SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, MET1 2020 VCAA 8

Part of the graph of  , where  , is shown below.
 


 

The graph of has a minimum at the point , as shown above.

  1. Find the coordinates of the point .   (2 marks)

  2. Using  , show that    has an antiderivative  .   (1 mark)

  3. Find the area of the region that is bounded by , the lines    and the horizontal axis for  , where is the -intercept of .   (2 marks)

  4. Let    for  .

     

    i. Find the value of for which    is a tangent to the graph of .   (1 mark)

    ii. Find all values of for which the graphs of and do not intersect.   (2 marks)

Show Answers Only
  4. i. 
  5. ii. 
Show Worked Solution

a.   

 

 
 

 

 

b.   
 
 

 

c.   

 
 
 
 

 

d.i.   

 

 

d.ii.

 


 

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)

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

a.   

 

b.i.   

 

 

b.ii.   


  

b.iii.   

 

  

b.iv.   

 

 

c.   

Calculus, MET1 2020 VCAA 6

Let  , where  .

  1. Find the domain and the rule for  , the inverse function of  .   (2 marks)

The graph of  , where  , is shown on the axes below.
 

     
 

  1. On the axes above, sketch the graph of    over its domain. Label the endpoints and point(s) of intersection with the function  , giving their coordinates.   (2 marks)

  2. Find the total area of the two regions: one region bounded by the functions  and , and the other region bounded by    and the line  . Give your answer in the form  , where  .   (4 marks)

Show Answers Only

  1.  

  2.  
       
  3.  ²
Show Worked Solution
a.   

 

 

 

b.     

 

c.     
 
 
 
 
 
  ²

Statistics, STD2 S1 2006 HSC 23c*

Vicki wants to investigate the number of hours spent on homework by students at her high school.

She asks each student how many hours (to the nearest hour) they usually spend on homework during one week. 

The responses are shown in the frequency table.

What is the mean amount of time spent on homework?   (2 marks)

Show Answers Only

Show Worked Solution

MARKER’S COMMENT: This “routine” exercise of finding a mean from grouped data was incorrectly answered by most students! The best responses copied the table and inserted a class-centre column (see solution).
      2UG-2006-23c Answer

 

 
 

Complex Numbers, EXT2 N1 2008 HSC 2b

  1. Write    in the form  , where and are real.  (2 marks)

  2. By expressing both    and    in  modulus-argument form, write    in modulus-argument form.   (3 marks)

  3. Hence find    in surd form.  (1 mark)

  4. By using the result of part (ii), or otherwise, calculate  .   (1 mark)

Show Answers Only
Show Worked Solution
i.     
   
   

 

ii.   


  


 


 

 
 

 

iii. 

 

 

iv.     
   
   

Proof, EXT2 P1 2020 HSC 8 MC

Consider the statement:

'If is even, then if is a multiple of 3, then is a multiple of 6'.

Which of the following is the negation of this statement?

  1. is odd and is not a multiple of 3 or 6.
  2. is even and is a multiple of 3 but not a multiple of 6.
  3. If is even, then is not a multiple of 3 and is not a multiple of 6.
  4. If is odd, then if is not a multiple of 3 then is not a multiple of 6.
Show Answers Only

Show Worked Solution

♦♦ Mean mark part 33%.


 

 

Calculus, EXT2 C1 2020 HSC 16b

Let 

  1. Prove that  .  (3 marks)

  2. Deduce that  .  (3 marks)

Let 

  1. Using the result of part (ii), or otherwise, show that  .  (3 marks)

  2. Prove that  .  (2 marks)

Show Answers Only
Show Worked Solution

i.   

♦ Mean mark (i) 48%.

 
 

 
♦ Mean mark (ii) 36%.

 

ii.    
   
   
   
     
 

 

 
 
 
♦♦♦ Mean mark (iii) 16%.

 

iii.   

 

   
 

 

 
 
 
 
 
 
♦♦♦ Mean mark (iv) 10%.

 

iv.   

  


 

 
 

 

Mechanics, EXT2 M1 2020 HSC 16a

Two masses, kg and kg, are attached by a light string. The string is placed over a smooth pulley as shown.

The two masses are at rest before being released and is the velocity of the larger mass at time seconds after they are released.
 


 

The force due to air resistance on each mass has magnitude , where is a positive constant.

