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Probability, MET2 2021 VCAA 6 MC

The probability of winning a game is 0.25

The probability of winning a game is independent of winning any other game.

If Ben plays 10 games, the probability that he will win exactly four rimes is closest to

  1. 0.1460
  2. 0.2241
  3. 0.9219
  4. 0.0781
  5. 0.7759
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Statistics, SPEC1 2021 VCAA 3

A company produces a particular type of light globe called Shiny. The company claims that the lifetime of these globes is normally distributed with a mean of 200 weeks and it is known that the standard deviation of the lifetime of Shiny globes is 10 weeks. Customers have complained, saying Shiny globes were lasting less than the claimed 200 weeks. It was decided to investigate the complaints. A random sample of 36 Shiny globes was tested and it was found that the mean lifetime of the sample was 195 weeks.

Use    and    to answer the following questions.

  1. Write down the null and alternative hypotheses for the one-tailed test that was conducted to investigate the complaints.   (1 mark)

    1. Determine the value, correct to three places decimal places, for the test.   (2 marks)

    2. What should the company be told if the test was carried out at the 1% level of significance?   (1 mark)

  3. The company decided to produce a new type of light globe called Globeplus.
    Find the approximate 95% confidence interval for the mean lifetime of the new globes if a random sample of 25 Globeplus globes is tested and the sample mean is found to be 250 weeks. Assume that the standard deviation of the population is 10 weeks. Give your answer correct to two decimal places.   (1 mark)

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a.   

 

b.i.   

♦ Mean mark part (b)(i) 48%.

 
 
 
 
 

 

b.ii.   

♦ Mean mark part (b)(ii) 49%.

 

 

 

c.   

 
 

Vectors, SPEC1 2021 VCAA 1

The net force acting on a body of mass 10 kg is    newtons.

  1. Find the acceleration of the body in .   (1 mark)

  2. The initial velocity of the body is  .
    Find the velocity of the body, in , at any time    seconds.   (2 marks)

  3. Find the momentum of the body, in kg , when    seconds.   (1 mark)

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a.  

 

 

b.   
   
   

 

 

 

c.   

 
 

CORE, FUR1 2021 VCAA 1-3 MC

The percentaged segmented bar chart below shows the age (under 55 years, 55 years and over) of visitors at a travel convention, segmented by preferred travel destination (domestic, international).
 

Part 1

The variables age (under 55 years, 55 years and over) and preferred travel destination (domestic, international) are

  1. both categorial variables.
  2. both numerical variables.
  3. a numerical variable and a categorical variable respectively.
  4. a categorical variable and a numerical variable respectively.
  5. a discrete variable and a continuous variable respectively.

 
Part 2

The data displayed in the percentaged segmented bar chart supports the contention that there is an association between preferred travel destination and age because

  1. more visitors favour international travel.
  2. 35% of visitors under 55 years favour international travel.
  3. 45% of visitors 55 years and over favour domestic travel.
  4. 65% of visitors under 55 years favour domestic travel while 45% of visitors 55 years and over favour domestic travel.
  5. the percentage of visitors who prefer domestic travel is greater than the percentage of visitors who prefer international travel.

 
Part 3

The results could also be summarised in a two-way frequency table.

Which one of the following frequency tables could match the pecentaged segmented bar chart?

 

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♦ Mean mark 50%.

Functions, MET1 2021 VCAA 3

Consider the function 

  1. State the range of .   (1 mark)

  2. State the period of .   (1 mark)

  3. Solve    for  .   (3 marks)

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a.   


 

b.   
 

c.   
 
 
 

 

Mechanics, EXT2 M1 2021 HSC 14b

An object of mass 5 kg is on a slope that is inclined at an angle of 60° to the horizontal. The acceleration due to gravity is   and the velocity of the object down the slope is .

As well as the force due to gravity, the object is acted on by two forces, one of magnitude newtons and one of magnitude newtons, both acting up the slope.

  1. Show that the resultant force down the slope is    newtons.  (2 marks)

  2. There is one value of such that the object will slide down the slope at a constant speed.
  3. Find this value of in , correct to 1 decimal place, given that  .  (2 marks)

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i.   

 
 
 

 

ii.   

 

 
 
 
 

Complex Numbers, EXT2 N2 2021 HSC 14c

  1. Using de Moivre’s theorem and the binomial expansion of , or otherwise, show that
  2.    .  (3 marks)

  3. By using part (i), or otherwise, show that  .  (3 marks)

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i.   


 

 
 
 
 

 

ii.   

♦♦ Mean mark 31%.

  
 
 

 

Calculus, EXT2 C1 2021 HSC 13c

  1. The integral    is defined by    for integers  .
    Show that    for   .  (2 marks)

  2. The diagram shows two regions.

Region is bounded by    and    between    and  .

Region is bounded by    and    between    and  .
 
   
 
The volume of the solid created when the region between the curve    and the  -axis, between    and  , is rotated about the -axis is given by  .

The volume of the solid of revolution formed when region is rotated about the -axis is .

The volume of the solid of revolution formed when region is rotated about the -axis is .

Using part (i), or otherwise, show that the ratio  is .  (4 marks)

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i.   
 
 

 

ii.   
   
   
   
   

 

 
 
 
 
 
 
 
 
 

Vectors, EXT2 V1 2021 HSC 12e

The diagram shows the pyramid    where    is a square. The diagonals of the square bisect each other at .
 

  1. Show that     (1 mark)

    Let be the point such that  .

  1. Using part (i), or otherwise, show that  .   (2 marks)

  2. Find the value of  λ  such that  λ   (1 mark)

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i.    


 

 
 
 
ii.    

 
 
 
iii. 
♦ Mean mark 48%.

 

λ

Proof, EXT2 P1 2021 HSC 12b

Consider Statement A.

Statement A: ‘If is even, then is even.’

  1. What is the converse of Statement A?.  (1 mark)

  2. Show that the converse of Statement A is true. (1 mark)

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i.     


 

ii.   

 
 
 

 

Complex Numbers, EXT2 N2 2021 HSC 11d

  1. Find the two square roots of  , giving the answers in the form  , where and are real numbers.  (2 marks)

  2. Hence, or otherwise, solve    giving your solutions in the form    where and are real numbers. (2 marks)

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i.     


 

 
 

 

 


 

 

ii.   

 
   
   

 

Proof, EXT2 P1 2021 HSC 4 MC

Consider the statement:

‘For all integers , if is a multiple of 6, then is a multiple of 2’.

Which of the following is the contrapositive of the statement?

  1. There exists an integer such that is a multiple of 6 and not a multiple of 2.
  2. There exists an integer such that is a multiple of 2 and not a multiple of 6.
  3. For all integers , if is not a multiple of 2, then is not a multiple of 6.
  4. For all integers , if is not a multiple of 6, then is not a multiple of 2.
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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.
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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.