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PHYSICS, M5 2021 HSC 14 MC

Which of the following statements correctly describes the gravitational interaction between the Earth and the Moon?

  1. The Earth accelerates towards the Moon.
  2. The net force acting on the Earth is zero.
  3. The Moon and Earth experience equal and opposite accelerations.
  4. The force acting on the Moon is smaller than the force acting on the Earth.
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→ According to Newton’s Third Law, the earth and moon experience equal forces in opposite directions. This causes them to accelerate towards each other.


♦ Mean mark 19%.

PHYSICS, M6 2021 HSC 10 MC

A strong magnet is moved past a copper block at a constant speed as shown.
 

What is the direction of the force acting on the copper block?

  1. To the left
  2. To the right
  3. Into the page
  4. Out of the page
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Show Worked Solution

→ Eddy currents will be induced in the copper block. According to Lenz’s Law, this will produce a force that opposes the motion of the magnet.

→ This is done by minimising the relative motion between the block and the magnet, producing a force on the copper block to the right.


♦♦♦ Mean mark 18%.

CHEMISTRY, M5 2021 HSC 11 MC

Consider this system in a fixed volume at constant temperature.

 

This system is initially at equilibrium. A small amount of solid is added.

Which statement is correct?

  1. The amount of will increase.
  2. The amount of will decrease.
  3. The amount of will not change.
  4. The amount of will increase then decrease.
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Show Worked Solution

The equilibrium constant expression is: 

→ As we can see from the equilibrium constant expression, the value of is only dependent on the concentration of .

→ As a result, the addition of solid will have no affect on the value of and thus has no impact on the equilibrium position.

→ Thus, the amount of will not change.


♦♦♦ Mean mark 19%.

CHEMISTRY, M6 2021 HSC 6 MC

Which row of the table describes what happens when a solution of a weak acid is diluted? (Assume constant temperature.)
 

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A weak acid has the following equilibrium:


 

→ The value of  is only affected by temperature, and thus the value of will remain the same.

→ When the solution is diluted, water is added. According to Le Chatelier’s Principle, the equilibrium will shift to the right to counteract the change.

→ Thus, the equilibrium will shift to the right and increase the extent of ionisation.


♦♦♦ Mean mark 25%.

BIOLOGY, M5 2021 HSC 10 MC

Cystic fibrosis is an autosomal recessive disorder caused by mutations in the CFTR gene. Many different recessive alleles cause cystic fibrosis.

The four most common alleles of the CFTR gene and their frequencies in the Australian population are shown in the table.
 

 

What will be the most common genotype of cystic fibrosis patients in Australia?

  1. a1/a1
  2. a1/a2
  3. A/a1
  4. A/A
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Show Worked Solution

→ A (dominant) where a single copy produces a normal phenotype.

→ a1 and a2 produce cystic fibrosis, with a1 having the highest frequency.

→ Most likely genotype of cystic fibrosis patient is a1/a1.


♦♦♦ Mean mark 22%.

Calculus, MET2 2020 VCAA 5

Let  .

Let    be the function representing the tangent to the graph of at  , where  .

Let be the -intercept of the graph of .

  1. Show that  .   (3 marks)

  2. State the values of for which does not exist.    (1 mark)

  3. State the nature of the graph of when does not exist.   (1 mark)

  4. i.  State all values of for which  . Give your answer correct to four decimal places.   (1 mark)

  5. ii. The graph of has an -intercept at (1, 0).
  6.      State the values of    for which  .
  7.      Give your answers correct to three decimal places.   (1 mark)

The coordinate is the horizontal axis intercept of .

Let be the function representing the tangent to the graph of at  , as shown in the graph below.
 
 
     
 

  1. Find the values of for which the graphs of and , where exists, are parallel and where  .   (3 marks)

Let  , where  .

  1. Show that    for all  .   (1 mark)

A property of the graphs of is that two distinct parallel tangents will always occur at and for all  .

  1. Find all values of such that a tangent to the graph of at , for some  , will have an -intercept at .   (1 mark)

  2. Let  , where    and  .
     
    State any restrictions on the values of , , , and , given that the image of under the transformation always has the property that parallel tangents occur at    and    for all  .   (1 mark)

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  4.  i.
  5. ii.
Show Worked Solution

a.   

 
 

  

 
 
 
 

 
b.   

♦ Mean mark part (b) 46%.

♦♦ Mean mark part (c) 23%.
 

c.   


 

d.i. 


  

d.ii. 

♦♦♦ Mean mark part (d)(ii) 13%.


 

e.   

 
 

 

♦♦♦ Mean mark part (e) 13%.
 
   

 


 

f.   
   
   
   

 
g. 

♦♦♦ Mean mark part (g) 3%.

 
 

 


 

h.   

♦♦♦ Mean mark part (h) 2%.

NETWORKS, FUR2 2020 VCAA 5

The Sunny Coast cricket clubroom is undergoing a major works project.

This project involves nine activities: to .

The table below shows the earliest start time (EST) and duration, in months, for each activity.

The immediate predecessor(s) is also shown.

The duration for activity is missing.
 

   

The information in the table above can be used to complete a directed network.

This network will require a dummy activity.

  1. Complete the following sentence by filling in the boxes provided.    (1 mark)

    This dummy activity could be drawn as a directed edge from the end of activity to the start of activity  
  1. What is the duration, in months, of activity  ?   (1 mark)

  2. Name the four activities that have a float time.   (1 mark)

  3. The project is to be crashed by reducing the completion time of one activity only.

     

    What is the minimum time, in months, that the project can be completed in?    (1 mark)

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Show Worked Solution

a.   

♦♦ Mean mark part (a) 26%.
   

b.   

♦ Mean mark part (b) 49%.

 


  

c.   

♦♦ Mean mark part (c) 23%.


  

d.   

♦♦♦ Mean mark part (d) 12%.

Statistics, MET2 2020 VCAA 3

A transport company has detailed records of all its deliveries. The number of minutes a delivery is made before or after its schedule delivery time can be modelled as a normally distributed random variable, , with a mean of zero and a standard deviation of four minutes. A graph of the probability distribution of is shown below.
 