  1. Show that  .   (2 marks)

  2. Given that  , show that when  , the velocity of the larger mass is  .  (3 marks)

Show Answers Only
Show Worked Solution

i. 

♦ Mean mark part (i) 37%.


  

  

 

 

 

ii.   
 
 
   

 

 

 

 

Vectors, EXT2 V1 2020 HSC 15b

The point divides the interval so that  . The position vectors of and are and respectively, as shown in the diagram.
 

  1. Show that  .  (2 marks)

     

  2. Prove that  .  (1 mark)

Let be a parallelogram with    and  . The point is the midpoint of and is the intersection of and , as shown in the diagram.
  
 
         
 

  1. Show that  .  (3 marks)

  2. Using parts (ii) and (iii), or otherwise, prove that is the point that divides the interval in the ratio 2 :1.   (1 mark)

Show Answers Only
Show Worked Solution

i.  

 

 

ii.   
   
   
   
   
   

 

iii. 

♦ Mean mark part (iii) 50%.
 

 


 

 
 
Mean mark part (iv) 51%.

 

iv.   

 

 

Proof, EXT2 P1 2020 HSC 15a

In the set of integers, let be the proposition:

'If    is divisible by 3, then    divisible by 3.'

  1. Prove that the proposition is true.  (2 marks)

  2. Write down the contrapositive of the proposition .  (1 mark)

  3. Write down the converse of the proposition and state, with reasons, whether this converse is true or false.  (3 marks)

Show Answers Only
Show Worked Solution

i.     

 
 
 
 

 

 

ii.    


 

♦♦ Mean mark part (iii) 36%.

iii.  


 

 

 
 
 

 

 
 
 
 

 

Statistics, STD1 S1 2020 HSC 24

  1. The ages in years, of ten people at the local cinema last Saturday afternoon are shown.

  1. The mean of this dataset is 47.1 years.
  2. How many of the ten people were aged between the mean age and the median age?  (2 marks)

  3. On Wednesday, ten people all aged 70 went to this same cinema.
  4. Would the standard deviation of the age dataset from Wednesday be larger than, smaller than or equal to the standard deviation of the age dataset given in part (a)? Briefly explain your answer without performing any calculations.  (2 marks)

Show Answers Only

a.    

b.   


 

Show Worked Solution

a.     

   

♦ Mean mark (a) 39%.

 
b.   


 

♦♦♦ Mean mark (b) 20%.

Financial Maths, STD1 F3 2020 HSC 30

Colin takes out a 5-year reducing balance loan of $19 000 with interest charged at 6% per annum. He uses this money to buy a car valued at $19 000.

The table shows some of the output from a spreadsheet used to model the reducing balance loan.
 


 

Colin's car is depreciated using the declining-balance method, with a depreciation rate of 20% per annum.

At the end of 3 years, after making the third repayment on the loan, Colin sells the car at its salvage value. He uses the money from the sale of the car to repay the amount owing on the loan at the end of the third year.

How much money will he have left over?  (4 marks)

Show Answers Only

Show Worked Solution

♦ Mean mark 23%.
 
 

 

 

Probability, STD1 S2 2020 HSC 26

Barbara plays a game of chance, in which two unbiased six-sided dice are rolled. The score for the game is obtained by finding the difference between the two numbers rolled. For example, if Barbara rolls a 2 and a 5, the score is 3.

The table shows some of the scores.
 


 

  1. Complete the six missing values in the table to show all possible scores for the game.   (1 mark)
  2. What is the probability that the score for a game is NOT 0?  (2 marks)

Show Answers Only
  1.  

     
Show Worked Solution

a.     

♦ Mean mark part (b) 47%.
b.      
   
   

 

b.      
   
   

Financial Maths, STD1 F2 2020 HSC 25

Tom is offered two different investment options.

Option A: 10% per annum simple interest.

Option B: 9% per annum interest, compounded annually.

Tom has $1000 to invest. The graph shows the future values over time of $1000 invested using Option B.
 


 

Tom wants to find the difference between the future values after 8 years using these two investment options.

By first drawing, on the grid above, the graph of the future values of $1000 invested using Option A, estimate the difference between the future values after 8 years.   (3 marks)

Show Answers Only

Show Worked Solution

♦♦ Mean mark 30%.

 

Measurement, STD1 M5 2020 HSC 23

The diagram shows a scale drawing of the floor of a room.
 


 

Carpet is to be laid to cover the entire floor. The cost of the carpet is $100 per square metre.

