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Probability, NAPX-p167447v01

Laura's country hut is visited by a possum twice every week.

What is the probability that the possum visits her hut today?

 
 `2/7`
 
 `2/5`
 
 `0.70`
 
 `text(25%)`

Show Answers Only

`2/7`

Show Worked Solution

`text{There are 7 days in a week.}`

`P` `= text(Favorable Events)/text(Total Possible Events)`
  `= 2/7`

Geometry, NAPX-p167443v02

Henry got lost on his way to visit his uncle’s house and made 3 U-turns before arriving.

In total, how many degrees does Henry turn through when making U-turns on his trip?

 
   150°
 
   270°
 
   540°
 
 1080°

Show Answers Only

`540^@`

Show Worked Solution

`text{One U-turn rotates the car by 180}^@`

`:. 3\ text(U-turns)` `= 3 xx 180`
  `= 540^@`

Number, NAPX-p167439v02

A pack of sugar weighs `1/4` of a kilogram.

Josh bought 6 packs for baking.

How many kilograms of sugar did he buy?

 
 `2/3`
 
 `1 1/2`
 
 `2 1/4`
 
 `3`

Show Answers Only

`1 1/2`

Show Worked Solution
`text{Weight of six packs}` `= 6 xx 1/4`
  `= 6/4`
  `= 1 1/2\ text(kg)`

Number, NAPX-p167439v01

A box of apples weighs `2/3` of a kilogram

Lou bought 3 boxes.

How many kilograms of apples did he bought?

 
    `1 frac{4}{9} \ text{kg}`
 
   `2 \ text{kg}`
 
   `frac{8}{9} \ text{kg}`
 
   `2 frac{2}{3} \ text{kg}`

Show Answers Only

`2\ text{kg}`

Show Worked Solution
`text(Total kilograms)` `=3 xx 2/3`  
  `=6/3`  
  `=2`  

Probability, NAPX-p116940v05

The arrow pictured below is spun once.
 


 

Which number is the spinner least likely to land on?

  1   2   3   1 or 2
 
 
 
 

Show Answers Only

`2`

Show Worked Solution

` text{Least likely to land on 2.}`

`text{It takes up the smallest area on the spinner}`

Probability, NAPX-p116940v04

A disk is thrown onto the table pictured below.
 


 

It has an equal chance of landing in any square.

Which numbered square is the disk least likely to land in?

  3   4   5 same chance for each number
 
 
 
 

Show Answers Only

`4`

Show Worked Solution

` text{Only 1 square is numbered 4 (all other numbers have 2 squares).}`

`therefore \ text{Least likely to land in square 4}`

Number, NAPX-p116689v02

The tallest living giraffe is measured at five thousand, seven hundred and eight millimetres tall.

Write this as a number in the box below

  millimetres

Show Answers Only

`5708 \ text{millimetres}`

Show Worked Solution

`5708 \ text{millimetres}`

Number, NAPX-p116689v05

The exact length of great white shark is measured as five thousand and ninety six millimetres.

Write this as a number in the box below

  millimetres

Show Answers Only

`5096 \ text{millimetres}`

Show Worked Solution

`5096 \ text{millimetres}`

Calculus, SPEC2 2020 VCAA 3

Let  `f(x) = x^2e^(−x)`.

  1. Find an expression for  `f′(x)`  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.   

Probability, NAPX-p116940v03

A spinning wheel has sections labelled with different numbers.
 


 

Which of the numbers in the wheel is the spinner most likely to land on?

 
   1 or 3
 
   1
 
   2 or 4
 
   All of the colours are equally likely
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Show Worked Solution

Geometry, NAPX-p167209v02

Yohan was driving from the hospital to his house.
 


 

What directions best describe Yohan’s travel from the hospital to his house?

 
 East, north-east, north
 
 West, north-west, north
 
 East, north-east, south
 
 West, north-west, south

Show Answers Only

Show Worked Solution


 

 
 

Geometry, NAPX-p167209v01

A man drives from his house to his office.
 


 

What directions best describe his way to the office?