  1. If  , find to the nearest minute.   (1 mark)

  2. Find the probability, correct to three decimal places, of a delivery being no later than three minutes after its scheduled delivery time, given that it arrives after its scheduled delivery time.   (2 marks)

  3. Using the model described above, the transport company can make 46.48% of its deliveries over the interval  .
  4. It has an improved delivery model with a mean of and a standard deviation of four minutes.
  5. Find the values of , correct to one decimal place, so that 46.48% of the transport company's deliveries can be made over the interval     (3 marks)

A rival transport company claims that there is a 0.85 probability that each delivery it makes will arrive on time or earlier.

Assume that whether each delivery is on time or earlier is independent of other deliveries.

  1. Assuming that the rival company's claim is true, find the probability that on a day in which the rival company makes eight deliveries, fewer than half of them arrive on time or earlier. Give your answer correct to three decimal places.   (2 marks)

  2. Assuming that the rival company's claim is true, consider a day in which it makes deliveries.
    1. Express, in terms of , the probability that one or more deliveries will not arrive on time or earlier.   (1 mark)

    2. Hence, or otherwise, find the minimum value of such that there is at least a 0.95 probability that one or more deliveries will not arrive on time or earlier.   (1 mark)
  3. An analyst from a government department believes the rival transport company's claim is only true for deliveries made before 4 pm. For deliveries made after 4 pm, the analyst believes the probability of a delivery arriving on time or earlier is , where 
  4. After observing a large number of the rival transport company's deliveries, the analyst believes that the overall probability that a delivery arrives on time or earlier is actually 0.75
  5. Let the probability that a delivery is made after 4 pm be .
  6. Assuming that the analyst's belief are true, find the minimum and maximum values of .   (2 marks)

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  5.  i.
  6. ii.

Show Worked Solution

a.   


 

b.   
   
   

 

c.   

 

d.   


 

e.i.   


 

e.ii.  

 
 

 

 

f.   

 
 
   

 

GEOMETRY, FUR2 2020 VCAA 3

Khaleda manufacturers the face cream in Dhaka, Bangladesh.

Dhaka is located at latitude 24° N and longitude 90° E.

Assume that the radius of Earth is 6400 km.

  1. Write a calculation that shows that the radius of the small circle of Earth at latitude 24° N is 5847 km, rounded to the nearest kilometre.   (1 mark)

Khaleda receives an order from Abu Dhabi, United Arab Emirates (24° N, 54° E).

  1. Find the shortest small circle distance between Dhaka and Abu Dhabi.
  2. Round your answer to the nearest kilometre.   (1 mark)

Khaleda sends the order by plane from Dhaka (24° N, 90° E) to Abu Dhabi (24° N, 54° E).

The flight departs Dhaka at 1.00 pm and arrives in Abu Dhabi 11 hours later.

The time difference between Dhaka and Abu Dhabi is two hours.

  1. What time did the flight arrive in Abu Dhabi?   (1 mark)

A helicopter takes the order from the airport to the customer's hotel.

The hotel is 27 km south and 109 km east of the airport.

  1. Show that the bearing of the hotel from the airport is 104°, correct to the nearest degree.   (1 mark)
  2. After the delivery to the hotel, the helicopter returns to its hangar.
  3. The hangar is located due south of the airport.
  4. The helicopter flies directly from the hotel to the hangar on a bearing of 282° .
  5. How far south of the airport is the hangar?
  6. Round your answer to the nearest kilometre.   (1 mark)

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Show Worked Solution

a.   
 

 
 

 

b.   

 

 

 
 

 

c.   

 

d.

  

 

e.

 

 

GEOMETRY, FUR2 2020 VCAA 2

Khaleda has designed a logo for her business.

The logo contains two identical equilateral triangles,

The side length of each triangle is 4.8 cm, shown in the diagram below.
 

  1. Write a calculation to show that the area of one of the triangles, rounded to the nearest centimetre, is 10 cm2.   (1 mark)

In the logo, the two triangles overlap, as shown below. Part of the logo is shaded and part of the logo is not shaded.
 

  1. What is the area of the entire logo?
  2. Round your answer to the nearest square centimetre.   (1 mark)
  3. What is the ratio of the area of the shaded region to the area of the non-shaded region of the logo?   (1 mark)
  4. The logo is enlarged and printed on the boxes for shipping.
  5. The enlarged logo and the original logo are similar shape.
  6. The area of the enlarged logo is four times the area of the original logo.
  7. What is the height, in centimetres, of the enlarged logo?   (1 mark)

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a.   

 
 
 

♦♦ Mean mark part (b) 32%.

 

b.   

♦ Mean mark part (c) 48%.

 

c.   
 
 

 

d.   

♦♦♦ Mean mark part (d) 17%.


 

Calculus, MET2 2020 VCAA 2

An area of parkland has a river running through it, as shown below. The river is shown shaded.

The north bank of the river is modelled by the function  .

The south bank of the river is modelled by the function  .

The horizontal axis points east and the vertical axis points north.

All distances are measured in metres.
 

A swimmer always starts at point , which has coordinates  (50, 30).

Assume that no movement of water in the river affects the motion or path of the swimmer, which is always a straight line.

  1. The swimmer swims north from point .
  2. Find the distance, in metres, that the swimmer needs to swim to get to get to the north bank of the river.   (1 mark)

  3. The swimmer swims east from point .
  4. Find the distance, in metres, that the swimmer needs to swim to get to the north bank of the river.   (2 marks)

  5. On another occasion, the swimmer swims the minimum distance from point to the north bank of the river.
  6. Find this minimum distance. Give your answer in metres, correct to one decimal place.   (2 marks)

  7. Calculate the surface area of the section of the river shown on the graph in square metres.   (1 mark)

  8. A horizontal line is drawn through point . The section of the river that is south of the line is declared a no "no swimming" zone.
  9. Find the area of the "no swimming" zone, correct to the nearest square metre.   (3 marks)

  10. Scientists observe that the north bank of the river is changing over time. It is moving further north from its current position. They model its predicted new location using the function with rule  , where .
  11. Find the values of  for which the distance north across the river, for all parts of the river, is strictly less than 20 m.   (2 marks)

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  5. ²
Show Worked Solution

a.   

 

b. 


 

c.  

♦ Mean mark part (c) 39%.

 

d.  

 

e.  

