advanced Trigonometry, 2ADV T2 2005 HSC 2a

Solve  \(\cos\,\theta = \dfrac{1}{\sqrt{2}}\)  for  \(0^{\circ} ≤ \theta ≤ 360^{\circ}\).   (2 marks)

Show Answers Only

\(45^{\circ}, 315^{\circ}`

Show Worked Solution

\(\cos\,\theta = \dfrac{1}{\sqrt{2}}\)  for  \(0^{\circ} ≤ \theta ≤ 360^{\circ}\)

\(\text{Reference angle:}\ \cos\,45^{\circ} = \dfrac{1}{\sqrt{2}}\)

\(\text{Since cos is positive in 1st/4th quadrants:}\)

\(\theta\) \(= 45^{\circ}, 360-45`
  \(= 45^{\circ}, 315^{\circ}\)

 

HMS, BM EQ-Bank 351

Compare the lactate levels experienced by a sprinter and a marathon runner during their respective competitions. Use examples to support your answer.   (6 marks)

Show Answers Only

Sample Answer 

Similarities:

  • Both athletes experience lactate production during competition.
  • Both start with low resting lactate levels before their events.
  • Both must manage lactate accumulation to maintain performance.
  • Both experience elevated lactate if they exceed their sustainable pace.

Differences:

  • Sprinters reach extreme lactate levels due to maximal anaerobic effort. Marathon runners maintain relatively low levels through aerobic metabolism.
  • Sprinters experience rapid lactate accumulation within seconds of starting. Marathon runners sustain steady lactate levels throughout their race.
  • Sprinters rely entirely on lactate tolerance for brief periods. Marathon runners train to maximise lactate clearance and aerobic efficiency.
  • Sprint events cause severe muscle burning from extreme lactate accumulation. Marathon runners experience minimal lactate-related discomfort until late stages.
  • Recovery differs significantly between events. Sprinters need extended time to clear high lactate levels post-race.

Examples:

  • A 400m sprinter experiences intense muscle burning in the final straight. Extreme lactate accumulation forces them to slow despite maximal effort.
  • Marathon runners maintain comfortable pacing for most of the race. Lactate only spikes during a final sprint or when hitting “the wall”.
Show Worked Solution

Sample Answer 

Similarities:

  • Both athletes experience lactate production during competition.
  • Both start with low resting lactate levels before their events.
  • Both must manage lactate accumulation to maintain performance.
  • Both experience elevated lactate if they exceed their sustainable pace.

Differences:

  • Sprinters reach extreme lactate levels due to maximal anaerobic effort. Marathon runners maintain relatively low levels through aerobic metabolism.
  • Sprinters experience rapid lactate accumulation within seconds of starting. Marathon runners sustain steady lactate levels throughout their race.
  • Sprinters rely entirely on lactate tolerance for brief periods. Marathon runners train to maximise lactate clearance and aerobic efficiency.
  • Sprint events cause severe muscle burning from extreme lactate accumulation. Marathon runners experience minimal lactate-related discomfort until late stages.
  • Recovery differs significantly between events. Sprinters need extended time to clear high lactate levels post-race.

Examples:

  • A 400m sprinter experiences intense muscle burning in the final straight. Extreme lactate accumulation forces them to slow despite maximal effort.
  • Marathon runners maintain comfortable pacing for most of the race. Lactate only spikes during a final sprint or when hitting “the wall”.