Find the total cost of the carpet required.  (3 marks)

Show Answers Only

Show Worked Solution


 

 

 

COMMENT: Note the image dimensions on the actual HSC paper were exactly 8 cm × 5 cm, making this ratio easy to calculate.

 

♦ Mean mark 41%.

 

Statistics, STD1 S3 2020 HSC 22

A group of students sat a test at the end of term. The number of lessons each student missed during the term and their score on the test are shown on the scatterplot.
 


 

  1. Describe the strength and direction of the linear association observed in this dataset.  (2 marks)

  2. Calculate the range of the test scores for the students who missed no lessons.  (1 mark)

  3. Draw a line of the best fit in the scatterplot above.  (1 mark)
  4. Meg did not sit the test. She missed five lessons.

     

    Use the line of the best fit drawn in part (c) to estimate Meg's score on this test. (1 mark)

  5. John also did not sit the test and he missed 16 lessons.

     

    Is it appropriate to use the line of the best fit to estimate his score on the test? Briefly explain your answer. (1 mark)

Show Answers Only

a.   

b.   
 

c.   

d. 


 

e.   

Show Worked Solution

a.   

♦ Mean mark (a) 45%.
♦♦ Mean mark (b) 31%.

b.   
 

c.   

d. 


 

 

e.   

♦ Mean mark (e) 38%.

Algebra, STD1 A2 2020 HSC 20

The weight of a bundle of A4 paper ( kg) varies directly with the number of sheets () of A4 paper that the bundle contains.

This relationship is modelled by the formula  , where    is a constant.

The weight of a bundle containing 500 sheets of A4 paper is 2.5 kilograms.

  1. Show that the value of    is 0.005.   (1 mark)

  2. A bundle of A4 paper has a weight of 1.2 kilograms. Calculate the number of sheets of A4 paper in the bundle.   (2 marks)

Show Answers Only
Show Worked Solution

a.     

 

 

b.     

♦ Mean mark 50%.
 

Algebra, STD1 A3 2020 HSC 19

Each year the number of fish in a pond is three times that of the year before.

  1. The table shows the number of fish in the pond for four years.

    Complete the table above showing the number of fish in 2021 and 2022.   (2 marks)
     

  2. Plot the points from the  table in part (a) on the grid.   (2 marks)
     
  3. Which model is more suitable for this dataset: linear or exponential? Briefly explain your answer.   (2 marks)

Show Answers Only

a.   

b.   
       

c.     The more suitable model is exponential.

A linear dataset would graph a straight line which is not the case here.

An exponential curve can be used to graph populations that grow at an increasing rate, such as this example.

Show Worked Solution

a.        

b.  

c.     The more suitable model is exponential.

A linear dataset would graph a straight line which is not the case here.

An exponential curve can be used to graph populations that grow at an increasing rate, such as this example.

♦ Mean mark (c) 31%.

Measurement, STD1 M2 2020 HSC 15

The time in Melbourne is 11 hours ahead of Coordinated Universal Time (UTC). The time in Honolulu is 10 hours behind UTC. A plane departs from Melbourne at 7 pm on Tuesday and lands in Honolulu 9 hours later.

What is the time and day in Honolulu when the plane lands?   (2 marks)

Show Answers Only

Show Worked Solution

♦♦ Mean mark 28%.


 

STRATEGY: 21 hours behind = 1 day behind + 3 hours.

Algebra, STD1 A3 2020 HSC 14

Adam travels on a straight road away from his home. His journey is shown in the distance – time graph.

  1. Describe the journey in the first 4 minutes by referring to change in speed and distance travelled.   (2 marks)

  2. After the 4 minutes shown on the graph. Adam rests for 2 minutes and then return home by travelling on the same road at a constant speed. Adam is away from home for a total of 10 minutes.

     

    On the above, complete the distance-time using the information provided.   (2 marks)

Show Answers Only

  2.  
Show Worked Solution

a.   

♦ Mean mark part (a) 35%.

Mean mark part (b) 51%.

 

b. 


 

Algebra, STD1 A3 2020 HSC 29

There are two tanks on a property, Tank A and Tank B. Initially, Tank A holds 1000 litres of water and Tank B is empty.

  1. Tank A begins to lose water at a constant rate of 20 litres per minute.

     

    The volume of water in Tank A is modelled by    where is the volume in litres and    is the time in minutes from when the tank begins to lose water.

