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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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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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♦ Mean mark 42%.
 
 
   

 

 
   
   

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)
     

 
                   

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a.   

 b.    

       

 

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a.   
 
 
 
   

 

 


♦ Mean mark (a) 49%.

b.   

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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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, 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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♦♦ 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)

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♦♦ Mean mark 30%.

Statistics, STD2 S4 2022 HSC 23

A teacher surveyed the students in her Year 8 class to investigate the relationship between the average number of hours of phone use per day and the average number of hours of sleep per day.

The results are shown on the scatterplot below.
 

  1. The data for two new students, Alinta and Birrani, are shown in the table below. Plot their results on the scatterplot.  (2 marks)

                               
  2. By first fitting the line of best fit by eye on the scatterplot, estimate the average number of hours of sleep per day for a student who uses the phone for an average of 2 hours per day.  (2 marks)

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a.   

b.   

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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a.   

 
   
   

 


♦ Mean mark 48%.

b.   

 
   

 

 
   

♦ Mean mark 43%.

Financial Maths, STD2 F5 2022 HSC 25

The table shows the future value of an annuity of $1.
 
     

Zal is saving for a trip and estimates he will need $15 000. He opens an account earning 3% per annum, compounded annually.

  1. How much does Zal need to deposit every year if he wishes to have enough money for the trip in 4 years time?  (2 marks)

  2. How much interest will Zal earn on his investment over the 4 years? Give your answer to the nearest dollar.  (2 marks)

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a.   

 
 
 
   

 

b.   

 
   
   
   

♦♦ Mean mark 33%.

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)

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a.   


 

b.   


♦ Mean mark part (b) 41%.

Statistics, STD2 S1 2022 HSC 15 MC

The cumulative frequency graph shows the distribution of the number of movie downloads made by 100 people in one month.
 

Which box-plot best represents the same data as displayed in the cumulative frequency graph?
 

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♦♦♦ Mean mark 30%.

Statistics, STD2 S4 2022 HSC 12 MC

For a particular course, the recorded data show a relationship between the number of hours of study per week and the marks achieved out of 100 .

A least-squares regression line is fitted to this dataset. The equation of this line is given by

where is the predicted mark and is the number of hours of study per week.

Based on this regression equation, which of the following is correct regarding the predicted mark of a student?

  1. It will be 3 for zero hours of study per week.
  2. It will be 20 for zero hours of study per week.
  3. It will increase by 20 for every additional hour of study per week.
  4. It will increase by 1 for every 3 additional hours of study per week.
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Statistics, STD2 S5 SM-Bank 7

800 participants auditioned for a stage musical. Each participant was required to complete a series of ability tests for which they received an overall score.

The overall scores were approximately normally distributed with a mean score of 69.5 points and a standard deviation of 6.5 points.

Only the participants who scored at least 76.0 points in the audition were considered successful.

How many of the participants were considered unsuccessful?   (2 marks)

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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.
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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, STD2 F5 2021 HSC 31

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

   
 

A bank lends Martina $500 000 to purchase a home, with interest charged at 1.5% per annum compounding monthly. She agrees to repay the loan by making equal monthly repayments over a 30-year period.

How much should the monthly payment be in order to pay off the loan in 30 years?

Give your answer correct to the nearest cent.  (2 marks)

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♦ Mean mark 43%.

 
   

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%.

 
 
 
 

Financial Maths, STD2 F5 2021 HSC 40

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 35%.

 
 
 
 

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.  
   
   

 

  

 
 
 

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%.
 
 

 

 
 

 

Statistics, STD2 S4 2021 HSC 28

A salesperson is interested in the relationship between the number of bottles of lemonade sold per day and the number of hours of sunshine on the day.

The diagram shows the dataset used in the investigation and the least-squares regression line.


 

  1. Find the equation of the least-squares regression line relating to the dataset.  (2 marks)

  2. Suppose a sixth data point was collected on a day which had 10 hours of sunshine. On that day 45 bottles of lemonade were sold.
  3. What would happen to the gradient found in part (a)?  (1 mark)

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a.   

♦ Mean mark part (a) 37%.


 


 

Mean mark part (b) 53%.

 
 b.  


 

Financial Maths, STD2 F5 2021 HSC 21

Julie invests $12 500 in a savings account. Interest is paid at a fixed monthly rate. At the end of each month, after the monthly interest is added, Julie makes a deposit of $500.

Julie has created a spreadsheet to show the activity in her savings account. The details for the first 6 months are shown.
 

   

By finding the monthly rate of interest, complete the final row above for the 7th month.  (3 marks)

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♦ Mean mark 43%.
 

 

 

 

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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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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♦♦ Mean mark 35%.

 

 
 
 

 

Financial Maths, STD2 F5 2020 HSC 37

Wilma deposited a lump sum into a new bank account which earns 2% per annum compound interest.

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


 

Wilma was able to make the following withdrawals from this account.

  • $1000 at the end of each year for twenty years (starting one year after the account is opened)
  • $3000 each year for ten years starting 21 years after the account is opened.

Calculate the minimum lump sum Wilma must have deposited when she opened the new account.   (3 marks)

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♦♦♦ Mean mark 23%.

 

 

 
 

 

 

Statistics, STD2 S5 2020 HSC 35

The intelligence Quotient (IQ) scores for adults in City A are normally distributed with a mean of 108 and a standard deviation of 10.

The IQ score for adults in City B are normally distributed with a mean of 112 and a standard deviation of 16.

  1. Yin is an adult who lives in City A and has an IQ score of 128.
    What percentage of the adults in this city have an IQ score higher than Yin's?   (2 marks) 

  2. There are 1 000 000 adults living in City B.
    Calculate the number of adults in City B that would be expected to have an IQ score lower than Yin's.  (2 marks)

  3. Simon, an adult who lives in City A, moves to City B. The  -score corresponding to his IQ score in City A is the same as the -score corresponding to his IQ score in City B.
    By first forming an equation, calculate Simon's IQ score. Give your answer correct to one decimal place.   (3 marks)

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a.   

♦ Mean mark 50%.


 


 


 

b.   

♦ Mean mark 44%.


 


 


 

c.   

♦♦♦ Mean mark 17%.

  

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%.

 
 
 

 

 

 

 
 

Statistics, STD2 S2 2020 HSC 28

Consider the following dataset.

Suppose a new value, , is added to this dataset, giving the following.

     

It is known that    is greater than 15. It is also known that the difference between the means of the two datasets is equal to ten times the difference between the medians of the two datasets.

Calculate the value of .    (4 marks)

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Mean mark 53%.