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Measurement, STD1 M4 2021 HSC 6 MC

Blood pressure is recorded as systolic pressure/diastolic pressure and measured in mmHg.

When standing upright, Mary’s blood pressure was 139/85. Three minutes after sitting down, her blood pressure was 118/74.

What was the change in Mary’s diastolic blood pressure?

  1. There was an increase of 11 mmHg.
  2. There was an increase of 21 mmHg.
  3. There was a decrease of 11 mmHg.
  4. There was a decrease of 21 mmH.
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♦ Mean mark 45%.

Calculus, 2ADV C4 2021 HSC 27

Kenzo has a solar powered phone charger. Its power, , can be modelled by the function

,

where    is the number of hours after sunrise.

  1. Sketch the graph of  for  .  (2 marks)

Power is the rate of change of energy. Hence the amount of energy, units, generated by the solar powered phone charger from    to  ,  where  is given by

.

  1. Show that  .  (2 marks)

  2. To make a phone call, a phone battery needs at least 300 units of energy. Kenzo woke up 3 hours after sunrise and found that his phone battery had no units of energy. He immediately began to use his solar powered charger to charge his phone battery.
  3. Find the least amount of time he needed to wait before he could make a phone call. Give your answer correct to the nearest minute.  (3 marks)

  4. The next day, Kenzo woke up 6 hours after sunrise and again found that his phone battery had no units of energy. He immediately began to use his solar powered charger to charge his phone battery.
  5. Would it take more time or less time or the same amount of time, compared to the answer in part (c), to charge his phone battery in order to make a phone call? Explain your answer by referring to the graph drawn in part (a).  (1 mark)

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  1.  
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a.

b.   
   
   
   

 

♦♦ Mean mark part (c) 29%
COMMENT: It is arguable that the nearest minute may also be 3h 58 m as the phone is not adequately charged at 3h 57 m.

c.   

 
 

 

 

♦♦ Mean mark part (d) 22%

d.   

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)

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a. 

b. 

♦♦♦ Mean mark part (b) 24%.

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

 

d.  

♦♦ Mean mark part (d) 25%.

Statistics, 2ADV S2 2021 HSC 17

For a sample of 17 inland towns in Australia, the height above sea level, (metres), and the average maximum daily temperature, (°C), were recorded.

The graph shows the data as well as a regression line.
 

     
 

The equation of the regression line is  .

The correlation coefficient is  .

  1. i.  By using the equation of the regression line, predict the average maximum daily temperature, in degrees Celsius, for a town that is 540 m above sea level. Give your answer correct to one decimal place.  (1 mark)

  2. ii. The gradient of the regression line is −0.011. Interpret the value of this gradient in the given context.  (2 marks)

  3. The graph below shows the relationship between the latitude, (degrees south), and the average maximum daily temperature, (°C), for the same 17 towns, as well as a regression line.
     
     
         
     
    The equation of the regression line is  .
  4. The correlation coefficient is  .
  5. Another inland town in Australia is 540 m above sea level. Its latitude is 28 degrees south.
  6. Which measurement, height above sea level or latitude, would be better to use to predict this town’s average maximum daily temperature? Give a reason for your answer.  (1 mark)

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  1. i. 
  2. ii.
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a.i.   
   
   
 

a.ii. 


 

b. 

Statistics, STD2 S4 2021 HSC 33

For a sample of 17 inland towns in Australia, the height above sea level, (metres), and the average maximum daily temperature, (°C), were recorded.

The graph shows the data as well as a regression line.
 

     
 

The equation of the regression line is  .

The correlation coefficient is  .

  1. i.  By using the equation of the regression line, predict the average maximum daily temperature, in degrees Celsius, for a town that is 540 m above sea level. Give your answer correct to one decimal place.  (1 mark)

  2. ii. The gradient of the regression line is −0.011. Interpret the value of this gradient in the given context.  (2 marks)

  3. The graph below shows the relationship between the latitude, (degrees south), and the average maximum daily temperature, (°C), for the same 17 towns, as well as a regression line.
     
