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Calculus, 2ADV C3 2021 HSC 13

Find the exact gradient of the tangent to the curve  `y = x tan x`  at the point where  `x = pi/3`.  (3 marks)

Show Answers Only

`sqrt3 + (4pi)/3`

Show Worked Solution

`y = x tan x`

`(dy)/(dx) = tan x + x sec^2 x`

`text(Find)\ \ m\ \ text(when)\ \ x = pi/3:`

`(dy)/(dx)` `= tan\ pi/3 + pi/3 · 1/(cos^2\ pi/3)`
  `= sqrt3 + pi/3 · 1/(1/4)`
  `= sqrt3 + (4pi)/3`

Financial Maths, STD2 F1 2021 HSC 22

The table shows the income tax rates for the 2020-2021 financial year.
 

   

William has a gross annual salary of $84 000. He has allowable tax deductions of $900 for home-office equipment and $474 for union fees. William must also pay a Medicare Levy of 2% of his taxable income.

Calculate the total tax payable by William including the Medicare Levy.  (3 marks)

Show Answers Only

`$18\ 972.97`

Show Worked Solution
`text(Taxable income)` `=84\ 000-900-474`  
  `=$82\ 626`  
`text(Income tax)` `=5092+0.325(82\ 626 – 45\ 000)`  
  `=$17\ 320.45`  
`text(Medicare Levy)` `=0.02 xx 82\ 626`  
  `=$1652.52`  

 

`:.\ text(Total tax payable)` `=17\ 320.45 + 1652.52`  
  `=$18\ 972.97`  

Measurement, STD2 M1 2021 HSC 16

The volume, `V`, of a sphere is given by the formula

`V = frac{4}{3} pi r^3,`

where `r` is the radius of the sphere.

A tank consists of the bottom half of a sphere of radius 2 metres, as shown.
 

Find the volume of the tank in cubic metres, correct to one decimal place.   (2 marks)

Show Answers Only

`16.8\ text{m}^3`

Show Worked Solution
 `V` `= frac{1}{2} times frac{4}{3} pi r^3`
  `= frac{1}{2} times frac{4}{3} times pi times 2^3`
  `= 16.755…`
  `= 16.8\ text{m}^3\ \ text{(1 d.p.)}`

Functions, 2ADV F1 2021 HSC 8 MC

The graph of  `y = f(x)`  is shown.

Which of the following could be the equation of this graph?

  1. `y = (1 - x)(2 + x)^3`
  2. `y = (x + 1)(x - 2)^3`
  3. `y = (x + 1)(2 - x)^3`
  4. `y = (x - 1)(2 + x)^3`
Show Answers Only

`C`

Show Worked Solution

`text(By elimination:)`

`text(A single negative root occurs when)\ \ x =–1`

`->\ text(Eliminate A and D)`

`text(When)\ \ x = 0, \ y > 0`

`->\ text(Eliminate B)`

`=> C`

Calculus, 2ADV C3 2021 HSC 7 MC

The diagram shows part of  `y = f(x)`  which has a local minimum at  `x = –2`  and a local maximum at  `x = 3`.
 

Which of the following shows the correct relationship between  `f^(″)(–2), \ f(0)`  and  `f^(′)(3)`?

  1. `f(0) < f^(′)(3) < f^(″)(–2)`
  2. `f(0) < f^(″)(–2) < f^(′)(3)`
  3. `f^(″)(–2) < f^(′)(3) < f(0)`
  4. `f^(″)(–2) < f(0) < f^(′)(3)`
Show Answers Only

`A`

Show Worked Solution

Mean mark 52%.
`f^(″)(–2)` `> 0\ \ \ (text(concave up at)\ \  x = –2)`
`f(0)` `< 0\ \ \ (text(see graph))`
`f^(′)(3)` `= 0\ \ \ (text(S.P.))`

 
`:. f(0) < f^(′)(3) < f^(″)(–2)`

`=> A`

Measurement, STD2 M1 2021 HSC 12 MC

A block of land is represented by the shaded region on the number plane. All measurements are in kilometres. 
 

Which of the following is the approximation for the area of this block of land in square kilometres, using two applications of the trapezoidal rule?

  1. 99
  2. 19.8
  3. 39.6
  4. 72
Show Answers Only

`B`

Show Worked Solution

`\text{Area}` `≈ \frac{6}{2} (1.2 + 2 \times 2 + 1.4)`
  `≈ 3 (6.6)`
  `≈ 19.8 \ \text{km}^2`

 
`=> B`

Probability, 2ADV S1 2021 HSC 6 MC

There are 8 chocolates in a box. Three have peppermint centres (P) and five have caramel centres (C).