♦♦ Mean mark part (e) 35%.

 
  ²²  

♦♦♦Mean mark part (f) 15%.

 

f.   

 
     
   

  

GRAPHS, FUR2 2020 VCAA 4

Another section of Kyla's business services and details cars and trucks.

Every vehicle is both serviced and detailed.

Each car takes two hours to service and one hour to detail.

Each truck takes three hours to service and three hours to detail.

Let represent the number of cars that are serviced and detailed each day.

Let represent the number of trucks that are serviced and detailed each day.

Past records suggest there are constraints on the servicing and detailing of vehicles each day.

These constraints are represented by Inequalities 1 to 4 below.

     
     
   
   
     
  1. Explain the meaning of Inequality 1 in the context of this problem.   (1 mark)
  2. Each employee at the business works eight hours per day.
  3. What is the maximum number of employees who can work in the servicing department each day?   (1 mark)

The graph below shows the feasible region (shaded) that satisfies Inequalities 1 to 4 .
 


 

  1. On a day when 20 cars are serviced and detailed, what is the maximum number of trucks that can be serviced and detailed?   (1 mark)
  2. When servicing and detailing, the business makes a profit of $150 per car and $225 per truck.
  3. List the points within the feasible region that will result in a maximum profit for the day.   (2 marks)

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Show Worked Solution

a.   

♦♦ Mean mark part (b) 34%.

 

b.   


 

c.   

♦ Mean mark part (c) 42%.

 

 

 

d.   

♦♦♦ Mean mark part (d) 19%.


 


 

GRAPHS, FUR1 2020 VCAA 10 MC

The feasible region for a linear programming problem is shaded in the diagram below.

The line through points and is horizontal and the line through points and is vertical.

The equation for the objective function for this problem is of the form

     where 

The value(s) of    such that the objective function is maximised only at point    is

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♦♦ Mean mark 31%.


 

 

NETWORKS, FUR1 2020 VCAA 7 MC

Four friends go to an ice-cream shop.

Akiro chooses chocolate and strawberry ice cream.

Doris chooses chocolate and vanilla ice cream.

Gohar chooses vanilla ice cream.

Imani chooses vanilla and lemon ice cream.

This information could be presented as a graph.

Consider the following four statements:

  • The graph would be connected.
  • The graph would be bipartite.
  • The graph would be planar.
  • The graph would be a tree.

How many of these four statements are true?

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
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Show Worked Solution

♦♦♦ Mean mark 11%.

Networks, STD2 N3 SM-Bank 50

Roadworks planned by the local council require 13 activities to be completed.

The network below shows these 13 activities and their completion times in weeks.
 

  1. What is the earliest start time, in weeks, of activity ?  (1 mark)

  2. How many of these activities have zero float time?  (2 marks)

  3. It is possible to reduce the completion time for activities and .
  4. The reduction in completion time for each of these five activities will incur an additional cost.
  5. The table below shows the five activities that can have their completion time reduced and the associated weekly cost, in dollars.
     
       
  6. The completion time for each these five activities can be reduced by a maximum of two weeks.
  7. The overall completion time for the roadworks can be reduced to 16 weeks.
  8. What is the minimum cost, in dollars, of this change in completion time?  (3 marks)

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Show Worked Solution

a.    


 

b.    


 

c.   


  


  


 


 

GRAPHS, FUR2 2021 VCAA 4

Health and training sessions are held each day at the new community centre.

  • Let be the number of sessions for children each day.
  • Let be the number of sessions for adults each day.
  • The total number of all sessions each day must be at least 10.
  • The number of sessions for adults must not be less than the number of sessions for children.
  • It takes 30 minutes for each session for children and 40 minutes for each session for adults.

The constraints on the health and training sessions each day can be represented by the following five inequalities.

Inequality 1     

Inequality 2     

Inequality 3     

Inequality 4       

Inequality 5       

  1. Explain the meaning of Inequality 5 in the context of this situation.  (1 mark)

The graph below shows the feasible region (shaded) that satisfies Inequality 1 to 5.

  1. What is the maximum number of sessions for children each day?  (1 mark)
  2. The community profits from these sessions.
  3. Each session for children makes a profit of $45 and each session for adults makes a profit of $60.
  4.  i.  Determine the maximum profit per day from all health and training sessions. (1 mark)
  5. ii. List all the points within the feasible region that result in this maximum profit. (1 mark)
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  3.  i. 
    ii.
Show Worked Solution

a.   


 

b.    


 

c.i. 

 
 

 

 

c.ii.

 

Statistics, SPEC2 2021 VCAA 6

The maximum load of a lift in a chocolate company's office building is 1000 kg. The masses of the employees who use the lift are normally distributed with a mean of 75 kg and a standard deviation of 8 kg. On a particular morning there are employees about to use the lift.

  1. What is the maximum possible value of for there to be less than a 1% chance of the lift exceeding the maximum load?   (2 marks)

Clare, who is one of the employees, likes to have a hot drink after she exits the lift. The time taken for the drink machine to dispense a hot drink is normally distributed with a mean of 2 minutes and a standard deviation of 0.5 minutes. Times taken to dispense successive hot drinks are independent.

  1. Clare has a meeting at 9.00 am and at 8.52 am she is fourth in the queue for a hot drink. Assume that the waiting time between hot drinks dispensed is negligible and that it takes Clare 0.5 minutes to get from the drink machine to the meeting room.
  2. What is the probability, correct to four decimal places, that Clare will get to her meeting on time?   (2 marks)

Clare is a statistician for the chocolate company. The number of chocolate bars sold daily is normally distributed with a mean of 60 000 and a standard deviation of 5000. To increase sales, the company decides to run an advertising campaign. After the campaign, the mean daily sales from 14 randomly selected days was found to be 63 500.

Clare has been asked to investigate whether the advertising campaign was effective, so she decides to perform a one-sided statistical test at the 1% level of significance.