HMS, BM EQ-Bank 349

Explain the significance of lactate levels for a 1500-metre runner during training. Provide an example to support your answer.   (5 marks)

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Sample Answer 

  • Lactate levels indicate exercise intensity and energy system usage for 1500-metre runners. This is significant because the event requires both aerobic and anaerobic contributions.
  • At the lactate threshold, blood lactate concentration increases sharply. The increase occurs when the shift from aerobic to anaerobic metabolism begins.
  • Training at or above this threshold improves lactate clearance ability. As a result, runners can maintain higher speeds before fatigue sets in.
  • Therefore, monitoring lactate helps coaches prescribe appropriate training intensities. Precise monitoring enables targeted development of specific energy systems for optimal performance.
  • Regular lactate testing allows precise adjustment of training zones. Consequently, runners avoid under-training or over-training at crucial intensities.
  • For example, a runner might perform 6 × 400m intervals at threshold pace. These intervals cause moderate lactate accumulation, which leads to improved lactate tolerance and clearance ability.
Show Worked Solution

Sample Answer 

  • Lactate levels indicate exercise intensity and energy system usage for 1500-metre runners. This is significant because the event requires both aerobic and anaerobic contributions.
  • At the lactate threshold, blood lactate concentration increases sharply. The increase occurs when the shift from aerobic to anaerobic metabolism begins.
  • Training at or above this threshold improves lactate clearance ability. As a result, runners can maintain higher speeds before fatigue sets in.
  • Therefore, monitoring lactate helps coaches prescribe appropriate training intensities. Precise monitoring enables targeted development of specific energy systems for optimal performance.
  • Regular lactate testing allows precise adjustment of training zones. Consequently, runners avoid under-training or over-training at crucial intensities.
  • For example, a runner might perform 6 × 400m intervals at threshold pace. These intervals cause moderate lactate accumulation, which leads to improved lactate tolerance and clearance ability.

HMS, BM EQ-Bank 348

Outline the relationship between exercise intensity and lactate levels during training.   (4 marks)

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Sample Answer 

  • At rest and during low-intensity exercise, lactate levels remain minimal. The aerobic system effectively meets energy demands without significant lactate accumulation.
  • As intensity increases to moderate levels, lactate production rises gradually. The body maintains balance between lactate production and removal.
  • At the lactate threshold during high-intensity exercise, production suddenly exceeds clearance ability. Blood lactate concentration increases sharply from this point.
  • During very high-intensity or maximal exercise, lactate accumulates rapidly in muscles and blood. This increasing acidity contributes to fatigue and limits performance duration.
Show Worked Solution

Sample Answer 

  • At rest and during low-intensity exercise, lactate levels remain minimal. The aerobic system effectively meets energy demands without significant lactate accumulation.
  • As intensity increases to moderate levels, lactate production rises gradually. The body maintains balance between lactate production and removal.
  • At the lactate threshold during high-intensity exercise, production suddenly exceeds clearance ability. Blood lactate concentration increases sharply from this point.
  • During very high-intensity or maximal exercise, lactate accumulates rapidly in muscles and blood. This increasing acidity contributes to fatigue and limits performance duration.

HMS, BM EQ-Bank 346 MC

During a 400-metre sprint, an athlete experiences a rapid increase in lactate levels. Which statement correctly explains what happens to the athlete's body during this activity?

  1. The aerobic energy system is the primary energy provider, resulting in decreased lactate levels
  2. Lactate production increases as the body relies on the ATP-PCr system for energy
  3. The athlete's muscles use more oxygen, which reduces lactate production in the bloodstream
  4. Lactate accumulates in the muscles as the glycolytic system produces energy without sufficient oxygen
Show Answers Only

\(D\)

Show Worked Solution
  • D is correct: Glycolytic system produces lactate during anaerobic sprinting

Other Options:

  • A is incorrect: Sprints use anaerobic not aerobic system
  • B is incorrect: ATP-PCr system doesn’t produce lactate
  • C is incorrect: Insufficient oxygen increases lactate production

HMS, BM EQ-Bank 343

Compare the immediate cardiac output response during high-intensity interval training with steady-state aerobic training.   (5 marks)

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Sample Answer

Similarities:

  • Both training methods increase cardiac output above resting levels.
  • Both responses result from increases in heart rate and stroke volume.
  • Both deliver increased oxygen to working muscles.
  • Both achieve significantly elevated cardiac output in trained individuals.