Kim randomly chooses a chocolate from the box and eats it. Sam then randomly chooses and eats one of the remaining chocolates.

A partially completed probability tree is shown.
 

What is the probability that Kim and Sam choose chocolates with different centres?

  1. `\frac{15}{64}`
  2. `\frac{15}{56}`
  3. `\frac{15}{32}`
  4. `\frac{15}{28}`
Show Answers Only

`D`

Show Worked Solution

 

`Ptext{(different centres)}` `= P text{(PC)} + P text{(CP)}`
  `=\frac{3}{8} · \frac{5}{7} + \frac{5}{8} · \frac{3}{7}`
  `= \frac{15}{56} + \frac{15}{56}`
  `= \frac{15}{28}`

 
`=> D`

L&E, 2ADV E1 2021 HSC 5 MC

Which of the following best represents the graph of  `y = 10 (0.8)^x`?
 

Show Answers Only

`A`

Show Worked Solution

`\text{By elimination:}`

`\text{When} \ x = 0 \ , \ y = 10(0.8) ^0 = 10`

`-> \ text{Eliminate B and D}`

`text(As)\ \ x→oo, \ y→0`

`-> \ text{Eliminate C}`

`=> A`

Algebra, STD1 A2 2021 HSC 8 MC

A student is thinking of a number. Let the number be `x`.

When the student subtracts 8 from this number and multiplies the result by 3, the answer is 2 more than `x`.

Which equation can be used to find `x`?

  1. `3(x-8)=2x`
  2. `3x-8=2x`
  3. `3(x-8)=x+2`
  4. `3x-8=x+2`
Show Answers Only

`C`

Show Worked Solution

`text(The description defines the following equation:)`

`(x-8) xx 3` `= x + 2`
`3(x-8)` `=x+2`

 
`=>C`

Financial Maths, STD1 F3 2021 HSC 4 MC

Three years ago an appliance was valued at $2467. Its value has depreciated by 15% each year, based on the declining-balance method.

What is the salvage value today, to the nearest dollar?

  1. $952
  2. $1110
  3. $1357
  4. $1515
Show Answers Only

`D`

Show Worked Solution

♦ Mean mark 50%.
`S` `= V_0 (1 – r)^n`
  `= 2467 (1 – 0.15)^3`
  `= 2467 (0.85)^3`
  `= $1515`

 
`=>  D`

Networks, STD2 N2 2021 HSC 2 MC

Consider the network diagram.
 

What is the sum of the degrees of all the vertices in this network?

  1.  5
  2.  8
  3.  14
  4.  16
Show Answers Only

`D`

Show Worked Solution

`text(Working from)\ A\ text(to)\ E:`

COMMENT: This simple question caused problems for many with mean mark just 58%.
`text{Sum of degrees}` `= 4 + 3 + 4 + 2 + 3`
  `= 16`

 
`=> D`

Measurement, STD2 M7 2021 HSC 27

The price and the power consumption of two different brands of television are shown.

The average cost for electricity is 25c/kWh. A particular family watches an average of 3 hours of television per day.

  1. The annual cost of electricity for Television A for this family is $48.18.
  2. For this family, what is the difference in the annual cost of electricity between Television A and Television B?  (2 marks)

  3. For this family, how many years will it take for the total cost of buying and using Television A to be equal to the cost of buying and using Television B?  (2 marks)

Show Answers Only
  1. `$4.38`
  2. `5\ text(years)`
Show Worked Solution
a.   `text{Annual power usage (B)}` `= 160 xx 3 xx 365`
    `=175\ 200`
    `=175.2\ text(kWh)`

♦ Mean mark part (a) 49%.
  `text{Annual cost (B)}` `= 175.2 xx 0.25`
    `=$43.80`

 

  `text{Difference in cost}` `= 48.18 – 43.80`
    `=$4.38`

♦♦ Mean mark part (b) 23%.
b.   `text{Difference in price}` `= 921.90 – 900`
    `=$21.90`

 

  `text(Years to even out cost)` `=21.90/4.38`
    `=5\ text{years}`

Financial Maths, STD2 F4 2021 HSC 26

Nina plans to invest $35 000 for 1 year. She is offered two different investment options.

Option A:  Interest is paid at 6% per annum compounded monthly.