 
 North, north-west, west, south-west
 
 North, north-east, east, south-east
 
 North, north-east, east, south
 
 North, north-west, east, south-east

Show Answers Only

Show Worked Solution


 


 

Statistics, NAPX-p167236v02

A store sells second hand mobile phones.

The graph below shows the price of 2 similar second-hand phones.
 


 

Which of the following is true based on the graph shown?

 
 Phone A is older and less expensive than Phone B
 
 Phone B is older and more expensive than phone A
 
 Phone A is newer and more expensive than Phone B
 
 Phone A is older and more expensive than Phone B

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Show Worked Solution

Statistics, NAPX-p167236v01

A man bought a plot of land in the past and now he is selling it.

The graph marks the price of the land when the man bought it and the price of the land now.
 


 

Which of the following is true based on the graph shown?

 
 The land is less expensive now than when it was purchased.
 
 The land is more expensive years ago than now.
 
 The land is more expensive now than years ago.
 
 The price of the land does not change with time.

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Show Worked Solution

NETWORKS, FUR1 2020 VCAA 10 MC

The directed network below shows the sequence of activities, to , that is required to complete an office renovation.

The time taken to complete each activity, in weeks, is also shown.
 


 

The project manager would like to complete the office renovation in less time.

The project manager asks all the workers assigned to activity to also work on activity .

This will reduce the completion time of activity to three weeks.

The workers assigned to activity cannot work on both activity and activity at the same time.

No other activity times will be changed.

This change to the network will result in a change to the completion time of the office renovation.

Which one of the following is correct?

  1. The completion time will be reduced by one week if activity is completed before activity is started.
  2. The completion time will be reduced by three weeks if activity is completed before activity is started.
  3. The completion time will be reduced by one week if activity is completed before activity is started.
  4. The completion time will be reduced by three weeks if activity is completed before activity is started.
  5. The completion time will be increased by three weeks if activity is completed before activity is started.
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Show Worked Solution

♦♦♦ Mean mark 24%.


 


 

NETWORKS, FUR1 2020 VCAA 9 MC

The flow of liquid through a series of pipelines, in litres per minute, is shown in the directed network below.
 


 

Five cuts labelled A to E are shown on the network.

The number of these cuts with a capacity equal to the maximum flow of liquid from the source to the sink, in litres per minute, is

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
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Show Worked Solution

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, 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)

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Show Worked Solution

i.   

♦ Mean mark (i) 48%.

 
 

 
♦ Mean mark (ii) 36%.

 

ii.    
   
   
   
     
 

 

 
 
 
♦♦♦ Mean mark (iii) 16%.

 

iii.   

 

   
 

 

 
 
 
 
 
 
♦♦♦ Mean mark (iv) 10%.

 

iv.   

  


 

 
 

 

Measurement, STD1 M5 2020 HSC 28

Two similar right-angled triangles are shown.
 


 

The length of side is 8 cm and the length of side is 4 cm.

The area of triangle is 20 cm2.

Calculate the length in centimetres of side in Triangle II, correct to two decimal places.   (4 marks)

Show Answers Only

Show Worked Solution

 

♦♦♦ Mean mark 11%.

 

 
 

Trigonometry, EXT1 T3 2020 HSC 14b

  1. Show that  .  (2 marks)

  2. By letting    in the cubic equation  .

     

    Show that  . (2 marks)

  3. Prove that  .  (3 marks)

Show Answers Only
Show Worked Solution

i.  

 
 
 
 
 

 

ii.   

 
♦♦♦ Mean mark (iii) 21%.

 

iii.   

 


 

αβγ

αβγ

αββγαγ

αβγαββγαγ

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, 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)

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 23%.

 

 

 
 

 

 

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) 

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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)

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Show Worked Solution

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)

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 18%.

 
 
 

 

 

 

 
 

Algebra, STD2 A4 2020 HSC 33

The graph shows the number of bacteria, , at time minutes. Initially (when ) the number of bacteria is 1000.
 