  1.   i. Write down suitable null and alternative hypotheses for this test.   (1 mark)

  2.  ii. Determine the value, correct for decimal places, for this test.   (1 mark)

  3. iii. Giving a reason, state whether there is any evidence for the success of the advertising campaign.   (1 mark)

  4. Find the range of values for the mean daily sales of another 14 randomly selected days that would lead to the null hypothesis being rejected when tested at the 1% level of significance. Give your answer correct to the nearest integer.   (1 mark)

  5. The advertising campaign has been successful to the extent that the mean daily sales is now 63 000.
  6. A statistical test is applied at the 5% level of significance.
  7. Find the probability that the null hypothesis would be incorrectly accepted, based on the sales of another 14 randomly selected days and assuming a standard deviation of 5000. Give your answer correct to three decimal places.   (2 marks)

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  3.   i.
  4.  ii. 
  5. iii. 
Show Worked Solution

a.   

♦♦ Mean mark 22%.


 


 


 

 

b.   

Mean mark part (b) 51%.


 

 

c.i. 


 

c.ii. 

 

 
 

c.iii. 

   

 

d.   

♦♦♦ Mean mark part (d) 16%.

 

e.   

♦♦♦ Mean mark part (e) 10%.


 


NETWORKS, FUR2 2021 VCAA 4

Roadworks planned by the local council require 13 activities to be completed.

The network below shows these 13 activities and their completion times in weeks.
 

  1. What is the earliest start time, in weeks, of activity ?   (1 mark)

  2. How many of these activities have zero float time?   (1 mark)

  3. It is possible to reduce the completion time for activities and .
  4. The reduction in completion time for each of these five activities will incur an additional cost.
  5. The table below shows the five activities that can have their completion time reduced and the associated weekly cost, in dollars.
     
       
  6. The completion time for each these five activities can be reduced by a maximum of two weeks.
  7. The overall completion time for the roadworks can be reduced to 16 weeks.
  8. What is the minimum cost, in dollars, of this change in completion time?   (1 mark)

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Show Worked Solution

a.    


 

b.    


 

c.   


  


  


 


 

NETWORKS, FUR2 2021 VCAA 3

The network diagram below shows the local road network of Town .

The number on the edges indicate the maximum number of vehicles per hour that can travel along each road in this network.

The arrows represent the permitted direction of travel.

The vertices and represent the intersections of the roads.
 

  1. Determine the maximum number of vehicles that can travel from the entrance to the exit per hour.   (1 mark)

  2. The local council plans to increase the number of vehicles per hour that can travel from the entrance to the exit by increasing the capacity of only one road.
  3.  i. Complete the following sentence by filling in the boxes provided.   (1 mark)

        The road that should have its capacity increased is the road from vertex  
     
    		  to 
     
  4. ii. What should be the minimum capacity of this road to maximise the flow of vehicles from the entrance to the exit?   (1 mark)

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  2.  i.
  3. ii.
Show Worked Solution

a.     

 

 

 

b.i. 


 

b.ii.

 
 


 

MATRICES, FUR2 2021 VCAA 4

Five staff members in Elena's office played a round-robin video game tournament, where each employee played each of the other employees once. In each game there was a winner and a loser.

A table of their one-step and two-step dominances was prepared to summarise the results.
 

   

Consider the results matrix shown below.

A '1' in this matrix shows that the player named in that row defeated the player named in that column.

A '0' in this matrix shows that the player named in that row lost to the player named in that column.

Use all of the information provided to complete the results matrix.   (2 marks)

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Show Worked Solution


 


 

MATRICES, FUR2 2021 VCAA 3

A market research study of shoppers showed that the buying preferences for the three olive oils, Carmani (), Linelli () and Ohana (), change from month to month according to the transition matrix  below.


 

The initial state matrix below shows the number of shoppers who bought each brand of olive oil in July 2021.

Let represent the state matrix describing the number of shoppers buying each brand months after July 2021.

  1. How many of these 8000 shoppers bought a different brand of olive oil in August 2021 from the brand bought in July 2021?   (1 mark)

  2. Using the rule , complete the matrix below.   (1 mark)

  1. Consider the shoppers who were expected to buy Carmani olive oil in August 2021.
  2. What percentage of these shoppers also bought Carmani olive oil in July 2021?
  3. Round your answer to the nearest percentage.   (1 mark)

  4. Write a calculation that shows Ohana olive oil is the brand bought by 50% of these shoppers in the long run.   (1 mark)

  5. Further research suggests more shoppers will buy olive oil in the coming months.
  6. A rule to model this situation is  , where represents the state matrix describing the number of shoppers months after July 2021.


   

  1. represents the extra number of shoppers expected to buy Ohana olive oil each month.
  2. If  , what is the value of ?   (1 mark)

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Show Worked Solution

a.   

 

 
b.    
 


 

c.    


 


 

d.   


 

e.      
 
   

 

CORE, FUR2 2021 VCAA 9

Sienna invests $152 431 into an annuity from which she will receive a regular monthly payment of $900 for 25 years. The interest rate for this annuity is 5.1% per annum, compounding monthly.

  1. Let be the balance of the annuity after monthly payments. A recurrence relation written in terms of and can model the value of this annuity from month to month.
  2. Showing recursive calculations, determine the value of the annuity after two months.
  3. Round your answer to the nearest cent.   (2 marks)

  4. After two years, the interest rate for this annuity will fail to 4.6%.
  5. To ensure that she will still receive the same number of $900 monthly payments, Sienna will add an extra one-pff amount into the annuity at this time.
  6. Determine the value of this extra amount that Sienna will add.
  7. Round your answer to the nearest cent.   (1 mark)

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Show Worked Solution

a.   

 
 
 

 

b.     

 

 

 

 

 

CORE, FUR2 2021 VCAA 8

For renovations to the coffee shop, Sienna took out a reducing balance loan of $570 000 with interest calculated fortnightly.

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

                 
 

  1. Calculate the balance of this loan after the first fortnightly repayment is made.   (1 mark)

  2. Show that the compound interest rate for this loan is 2.6% per annum.   (1 mark)

  3. For the loan to be fully repaid, to the nearest cent, Sienna's final repayment will be a larger amount.
  4. Determine this final repayment amount.
  5. Round your answer to the nearest cent.   (1 mark)

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Show Worked Solution
a.  
   


b. 


 

 

 

c. 

 

 

 

 

 

Mechanics, SPEC2 2021 VCAA 5

A mass of kilograms is placed on a plane inclined at 30° to the horizontal. It is connected by a light inextensible string to a second mass of kilograms that hangs below a frictionless pulley situated at the top end of the incline, over which the string passes.
 