Option B:  Interest is paid at `r` % per annum simple interest.

  1. Calculate the future value of Nina's investment after 1 year if she chooses Option A.  (2 marks)

  2. Find the value of `r` in Option B that would give Nina the same future value after 1 year as for Option A. Give your answer correct to two decimal places.  (2 marks)

Show Answers Only
  1. `$37\ 158.72`
  2. `6.17text(%)`
Show Worked Solution
a.   `r` `= text(6%)/12= text(0.5%) = 0.005\ text(per month)`
  `n` `=12`

 

`FV` `= PV(1 + r)^n`
  `= 35\ 000(1 + 0.005)^(12)`
  `= $37\ 158.72`

♦♦ Mean mark part (b) 36%.
b.   `I` `=Prn`
  `2158.72` `=35\ 000 xx r xx 1`
  `r` `=2158.72/(35\ 000)`
    `=0.06167…`
    `=6.17 text{% (to 2 d.p.)}`

Measurement, STD2 M2 2021 HSC 20

City A is in Sweden and is located at (58°N, 16°E). Sydney, in Australia, is located at (33°S, 151°E).

Robert lives in Sydney and needs to give an online presentation to his colleagues in City A starting at 5:00 pm Thursday, local time in Sweden.

What time and day, in Sydney, should Robert start his presentation?

It is given that 15° = 1 hour time difference. Ignore daylight saving.  (3 marks)

Show Answers Only

`text(2 am Friday)`

Show Worked Solution

`text{Angular difference}\ = 151 – 16 = 135°`

Mean mark 52%.

`=>\ text{Time difference}\ = 135/15 = 9\ text(hours)`

`text(Sydney is east of Sweden → ahead)`
 

`text{Presentation time (Sydney)}` `=\ text(5 pm Thurs + 9 hours)`  
  `=\ text(2 am Friday)`  

Statistics, STD2 S1 2021 HSC 17

The five-number summary of a dataset is given.

Lowest score = 1

Lowest quartile (`Q_1`) = 4

Median (`Q_2`) = 7

Upper quartile (`Q_3`) = 10

Highest score = 20

Is 20 an outlier? Justify your answer with calculations.  (2 marks)

Show Answers Only

`text(20 is an outlier – See Worked Solution)`

Show Worked Solution

`IQR = Q_3 – Q_1 = 10-4=6`

COMMENT: Stating 20 > 19 is necessary to justify this answer.
`text(Upper Fence)` `= Q_3 + 1.5 xx IQR`
  `=10 + 1.5 xx 6`
  `=19`

  
`text{S}text{ince 20 > 19, 20 is an outlier.}`

Algebra, STD2 A2 2021 HSC 9 MC

A student is thinking of a number. Let the number be `x`.

When the student subtracts 8 from this number and multiplies the result by 3, the answer is 2 more than `x`.

Which equation can be used to find `x`?

  1. `3(x-8)=2x`
  2. `3x-8=2x`
  3. `3(x-8)=x+2`
  4. `3x-8=x+2`
Show Answers Only

`C`

Show Worked Solution

`text(The description defines the following equation:)`

`(x-8) xx 3` `= x + 2`
`3(x-8)` `=x+2`

 
`=>C`

Statistics, STD2 S5 2021 HSC 8 MC

On a test, Zac's mark corresponded to a `z`-score of 2. The test scores had a mean of 63 and a standard deviation of 8.

What was Zac's actual mark on the test?

  1. 65
  2. 67
  3. 73
  4. 79
Show Answers Only

`D`

Show Worked Solution

`text{Method 1 (quicker)}`

`text(Actual mark)` `= mu + 2 xx sigma`
  `= 63 + 2 xx 8`
  `= 79`

 

`text(Method 2)`

`ztext(-score)` `= (x – mu)/sigma`
`2` `= (x-63)/8`
`16` `=x-63`
`x` `=79`

 
`=>D`

Measurement, STD2 M1 2021 HSC 6 MC

Suppose  `a=b/7`, where  `b=22.`

What is the value of  `a`, correct to three significant figures?

  1. 3.14
  2. 3.15
  3. 3.142
  4. 3.143
Show Answers Only

`A`

Show Worked Solution

`a=b/7=22/7=3.1428…`

`3.1428 = 3.14\ text{(to 3 sig fig)}`

`=>  A`

Financial Maths, STD2 F2 2021 HSC 5 MC

Peter currently earns $21.50 per hour. His hourly wage will increase by 2.1% compounded each year for the next four years.

