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EXAMCOPY Algebra, STD2 A4 2022 HSC 24

A student believes that the time it takes for an ice cube to melt ( minutes) varies inversely with the room temperature . The student observes that at a room temperature of it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating and .   (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from to    (2 marks)

 
           

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a.   
 
 
 
   

 

b.   

 

     

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a.   
 
 
 
   

 


♦♦ Mean mark part (a) 29%.

b.   

 

     


♦ Mean mark 44%.

Statistics, 2ADV S2 2024 HSC 16

Flowers were planted in two gardens (Garden A and Garden B).

On a particular day, 25 flowers were randomly selected from each garden and their heights measured in millimetres.

The data are represented in parallel box-plots.
 

Compare the two datasets by examining the skewness of the distributions, and the measures of central tendency and spread.   (3 marks)

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Show Worked Solution


  


   
 


 


L&E, 2ADV E1 2024 HSC 13

The graph shows the populations of two different animals, and , in a conservation park over time. The -axis is the size of the population and the -axis is the number of years since 1985 .

Population is modelled by the equation  .

Population is modelled by the equation  .
 

Complete the table using the information provided.   (3 marks)

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Algebra, STD2 A4 2024 HSC 22

The graph shows the populations of two different animals, and , in a conservation park over time. The -axis is the size of the population and the -axis is the number of years since 1985 .

Population is modelled by the equation  .

Population is modelled by the equation  .
 

Complete the table using the information provided.   (3 marks)

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Show Worked Solution

Mean mark 51%.

Financial Maths, 2ADV M1 2024 HSC 26

Twenty-five years ago, Phoenix deposited a single sum of money into a new bank account, earning 2.4% interest per annum compounding monthly.

Present value interest factors for an annuity of $1 for various interest rates and numbers of periods are given in the table.

Phoenix made the following withdrawals from this account.

  • $2000 at the end of each month for the first 15 years, starting at the end of the first month.
  • $1200 at the end of each month for the next 10 years, starting at the end of the 181st month after the account was opened.

Calculate the minimum sum that Phoenix could have deposited in order to make these withdrawals.   (4 marks)

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♦ Mean mark 51%.

 
   

 

Financial Maths, STD2 F5 2024 HSC 41

Twenty-five years ago, Phoenix deposited a single sum of money into a new bank account, earning 2.4% interest per annum compounding monthly.

Present value interest factors for an annuity of $1 for various interest rates and numbers of periods are given in the table.

Phoenix made the following withdrawals from this account.

  • $2000 at the end of each month for the first 15 years, starting at the end of the first month.
  • $1200 at the end of each month for the next 10 years, starting at the end of the 181st month after the account was opened.

Calculate the minimum sum that Phoenix could have deposited in order to make these withdrawals.   (4 marks)

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♦♦ Mean mark 39%.

 
   

 

Statistics, 2ADV S2 2024 HSC 21

A researcher is studying anacondas (a type of snake).

A dataset recording the age (in years) and length (in cm) of female and male anacondas is displayed on the graph.

Anacondas reach maturity at about 4 years of age.
 

Write THREE observations about anacondas that may be made from the scatterplot. (Note: No calculations are required.)   (3 marks)

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Statistics, 2ADV S3 2024 HSC 23

A random variable is normally distributed with mean 0 and standard deviation 1. The table gives the probability that this random variable is less than .

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

The scores in a university examination with a large number of candidates are normally distributed with mean 58 and standard deviation 15.

  1. By calculating a -score, find the percentage of scores that are between 58 and 70.   (2 marks)
  2. Explain why the percentage of scores between 46 and 70 is twice your answer to part (a).   (1 mark)
  3. By using the values in the table above, find an approximate minimum score that a candidate would need to be placed in the top 10% of the candidates.   (2 marks)
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a.   

b.  

c.   

Show Worked Solution

a.   


 

b.  


 

c.   

 
   
   

 

Statistics, STD2 S5 2024 HSC 35

A random variable is normally distributed with mean 0 and standard deviation 1. The table gives the probability that this random variable is less than .

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

The scores in a university examination with a large number of candidates are normally distributed with mean 58 and standard deviation 15.

  1. By calculating a -score, find the percentage of scores that are between 58 and 70.   (2 marks)
  2. Explain why the percentage of scores between 46 and 70 is twice your answer to part (a).   (1 mark)
  3. By using the values in the table above, find an approximate minimum score that a candidate would need to be placed in the top 10% of the candidates.   (2 marks)
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a.   

b.  

c.   