Differences:

  • Steady-state aerobic training produces gradual cardiac output increases. These stabilise at sustainable levels throughout exercise.
  • HIIT causes rapid cardiac output spikes during work intervals. Partial recovery occurs between intervals.
  • Steady-state maintains consistent cardiac output throughout exercise duration. This allows efficient oxygen delivery.
  • HIIT creates fluctuating cardiac output between work and recovery periods. Peak demands exceed steady-state levels.
  • Steady-state allows the cardiovascular system to reach equilibrium. HIIT repeatedly stresses the system without achieving steady state.
  • HIIT produces greater cardiovascular variability and higher peak demands than steady-state training.
Show Worked Solution

Sample Answer

Similarities:

  • Both training methods increase cardiac output above resting levels.
  • Both responses result from increases in heart rate and stroke volume.
  • Both deliver increased oxygen to working muscles.
  • Both achieve significantly elevated cardiac output in trained individuals.

Differences:

  • Steady-state aerobic training produces gradual cardiac output increases. These stabilise at sustainable levels throughout exercise.
  • HIIT causes rapid cardiac output spikes during work intervals. Partial recovery occurs between intervals.
  • Steady-state maintains consistent cardiac output throughout exercise duration. This allows efficient oxygen delivery.
  • HIIT creates fluctuating cardiac output between work and recovery periods. Peak demands exceed steady-state levels.
  • Steady-state allows the cardiovascular system to reach equilibrium. HIIT repeatedly stresses the system without achieving steady state.
  • HIIT produces greater cardiovascular variability and higher peak demands than steady-state training.

HMS, BM EQ-Bank 342

Describe how cardiac output changes during a moderate aerobic training session and explain its relationship to oxygen delivery to muscles.   (4 marks)

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Sample Answer

  • Cardiac output increases immediately from 5 L/min at rest to 15-20 L/min during moderate aerobic training.
  • Heart rate rises from 70 bpm to 120-150 bpm during exercise.
  • Stroke volume increases from 70 mL to 100-120 mL per beat.
  • This occurs because working muscles demand more oxygen for energy production.
  • The increased cardiac output enables greater oxygen delivery through increased blood flow.
  • Therefore, muscles receive sufficient oxygen to support aerobic ATP production, which allows sustained exercise performance.
Show Worked Solution

Sample Answer

  • Cardiac output increases immediately from 5 L/min at rest to 15-20 L/min during moderate aerobic training.
  • Heart rate rises from 70 bpm to 120-150 bpm during exercise.
  • Stroke volume increases from 70 mL to 100-120 mL per beat.
  • This occurs because working muscles demand more oxygen for energy production.
  • The increased cardiac output enables greater oxygen delivery through increased blood flow.
  • Therefore, muscles receive sufficient oxygen to support aerobic ATP production, which allows sustained exercise performance.

HMS, BM EQ-Bank 339 MC

During a moderate-intensity training session, a swimmer has a heart rate of 130 beats per minute and a stroke volume of 120 mL. What is her cardiac output during this exercise?

  1. 120 L/min
  2. 15.6 L/min
  3. 1.56 L/min
  4. 250 L/min
Show Answers Only

\(B\)

Show Worked Solution

Consider Option B: 15.6 L/min

\(\text{Cardiac output}\) \(=\text{Heart rate}\times\ \text{Stroke volume}\)
  \(=130\ \text{beats/min}\times 120\ \text{mL/beat}\)
  \(=15\,600\ \text{mL/min}\)
  \(=15.6\ \text{L/min}\)

Other Options:

  • A is incorrect: Failed to multiply HR and SV
  • C is incorrect: Decimal conversion error
  • D is incorrect: Added instead of multiplying values

Calculus, SPEC2 2024 VCAA 3

A pollutant, at time \(t=0\) days, begins to enter a pond of still, unpolluted water at a rate of  \(\dfrac{d V}{d t}=\dfrac{8 t}{240+5 t^4}\), where \(V\) is the volume of pollutant, in cubic metres, in the pond after \(t\) days.