What will his hourly wage be after four years?

  1. `21.50(1.21)^4`
  2. `21.50(1.021)^4`
  3. `21.50 + 21.50 xx 0.21 xx 4`
  4. `21.50 + 21.50 xx 0.021 xx 4`
Show Answers Only

`B`

Show Worked Solution

`text(Wage after 1 year) = 21.50 xx 1.021`

`text(Wage after 2 years) = 21.50 xx 1.021 xx 1.021 = 21.50(1.021)^2`

`vdots`

`text(Wage after 4 years) = 21.50(1.021)^4`

`=>  B`

Measurement, NAP-E3-NC03v1

At the start of the day, a farmer drove around his property inspecting the fences and livestock.

He started the drive at 6:15 and finished at 7:04.

How long did the farmer drive for?

`text(49 minutes)` `text(51 minutes)` `text(71 minutes)` `text(89 minutes)`
 
 
 
 
Show Answers Only

`text(49 minutes)`

Show Worked Solution

`text(49 minutes)`

Measurement, NAP-K3-CA01v1

Fiona is nurse who is administering vaccines to patients using a needle

Which unit would be the most appropriate to measure the volume of vaccine she needs to inject?

milligrams joules millimetres millilitres
 
 
 
 
Show Answers Only

`text(millilitres)`

Show Worked Solution

`text(A needle injects liquid vaccine into people.)`

`text(The most appropriate measure: millilitres)`

Measurement, NAP-E2-05v1

Helene started her walk to work at 2:15 pm.

She arrived at her work at 3:03 pm.

How long did Helene walk for?

`text(12 minutes)` `text(48 minutes)` `text(72 minutes)` `text(88 minutes)`
 
 
 
 
Show Answers Only

`text(48 minutes)`

Show Worked Solution

`text(One Strategy:)`

`text(2:15 pm to 3:00 pm = 45 minutes)`

`text(3:00 pm to 3:03 pm = 3 minutes)`

`text(Walking time = 45 + 3 = 48 minutes)`

Number and Algebra, NAP-I2-01v1

Kranskie delivers brochures by hand and is paid 10 cents for every brochure he delivers.

Kranskie delivers 79 brochures in his first hour of work.

How much money will he be paid for this?

`79¢` `$7.90` `$79` `$790`
 
 
 
 
Show Answers Only

`$7.90`

Show Worked Solution

`79 xx 10\ text(cents)`

`= 790\ text(cents)`

`= $7.90`

Number and Algebra, NAP-C1-13v1

Ernie has collected 93 bottles for recycling.

Grover has collected 88 bottles for recycling.

In total, how many bottles can Ernie and Grover deliver to the recycling depot?

`165` `171` `177` `181`
 
 
 
 
Show Answers Only

`181`

Show Worked Solution

`text(One strategy:)`

`text(Total bottles)` `=93+88`
  `=90 + 80 + 3 + 8`
  `=170 + 11`
  `=181`

Networks, STD2 N3 FUR2 4

Training program 1 has the cricket team starting from exercise station `S` and running to exercise station `O`.

For safety reasons, the cricket coach has placed a restriction on the maximum number of people who can use the tracks in the fitness park.

The directed graph below shows the capacity of the tracks, in number of people per minute.
 


 

  1. Determine the capacity of Cut 1, shown above.  (1 mark)

  2. What is the maximum flow from `S` to `O`, in number of people per minute?  (1 mark)

Show Answers Only
  1. `52`
  2. `50`
Show Worked Solution
a.   `text{Capacity (Cut 1)}` `= 20 + 12 + 20`
    `= 52`

 

b.   `text(Max flow/minimum cut)`

♦♦ Mean mark part (c) 32%.

`= 20 + 10 + 20`

`= 50`
 

NETWORKS, FUR2 2020 VCAA 3

A local fitness park has 10 exercise stations: `M` to `V`.

The edges on the graph below represent the tracks between the exercise stations.

The number on each edge represents the length, in kilometres, of each track.
 


 

The Sunny Coast cricket coach designs three different training programs, all starting at exercise station `S`.

  Training program 
number		 Training details
  1 The team must run to exercise station `O`.
  2 The team must run along all tracks just once.
  3 The team must visit each exercise station and return to  exercise station `S`.