 

  1. Find the number of bacteria at 40 minutes.   (1 mark)

  2. The number of bacteria can be modelled by the equation  , where and are constants.
    Use the guess and check method to find, to two decimal places, an upper and lower estimate for the value of . The upper and lower estimates must differ by 0.01. (2 marks)

Show Answers Only
Show Worked Solution

a.   


 

b.   

♦♦♦ Mean mark 14%.


 

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)

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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. 
   
   
 

 

 

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%.
 

 

Algebra, STD2 A1 2020 HSC 13 MC

When Jake stops drinking alcohol at 10:30 pm, he has a blood alcohol content (BAC) of 0.08375.

The number of hours required for a person to reach zero BAC after they stop consuming alcohol is given by the formula

.

At what time on the next day should Jake expect his BAC to be 0.05?

  1.  12:45 am
  2.  1:50 am
  3.  2:15 am
  4.  4:05 am
Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 17%.
COMMENT: The rates aspect of this question proved extremely challenging.


 

 


 

 

 

Mechanics, EXT2 M1 EQ-Bank 2

A particle with mass moves horizontally against a resistance force , equal to    where is the particle's velocity.

Initially, the particle is travelling in a positive direction from the origin at velocity .

  1. Show that the particle's displacement from the origin, , can be expressed as
     
           (2 marks)

     

  2. Show that the time, , when the particle is travelling at velocity , is given by
     
           (4 marks)

     

  3. Express as a function of , and hence find the limiting values of and .  (2 marks)

Show Answers Only
  2. `text(See Worked Solutions)“
Show Worked Solution

i.   

 

 


 

 
 

 

ii.   


 

     
     

 

 
 
 
 

 


 

 
 

 

iii.
 
 
 
 
 

 

GEOMETRY, FUR1-NHT 2019 VCAA 8 MC

A young tree is protected by a tree guard in the shape of a square-based pyramid.

The height of the tree guard is 54 cm, as shown in the diagram below.
 
 

         
 

The top section of the tree guard is removed along the dotted line to allow the tree to grow.

Removing this top section decreases the height of the tree guard to 45 cm, as shown in the diagram below.
 
 

         
 

The ratio of the volume of the tree guard that is removed to the volume of the tree guard that remains is

  1.  
  2.  
  3.  
  4.  
  5.  
Show Answers Only

Show Worked Solution

 
 

 

 

 


 

Calculus, MET1-NHT 2018 VCAA 9

Let diagram below shows a trapezium with vertices at    and  , where    is a real number and 

On the same axes as the trapezium, part of the graph of a cubic polynomial function is drawn. It has the rule  , where    is a non-zero real number and  .

  1. At the local maximum of the graph, .
  2. Find in terms of .   (3 marks)

The area between the graph of the function and the -axis is removed from the trapezium, as shown in the diagram.
 

  1. Show that the expression for the area of the shaded region is    square units.   (3 marks)

  2. Find the value of    for which the area of the shaded region is a maximum and find this maximum area.   (2 marks)

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Show Worked Solution
a.     
 
 

 


 

 

 

 

b.   

 
 
 
 
 
 

 


 

 

 

c.   


 

 
 

CORE, FUR1-NHT 2019 VCAA 24 MC

Robyn has a current balance of $347 283.45 in her superannuation account.

Robyn’s employer deposits $350 into this account every fortnight.

This account earns interest at the rate of 2.5% per annum, compounding fortnightly.

Robyn will stop work after 15 years and will no longer receive deposits from her employer.

The balance of her superannuation account at this time will be invested in an annuity that will pay interest at the rate of 3.6% per annum, compounding monthly.

After 234 monthly payments there will be no money left in Robyn’s annuity.

The value of Robyn’s monthly payment will be closest to

  1. $3993
  2. $5088
  3. $6664
  4. $8051
  5. $9045
Show Answers Only

Show Worked Solution

 

 

 

CORE, FUR2-NHT 2019 VCAA 8

A record producer gave the band $50 000 to write and record an album of songs.

This $50 000 was invested in an annuity that provides a monthly payment to the band.

The annuity pays interest at the rate of 3.12% per annum, compounding monthly.