Show Worked Solution

a.   


 

♦♦ Mean mark (a) 36%.

b.  


 

♦♦♦ Mean mark (b) 19%.

c.   

 
   
   

 

♦♦ Mean mark (c) 30%.

Statistics, STD2 S4 2024 HSC 30

A researcher is studying anacondas (a type of snake).

A dataset recording the age (in years) and length (in cm) of female and male anacondas is displayed on the graph.

Anacondas reach maturity at about 4 years of age.
 

Write THREE observations about anacondas that may be made from the scatterplot. (Note: No calculations are required.)   (3 marks)

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Show Worked Solution

Statistics, STD2 S1 2024 HSC 28

Flowers were planted in two gardens (Garden A and Garden B).

On a particular day, 25 flowers were randomly selected from each garden and their heights measured in millimetres.

The data are represented in parallel box-plots.
 

Compare the two datasets by examining the skewness of the distributions, and the measures of central tendency and spread.   (3 marks)

Show Answers Only


  


 


 

Show Worked Solution


  


 


 

♦ Mean mark 41%.

Statistics, 2ADV S3 2023 HSC 23

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

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

The weights of adult male koalas form a normal distribution with mean = 10.40 kg, and standard deviation = 1.15 kg.

In a group of 400 adult male koalas, how many would be expected to weigh more than 11.93 kg?  (4 marks)

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Statistics, 2ADV S2 2023 HSC 18

A university uses gas to heat its buildings. Over a period of 10 weekdays during winter, the gas used each day was measured in megawatts (MW) and the average outside temperature each day was recorded in degrees Celsius (°C).

Using as the average daily outside temperature and as the total daily gas usage, the equation of the least-squares regression line was found.

The equation of the regression line predicts that when the temperature is 0°C, the daily gas usage is 236 MW.

The ten temperatures measured were: 0°, 0°, 0°, 2°, 5°, 7°, 8°, 9°, 9°, 10°,

The total gas usage for the ten weekdays was 1840 MW.

In any bivariate dataset, the least-squares regression line passes through the point , where is the sample mean of the -values and is the sample mean of the -values.

  1. Using the information provided, plot the point and the -intercept of the least-squares regression line on the grid.  (3 marks)
     

 

  1. What is the equation of the regression line?  (2 marks)

  2. In the context of the dataset, identify ONE problem with using the regression line to predict gas usage when the average outside temperature is 23°C.  (1 mark)

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a.    
       

b.   

c.   

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a.   


 

 

b.   

 
 
 

 
c.   

Trigonometry, 2ADV T1 2023 HSC 16

The diagram shows a shape . The shape consists of a rectangle with an arc on side and with side lengths = 3.6 m and = 8.0 m.

The arc is an arc of a circle with centre and radius 2.1 m and .

 

What is the perimeter of the shape ? Give your answer correct to one decimal place.  (4 marks)

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Show Worked Solution
 
   

 

 
 

 
 
   

 

 
   
   

Financial Maths, 2ADV M1 2023 HSC 15

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

  1. Micky wants to save $450 000 over the next 10 years.
  2. If the interest rate is 6% per annum compounding annually, how much should Micky contribute each year? Give your answer to the nearest dollar.  (2 marks)

  3. Instead, Micky decides to contribute  $8535 every three months for 10 years to an annuity paying 6% per annum, compounding quarterly.
  4. How much will Micky have at the end of 10 years?  (3 marks)

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Show Worked Solution

a.   

 
   

 
b.   

 
   

Statistics, STD2 S5 2023 HSC 38

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

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

The weights of adult male koalas form a normal distribution with mean = 10.40 kg, and standard deviation = 1.15 kg.

In a group of 400 adult male koalas, how many would be expected to weigh more than 11.93 kg?  (4 marks)

Show Answers Only

Show Worked Solution
 
   
   

 

 
   
♦ Mean mark 39%.
 
   
   

 

Statistics, STD2 S4 2023 HSC 34

A university uses gas to heat its buildings. Over a period of 10 weekdays during winter, the gas used each day was measured in megawatts (MW) and the average outside temperature each day was recorded in degrees Celsius (°C).

Using as the average daily outside temperature and as the total daily gas usage, the equation of the least-squares regression line was found.

The equation of the regression line predicts that when the temperature is 0°C, the daily gas usage is 236 MW.