The pollutant does not dissolve or mix, and spreads across the pond, maintaining the shape of a thin circular disc of radius \(r(t)\) metres and constant depth of 1 millimetre.

  1. What is the maximum rate, in cubic metres per day, at which the pollutant will enter the pond, and for what value of \(t\) will this rate occur?   (1 mark)

  2. At what rate is the radius of the disc increasing after \(t=4\) days, where it may be assumed that the radius of the disc is 6.54 m ?
  3. Give your answer in metres per day correct to two decimal places.   (3 marks)

    1. Use the substitution  \(u=\sqrt{5} \, t^2\)  to express  \(\displaystyle \int \frac{8 t}{240+5 t^4} d t\)  as an integral involving only the variable \(u\).   (1 mark)

    2. Hence, or otherwise, find, in terms of \(t\), the total volume \(V \) m\(^3\) of pollutant that has entered the pond after \(t\) days.
    3. Give your answer in the form \(\dfrac{1}{a \sqrt{b}} \arctan \left(\dfrac{t^c}{d \sqrt{b}}\right)\),  where  \(a, b, c, d \in Z^{+}\).    (1 mark)

  5. What surface area of the pond would the coverage of the pollutant approach?
  6. Give your answer in square metres correct to two decimal places.  (2 marks)

  7. The clean-up of the pond begins after five days, where the pollutant is removed at a constant rate of 0.05 cubic metres per day until the pond is free of pollutant. However, efforts to stem the flow are unsuccessful and the pollutant continues to enter the pond at a rate of  \(\dfrac{8 t}{240+5 t^4}\)  cubic metres per day.
  8. After how many days, from the start of the clean-up, will the pond be free of pollutant? Give your answer in days correct to one decimal place.  (2 marks)

Show Answers Only

a.    \\(\dfrac{dV}{dt} \text{(max)}=\dfrac{1}{20} \ \text{m\(^3\)/day.}\)

b.  \(\dfrac{dr}{dt}=\dfrac{2}{95} \times \dfrac{500}{\pi \times 6.54}=0.51 \ \text{m/day}\)

c.i.  \(\displaystyle \int \frac{8t}{240+5 t^4} \, dt=\int \frac{8}{240+u^2} \cdot \frac{du}{2 \sqrt{5}}=\frac{4}{\sqrt{5}} \int \frac{du}{240+u^2}\)

c.ii.  \(V=\displaystyle \frac{1}{5 \sqrt{3}} \tan ^{-1}\left(\frac{t^2}{4 \sqrt{3}}\right)\)

d.   \(\dfrac{100\pi}{\sqrt{3}}\)

e.   \(t=3.4 \ \text{days}\)

Show Worked Solution

a.    \(\dfrac{dV}{dt}=\dfrac{8t}{240+5t^4}\)

\(\text {Max rate when} \ \ \dfrac{d^2V}{dt^2}=0\)

\(\text {Solve:  \(\ \dfrac{d^{2}V}{dt^2}=\dfrac{1920-120 t^4}{\left(240+5t^4\right)^2}=0 \ \)  (by CAS)}\)

\(\Rightarrow t=2 \quad (t>0)\)

\(\dfrac{dV}{dt} \text{(max)}=\dfrac{16}{240+5 \times 2^4}=\dfrac{1}{20} \ \text{m\(^3\)/day.}\)
 

b.    \(r(t) \ \text{is measured in metres,} \ \  h=1 \ \text{mm}=\dfrac{1}{1000} \ \text{m}\)

\(V=\pi r^2 h=\dfrac{\pi r^2}{1000} \Rightarrow \dfrac{dV}{dr}=\dfrac{\pi r}{500}\)

\(\text{At} \ \ t=4, \ \ \dfrac{dV}{dt}=\dfrac{8 \times 4}{240+5 \times 4^4} =\dfrac{2}{95}\)

\(\dfrac{dV}{dt}=\dfrac{dV}{dr} \cdot \dfrac{dr}{dt}\)

