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Statistics, STD1 S3 2021 HSC 18

People are placed into groups to complete a puzzle. There are 9 different groups.

The table shows the number of people in each group and the amount of time, in minutes, for each group to complete the puzzle.

  1. Complete the scatterplot by adding the last four points from the table.  (2 marks)
     
       
  2. Add a line of best fit by eye to the graph in part (a).  (1 mark)
  3. The graph in part (a) shows the association between the time to complete the puzzle and the number of people in the group.
  4. Identify the form (linear or non-linear), the direction and the strength of the association.  (3 marks)

  5. Calculate the mean of the time taken to complete the puzzle for the three groups of size 7 observed in the dataset.  (1 mark)

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  1.  
       
  2.  
       
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a.

b.


 

c.   

♦ Mean mark (c) 50%.


 

d.
   

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. 

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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a.   
 
   
   

 

b.   
   
   

 

 
   
   

 

 
   
   

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. 

Financial Maths, STD2 F1 2021 HSC 19

Adam purchased some office furniture five years ago. It depreciated by $2300 each year based on the straight-line method of depreciation. The salvage value of the furniture is now $7500.

Find the initial value of the office furniture.  (2 marks)

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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 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.    
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a. 

♦ Mean mark part (b) 45%.
b.  
 
   

 

c.i.  

 

c.ii.  
 

 

d. 

♦♦ Mean mark part (e) 24%.
 

e.  
 

CORE, FUR2 2020 VCAA 8

 

Samuel has a reducing balance loan.

The first five lines of the amortisation table for Samuel’s loan are shown below.
 


 

Interest is calculated monthly and Samuel makes monthly payments of $1600.

Interest is charged on this loan at the rate of 3.6% per annum.

  1. Using the values in the amortisation table
    1. calculate the principal reduction associated with payment number 3.   (1 mark)
    2. Calculate the balance of the loan after payment number 4 is made.
    3. Round your answer to the nearest cent.   (1 mark)
  2. Let be the balance of Samuel’s loan after months.
  3. Write down a recurrence relation, in terms of   and  , that could be used to model the month-to-month balance of the loan.   (1 mark)

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  1.  i.
  2. ii.

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a.i.  
   
   

 

♦ Mean mark part a.ii. 39%.

a.ii.  
   

 

 

♦♦ Mean mark part b. 30%.

b.  
 

CORE, FUR2 2020 VCAA 6

The table below shows the mean age, in years, and the mean height, in centimetres, of 648 women from seven different age groups.
 


 

  1. What was the difference, in centimetres, between the mean height of the women in their twenties and the mean height of the women in their eighties?  (1 mark)

A scatterplot displaying this data shows an association between the mean height and the mean age of these women. In an initial analysis of the data, a line is fitted to the data by eye, as shown.
 

 

  1. Describe this association in terms of strength and direction.  (1 mark)

  2. The line on the scatterplot passes through the points (20,168) and (85,157).

     

    Using these two points, determine the equation of this line. Write the values of the intercept and the slope in the appropriate boxes below.

     

    Round your answers to three significant figures.  (1 mark)

mean height = 
 
   +  
 
   × mean age

 

  1. In a further analysis of the data, a least squares line was fitted.

     

    The associated residual plot that was generated is shown below.

     
     

          

     

    The residual plot indicates that the association between the mean height and the mean age of women is non-linear.

     

    The data presented in the table in part a is repeated below. It can be linearised by applying an appropriate transformation to the variable mean age.

     

      

     

    Apply an appropriate transformation to the variable mean age to linearise the data. Fit a least squares line to the transformed data and write its equation below.

     

    Round the values of the intercept and the slope to four significant figures.  (2 marks)


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a.  
   

 

Mean mark part b. 51%.

b. 

 

♦♦ Mean mark part c. 23%.

c.   

 

 

D.    

CORE, FUR2 2020 VCAA 4

The age, in years, body density, in kilograms per litre, and weight, in kilograms, of a sample of 12 men aged 23 to 25 years are shown in the table below.
 

          Age       
(years)

        Body density        
(kg/litre)

        Weight        
(kg)
  23 1.07 70.1
  23 1.07 90.4
  23 1.08 73.2
  23 1.08 85.0
  24 1.03 84.3
  24 1.05 95.6
  24 1.07 71.7
  24 1.06 95.0
  25 1.07 80.2
  25 1.09 87.4
  25 1.02 94.9
  25 1.09 65.3
     
  1. For these 12 men, determine
  2.  i. their median age, in years.   (1 mark)

  3. ii. the mean of their body density, in kilograms per litre.   (1 mark)

  4. A least squares line is to be fitted to the data with the aim of predicting body density from weight.
  5.  i. Name the explanatory variable for this least squares line.   (1 mark)

  6. ii. Determine the slope of this least squares line.
  7.     Round your answer to three significant figures.   (1 mark)

  8. What percentage of the variation in body density can be explained by the variation in weight?
  9. Round your answer to the nearest percentage.   (1 mark)

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  1.  i.
  2. ii.
  3. i.
  4. ii.
Show Worked Solution
a.i.    
 