The ten temperatures measured were: 0°, 0°, 0°, 2°, 5°, 7°, 8°, 9°, 9°, 10°,

The total gas usage for the ten weekdays was 1840 MW.

In any bivariate dataset, the least-squares regression line passes through the point , where is the sample mean of the -values and is the sample mean of the -values.

  1. Using the information provided, plot the point and the -intercept of the least-squares regression line on the grid.  (3 marks)
     

 

  1. What is the equation of the regression line?  (2 marks)

  2. In the context of the dataset, identify ONE problem with using the regression line to predict gas usage when the average outside temperature is 23°C.  (1 mark)

Show Answers Only

a.    
         

b.   

c.   

Show Worked Solution

a.   


 

♦ Mean mark (a) 44%.

b.   

 
 
 
♦♦♦ Mean mark (b) 21%.

 
c.   

♦♦♦ Mean mark 23%.

Measurement, STD2 M6 2023 HSC 33

The diagram shows a shape . The shape consists of a rectangle with an arc on side and with side lengths = 3.6 m and = 8.0 m.

The arc is an arc of a circle with centre and radius 2.1 m and .

 

What is the perimeter of the shape ? Give your answer correct to one decimal place.  (4 marks)

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Show Worked Solution
 
   

 

 

♦ Mean mark 42%.
 
 
   

 

 
   
   

Financial Maths, STD2 F5 2023 HSC 25

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

  1. Micky wants to save $450 000 over the next 10 years.
  2. If the interest rate is 6% per annum compounding annually, how much should Micky contribute each year? Give your answer to the nearest dollar.  (2 marks)

  3. Instead, Micky decides to contribute  $8535 every three months for 10 years to an annuity paying 6% per annum, compounding quarterly.
  4. How much will Micky have at the end of 10 years?  (3 marks)

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Show Worked Solution

a.   

 
   

 
b.   

 
   
Mean mark (b) 53%.
 

Functions, 2ADV F1 2022 HSC 12

A student believes that the time it takes for an ice cube to melt ( minutes) varies inversely with the room temperature . The student observes that at a room temperature of it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating and .    (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from to .   (2 marks)
     

 
                   

Show Answers Only

a.   

 b.    

       

 

Show Worked Solution
a.   
 
 
 
   

 

 


♦ Mean mark (a) 49%.

b.   

Algebra, STD2 A4 2022 HSC 24

A student believes that the time it takes for an ice cube to melt ( minutes) varies inversely with the room temperature . The student observes that at a room temperature of it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating and .   (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from to    (2 marks)

 
           

Show Answers Only
a.   
 
 
 
   

 

b.   

 

     

Show Worked Solution
a.   
 
 
 
   

 


♦♦ Mean mark part (a) 29%.

b.   

 

     


♦ Mean mark 44%.

Statistics, 2ADV S3 2022 HSC 26

The life span of batteries from a particular factory is normally distributed with a mean of 840 hours and a standard deviation of 80 hours.

It is known from statistical tables that for this distribution approximately 60% of the batteries have a life span of less than 860 hours.

What is the approximate percentage of batteries with a life span between 820 and 920 hours?  (3 marks)

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Show Worked Solution

 

 


 


 

Statistics, 2ADV S2 2022 HSC 24

Jo is researching the relationship between the ages of teenage characters in television series and the ages of actors playing these characters.

After collecting the data, Jo finds that the correlation coefficient is 0.4564.

A scatterplot showing the data is drawn. The line of best fit with equation  , is also drawn.
 


 

Describe and interpret the data and other information provided, with reference to the context given.  (4 marks)

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Financial Maths, 2ADV M1 2022 HSC 21

Eli is choosing between two investment options.

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

  1. What is the value of Eli's investment after 10 years using Option 1 ?  (2 marks)

  2. What is the difference between the future values after 10 years using Option 1 and Option 2?  (2 marks)

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a.   

 
   
   

 

b.   

 
   

 

 
   

Statistics, 2ADV S2 2022 HSC 11

The table shows the types of customer complaints received by an online business in a month.

  1. What are the values of and ?  (2 marks)

  2. The data from the table are shown in the following Pareto chart.
     
     
  3. The manager will address 80% of the complaints.
  4. Which types of complaints will the manager address?  (1 mark)

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a.   


 

b.   

Statistics, STD2 S5 2022 HSC 37

The life span of batteries from a particular factory is normally distributed with a mean of 840 hours and a standard deviation of 80 hours.