 

  1. Given that the inclined plane is smooth, find the relationship between and if the mass moves down the plane at constant speed.  (2 marks)

The masses are now placed on a rough plane inclined at 30°, with the light inextensible string passing over a frictionless pulley in the same way, as shown in the diagram above. Let be the magnitude of the normal force exerted on the mass by the plane. A resistance force of magnitude acts on and opposes the motion of the mass .

  1. The mass moves up the plane.
  2.   i. Mark and label all forces acting on this mass on the diagram above.  (1 mark)
  3.  ii. Taking the direction up the plane as positive, find the acceleration of the mass in terms of , and .  (2 marks)

Some time after the masses have begun to move, the mass hits the ground at 4.5 ms and the string becomes slack. At this instant, the mass is at the point on the plane, which is 2 m from the pulley. Take the value of to be 0.1

  1. How far from point does the mass travel before it starts to slide back down the plane?
  2. Give your answer in metres, correct to two decimal places.  (2 marks)
  3. Find the time taken, from when the string becomes slack, for the mass to return to point .
  4. Give your answer correct to the nearest tenth of a second.  (3 marks)
Show Answers Only
  2. i. 
       
     
  3. ii.
Show Worked Solution

a.   

 
b.i.   

 

b.ii.   

♦ Mean mark part (b)(ii) 43%.

 

c.   

♦♦♦ Mean mark part (c) 17%.
 

 

 

d.   

♦♦♦ Mean mark part (d) 7%.


 

 

 
 

Vectors, SPEC2 2021 VCAA 4

A car that performs stunts moves along a track, as shown in the diagram below. The car accelerates from rest at point , is launched into the air by the ramp and lands on a second section of track at or beyond point .  This second section of track is inclined at to the horizontal.

Due to tailwind, the effect of air resistance is negligible. Point is taken as the origin of a cartesian coordinate system and all displacements are measured in metres. Point has the coordinates .

At point , the speed of the car is ms and it takes off at an angle of to the horizontal direction. After the car passes point , it follows a trajectory where the position of the car's rear wheels relative to point , is given by

  until the car lands on the second section of track that starts at point .
 

  1. Show that the path of the rear wheels of the car, while in the air, is given in cartesian form by
  2.    .   (1 mark)

  3. If  , find the minimum speed, in ms, that the car must reach at point for the rear wheels to land on the second section of track at or beyond point . Give your answer correct to two decimal places.   (2 marks)

  4. The ramp    is constructed so that the angle can be varied.
  5. For what values of and will the path of the rear wheels of the car join up smoothly with the beginning of the second section of track at point ? Give your answer for in degrees, correct to the nearest degree, and give your answer for in ms, correct to one decimal place.   (3 marks)

The car accelerates from rest along the horizontal section of track , where its acceleration,   ms, after it has travelled metres from point , is given by  , where is its speed at metres.

  1. Show that in terms of given by  .   (2 marks)

  2. After the car leaves point , it accelerates to reach a speed of 20 ms at point . However, if the stunt is called off, the car immediately brakes and reduces its speed at a rate of 9 ms.  It is only safe to call off the stunt if the car can come to rest at or before point .  Point is the furthest point along the section at which the stunt can be called off.
  3. How far is point from point ?  Give your answer in metres, correct to one decimal place.   (3 marks)

Show Answers Only
Show Worked Solution

a.   

 

 

b.   

♦ Mean mark part (b) 45%.

 

c.   

♦♦♦ Mean mark part (c) 12%.

 

d.   

♦ Mean mark part (d) 40%.

 

e.   

♦♦♦ Mean mark part (e) 8%.

 

 

 

 

Calculus, SPEC2 2021 VCAA 3

A thin-walled vessel is produced by rotating the graph of    about the -axis for  .

All lengths are measured in centimetres.

    1. Write down a definite integral in terms of and for the volume of the vessel in cubic centimetres.   (1 mark)

    2. Hence, find an expression for the volume of the vessel in terms of .   (1 mark)

Water is poured into the vessel. However, due to a crack in the base, water leaks out at a rate proportional to the square root of the depth of water in the vessel, that is  , where is the volume of water remaining in the vessel, in cubic centimetres, after minutes.

    1. Show that  .   (2 marks)

    2. Find the maximum rate, in centimetres per minute, at which the depth of water in the vessel decreases, correct to two decimal places, and find the corresponding depth in centimetres.   (2 marks)

    3. Let    for a particular vessel. The vessel is initially full and water continues to leak out at a rate of    cm³ min.
    4. Find the maximum rate at which water can be added, in cubic centimetres per minute, without the vessel overflowing.   (1 mark)

  2. The vessel is initially full where    and water leaks out at a rate of    cm³ min. When the depth of the water drops to 25 cm, extra water is poured in at a rate of    cm³ min.
  3. Find how long it takes for the vessel to refill completely from a depht of 25 cm. Give your answer in minutes, correct to one decimal place.   (3 marks)

Show Answers Only
    3. ³
Show Worked Solution

a.i.   

 

a.ii.   

 

b.i.   

 

 

b.ii.   

♦ Mean mark part (b)(ii) 42%.

♦♦ Mean mark part (b)(iii) 24%.

 

b.iii.   ³

³

 

c.   

♦♦♦ Mean mark part (c) 16%.

 

 

Calculus, SPEC2 2021 VCAA 1

Let  .

  1. Express in the form  , where , an are real constants.   (1 mark)

  2. State the equation of the asymptotes of the graph of .   (2 marks)

  3. Sketch the graph of on the set of axes below. Label the asymptotes with their equations, and label the maximum turning point and the point of inflection with their coordinates, correct to two decimal places. Label the intercepts with the coordinate axes.   (3 marks)

     

     

  4. Let  , where is a real constant.
  5.  i. For what values of will the graph of , have two asymptotes?   (2 marks)

  6. ii. Given that the graph of has more than two asymptotes, for what values of will the graph of have no stationary points?   (2 marks)

Show Answers Only

  3.  
  4.  i.
  5. ii.
Show Worked Solution

a.   


 

b.   

♦ Mean mark part (c) 48%.

 
c.   

 

d.i.   