After six months of writing and recording, the band has $32 667.68 remaining in the annuity.

  1. What is the value, in dollars, of the monthly payment to the band?   (1 mark)

  2. After six months of writing and recording, the band decided that it needs more time to finish the album.

     

    To extend the time that the annuity will last, the band will work for three more months without withdrawing a payment.

     

    After this, the band will receive monthly payments of $3800 for as long as possible.

     

    The annuity will end with one final monthly payment that will be smaller than all of the others.

     

    Calculate the total number of months that this annuity will last.   (2 marks)

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Show Worked Solution

a.   

 
 
 
 
 
 

 

 

b.   

 
 
 
 
 
 

 

 

 
 
 
 
 
 

 

 

CORE, FUR2-NHT 2019 VCAA 5

A random sample of 12 mammals drawn from a population of 62 types of mammals was categorized according to two variables.

likelihood of attack (1 = low, 2 = medium, 3 = high)

exposure to attack during sleep (1 = low, 2 = medium, 3 = high)

The data is shown in the following table.
 


 

  1. Use this data to complete the two-way frequency table below.   (1 mark)

      
         
     

The following two-way frequency table was formed from the data generated when the entire population of 62 types of mammals was similarly categorized.
 
     

    1. How many of these 62 mammals had both a high likelihood of attack and a high exposure to attack during sleep?   (1 mark)

    2. Of those mammals that had a medium likelihood of attack, what percentage also had a low exposure to attack during sleep?   (1 mark)

    3. Does the information in the table above support the contention that likelihood of attack is associated with exposure to attack during sleep? justify your answer by quoting appropriate percentages. It is sufficient to consider only one category of likelihood of attack when justifying your answer.   (2 marks)

Show Answers Only

  1.  

    2.                      


    3.  

         
       

         
       

         

Show Worked Solution

a.     

 

b.    i.

ii.   
 

 

iii.

MATRICES, FUR1-NHT 2019 VCAA 7-8 MC

A farm contains four water sources, , , and .

Part 1

Cows on the farm are free to move between the four water sources.

The change in the number of cows at each of these water sources from week to week is shown in the transition diagram below.
 

Let be the state matrix for the location of the cows in week of 2019.

The state matrix for the location of the cows in week 23 of 2019 is
 

The state matrix for the location of the cows in week 24 of 2019 is

Of the cows expected to be at in week 24 of 2019, the percentage of these cows at in week 23 of 2019 is closest to

  1.   8%
  2.   9%
  3. 20%
  4. 22%
  5. 25%

 

Part 2

Sheep on the farm are also free to move between the four water sources.

The change in the number of sheep at each water source from week to week is shown in matrix below.
 


 

In the long term, 635 sheep are expected to be at each week.

In the long term, the number of sheep expected to be at each week is closest to

  1. 371
  2. 493
  3. 527
  4. 607
  5. 635
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Show Worked Solution


 


 

 
 

 

 

 

 

GRAPHS, FUR1-NHT 2019 VCAA 8 MC

Jamie sold bottles of homemade lemonade to his neighbours on Saturday.

The revenue, in dollars, he made from selling bottles of lemonade is given by

revenue = 3.5

The cost, in dollars, of making bottles of lemonade is given by

cost = 60 +

The profit made by Jamie on Saturday could have been

  1. $162
  2. $165
  3. $168
  4. $173
  5. $177
Show Answers Only

Show Worked Solution
 
 

 

 

NETWORKS, FUR1-NHT 2019 VCAA 8 MC

A landscape gardener is planning the construction of a garden.

This project requires 13 activities, to .

The directed graph below shows each of these activities represented by edges.

The numbers on the edges are the durations of the activities, in days.
 


  

The cost of reducing the completion time of any activity in this project is $1000 per day.

The landscape gardener has a maximum of $3000 to spend on reducing the minimum completion time of this project.