     
         
     
    The equation of the regression line is  .
  4. The correlation coefficient is  .
  5. Another inland town in Australia is 540 m above sea level. Its latitude is 28 degrees south.
  6. Which measurement, height above sea level or latitude, would be better to use to predict this town’s average maximum daily temperature? Give a reason for your answer.  (1 mark)

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  1. i. 
  2. ii.
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a.i.   
   
   
♦♦ Mean mark part a.ii. 28%.

 
a.ii. 


 

♦♦♦ Mean mark part b 18%.

b. 

Financial Maths, 2ADV M1 2021 HSC 29

  1. On the day that Megan was born, her grandfather deposited $5000 into an account earning 3% per annum compounded annually. On each birthday after this, her grandfather deposited $1000 into the same account, making his final deposit on Megan’s 17th birthday. That is, a total of 18 deposits were made.
  2. Let be the amount in the account on Megan’s th birthday, after the deposit is made.
  3. Show that  .  (2 marks)

  4. On her 17th birthday, just after the final deposit is made, Megan has $30 025.83 in her account. You are NOT required to show this.
  5. Megan then decides to leave all the money in the same account continuing to earn interest at 3% per annum compounded annually. On her 18th birthday, and on each birthday after this, Megan withdraws $2000 from the account.
  6. How many withdrawals of $2000 will Megan be able to make?  (3 marks)

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a.   

 
 
 

 

b.   

 
 

 

 
 

 

Measurement, STD2 M6 2021 HSC 32

A right-angled triangle    is cut out from a semicircle with centre . The length of the diameter    is 16 cm and    = 30°, as shown on the diagram.
 


 

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

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

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a.   
 
   
   

 

b.   
   
   

♦ Mean mark part (b) 36%.
 
   
   

 

 
   
   

Financial Maths, 2ADV M1 2021 HSC 25

A table of future value interest factors for an annuity of $1 is shown.
 

   

Simone deposits $1000 into a savings account at the end of each year for 8 years. The interest rate for these 8 years is 0.75% per annum, compounded annually.

After the 8th deposit, Simone stops making deposits but leaves the money in the savings account. The money in her savings account then earns interest at 1.25% per annum, compounded annually, for a further two years.

Find the amount of money in Simone's savings account at the end of ten years.  (3 marks)

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Mean mark 52%.

 
 
 
 

Statistics, STD2 S5 2021 HSC 38

A random variable is normally distributed with mean 0 and standard deviation 1. The table gives the probability that this random variable lies between 0 and for different values of .

 

The probability values given in the table for different values of are represented by the shaded area in the following diagram.
 

  1. Using the table, show that the probability that a value from a random variable that is normally distributed with mean 0 and standard deviation 1 is greater than 0.3 is equal to 0.3821.  (1 mark)

  2. Birth weights are normally distributed with a mean of 3300 grams and a standard deviation of 570 grams. By first calculating a -score, find how many babies, out of 1000 born, are expected to have a birth weight greater than 3471 grams.  (3 marks)

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a.   

♦♦♦ Mean mark part (a) 1%.
COMMENT: Note the Std2 and Advanced questions varied slightly but used the same table and graph.


 

b.  
   
   

 

  

 
 
 

Calculus, 2ADV C3 2021 HSC 26

A particle is shot vertically upwards from a point 100 metres above ground level.

The position of the particle, metres above the ground after seconds, is given by

.

  1. Find the maximum height above ground level reached by the particle.  (2 marks)

  2. Find the velocity of the particle, in metres per second, immediately before it hits the ground, leaving your answer in the form  ,  where    and    are integers.  (3 marks)

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a.   

 

 

 

b.   

♦ Mean mark 38%.

 

 
 

 

 

 

Algebra, STD2 A4 2021 HSC 24

A population,  ,  is to be modelled using the function  ,  where is the time in years.