♦♦ Mean mark part (d)(i) 24%.

 

d.ii.   

♦♦♦ Mean mark part (d)(ii) 16%.


 

Calculus, MET2 2021 VCAA 5

Part of the graph of    is shown below.
 

  1. State the period of .   (1 mark)

  2. State the minimum value of , correct to three decimal places.   (1 mark)

  3. Find the smallest positive value of for which  .   (1 mark)
Consider the set of functions of the form  , where is a positive integer.
  1. State the value of such that    for all .   (1 mark)

  2. i.  Find an antiderivative of in terms of .   (1 mark)

  3. ii. Use a definite integral to show that the area bounded by and the -axis over the interval    is equal above and below the -axis for all values of .  (3 marks)

  4. Explain why the maximum value of cannot be greater than 2 for all values of and why the minimum value of cannot be less than –2 for all values of .   (1 mark)

  5. Find the greatest possible minimum value of .   (1 mark)

Show Answers Only
  5. i. 
    ii.
Show Worked Solution

a.   


 

b.   

♦ Mean mark part (c) 21%.
 

c.     


 

d.   


 

e.i. 

♦ Mean mark part (e)(i) 50%.


 

e.ii.   

♦ Mean mark part (e)(ii) 29%.


 


 

f.   

♦ Mean mark part (f) 13%.


 

g.   

♦ Mean mark part (g) 2%.

Probability, MET2 2021 VCAA 4

A teacher coaches their school's table tennis team.

The teacher has an adjustable ball machine that they use to help the players practise.

The speed, measured in metres per second, of the balls shot by the ball machine is a normally distributed random variable .

The teacher sets the ball machine with a mean speed of 10 metres per second and standard deviation of 0.8 metres per second.

  1. Determine  , correct to three decimal places.   (1 mark)

  2. Find the value of , in metres per second, which 80% of ball speeds are below. Give your answer in metres per second, correct to one decimal place.   (1 mark) 

The teacher adjusts the height setting for the ball machine. The machine now shoots balls high above the table tennis table.

Unfortunately, with the new height setting, 8% of balls do not land on the table.

Let    be the random variable representing the sample proportion of the balls that do not land on the table in random samples of 25 balls.

  1. Find the mean and the standard deviation of  .   (2 marks)

  2. Use the binomial distribution to find  , correct to three decimal places.   (2 marks) 

The teacher can also adjust the spin setting on the ball machine.

The spin, measured in revolutions per second, is a continuous random variable  with the probability density function
 

       
 

  1. Find the maximum possible spin applied by the ball machine, in revolutions per second.   (1 mark)

Show Answers Only

  6. This content is no longer in the Study Design.
  7. This content is no longer in the Study Design.
  8. This content is no longer in the Study Design.

Show Worked Solution

a.   


 

b.   
 

c.   

♦ Mean mark part (c) 45%.


 

d.   

♦ Mean mark part (d) 49%.

 
 
 

 

e.   

♦♦ Mean mark part (e) 21%.

 

f.    This content is no longer in the Study Design.

g.    This content is no longer in the Study Design.

h.    This content is no longer in the Study Design.

Calculus, MET2 2021 VCAA 3

Let  .

  1. State the maximal domain and the range of .   (2 marks)

  2.  i. Find the equation of the tangent to the graph of when  .   (1 mark)

  3. ii. Find the equation of the line that is perpendicular to the graph of when    and passes through the point  (-2, 0).   (1 mark)

Let 

  1. Explain why is not a one-to one function.   (1 mark)

  2. Find the gradient of the tangent to the graph of at  .   (1 mark)

The diagram below shows parts of the graph of and the line  .
 
 
                     
 
The line    and the tangent to the graph of at    intersect with an acute angle of between them.

  1. Find the value(s) of for which  . Give your answer(s) correct to two decimal places.   (3 marks)

  2. Find the -coordinate of the point of intersection between the line  and the graph of , and hence find the area bounded by  , the graph of and the -axis, both correct to three decimal places.   (3 marks)

Show Answers Only

  2. i. 
    ii.
Show Worked Solution

a.   


 

 


 
 

b.     i.   


 

ii.   


  

c.    


 

d.  


 
 

e. 

 


 


 

f.    


 

 

Calculus, MET2 2021 VCAA 2

Four rectangles of equal width are drawn and used to approximate the area under the parabola    from    to  .

The heights of the rectangles are the values of the graph of  at the right endpoint of each rectangle, as shown in the graph below.
 

  1. State the width of each of the rectangles shown above.  (1 mark)
  2. Find the total area of the four rectangles shown above.  (1 mark)
  3. Find the area between the graph of  , the -axis and the line  .  (2 marks)
  4. The graph of is shown below.
     
         

    Approximate    using four rectangles of equal width and the right endpoint of each rectangle.  (1 mark)

Parts of the graphs of    and    are shown below.
 
     

  1. Find the area of the shaded region.  (1 mark)
  2. The graph of  is transformed to the graph of  , where  .
  3. Find the values of such that the area defined by region(s) bounded by the graphs of    and    and the lines    and    is equal to . Give your answer correct to two decimal places.  (4 marks)
Show Answers Only
Show Worked Solution

a.   

 

b.    
   

 

c.    
   

 

d.

♦♦♦ Mean mark part (d) 16%.

 

 

e.  
   

 
f.   

♦♦♦ Mean mark part (f) 18%.


  

Vectors, SPEC1 2021 VCAA 9

Let    and    be the position vectors relative to a fixed point of particle and particle respectively for  , where is a positive real constant.

    1. Show that the cartesian equation of the path of particle is  .   (1 mark)

    2. Show that the cartesian equation of the path of particle in the first quadrant can be written as  .   (1 mark)

    1. Show that the particles and will collide.   (1 mark)

    2. Hence, find the coordinates of the point of collision of the two particles.   (1 mark)

    1. Show that  .   (2 marks)


    2.    

      Hence, find the area bounded by the graph of  ,  the -axis and the lines    and  ,  as shown in the diagram above. Give your answer in the form  , where and are positive integers.   (2 marks)

Show Answers Only
Show Worked Solution

a.i.   

 

 

a.ii. 

♦ Mean mark part (a)(ii) 41%.

 

 
 

 

b.i.   