     

    On the grid below, draw the graph of this model and label it as Tank A.   (1 mark)
     

     

  2. Tank B remains empty until    when water is added to it at a constant rate of 30 litres per minute.
     By drawing a line on the grid (above), or otherwise, find the value of    when the two tanks contain the same volume of water.  (2 marks)
  3. Using the graphs drawn, or otherwise, find the value of    (where  ) when the total volume of water in the two tanks is 1000 litres.  (1 mark)

Show Answers Only
  1.   
      

     
Show Worked Solution

a.    

♦ Mean mark part (a) 45%.
 


 

b.  

♦♦ Mean mark part (b) 24%.
 

   

 
c.  

♦♦♦ Mean mark part (c) 5%.


  

Financial Maths, STD1 F1 2020 HSC 27

The table shows the income tax rates for the 2019 – 2020 financial year.

For the 2019 – 2020 financial year, Wally had a taxable income of $122 680. During the year, he paid $3000 per month in Pay As You Go (PAYG) tax.

Calculate Wally's tax refund, ignoring the Medicare levy.   (3 marks)

Show Answers Only

Show Worked Solution
 

 

♦ Mean mark 35%.


 

 
   

Networks, STD1 N1 2020 HSC 21

The diagram represents a network with weighted edges.
 


 

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

  2. The network is revised by adding another vertex, . Edges and have weights of 12 and 10 respectively, as shown.
     

   
 

What is the length of the minimum spanning tree for this revised network?   (1 mark)

Show Answers Only
  1.  

     

     
     

     

Show Worked Solution

a.     

♦♦ Mean mark part (a) 32%.








 


 

 
♦ Mean mark part (b) 45%.

 

b.     

 

Measurement, STD1 M3 2020 HSC 11

Consider the triangle shown.
 


 

  1. Find the value of , correct to the nearest degree.   (2 marks)

  2. Find the value of , correct to one decimal place.   (2 marks)

Show Answers Only
Show Worked Solution
a.     
 
   
   

♦ Mean mark 44% and 39% for part (a) and (b) respectively.

 

b.     

 
 

Networks, STD1 N1 2020 HSC 9 MC

Team and Team have entered a chess competition.

Team and have three members each. Each member of Team must play each member of Team once.

Which of the following network diagrams could represent the chess games to be played?
 

 

 

  
Show Answers Only

Show Worked Solution

♦♦ Mean mark 34%.

Calculus, EXT1 C2 2020 HSC 13c

Suppose    and  .

The graph of    is given.
 

  1. Show that  .  (4 marks)

  2. Using part (i), or otherwise, show that  .  (3 marks) 

Show Answers Only
Show Worked Solution

i.   

♦ Mean mark (i) 50%.
 
 

 

 
 
 
 
 

 

ii.   

♦♦♦ Mean mark (ii) 15%.


 

 
 

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)

Show Answers Only
Show Worked Solution

a.   

♦ Mean mark 50%.


 


 


 

b.   

♦ Mean mark 44%.


 


 


 

c.   

♦♦♦ Mean mark 17%.

  

Measurement, STD2 M6 2020 HSC 32

The diagram shows a regular decagon (ten-sided shape with all sides equal and all interior angles equal). The decagon has centre .
 

The perimeter of the shape is 80 cm.

By considering triangle , calculate the area of the ten-sided shape. Give your answer in square centimetres correct to one decimal place.  (4 marks)

Show Answers Only

²

Show Worked Solution

♦♦ Mean mark 33%.


 

 

 
  ²

Measurement, STD2 M6 2020 HSC 31

Mr Ali, Ms Brown and a group of students were camping at the site located at . Mr Ali walked with some of the students on a bearing of 035° for 7 km to location . Ms Brown, with the rest of the students, walked on a bearing of 100° for 9 km to location .
 


 

  1. Show that the angle is 65°.  (1 mark)

  2. Find the distance .  (2 marks)

  3. Find the bearing of Ms Brown's group from Mr Ali's group. Give your answer correct to the nearest degree.  (2 marks)

Show Answers Only
Show Worked Solution
a.   
   

 

b.   

Mean mark 53%.
 
 
 

 
c.

 

♦♦ Mean mark 22%.

 
   
 
   

 

 

Networks, STD2 N3 2020 HSC 30

The network diagram shows a series of water channels and ponds in a garden. The vertices , , , , ,  and represent six ponds. The edges represent the water channels which connect the ponds. The numbers on the edges indicate the maximum capacity of the channels.
 