   
   

 

a.ii.  
   

 

b.i.   

♦ Mean mark b.ii. 29%.
MARKER’S COMMENT: Most students did not round correctly.

b.ii.   

 

c.  
 

 

CORE, FUR2 2020 VCAA 3

In a study of the association between BMI and neck size, 250 men were grouped by neck size (below average, average and above average) and their BMI recorded.

Five-number summaries describing the distribution of BMI for each group are displayed in the table below along with the group size.

The associated boxplots are shown below the table.
 

  1. What percentage of these 250 men are classified as having a below average neck size?   (1 mark)

  2. What is the interquartile range (IQR) of BMI for the men with an average neck size?   (1 mark)

  3. People with a BMI of 30 or more are classified as being obese.
  4. Using this criterion, how many of these 250 men would be classified as obese? Assume that the BMI values were all rounded to one decimal place.   (1 mark)

  5. Do the boxplots support the contention that BMI is associated with neck size? Refer to the values of an appropriate statistic in your response.   (2 marks)

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a.   
   

 

b.   
   

 

c.   

♦♦ Mean mark part c. 22%.
COMMENT: Many students incorrectly counted the two “above average” outliers twice.

 

d.   

♦ Mean mark 49%.
MARKER’S COMMENT: General statement of change = 1 mark. Median or IQR values need to be quoted directly for the second mark.

CORE, FUR2 2020 VCAA 1

Body mass index (BMI), in kilograms per square metre, was recorded for a sample of 32 men and displayed in the ordered stem plot below.
  

  1. Describe the shape of the distribution.   (1 mark)

  2. Determine the median BMI for this group of men.   (1 mark)

  3. People with a BMI of 25 or over are considered to be overweight.
  4. What percentage of these men would be considered to be overweight?   (1 mark)

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a.   
 

b.   

 
 

 

c.   
   

Geometry, NAPX-p124059v02

Nancy is making a rectangular prism using plastic balls and sticks.
  

     
 

How many more sticks does Nancy need to finish the rectangular prism?

  1   2   4   5   6
 
 
 
 
 
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Calculus, SPEC2 2020 VCAA 3

Let  .

  1. Find an expression for    and state the coordinates of the stationary points of  .  (2 marks)
  2. State the equation(s) of any asymptotes of  .  (1 mark)
  3. Sketch the graph of    on the axes provided below, labelling the local maximum stationary point and all points of inflection with their coordinates, correct to two decimal places.  (3 marks)
     
         

Let  , where  .

  1. Write down an expression for  .  (1 mark)
  2.  i. Find the non-zero values of for which  .  (1 mark)
  3. ii. Complete the following table by stating the value(s) of for which the graph of    has the given number of points of inflection.  (2 marks)
     
       
         
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  3.   
  5.  i.
  6. ii.
       
Show Worked Solution

a.   

 

 

b.   

 

c.   

 

d.   

 

e.i.   

♦♦♦ Mean mark (e)(ii) 9%.

 

e.ii.   

GRAPHS, FUR1 2020 VCAA 3 MC

The delivery fee for a parcel, in dollars, charged by a courier company is based on the weight of the parcel, in kilograms.

This relationship is shown in the step graph below for parcels that weigh up to 20 kg.
 

Which one of the following statements is not true?

  1. The delivery fee for a 4 kg parcel is $20.
  2. The delivery fee for a 12 kg parcel is $26.
  3. The delivery fee for a 13 kg parcel is the same as the delivery fee for a 20 kg parcel.
  4. The delivery fee for a 10 kg parcel is $14 more than the delivery fee for a 2 kg parcel.
  5. The delivery fee for a 12 kg parcel is $18 more than the delivery fee for a 2 kg parcel.
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GRAPHS, FUR1 2020 VCAA 1 MC

The graph below shows the number of people who attended a market over a seven-hour time period.
 

For how many hours were there at least 600 people at the market?

  1. 2
  2. 3
  3. 4
  4. 6
  5. 7
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GEOMETRY, FUR1 2020 VCAA 2 MC

A flag consists of three different coloured sections: red, white and blue.

The flag is 3 m long and 2 m wide, as shown in the diagram below.
 

The blue section is an isosceles triangle that extends to half the length of the flag.