\(\text{Find } \dfrac{dr}{dt} \ \text {when}\ \  r=6.54:\)

\(\dfrac{2}{95}=\dfrac{\pi r}{500} \times \dfrac{dr}{dt}\)

\(\dfrac{dr}{dt}=\dfrac{2}{95} \times \dfrac{500}{\pi \times 6.54}=0.51 \ \text{m/day}\)
 

c.i.  \(\displaystyle \int \frac{8t}{240+5t^4} \, dt\)

\(u=\sqrt{5} t^2, \ \dfrac{du}{dt}=2 \sqrt{5} t \ \Rightarrow \ dt=\dfrac{du}{2 \sqrt{5} t}\)

\(\displaystyle \int \frac{8t}{240+5 t^4} \, dt=\int \frac{8}{240+u^2} \cdot \frac{du}{2 \sqrt{5}}=\frac{4}{\sqrt{5}} \int \frac{du}{240+u^2}\)
 

c.ii.  \(V\) \(=\displaystyle\frac{4}{\sqrt{5}} \int_0^{\sqrt{5} t^2} \frac{1}{240+u^2} \, du\)
    \(=\displaystyle\frac{4}{\sqrt{5 \times 240}} \int_0^{\sqrt{5} t^2} \frac{\sqrt{240}}{240+u^2} \, du\)
    \(=\displaystyle \frac{1}{\sqrt{75}}\left[\tan ^{-1}\left(\frac{u}{\sqrt{240}}\right)\right]_0^{\sqrt{5} t^{2}}\)
    \(=\displaystyle \frac{1}{5 \sqrt{3}} \tan ^{-1}\left(\frac{t^2}{4 \sqrt{3}}\right)\)

 

d.    \(V=\text{S.A.} \times \text{height}=\text{S.A.} \times \dfrac{1}{1000}\)

\(\text{S.A.}=1000 \times V=\dfrac{200}{\sqrt{3}} \tan^{-1} \left(\dfrac{t^2}{4 \sqrt{3}}\right)\)

\(\text{As} \ t \rightarrow \infty, \tan^{-1}\left(\dfrac{t^2}{4 \sqrt{3}}\right) \rightarrow \dfrac{\pi}{2}\)

\(\text{As} \ t \rightarrow \infty, \ \text{S.A.} \ \rightarrow \dfrac{200}{\sqrt{3}} \cdot \dfrac{\pi}{2} = \dfrac{100\pi}{\sqrt{3}}\)

♦♦♦ Mean mark (d) 22%.

e.    \(\text{After 5 days}\)

\(V=\dfrac{1}{5 \sqrt{3}} \tan^{-1}\left(\dfrac{25}{4 \sqrt{3}}\right)\)

\(\text{Pollutant removed (after day 5)}=\displaystyle \int_{5}^{t+5}\left(\dfrac{8 x}{240+5x^4}-0.05\right)\)

\(\text{Find \(t\) such that:}\)

\(\displaystyle \int_5^{t+5}\left(\frac{8 x}{240+5 x^4}-0.05\right)+V(5)=0\)

\(t=3.4 \ \text{days}\)

♦♦♦ Mean mark (e) 13%.

Complex Numbers, SPEC2 2024 VCAA 2

  1. Express the relation  \(\abs{z-z_1}=\abs{z-z_2}\)  in the form  \(y=m x+c\), where  \(x, y, m, c \in R\), \(z=x+i y, \ z_1=1+2 i\)  and  \(z_2=4\).   (2 marks)

  2. The line segment from  \(z_1=1+2 i\)  to  \(z_2=4\)  is the diameter of a circle.
  3. Find the equation of this circle in the form \(\abs{z-z_c}=r\), where \(z_c\) is the centre of the circle and \(r\) is the radius.   (2 marks)

  4. A second circle is given by  \(\abs{z-(1+2 i)}=2\).
  5. Sketch this circle on the Argand diagram below, labelling the imaginary axis intercepts with their values.  (2 marks)
     