 

  1. Determine the maximum flow of the network.   (2 marks)

  2. A cut is added to the network, as shown.
     
       
    Is the cut shown a minimum cut? Give a reason for your answer.  (1 mark)

Show Answers Only
Show Worked Solution

a.   

♦ Mean mark 41%.

 

 
   

♦♦ Mean mark 28%.
b.     
   

 

Combinatorics, EXT1 A1 2020 HSC 14a

  1. Use the identity

     

    to show that
     
        ,
     
    where is a positive integer.  (2 marks)

  2. A club has members, with women and men.

     

    A group consisting of an even number of members is chosen, with the number of men equal to the number of women.
     
    Show, giving reasons, that the number of ways to do this is .  (2 marks)

  3. From the group chosen in part (ii), one of the men and one of the women are selected as leaders.

     

    Show, giving reasons, that the number of ways to choose the even number of people and then the leaders is
     

     

        .  (2 marks)

  4. The process is now reversed so that the leaders, one man and one woman, are chosen first. The rest of the group is then selected, still made up of an equal number of women and men.

     

    By considering this reversed process and using part (ii), find a simple expression for the sum in part (iii).  (2 marks)

Show Answers Only
Show Worked Solution

i.   

♦♦ Mean mark part (i) 26%.


 


 

♦♦ Mean mark part (ii) 23%.

 

ii.   

 
 


 

 

iii.   

♦♦ Mean mark part (iii) 26%.


 

♦♦♦ Mean mark part (iv) 16%.

 

iv. 
   
   
 

 

 

Networks, STD2 N3 2020 HSC 26

The preparation of a meal requires the completion of all activities to . The network diagram shows the activities and their completion times in minutes.
 


 

  1. What is the minimum time needed to prepare the meal?   (1 mark)

  2. List the activities which make up the critical path for this network.  (2 marks)

  3. Complete the table below, showing the earliest start time and float time for activities and   (2 marks)
     

Show Answers Only
  3.  
Show Worked Solution

a.     

♦♦ Mean mark part (a) 31%.
 


 


 

b.     

Mean mark part (b) 51%.

 

c.  

♦♦ Mean mark part (c) 22%.
  

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)

Show Answers Only

Show Worked Solution

Mean mark 53%.


 


  

 


 

 

Measurement, STD2 M1 2020 HSC 27

The shaded region on the diagram represents a garden. Each grid represents 5 m × 5 m.
 


 

  1. Use two applications of the trapezoidal rule to calculate the approximate area of the garden.   (3 marks)

  2. Should the answer to part (a) be more than, equal to or less than the actual area of the garden? Referring to the diagram above, briefly explain your answer.   (2 marks)

Show Answers Only
Show Worked Solution

a.     

♦ Mean mark 47%.
   
 
 

 

♦♦ Mean mark 23%.

b.     

Algebra, STD2 A4 2020 HSC 24

There are two tanks on a property, Tank A and Tank B. Initially, Tank A holds 1000 litres of water and Tank B is empty.

  1. Tank A begins to lose water at a constant rate of 20 litres per minute.

     

    The volume of water in Tank A is modelled by    where is the volume in litres and    is the time in minutes from when the tank begins to lose water.

     

    On the grid below, draw the graph of this model and label it as Tank A.   (1 mark)
     

     

  2. Tank B remains empty until    when water is added to it at a constant rate of 30 litres per minute.
     By drawing a line on the grid (above), or otherwise, find the value of    when the two tanks contain the same volume of water.  (2 marks)

  3. Using the graphs drawn, or otherwise, find the value of    (where  ) when the total volume of water in the two tanks is 1000 litres.  (1 mark)

Show Answers Only
  1.   
      
Show Worked Solution

a.      
 

 

b.     
 

   

 
c.  

♦♦ Mean mark part (c) 22%.


  

Trigonometry, 2ADV T3 2020 HSC 31

The population of mice on an isolated island can be modelled by the function.

,

where    is the time in weeks and  . The population of mice reaches a maximum of 35 000 when    and a minimum of 5000 when  . The graph of    is shown.
 

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

  2. On the same island, the population of cats can be modelled by the function
     

     
    Consider the graph of    and the graph of  .