  1. What is the initial population?  (1 mark)

  2. Find the population after 5 years.  (1 mark)

  3. On the axes below, draw the graph of the population against time, showing the points at    and at  .  (2 marks)
      

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a.   
 
   

 

b.   

 
   
   

 

♦ Mean mark (c) 48%.

c. 

Networks, STD1 N1 2021 HSC 17

The network diagram shows the travel times in minutes along roads connecting a number of different towns.
 


 

  1. Draw a minimum spanning tree for this network and determine its length.  (3 marks)

  2. How long does it take to travel from Queentown to Underwood using the fastest route?  (1 mark)

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a.  

♦ Mean mark part (a) 40%.

 

 
 
b.  
   

Networks, STD2 N2 2021 HSC 23

The network diagram shows the travel times in minutes along roads connecting a number of different towns.
 


 

  1. Draw a minimum spanning tree for this network and determine its length.  (3 marks)

  2. How long does it take to travel from Queentown to Underwood using the fastest route?  (1 mark)

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a.  

 

 
 
b.  
   

Financial Maths, STD2 F1 2021 HSC 22

The table shows the income tax rates for the 2020-2021 financial year.
 

   

William has a gross annual salary of $84 000. He has allowable tax deductions of $900 for home-office equipment and $474 for union fees. William must also pay a Medicare Levy of 2% of his taxable income.

Calculate the total tax payable by William including the Medicare Levy.  (3 marks)

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Show Worked Solution
 
   
 
   
 
   

 

 
   

Measurement, STD2 M1 2021 HSC 16

The volume, , of a sphere is given by the formula

where is the radius of the sphere.

A tank consists of the bottom half of a sphere of radius 2 metres, as shown.
 

Find the volume of the tank in cubic metres, correct to one decimal place.   (2 marks)

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Probability, 2ADV S1 2021 HSC 6 MC

There are 8 chocolates in a box. Three have peppermint centres (P) and five have caramel centres (C).

Kim randomly chooses a chocolate from the box and eats it. Sam then randomly chooses and eats one of the remaining chocolates.

A partially completed probability tree is shown.
 

What is the probability that Kim and Sam choose chocolates with different centres?

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Measurement, STD2 M7 2021 HSC 27

The price and the power consumption of two different brands of television are shown.

The average cost for electricity is 25c/kWh. A particular family watches an average of 3 hours of television per day.

  1. The annual cost of electricity for Television A for this family is $48.18.
  2. For this family, what is the difference in the annual cost of electricity between Television A and Television B?  (2 marks)

  3. For this family, how many years will it take for the total cost of buying and using Television A to be equal to the cost of buying and using Television B?  (2 marks)

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a.  
   
   

♦ Mean mark part (a) 49%.
 
   

 

 
   

♦♦ Mean mark part (b) 23%.
b.  
   

 

 
   

Financial Maths, STD2 F4 2021 HSC 26

Nina plans to invest $35 000 for 1 year. She is offered two different investment options.

Option A:  Interest is paid at 6% per annum compounded monthly.

Option B:  Interest is paid at % per annum simple interest.

  1. Calculate the future value of Nina's investment after 1 year if she chooses Option A.  (2 marks)

  2. Find the value of in Option B that would give Nina the same future value after 1 year as for Option A. Give your answer correct to two decimal places.  (2 marks)

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a.  
 

 

 
 

♦♦ Mean mark part (b) 36%.
b.  
 
 
   
   

Measurement, STD2 M2 2021 HSC 20

City A is in Sweden and is located at (58°N, 16°E). Sydney, in Australia, is located at (33°S, 151°E).

Robert lives in Sydney and needs to give an online presentation to his colleagues in City A starting at 5:00 pm Thursday, local time in Sweden.

What time and day, in Sydney, should Robert start his presentation?