 

  1. What is the shortest distance, in kilometres, covered in training program 1?  (1 mark)

  2.  i. What mathematical term is used to describe training program 2?   (1 mark)

  3. ii. At which exercise station would training program 2 finish?  (1 mark)

  4. To complete training program 3 in the minimum distance, one track will need to be repeated.

     

    Complete the following sentence by filling in the boxes provided.  (1 mark)


    This track is between exercise station   
     
    		 and exercise station   
     
Show Answers Only
  1. `3.2\ text(km)`
  2.  i. `text(Eulerian trail)`
  3. ii. `text(Station)\ P`
  4. `S and T`
Show Worked Solution
a.   `text(Shortest distance)` `= STUVO`
    `= 0.6 + 1.2 + 0.6 + 0.8`
    `= 3.2\ text(km)`

 

b.i.  `text(Eulerian trail)`
 

b.ii.  `text(Station)\ P\ text{(only other vertex with}\ S\ text{to have odd degree)} `

♦♦ Mean mark part (c) 25%.

 

c.   `S and T`

NETWORKS, FUR2 2020 VCAA 2

A cricket team has 11 players who are each assigned to a batting position.

Three of the new players, Alex, Bo and Cameron, can bat in position 1, 2 or 3.

The table below shows the average scores, in runs, for each player for the batting positions 1, 2 and 3.

 

      Batting position
      1 2 3
     Player          Alex              22                     24                     24          
    Bo 25 25 21
    Cameron    24 25 19

 

Each player will be assigned to one batting position.

To which position should each player be assigned to maximise the team’s score? Write your answer in the table below.   (1 mark)

 

       Player           Batting position     
     Alex  
     Bo  
     Cameron  

 

Show Answers Only

`text{Bo (1), Cameron (2), Alex (3)}`

Show Worked Solution

`text(Test different combinations:)`

`text(CBA) = 24 + 25 + 24 = 73`

`text(BCA) = 25 + 25 + 24 = 74`

`text(BAC) = 25 + 24 + 19 = 68`

`:.\ text{Combination for max score: Bo (1), Cameron (2), Alex (3)}`
 

       Player           Batting position     
     Alex 3
     Bo 1
     Cameron 2

NETWORKS, FUR2 2020 VCAA 1

The Sunny Coast Cricket Club has five new players join its team: Alex, Bo, Cameron, Dale and Emerson.

The graph below shows the players who have played cricket together before joining the team.

For example, the edge between Alex and Bo shows that they have previously played cricket together.
 

  1. How many of these players had Emerson played cricket with before joining the team?   (1 mark)

  2. Who had played cricket with both Alex and Bo before joining the team?   (1 mark)

  3. During the season, another new player, Finn, joined the team.

      

    Finn had not played cricket with any of these players before.

      

    Represent this information on the graph above.   (1 mark)

Show Answers Only
  1. `text(one)`
  2. `text(Dale)`
  3.  
Show Worked Solution

a.  `text{One (one edge connected to Emerson.)}`
  

b.  `text{Dale (he has edges directly connected to Alex and Bo.)}`

 

c.  

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  `S_n`  be the state matrix that shows the expected number of shoppers at each of the three centres  `n`  weeks after Westmall is sold.

A matrix recurrence relation that generates values of  `S_n`  is

`S_(n+1) = T xx S_n`

`{:(quad qquad qquad qquad qquad qquad qquad qquad text(this week)),(qquad qquad qquad qquad qquad qquad quad \ W qquad quad G qquad quad \ E),(text(where)\ T = [(quad 0.80, 0.09, 0.10),(quad 0.12, 0.79, 0.10),(quad 0.08, 0.12, 0.80)]{:(W),(G),(E):}\ text(next week,) qquad qquad  S_0 = [(250\ 000), (230\ 000), (200\ 000)]{:(W),(G),(E):}):}`
 

  1. Calculate the state matrix, `S_1`, 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)

Show Answers Only
  1. `S_1 =[(240\ 700),(231\ 700),(207\ 600)]`
  2. `30\ 000`
  3. `S_6 = T^6S_0 =  [(text(__)), (233\ 708), (text(__))]`
  4. `218\ 884`
Show Worked Solution
a.   `S_1` `= TS_0`
    `= [(0.80, 0.09, 0.10),(0.12, 0.79, 0.10),(0.08, 0.12, 0.80)][(250\ 000),(230\ 000),(200\ 000)]=[(240\ 700),(231\ 700),(207\ 600)]`

 

b.   `text(Using the graph)`
  `text(Difference)` `= 250\ 000-220\ 000`
    `= 30\ 000`