     

    Find the values of  , for which both populations are increasing.  (3 marks)

  3. Find the rate of change of the mice population when the cat population reaches a maximum.  (2 marks)

Show Answers Only
Show Worked Solution
a.   
   

 

 
 

 

b.   

♦♦ Mean mark part (b) 30%.

 

c.   

♦♦♦ Mean mark part (c) 27%.
 

 

Measurement, STD2 M7 2020 HSC 23

In a tropical drink, the ratio of pineapple juice to mango juice to orange juice is 15 : 9 : 4 .

  1. How much orange juice is needed if the tropical drink is to contain 3 litres of pineapple juice?  (2 marks)

  2. The internal dimensions of a drink container, in the shape of a rectangular prism, are shown.
     
     
         
     
    To completely fill the container with the tropical drink, how many litres of mango juice are required. (3 marks)

Show Answers Only
Show Worked Solution
a.   
 
   

 

 
   

 

Mean mark 51%.
b.     
   

 

 
 
 

Statistics, 2ADV S2 2020 HSC 27

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,

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)

Show Answers Only

Show Worked Solution

♦ Mean mark 46%.

 

 

 
 

 

 

 

 
 

Statistics, 2ADV S3 2020 HSC 28

In a particular country, the hourly rate of pay for adults who work is normally distributed with a mean of $25 and a standard deviation of $5.

  1. Two adults who both work are chosen at random.

     

    Find the probability that at least one of them earns between $15 and $30 per hour.  (3 marks)

  2. The number of adults who work is equal to three times the number of adults who do not work.

     

    One adult is chosen at random.

     

    Find the probability that the chosen adult works and earn more than $25 per hour.  (2 marks)

Show Answers Only
Show Worked Solution

♦ Mean mark part (a) 40%.


 


 


b.   

Functions, 2ADV F1 2020 HSC 24

The circle of    is reflected in the -axis.

Sketch the reflected circle, showing the coordinates of the centre and the radius.  (3 marks)

Show Answers Only

Show Worked Solution

 

♦ Mean mark 48%.


 

Calculus, 2ADV C4 2020 HSC 30

The diagram shows two parabolas    and  , where  . The two parabolas intersect at the origin, , and at .
 


 

  1. Show that the -coordinate of    is  .  (2 marks)

  2. Find the value of    such that the shaded area is .  (4 marks)

Show Answers Only
Show Worked Solution

a.   
 
 

♦ Mean mark part (b) 48%.

 

b.   
 
 
 
 
 

Calculus, 2ADV C3 2020 HSC 29

The diagram shows the graph of  .
 

  1. Show that the equation of the tangent to    at  , where  , is
     
    .  (2 marks)

  2. Find the value of such that the tangent from part (a) has a gradient of 1 and passes through the origin.  (2 marks)

Show Answers Only
Show Worked Solution

a.   

 

 

b.   

♦ Mean mark part (b) 40%.

 

 

Financial Maths, STD2 F5 2020 HSC 34

Tina inherits $60 000 and invests it in an account earning interest at a rate of 0.5% per month. Each month, immediately after the interest has been paid, Tina withdraws $800.

The amount in the account immediately after the th withdrawal can be determined using the recurrence relation

,

where   and 

  1. Use the recurrence relation to find the amount of money in the account immediately after the third withdrawal.  (2 marks)

  2. Calculate the amount of interest earned in the first three months.  (2 marks)

Show Answers Only
Show Worked Solution

♦ Mean mark part (a) 41%.
a.   
 
 

 

b.   

♦♦ Mean mark part (b) 33%.


 

Financial Maths, 2ADV M1 2020 HSC 26

Tina inherits $60 000 and invests it in an account earning interest at a rate of 0.5% per month. Each month, immediately after the interest has been paid, Tina withdraws $800.

The amount in the account immediately after the th withdrawal can be determined using the recurrence relation

,

where   and 

  1. Use the recurrence relation to find the amount of money in the account immediately after the third withdrawal.  (2 marks)

  2. Calculate the amount of interest earned in the first three months.  (2 marks)

  3. Calculate the amount of money in the account immediately after the 94th withdrawal.  (3 marks) 

Show Answers Only
Show Worked Solution
a.   
 
 

 

b.   


 


 

c.   

 
 
 
 
 

Algebra, STD2 A4 2020 HSC 19

A fence is to be built around the outside of a rectangular paddock. An internal fence is also to be built.

The side lengths of the paddock are metres and metres, as shown in the diagram.
 