 

 

 

 

b.ii.   

 
♦ Mean mark part (c)(i) 37%.

 

c.i.  

♦ Mean mark part (c)(ii) 45%.

 

c.ii.   
   
   
   
   
   
   
   

GRAPHS, FUR1 2021 VCAA 8 MC

A company produces two types of tennis racquets: the Smash and the Volley.

Let    be the number of Smash tennis racquets that are produced each week.

Let  be the number of Volley tennis racquets that are produced each week.

The constraints on the production of these tennis racquets are given by the following set of inequalities.

 

The graph below shows the lines that represent the boundaries of these inequalities.
 

The profit, , that the company can make each week from the sale of these tennis racquets is in the form  .

The maximum profit will occur when 20 Smash tennis racquets and 40 Volley tennis racquets are sold each week.

If  , the maximum profit is

  1.   600
  2.   900
  3. 1000
  4. 1200
  5. 1500
Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 29%.

 

 
 

 

  

NETWORKS, FUR1 2021 VCAA 8 MC

A network of roads connecting towns in an alpine region is shown below.

The distances between neighbouring towns, represented by vertices, are given in kilometres.
 

The region receives a large snowfall, leaving all roads between the towns closed to traffic.

To ensure each town is accessible by car from every other town, some roads will be cleared.

The minimal total length of road, in kilometres, that needs to be cleared is

  1. 361 if  = 50 and  = 55
  2. 361 if  = 50 and  = 60
  3. 366 if  = 55 and  = 55
  4. 366 if  = 55 and  = 60
  5. 371 if  = 55 and  = 65
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Show Worked Solution


 

♦ Mean mark 48%.

   

   

MATRICES, FUR1 2021 VCAA 8 MC

A new colony of endangered marsupials is established on a remote island.

For one week, the marsupials can feed from only one of three feeding stations: , or .

On Monday, 50% of the marsupials were observed feeding at station and 50% were observed feeding at station . No marsupials were observed feeding .

The marsupials are expected to change their feeding stations each day this week according to the transition matrix .


 

Let  represent the state matrix showing the percentage of marsupials observed feeding at each feeding station   days after Monday of this week.

The matrix recurrence rule    is used to model this situation.

From Tuesday to Wednesday, the percentage of marsupials who are not expected to change their feeding location is

  1. 44.5%
  2. 45%
  3. 50%
  4. 51.5%
  5. 52
Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 29%.


  


 

Functions, MET1 2021 VCAA 9

Consider the unit circle    and the tangent to the circle at the point , shown in the diagram below.
  

  1. Show that the equation of the line that passes through the points and is given by  .   (2 marks)

Let  ,  where  , and let the graph of the function be the transformation of the line that passes through the points and under .

    1. Find the values of for which the graph of intersects with the unit circle at least once.   (1 mark)

    2. Let the graph of intersect the unit circle twice.
    3. Find the values of for which the coordinates of the points of intersection have only positive values.   (1 mark)

  2. For  , let  be the point of intersection of the graph of with the unit circle, where  is always the point of intersection that is closest to , as shown in the diagram below.
     
         

Let be the function that gives the area of triangle in terms of .

    1. Define the function .   (2 marks)

    2. Determine the maximum possible area of triangle .   (2 marks)

Show Answers Only
  2.  i.
  3. ii.
  4. i.
  5. ii. ²
Show Worked Solution

a.   

♦♦♦ Mean mark part (a) 18%.

 
 

 

♦♦♦ Mean mark part (b)(i) 6%.
bi. 

 

b.ii.

 

♦♦♦ Mean mark part (b)(ii) 4%.


 

c.i.   `text(If)\ \ q=1, \ P^{′} = P and theta=pi/3`

♦♦♦ Mean mark part (c)(i) 10%.
 
   

 
c.ii. 

♦♦♦ Mean mark part (c)(ii) 12%.

²

Probability, MET1 2021 VCAA 6

An online shopping site sells boxes of doughnuts.

A box contains 20 doughnuts. There are only four types of doughnuts in the box. They are:

    • glazed, with custard
    • glazed, with no custard
    • not glazed, with custard
    • not glazed, with no custard

It is known that, in the box:

    •   of the doughnuts are with custard
    •   of the doughnuts are not glazed
    •   of the doughnuts are glazed, with custard
  1. A doughnut is chosen at random from the box.
  2. Find the probability that it is not glazed, with custard.  (1 mark)

  3. The 20 doughnuts in the box are randomly allocated to two new boxes, Box A and Box B.
  4. Each new box contains 10 doughnuts.
  5. One of the two new boxes is chosen at random and then a doughnut from that box is chosen at random.
  6. Let be the number of glazed doughnuts in Box A.
  7. Find the probability, in terms of , that the doughnut comes from box B given that it is glazed.  (2 marks)

  8. The online shopping site has over one million visitors per day.
  9. It is know that half of these visitors are less than 25 years old.
  10. Let    be the random variable representing the proportion of visitors who are less than 25 years old in a random sample of five visitors.
  11. Find  . Do not use a normal approximation.  (3 marks)

Show Answers Only

Show Worked Solution

a.   

 
   

 
b.   

♦♦♦ Mean mark part (b) 20%.

 
   
   
   

 
c.   

♦ Mean mark part (c) 40%.

 
   
   
   

Vectors, EXT2 V1 2021 HSC 16a

  1. The point    lies on the sphere of radius 1 centred at the origin .
  2. Using the position vector of , and the triangle inequality, or otherwise, show that .  (2 marks)

  3. Given the vectors    and  , show that 
  4.    .  (3 marks)

  5. As in part (i), the point    lies on the sphere of radius 1 centred at the origin .
  6. Using part (ii), or otherwise, show that  .  (2 marks)

Show Answers Only
Show Worked Solution
i.     
♦ Mean mark (i) 50%.
 
 
 

 

ii.   

♦ Mean mark (ii) 42%.

 

 

 

iii. 

♦♦♦ Mean mark (iii) 14%.

Proof, EXT2 P1 2021 HSC 15b

For integers  , the triangular numbers    are defined by  ,  giving    and so on.

For integers  ,  the hexagonal numbers    are defined by  ,  giving    and so on.