 

♦♦ Mean mark part (c) 27%.

c.  `text(Testing options:)`

`S_6 = T^6S_0 = [(0.80, 0.09, 0.10),(0.12, 0.79, 0.10),(0.08, 0.12, 0.80)]^6[(250\ 000),(230\ 000),(200\ 000)] = [(text(__)), (233\ 708), (text(__))]`
 

`:.\ text(Maximum shoppers in Grandmall expected in week 6.)`


♦ Mean mark part (d) 39%.

d.  `text(Test with high integer)\ n:`

`S_50 = T^50S_0 -> text(Westmall) = 218\ 884`

MATRICES, FUR2 2020 VCAA 2

The preferred number of cafes `(x)` and sandwich bars `(y)` in Grandmall’s food court can be determined by solving the following equations written in matrix form.
 

`[(5, -9),(4, -7)][(x),(y)]=[(7), (6)]`
 

  1. The value of the determinant of the 2 × 2 matrix is 1.
  2. Use this information to explain why this matrix has an inverse.   (1 mark)

  3. Write the three missing values of the inverse matrix that can be used to solve these equations.   (1 mark)

 
`[(text( __), 9),(text( __), text( __)\ )]`
 

  1. Determine the preferred number of sandwich bars for Grandmall’s food court.   (1 mark)

Show Answers Only
  1. `text(S) text(ince determinant) = 1 != 0,`

     

    `text(the matrix has an inverse)`

  2. `[(-7, 9),(-4, 5)]`
  3. `2`
Show Worked Solution

a.  `text(S) text(ince determinant) = 1 != 0,`

♦ Mean mark part (a) 37%.

`->\ text(the matrix has an inverse)`


b.  `[(-7, 9),(-4, 5)]`

 

Mean mark part (c) 51%.
c.   `[(x), (y)] = [(-7, 9), (-4, 5)][(7), (6)] = [(7),(2)]`

 
`:.\ text(Preferred number of sandwich bars) = 2`

MATRICES, FUR2 2020 VCAA 1

The three major shopping centres in a large city, Eastmall `(E)`, Grandmall `(G)` and Westmall `(W)`, 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 `V`.

`qquad qquad qquad {:(qquad qquad qquad \ E qquad qquad G qquad qquad \  W),(V = [(2300,2700,2200)]):}`

  1. Write down the order of matrix `V`.   (1 mark)

Each of these centres has three major shopping areas: food `(F)`, clothing `(C)` and merchandise `(M)`.

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 `P` 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  `Q = P xx V`  is shown below. Two of the elements of this matrix are missing.
     
    `{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, \ text{___}, 1056 ), (621,\ text{___}, 594), (575, 675, 550)]{:(F),(C), (M):}):}`
     
    1. Complete matrix `Q` above by filling in the missing elements.   (1 mark)

    2. The element in row `i` and column `j` of matrix `Q` is `q_(ij)`.
    3. What does the element `q_23` 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  `A_2019`  below.

`qquad qquad A_2019 = [(21.30), (34.00), (14.70)] {:(F),(C),(M):}`

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  `A_2019`, 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

  3. `qquad qquad A_2020 = K xx A_2019`
  4. Write down matrix `K`.    (1 mark)

Show Answers Only
  1. `1 xx 3`
  2. `2800`
  3.  i.   `{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, 1296, 1056 ), (621, 729, 594), (575, 675, 550)]{:(F),(C), (M):}):}`
     
  4. ii. `q_23\ text(represents the number of people)`
  5.    `text(in the clothing area of Westmall.)`
  6. `text(Total spent) = [(135, 143, 131)][(21.30), (34.00), (14.70)] = [9663.20]`
  7. `K = [(1.05, 0, 0),(0, 0.85, 0),(0, 0, 0.99)]`
Show Worked Solution

a.  `1 xx 3`

♦ Mean mark part (b) 45%.
b.   `0.25 xx G\ text(shoppers in)\ M` `= 700`
  `:. G\ text(shoppers in)\ M` `= 700/0.25`
    `= 2800`

 

c.i.   `{:(quad qquad qquad qquad \ E qquad qquad G qquad qquad W), (Q = [(1104, 1296, 1056 ), (621, 729, 594), (575, 675, 550)]{:(F),(C), (M):}):}`

 

c.ii.   `q_23\ text(represents the number of people)`
  `text(in the clothing area of Westmall.)`

 

d.  `text(Total spent) = [(135, 143, 131)] [(21.30), (34.00), (14.70)] = [9663.20]`

♦♦ Mean mark part (e) 24%.
 

e.   `A_2020` `= K xx [(21.30), (34.00), (14.70)]`
  `:. K` `= [(1.05, 0, 0),(0, 0.85, 0),(0, 0, 0.99)]`

NETWORKS, FUR2 2020 VCAA 4

Training program 1 has the cricket team starting from exercise station `S` and running to exercise station `O`.