The area of the blue section, in square metres, is

  1.  0.75
  2.  1.5
  3.  2
  4.  3
  5.  6
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  ²

 

MATRICES, FUR1 2020 VCAA 5 MC

The diagram below shows the direct communication links that exist between Sam (S), Tai (T), Umi (U) and Vera (V). For example, the arrow from Umi to Vera indicates that Umi can communicate directly with Vera.
 


 

A communication matrix can be used to convey the same information.

In this matrix:

  • a ‘1’ indicates that a direct communication link exists between a sender and a receiver
  • a ‘0’ indicates that a direct communication link does not exist between a sender and a receiver.

The communication matrix could be

A.  

 

 B.  

 
C.  

 

 D.  

 
E.  

 

    
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MATRICES, FUR1 2020 VCAA 4 MC

In a particular supermarket, the three top-selling magazines are Angel (A), Bella (B) and Crystal (C).

The transition diagram below shows the way shoppers at this supermarket change their magazine choice from week to week.
 

A transition matrix that provides the same information as the transition diagram is

A.  

 

 B.  

 
C.  

 

 D.  

 
E.  

 

    
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CORE, FUR1 2020 VCAA 24 MC

Manu invests $3000 in an account that pays interest compounding monthly.

The balance of his investment after months, , can be determined using the recurrence relation

The total interest earned by Manu’s investment after the first five months is closest to

  1. $57.60
  2. $58.02
  3. $72.00
  4. $72.69
  5. $87.44
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CORE, FUR1 2020 VCAA 17-18 MC

Table 4 below shows the monthly rainfall for 2019, in millimetres, recorded at a weather station, and the associated long-term seasonal indices for each month of the year.
 


 

Part 1

The deseasonalised rainfall for May 2019 is closest to

  1.   71.3 mm
  2.   75.8 mm
  3.   86.1 mm
  4.   88.1 mm
  5. 113.0 mm

 
Part 2

The six-mean smoothed monthly rainfall with centring for August 2019 is closest to

  1. 67.8 mm
  2. 75.9 mm
  3. 81.3 mm
  4. 83.4 mm
  5. 86.4 mm
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Part 1


 


 

Part 2


 

CORE, FUR1 2020 VCAA 1-3 MC

The times between successive nerve impulses (time), in milliseconds, were recorded.

Table 1 shows the mean and the five-number summary calculated using 800 recorded data values.
 


 

Part 1

The difference, in milliseconds, between the mean time and the median time is

  1.  10
  2.  70
  3.  150
  4.  220
  5.  230

 
Part 2

Of these 800 times, the number of times that are longer than 300 milliseconds is closest to

  1. 20
  2. 25
  3. 75
  4. 200
  5. 400

 
Part 3

The shape of the distribution of these 800 times is best described as

  1. approximately symmetric.
  2. positively skewed.
  3. positively skewed with one or more outliers.
  4. negatively skewed.
  5. negatively skewed with one or more outliers.
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Part 1

 


 

Part 2


 

Part 3

♦ Mean mark 50%.

 
 

 

Calculus, EXT1 C2 2020 SPEC1 6

Let  .

  1. Show that  .  (1 mark)

  2. Hence, show that the graph of    has a point of inflection at  .  (2 marks)

  3. Sketch the graph of    on the axes provided below. Label any asymptotes with their equations and the point of inflection with its coordinates.   (2 marks)

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a.   
   

 

b.   

 

c.   

Calculus, SPEC1 2020 VCAA 6

Let  .

  1. Show that  .  (1 mark)

  2. Hence, show that the graph of    has a point of inflection at  .  (2 marks)

  3. Sketch the graph of    on the axes provided below. Label any asymptotes with their equations and the point of inflection with its coordinates.   (2 marks)

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a.   
   

 

b.   

♦ Mean mark part (b) 42%.

 

c.   

Calculus, MET1 2020 VCAA 7

Consider the function    and the point  . Part of the graph    is shown below.
 

   
 

  1. Show the point is not on the graph of  .   (1 mark)

  2. Consider a point    to be a point on the graph of .

     

      i. Find the slope of the line connecting points and in terms of .   (1 mark)

     ii. Find the slope of the tangent to the graph of at point in terms of .   (1 mark)

    iii. Let the tangent to the graph of at    pass through point .

     

    Find the values of .   (2 marks)

     iv. Give the equation of one of the lines passing through point that is tangent to the graph of .   (1 mark)

  3. Find the value of , that gives the shortest possible distance between the graph of the function of    and point .   (2 marks)

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  2.   i. 
     ii. 
    iii. 
    iv. 
Show Worked Solution

a.   

 

b.i.   

 

 

b.ii.   


  

b.iii.   

 

  

b.iv.   

 

 

c.