The total completion time of the project can be reduced by three days by reducing

  1. activity by one day and activity by two days.
  2. activity by two days and activity by one day.
  3. activity by two days and activity by one day.
  4. activity by one day and activity by two days.
  5. activity by one day, activity by one day and activity by one day.
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Show Worked Solution


 


 

Calculus, MET2-NHT 2019 VCAA 5

Let    and  .

  1. Find  .   (1 mark)

  2. Find the coordinates of point  , where the tangent to the graph of  at  is parallel to the graph of  .   (2 marks)

  3. Show that the equation of the line that is perpendicular to the graph of    and goes through point  is  .   (1 mark)

Let be the point of intersection of the graphs of and  , as shown in the diagram below.
 

               
 

  1. Determine the coordinates of point .   (1 mark)

  2. The shaded region below is enclosed by the axes, the graphs of  and , and the line  .
     
     
               
     
    Find the area of the shaded region.   (2 marks)

Let    and  .

  1. The graphs of , and    are shown in the diagram below. The graphs of and touch but do not cross.
     
     
               
     

     Find the value of  .   (2 marks)

  2. Find the value of  , for which the tangent to the graph of at its -intercept and the tangent to the graph of at its -intercept are parallel.   (1 mark)

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Calculus, MET2-NHT 2019 VCAA 4

A mining company has found deposits of gold between two points, and , that are located on a straight fence line that separates Ms Pot's property and Mr Neg's property. The distance between and is 4 units.

The mining company believes that the gold could be found on both Ms Pot's property and Mr Neg's property.

The mining company initially models he boundary of its proposed mining area using the fence line and the graph of 

where is the number of units from point in the direction of point and is the number of units perpendicular to the fence line, with the positive direction towards Ms Pot's property. The mining company will only mine from the boundary curve to the fence line, as indicated by the shaded area below.
 

  1. Determine the total number of square units that will be mined according to this model.   (2 marks)

The mining company offers to pay Mr Neg $100 000 per square unit of his land mined and Ms Pot $120 000 per square unit of her land mined.

  1. Determine the total amount of money that the mining company offers to pay.   (1 mark)

The mining company reviews its model to use the fence line and the graph of

       where .

  1. Find the value of    for which    for all .   (1 mark)

  2. Solve    for in terms of .   (2 marks)

Mr Neg does not want his property to be mined further than 4 units measured perpendicular from the fence line.

  1. Find the smallest value of , correct to three decimal places, for this condition to be met.   (2 marks)

  2. Find the value of for which the total area of land mined is a minimum.   (3 marks)

  3. The mining company offers to pay Ms Pot $120 000 per square unit of her land mined and Mr Neg $100 000 per square unit of his land mined.
  4. Determine the value of that will minimize the total cost of the land purchase for the mining company. Give your answer correct to three decimal places.   (2 marks)

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Trigonometry, MET2-NHT 2019 VCAA 2

The wind speed at a weather monitoring station varies according to the function

where is the speed of the wind, in kilometres per hour (km/h), and    is the time, in minutes, after 9 am.

  1. What is the amplitude and the period of  ?   (2 marks)

  2. What are the maximum and minimum wind speeds at the weather monitoring station?   (1 mark)

  3. Find  , correct to four decimal places.   (1 mark)

  4. Find the average value of    for the first 60 minutes, correct to two decimal places.   (2 marks)

A sudden wind change occurs at 10 am. From that point in time, the wind speed varies according to the new function

                   

where    is the speed of the wind, in kilometres per hour, is the time, in minutes, after 9 am and  . The wind speed after 9 am is shown in the diagram below.
 

  1. Find the smallest value of , correct to four decimal places, such that    and    are equal and are both increasing at 10 am.   (2 marks)

  2. Another possible value of was found to be 31.4358

     

    Using this value of , the weather monitoring station sends a signal when the wind speed is greater than 38 km/h.

     

    i.  Find the value of at which a signal is first sent, correct to two decimal places.   (1 mark)

     

    ii. Find the proportion of one cycle, to the nearest whole percent, for which  .   (2 marks)

  3. Let    and  .
     
    The transformation    maps the graph of    onto the graph of  .

     

    State the values of  , , and , in terms of where appropriate.   (3 marks)

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