 
A total of 900 metres of fencing is to be used. Therefore  .
 
The area, , in square metres, of the rectangular paddock is given by  .

The graph of this equation is shown.
  

  1. If the area of the paddock is , what is the largest possible value of ?   (1 mark)

  2. Find the values of and so that the area of the paddock is as large as possible.   (2 marks)

  3. Using your value from part (b), find the largest possible area of the paddock.   (1 mark)

Show Answers Only
Show Worked Solution

a.     

♦ Mean mark part (a) 39%.

 


 

b.   

♦♦ Mean mark part (b) 34%.

 

♦ Mean mark part (c) 40%.
c.   
   
   

Probability, 2ADV S1 2020 HSC 14

History and Geography are two of the subjects students may decide to study. For a group of 40 students, the following is known.

    • 7 students study neither History nor Geography
    • 20 students study History
    • 18 students study Geography
  1. A student is chosen at random. By a using a Venn diagram, or otherwise, find the probability that the student studies both History and Geography.  (2 marks)

  2. A students is chosen at random. Given that the student studies Geography, what is the probability that the student does NOT study History?  (1 mark)

  3. Two different students are chosen at random, one after the other. What is the probability that the first student studies History and the second student does NOT study History?  (2 marks)

Show Answers Only
  1.  
Show Worked Solution
a.   
 

 

♦ Mean mark (b) 49%.
b.   
   

 

c.   
   

Calculus, 2ADV C3 2020 HSC 25

A landscape gardener wants to build a garden in the shape of a rectangle attached to a quarter-circle. Let and be the dimensions of the rectangle in metres, as shown in the diagram.
 


 

The garden bed is required to have an area of 36 m² and to have a perimeter which is as small as possible. Let metres be the perimeter of the garden bed.

  1. Show that  .  (3 marks)

  2. Find the smallest possible perimeter of the garden bed, showing why this is the minimum perimeter.  (4 marks)

Show Answers Only
Show Worked Solution
a.   
 
 
 

♦ Mean mark part (a) 47%.

 

 
 
 

 

b.   

 

 

 

 

Statistics, 2ADV S3 2020 HSC 23

A continuous random variable, , has the following probability density function.
 


 

  1. Find the value of .  (2 marks)

  2. Find  . Give your answer correct to four decimal places.  (2 marks)

Show Answers Only
Show Worked Solution
a.   
 
 
 
 
 

 

♦ Mean mark part (b) 44%.
b.   
   
   
   
   
   

Calculus, 2ADV C3 2020 HSC 21

Hot tea is poured into a cup. The temperature of tea can be modelled by  , where is the temperature of the tea, in degrees Celsius, minutes after it is poured.

  1. What is the temperature of the tea 4 minutes after it has been poured?  (1 mark)

  2. At what rate is the tea cooling 4 minutes after it has been poured?  (2 marks)

  3. How long after the tea is poured will it take for its temperature to reach 55°C?  (3 marks)

Show Answers Only
Show Worked Solution
a.   
   
   

 

b.   
   

 

 
 

 

c.   

♦ Mean mark part (c) 44%.
 
 

Probability, STD2 S2 2020 HSC 15 MC

The top of a rectangular table is divided into 8 equal sections as shown.
 

A standard die with faces labelled 1 to 6 is rolled onto the table. The die is equally likely to land in any of the 8 sections of the table. If the die does not land entirely in one section of the table, it is rolled again.

A score is calculated by multiplying the value shown on the top face of the die by the number shown in the section of the table where the die lands.

What is the probability of getting a score of 6?

Show Answers Only

Show Worked Solution

♦ Mean mark 40%.

 

 

 
 
 

 

Financial Maths, STD2 F5 2020 HSC 14 MC

An annuity consists of ten payments, each equal to $1000. Each payment is made on 30 June each year from 2021 through to 2030 inclusive.

The rate of compound interest is 5% per annum.

The present value of the annuity is calculated at 30 June 2020.

The future value of the annuity is calculated at 30 June 2030.

Without performing any calculations, which of the following statements is true?

  1. Present value of the annuity  <  $10 000  <  future value of the annuity
  2. $10 000  <  present value of the annuity  <  future value of the annuity
  3. Future value of the annuity  <  $10 000  <  present value of the annuity
  4. $10 000  <  future value of the annuity  <  present value of the annuity
Show Answers Only

Show Worked Solution

Mean mark 53%.