It is known from statistical tables that for this distribution approximately 60% of the batteries have a life span of less than 860 hours.

What is the approximate percentage of batteries with a life span between 820 and 920 hours?  (3 marks)

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Show Worked Solution

 

 


 


 


♦♦ Mean mark 28%.

Statistics, STD2 S4 2022 HSC 35

Jo is researching the relationship between the ages of teenage characters in television series and the ages of actors playing these characters.

After collecting the data, Jo finds that the correlation coefficient is 0.4564.

A scatterplot showing the data is drawn. The line of best fit with equation  , is also drawn.
 


 

Describe and interpret the data and other information provided, with reference to the context given.  (4 marks)

Show Answers Only



  



Show Worked Solution



  




♦♦ Mean mark 30%.

Financial Maths, STD2 F5 2022 HSC 30

Eli is choosing between two investment options.

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

  1. What is the value of Eli's investment after 10 years using Option 1 ?  (2 marks)

  2. What is the difference between the future values after 10 years using Option 1 and Option 2?  (2 marks)

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Show Worked Solution

a.   

 
   
   

 


♦ Mean mark 48%.

b.   

 
   

 

 
   

♦ Mean mark 43%.

Statistics, STD2 S1 2022 HSC 19

The table shows the types of customer complaints received by an online business in a month.
 

  1. What are the values of and ?  (2 marks)

  2. The data from the table are shown in the following Pareto chart.
     
     
  3. The manager will address 80% of the complaints.
  4. Which types of complaints will the manager address?  (1 mark)

Show Answers Only
Show Worked Solution

a.   


 

b.   


♦ Mean mark part (b) 41%.

Statistics, STD2 S5 2021 HSC 41

In a particular city, the heights of adult females and the heights of adult males are each normally distributed.

Information relating to two females from that city is given in Table 1.
 

The means and standard deviations of adult females and males, in centimetres, are given in Table 2.
 


 

A selected male is taller than 84% of the population of adult males in this city.

By first labelling the normal distribution curve below with the heights of the two females given in Table 1, calculate the height of the selected male, in centimetres, correct to two decimal places.  (4 marks)

 

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♦♦♦ Mean mark 16%.


 

 
 
 
 
   
 

 

 

 
   

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.
Show Worked Solution
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)

Show Answers Only
  1. i. 
  2. ii.
Show Worked Solution
a.i.   
   
   
♦♦ Mean mark part a.ii. 28%.

 
a.ii. 


 

♦♦♦ Mean mark part b 18%.

b. 

Trigonometry, 2ADV T1 2021 HSC 12

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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Show Worked Solution

 

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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Show Worked Solution

 

a.   
 
   
   

 

b.   
   
   

♦ Mean mark part (b) 36%.
 
   
   

 

 
   
   

Statistics, 2ADV S3 2021 HSC 22

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, find the probability that a value from a random variable that is normally distributed with a mean of 0 and standard deviation 1 lies between 0.1 and 0.5.  (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 3528 grams.  (3 marks)

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♦♦ Mean mark part (a) 29%.
COMMENT: Note the Advanced and Std2 questions varied slightly but used the same table and graph.
a.   
   
 

b.   

♦ Mean mark part (b) 48%.
 
 

 

 
 

 

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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Show Worked Solution

 

 
 
 

 

Probability, STD2 S2 2021 HSC 11 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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Show Worked Solution

♦♦ Mean mark 35%.

 

 
 
 

 

Statistics, STD2 S4 2020 HSC 36

A cricket is an insect. The male cricket produces a chirping sound.

A scientist wants to explore the relationship between the temperature in degrees Celsius and the number of cricket chirps heard in a 15-second time interval.

Once a day for 20 days, the scientist collects data. Based on the 20 data points, the scientist provides the information below.

  • A box-plot of the temperature data is shown.
     
       
  • The mean temperature in the dataset is 0.525°C below the median temperature in the dataset.
  • A total of 684 chirps was counted when collecting the 20 data points.

The scientist fits a least-squares regression line using the data , where is the temperature in degrees Celsius and is the number of chirps heard in a 15-second time interval. The equation of this line is

,

where is the slope of the regression line.

The least-squares regression line passes through the point  , where    is the sample mean of the temperature data and    is the sample mean of the chirp data.

Calculate the number of chirps expected in a 15-second interval when the temperature is 19° Celsius. Give your answer correct to the nearest whole number.  (5 marks)

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♦♦♦ Mean mark 18%.