For safety reasons, the cricket coach has placed a restriction on the maximum number of people who can use the tracks in the fitness park.

The directed graph below shows the capacity of the tracks, in number of people per minute.
 


 

  1. How many different routes from `S` to `O` are possible?   (1 mark)

When considering the possible flow of people through this network, many different cuts can be made.

  1. Determine the capacity of Cut 1, shown above.   (1 mark)

  2. What is the maximum flow from `S` to `O`, in number of people per minute?   (1 mark)

Show Answers Only
  1. `10`
  2. `52`
  3. `50`
Show Worked Solution
a.   `text(Routes: )` `SMO, STUNO, STUVO, STUVPO, SRQUNO,SRQUVO,`
    `SRQUVPO, SRQVO, SRQVPO, SRQPO`

♦♦ Mean mark part (a) 30%.

 
`:. 10\ text(routes)`

 

b.   `text{Capacity (Cut 1)}` `= 20 + 12 + 20`
    `= 52`

 

c.   `text(Max flow/minimum cut)`

♦♦ Mean mark part (c) 32%.

`= 20 + 10 + 20`

`= 50`
 

CORE, FUR2 2020 VCAA 10

Samuel now invests $500 000 in an annuity from which he receives a regular monthly payment.

The balance of the annuity, in dollars, after  `n`  months,  `A_n` , can be modelled by a recurrence relation of the form

`A_0 = 500\ 000, qquad A_(n+1) = kA_n - 2000`

  1. Calculate the balance of this annuity after two months if  `k = 1.0024`.   (1 mark)

  2. Calculate the annual compound interest rate percentage for this annuity if  `k = 1.0024`.   (1 mark)

  3. For what value of  `k`  would this investment act as a simple perpetuity?   (1 mark)

Show Answers Only
  1. `$498\ 398.08`
  2. `2.88 text(%)`
  3. `1.004`
Show Worked Solution
a.   `A_1 = 1.0024 xx 500\ 000-2000 = $499\ 200`
  `A_2 = 1.0024 xx 499\ 200-2000 = $498\ 398.08`

 

♦ Mean mark 48%.
b.   `text(Monthly interest rate)` `= (1.0024-1) xx 100 = 0.24text(%)`
  `text(Annual interest rate)` `= 12 xx 0.24 = 2.88text(%)`

 

♦ Mean mark 36%.
c.   `text(Perpetuity would occur when)`
  `k xx 500\ 000-2000` `= 500\ 000`
  `k` `= (502\ 000)/(500\ 000)`
    `= 1.004`

CORE, FUR2 2020 VCAA 9

Samuel opens a savings account.

Let `B_n` be the balance of this savings account, in dollars, `n` months after it was opened.

The month-to-month value of `B_n` can be determined using the recurrence relation shown below.

`B_0 = 5000, qquad B_(n+1) = 1.003B_n`

  1. Write down the value of `B_4`, the balance of the savings account after four months.
  2. Round your answer to the nearest cent.   (1 mark)

  3. Calculate the monthly interest rate percentage for Samuel’s savings account.   (1 mark)

  4. After one year, the balance of Samuel’s savings account, to the nearest dollar, is $5183.

     

    If Samuel had deposited an additional $50 at the end of each month immediately after the interest was added, how much extra money would be in the savings account after one year?

     

    Round your answer to the nearest dollar.   (1 mark)

Show Answers Only
  1. `$5060.27`
  2. `0.3 text(%)`
  3. `$610`
Show Worked Solution
a.   `B_1` `= 1.003 (5000)`
  `B_2` `= 1.003^2 (5000)`

`vdots`

`:. B_4` `= 1.003^4 (5000)`
  `= $5060.27`

 

b.  `text(Monthly interest rate)`

`= (1.003-1) xx 100`

`= 0.3%`

 

c.   `text(Extra)\ =\ text(value of annuity after 12 months)`

`text(By TVM solver:)`

`N` `= 12`
`I(%)` `= 3.6`
`PV` `= 0`
`PMT` `= 50`
`FV` `= ?`
`text(PY)` `= text(CY) = 12`

 
`FV = 609.84`

`:.\ text(Extra money) = $610`

CORE, FUR2 2020 VCAA 7

Samuel owns a printing machine.