  1. Show that the triangular numbers  , and so on, are also hexagonal numbers.  (2 marks)

  2. Show that the triangular numbers  , and so on, are not hexagonal numbers.  (1 mark)

Show Answers Only
Show Worked Solution

i.   

♦ Mean mark part (i) 50%.

 
 

 

 

ii.   

♦ Mean mark part (ii) 1%!

 

 

 

Proof, EXT2 P1 2021 HSC 15a

For all non-negative real numbers and .  (Do NOT prove this.)

  1. Using this fact, show that for all non-negative real numbers , and ,
  2.     .  (2 marks)

  3. Using part (i), or otherwise, show that for all non-negative real numbers , and ,  
  4.     .  (2 marks)

Show Answers Only
Show Worked Solution

i.   

♦ Mean mark part (i) 50%.


 

 

 

ii.   

♦♦♦ Mean mark part (ii) 26%.

 

 

Mechanics, EXT2 M1 2021 HSC 13d

An object is moving in simple harmonic motion along the -axis. The acceleration of the object is given by    where is its displacement from the origin, measured in metres, after seconds.

Initially, the object is 5.5 metres to the right of the origin and moving towards the origin. The object has a speed of 8 m s as it passes through the origin.

  1. Between which two values of is the particle oscillating?  (2 marks)

  2. Find the first value of for which  ,  giving the answer correct to 2 decimal places.  (2 marks)

Show Answers Only
Show Worked Solution

i.   

♦ Mean mark 47%.

 


 

 

 

ii.   

♦♦ Mean mark 24%.


 

  

  

 

Calculus, EXT1 C3 2021 HSC 12a

The direction field for a differential equation is shown below.

The graph of a particular solution to the differential equation passes through the point .

On the graph, sketch the graph of this particular solution.  (1 mark)

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Show Worked Solution

♦♦♦ Mean mark 18%!
MARKER COMMENT: A solution curve does not cross any tangent line.

Financial Maths, STD1 F3 2021 HSC 30

Blake opens a new credit card account on 1 May. He uses it, for the first time, on 4 May to buy concert tickets for $850.

He makes no further purchases or repayments during the month of May.

A statement for the credit card is issued on the last day of each month.

The statement for May shows that interest is charged at 19.75% per annum, compounding daily, from 20 May (included) until 31 May (included).

  1. What is the compound interest shown on the statement issued on 31 May?  (3 marks)

  2. The minimum payment is calculated as 3% of the closing balance on 31 May. Calculate the minimum payment.  (1 mark)

Show Answers Only
Show Worked Solution

a.   

♦♦♦ Mean mark part (a) 14%.
 
 

 

 

 

♦♦♦ Mean mark part (b) 15%.
b.   
   

Measurement, STD1 M3 2021 HSC 28

A right-angled triangle is shown. The length of is 16 cm and .
 


 

  1. Find the side length, , of the triangle in centimetres, correct to two decimal places.  (2 marks)

  2. Hence, find the area of triangle in square centimetres, correct to one decimal place.  (3 marks)

Show Answers Only
Show Worked Solution
♦♦ Mean mark part (a) 24%.
a.   
 
   
   

 

b. 

♦♦♦ Mean mark part (b) 14%.
 
   
   
   

Financial Maths, STD1 F3 2021 HSC 27

Tracy takes out a 30-year reducing balance loan of $680 000 to buy a house. Interest is charged at 0.25% per month. The loan is to be repaid in equal monthly instalments of $2866.91 over a term of 30 years.

Part of a spreadsheet used to model the reducing balance loan is shown.
 

   

  1. Find the amount owing at the end of the second month.  (2 marks)

  2. Suppose that the interest rate reduces to 0.15% per month and the monthly instalments remain as $2866.91.
  3. What will happen to the term of the loan? Explain your answer without using calculations.  (2 marks)

Show Answers Only


Show Worked Solution

a.   

♦♦ Mean mark part (a) 31%.

 

 

b.   

♦♦♦ Mean mark part (b) 18%.

Financial Maths, STD1 F1 2021 HSC 19

Yin purchased a car for $20 000. The value of the car decreases according to a linear model. The graph shows the value of the car, $, against the time, t months, since it was purchased.
 


 

  1. By how much does the value of the car decrease every 10 months?  (1 mark)

  2. Find the value of the car after 5 years.  (1 mark)

  3. Identify ONE problem with using this model to determine the value of Yin’s car over time.  (1 mark)

Show Answers Only
Show Worked Solution
a.   
   

 

b.   

♦♦♦ Mean mark part (c) 11%.

 

c.   

Calculus, 2ADV C4 2021 HSC 28

The region bounded by the graph of the function    and the coordinate axes is shown
 

  1. Show that the exact area of the shaded region is given by  .  (3 marks)

  2. A new function    is found by taking the graph of     and translating it by 5 units to the right.
  3. Sketch the graph of    showing the -intercept and the asymptote.  (2 marks)

  4. Hence, find the exact value of  .  (1 mark)

Show Answers Only
  2.  
Show Worked Solution

a.   

 
 
 
  ²

♦ Mean mark part (b) 48%.

b.


 

♦♦♦ Mean mark part (c) 13%.

c.   

Statistics, 2ADV S3 2021 HSC 33

People are given a maximum of six hours to complete a puzzle. The time spent on the puzzle, in hours, can be modelled using the continuous random variable which has probability density function

The graph of the probability density function is shown below. The graph has a local maximum.
 

  1. Show that  .  (2 marks)

  2. Show that the mode of is two hours.  (2 marks)

  3. Show that  .  (2 marks)

  4. The Intelligence Quotient (IQ) scores of people are normally distributed with a mean of 100 and standard deviation of 15.
  5. It has been observed that the puzzle is generally completed more quickly by people with a high IQ.
  6. It is known that 80% of people with an IQ greater than 130 can complete the puzzle in less than two hours.
  7. A person chosen at random can complete the puzzle in less than two hours.
  8. What is the probability that this person has an IQ greater than 130? Give your answer correct to three decimal places.  (2 marks)

Show Answers Only
Show Worked Solution

a. 

b. 

♦♦♦ Mean mark part (b) 24%.

♦♦ Mean mark part (c) 30%.
c.   
   
   
   
   
   

 

d.  

♦♦ Mean mark part (d) 25%.