It is given that 15° = 1 hour time difference. Ignore daylight saving.  (3 marks)

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Mean mark 52%.


 

 
   

Statistics, STD2 S1 2021 HSC 17

The five-number summary of a dataset is given.

Lowest score = 1

Lowest quartile () = 4

Median () = 7

Upper quartile () = 10

Highest score = 20

Is 20 an outlier? Justify your answer with calculations.  (2 marks)

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COMMENT: Stating 20 > 19 is necessary to justify this answer.
 
 

  

Measurement, NAP-E3-NC03v1

At the start of the day, a farmer drove around his property inspecting the fences and livestock.

He started the drive at 6:15 and finished at 7:04.

How long did the farmer drive for?

 
 
 
 
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Networks, STD2 N3 FUR2 4

Training program 1 has the cricket team starting from exercise station and running to exercise station .

For safety reasons, the cricket coach has placed a restriction on the maximum number of people who can use the tracks in the fitness park.

The directed graph below shows the capacity of the tracks, in number of people per minute.
 


 

  1. Determine the capacity of Cut 1, shown above.  (1 mark)

  2. What is the maximum flow from to , in number of people per minute?  (1 mark)

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a.  
   

 

b.   

♦♦ Mean mark part (c) 32%.


 

NETWORKS, FUR2 2020 VCAA 3

A local fitness park has 10 exercise stations: to .

The edges on the graph below represent the tracks between the exercise stations.

The number on each edge represents the length, in kilometres, of each track.
 


 

The Sunny Coast cricket coach designs three different training programs, all starting at exercise station .

  Training program 
number		 Training details
  1 The team must run to exercise station .
  2 The team must run along all tracks just once.
  3 The team must visit each exercise station and return to  exercise station .

 

  1. What is the shortest distance, in kilometres, covered in training program 1?  (1 mark)

  2.  i. What mathematical term is used to describe training program 2?   (1 mark)

  3. ii. At which exercise station would training program 2 finish?  (1 mark)

  4. To complete training program 3 in the minimum distance, one track will need to be repeated.

     

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


    This track is between exercise station   
     
    		 and exercise station   
     
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  2.  i.
  3. ii.
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a.  
   
   

 

b.i. 
 

b.ii. 

♦♦ Mean mark part (c) 25%.

 

c.   

NETWORKS, FUR2 2020 VCAA 2

A cricket team has 11 players who are each assigned to a batting position.

Three of the new players, Alex, Bo and Cameron, can bat in position 1, 2 or 3.

The table below shows the average scores, in runs, for each player for the batting positions 1, 2 and 3.

 

      Batting position
      1 2 3
     Player          Alex              22                     24                     24          
    Bo 25 25 21
    Cameron    24 25 19

 

Each player will be assigned to one batting position.

To which position should each player be assigned to maximise the team’s score? Write your answer in the table below.   (1 mark)

 

       Player           Batting position     
     Alex  
     Bo  
     Cameron  

 

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       Player           Batting position     
     Alex 3
     Bo 1
     Cameron 2

NETWORKS, FUR2 2020 VCAA 1

The Sunny Coast Cricket Club has five new players join its team: Alex, Bo, Cameron, Dale and Emerson.

The graph below shows the players who have played cricket together before joining the team.

For example, the edge between Alex and Bo shows that they have previously played cricket together.
 

  1. How many of these players had Emerson played cricket with before joining the team?   (1 mark)

  2. Who had played cricket with both Alex and Bo before joining the team?   (1 mark)

  3. During the season, another new player, Finn, joined the team.

      

    Finn had not played cricket with any of these players before.

      

    Represent this information on the graph above.   (1 mark)

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  3.  
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a. 
  

b. 

 

c.  

MATRICES, FUR2 2020 VCAA 3

An offer to buy the Westmall shopping centre was made by a competitor.

One market research project suggested that if the Westmall shopping centre were sold, each of the three centres (Westmall, Grandmall and Eastmall) would continue to have regular shoppers but would attract and lose shoppers on a weekly basis.