     

  1. A ray originating at the point  \(z=2-i\)  passes through the point  \(z=-2+3 i\),  cutting the second circle into two segments.
    1. Sketch the ray on the Argand diagram provided in part c.  (1 mark)

    2. Find the equation of the ray in the form \(\operatorname{Arg}\left(z-z_0\right)=\theta\)  where  \(z_0 \in C\) and \(\theta\) is measured in radians in terms of \(\pi\).  (1 mark)

  3. Find the area of the minor segment formed by the intersection of the ray and the circle.  (2 marks)

Show Answers Only

a.    \(y=\dfrac{3}{2} x-\dfrac{11}{4}\)

b.    \(\abs{z-\left(\dfrac{5}{2}+i\right)} = \dfrac{\sqrt{13}}{2}\)

c.    

d.i.    \(\text{See image above}\)

d.ii.   \(\operatorname{Arg}(z-2+i)=\dfrac{3 \pi}{4}\)

e.   \(A=\pi-2\)

Show Worked Solution

a.    \(\text{Method 1}\)

\(z_1=1+2i, \quad z_2=4\)

\(\abs{z-z_1}=\abs{z-z_2}\)

\(\text{Equation can be written:}\)

\((x-1)^2+(y-2)^2\) \(=(x-4)^2+y^2\)
\(-2x-4y+5\) \(=-8x+16 \ \ \ \text{(all squares cancel)}\)
\(6x+4y\) \(=11\)
\(y\) \(=\dfrac{3}{2} x-\dfrac{11}{4}\)

 
\(\text{Method 2}\)

\(\text {Find line of points equidistant from \(\ z_1\)  and  \(z_2\)}\)

\(m_{\text{line}\ z_1 z_2}=\dfrac{0-2}{4-1}=\dfrac{-2}{3}\)

\(m_{\perp}=\dfrac{3}{2}\)

\(\text{Mid point} \  z_1 z_2 =\dfrac{5+2i}{2}=\left(\dfrac{5}{2},1\right)\)

\(\perp \ \text{bisector:} \ m=\dfrac{3}{2}, \ \text{passes through} \ \left(\dfrac{5}{2}, 1\right)\)

\(y-1\) \(=\dfrac{3}{2}\left(x-\dfrac{5}{2}\right)\)
\(y\) \(=\dfrac{3}{2} x-\dfrac{11}{4}\)

 

b.    \(\text{Centre of circle }=\text {midpoint}\left(z_1 z_2\right)\)

\(z_c=\dfrac{z_1+z_2}{2}=\dfrac{5+2 i}{2}=\dfrac{5}{2}+i\)
 

\(\text{Radius}(r)=\dfrac{1}{2} \times \text{diameter}\)

\(r=\dfrac{1}{2} \sqrt{(4-1)^2+(0-2)^2}=\dfrac{\sqrt{13}}{2}\)
 

\(\text{Circle equation:}\)

\(\abs{z-\left(\dfrac{5}{2}+i\right)} = \dfrac{\sqrt{13}}{2}\)
 

c.    \(\text{Find \(y\)-axis intercepts \((z=y i)\):}\)

\(4=\abs{z-(1+2i)}^2=\abs{yi-(1+2i)}^2=(-1)^2+(y-2)^2\)

\(y^2-4y+1=0 \ \Rightarrow \ =2 \pm \sqrt{3}\)
 

d.i.    \(\text{See image above}\)

d.ii.   \(\operatorname{Arg}(z-2+i)=\dfrac{3 \pi}{4}\)

Mean mark (d.ii.) 56%.

e.    \(\text{Circle radius}=2\)

\(\text{Angle subtending minor segment}=\dfrac{\pi}{2}\)

\(A=\dfrac{r^2}{2}(\theta-\sin \theta)=2\left(\dfrac{\pi}{2}-1\right)=\pi-2\)