Let    be the state matrix that shows the expected number of shoppers at each of the three centres    weeks after Westmall is sold.

A matrix recurrence relation that generates values of    is


 

  1. Calculate the state matrix, , to show the expected number of shoppers at each of the three centres one week after Westmall is sold.   (1 mark)

Using values from the recurrence relation above, the graph below shows the expected number of shoppers at Westmall, Grandmall and Eastmall for each of the 10 weeks after Westmall is sold.
 


 

  1. What is the difference in the expected weekly number of shoppers at Westmall from the time Westmall is sold to 10 weeks after Westmall is sold?
  2. Give your answer correct to the nearest thousand.   (1 mark)

  3. Grandmall is expected to achieve its maximum number of shoppers sometime between the fourth and the tenth week after Westmall is sold.
  4. Write down the week number in which this is expected to occur.   (1 mark)

  5. In the long term, what is the expected weekly number of shoppers at Westmall?
  6. Round your answer to the nearest whole number.   (1 mark)

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a.  
   

 

b.  
 
   

 

♦♦ Mean mark part (c) 27%.

c. 


 


♦ Mean mark part (d) 39%.

d. 

MATRICES, FUR2 2020 VCAA 2

The preferred number of cafes and sandwich bars in Grandmall’s food court can be determined by solving the following equations written in matrix form.
 


 

  1. The value of the determinant of the 2 × 2 matrix is 1.
  2. Use this information to explain why this matrix has an inverse.   (1 mark)

  3. Write the three missing values of the inverse matrix that can be used to solve these equations.   (1 mark)

 

 

  1. Determine the preferred number of sandwich bars for Grandmall’s food court.   (1 mark)

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  1.  

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a. 

♦ Mean mark part (a) 37%.


b. 

 

Mean mark part (c) 51%.
c.  

 

MATRICES, FUR2 2020 VCAA 1

The three major shopping centres in a large city, Eastmall , Grandmall and Westmall , are owned by the same company.

The total number of shoppers at each of the centres at 1.00 pm on a typical day is shown in matrix .

  1. Write down the order of matrix .   (1 mark)

Each of these centres has three major shopping areas: food , clothing and merchandise .

The proportion of shoppers in each of these three areas at 1.00 pm on a typical day is the same at all three centres and is given in matrix below

`qquad qquad qquad P = [(0.48), (0.27), (0.25)] {:(F),(C),(M):}

  1. Grandmall’s management would like to see 700 shoppers in its merchandise area at 1.00 pm.

     

    If this were to happen, how many shoppers, in total, would be at Grandmall at this time?   (1 mark)

  2. The matrix    is shown below. Two of the elements of this matrix are missing.
     

     
    1. Complete matrix above by filling in the missing elements.   (1 mark)

    2. The element in row and column of matrix is .
    3. What does the element represent?   (1 mark)

The average daily amount spent, in dollars, by each shopper in each of the three areas at Grandmall in 2019 is shown in matrix    below.

On one particular day, 135 shoppers spent the average daily amount on food, 143 shoppers spent the average daily amount on clothing and 131 shoppers spent the average daily amount on merchandise.

  1. Write a matrix calculation, using matrix  , showing that the total amount spent by all these shoppers is $9663.20   (1 mark)

  2. In 2020, the average daily amount spent by each shopper was expected to change by the percentage shown in the table below.
      

     

      Area food clothing merchandise
      Expected change     increase by 5%       decrease by 15%       decrease by 1%   

     

     

    The average daily amount, in dollars, expected to be spent in each area in 2020 can be determined by forming the matrix product

  4. Write down matrix .    (1 mark)

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

a. 

♦ Mean mark part (b) 45%.
b.  
 
   

 

c.i.  

 

c.ii.  
 

 

d. 

♦♦ Mean mark part (e) 24%.
 

e.