The printing machine is depreciated in value by Samuel using flat rate depreciation.

The value of the machine, in dollars, after `n` years, `Vn` , can be modelled by the recurrence relation

`V_0 = 120\ 000, qquad V_(n+1) = V_n-15\ 000`

  1. By what amount, in dollars, does the value of the machine decrease each year?   (1 mark)

  2. Showing recursive calculations, determine the value of the machine, in dollars, after two years.   (1 mark)

  3. What annual flat rate percentage of depreciation is used by Samuel?   (1 mark)

  4. The value of the machine, in dollars, after `n` years, `V_n`, could also be determined using a rule of the form `V_n = a + bn`.

     

    Write down this rule for `V_n`.   (1 mark)

Show Answers Only
  1. `$15\ 000`
  2. `$90\ 000`
  3. `12.5%`
  4. `V_n = 120\ 000-15\ 000n, n = 0, 1, 2, …`
Show Worked Solution

a. `$15\ 000`
  

b.   `V_1` `= 120\ 000-15\ 000 = $105\ 000`
  `V_2` `= 105\ 000-15\ 000 = $90\ 000`

 

c.   `text(Flat rate percentage` `= (15\ 000)/ (120\ 000) xx 100`
    `= 12.5 text(%)`

 

♦ Mean mark part d. 44%.

d.  `V_n = 120\ 000-15\ 000n, \ n = 0, 1, 2, …`

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)


Show Answers Only
  1. `10.4\ text(cm)`
  2. `text(Strong and negative)`
  3. `text(mean height) = 171 – 0.169 xx text(mean age)`
  4. `text(mean height) = 167.9 – 0.001621 xx text{(mean age)}^2`
Show Worked Solution
a.   `text(Difference)` `= 167.1 – 156.7`
    `= 10.4\ text(cm)`

 

Mean mark part b. 51%.

b.  `text(Strong and negative.)`

 

♦♦ Mean mark part c. 23%.

c.   `text(Slope) = (157 – 168)/(85 – 20) = -0.169`

`text(Equation of line)`

`y – 168` `= -0.1692 (x – 20)`
`y` `= -0.169x + 171`

 
`:.\ text(mean height) = 171 – 0.169 xx text(mean age)`

 

D.    `text(By CAS)`

`text(mean height) = 167.9 – 0.001621 xx text{(mean age)}^2`

CORE, FUR2 2020 VCAA 5

The scatterplot below shows body density, in kilograms per litre, plotted against waist measurement, in centimetres, for 250 men.

When a least squares line is fitted to the scatterplot, the equation of this line is

body density = 1.195 – 0.001512 × waist measurement

  1. Draw the graph of this least squares line on the scatterplot above.   (1 mark)

  2. Use the equation of this least squares line to predict the body density of a man whose waist measurement is 65 cm.
  3. Round your answer to two decimal places.   (1 mark)

  4. When using the equation of this least squares line to make the prediction in part b., are you extrapolating or interpolating?   (1 mark)

  5. Interpret the slope of this least squares line in terms of a man’s body density and waist measurement.   (1 mark)

  6. In this study, the body density of the man with a waist measurement of 122 cm was 0.995 kg/litre.
  7. Show that, when this least squares line is fitted to the scatterplot, the residual, rounded to two decimal places, is –0.02   (1 mark)

  8. The coefficient of determination for this data is 0.6783
  9. Write down the value of the correlation coefficient `r`.
  10. Round your answer to three decimal places.   (1 mark)

  11. The residual plot associated with fitting a least squares line to this data is shown below.
     
       

     

    Does this residual plot support the assumption of linearity that was made when fitting this line to this data? Briefly explain your answer.   (1 mark)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `1.10\ text(kg/litre)`
  3. `text(extrapolating)`
  4. `text(See Worked Solutions)`
  5. `-0.02`
  6. `-0.824`
  7. `text(See Worked Solutions)`
Show Worked Solution

a.   `text(LSRL passes through)\ (60, 1.1043) and (130, 0.998)`

♦ Mean mark part a. 41%.

b.   `text(body density)` `= 1.195-0.001512 xx 65`
    `= 1.09672`
    `= 1.10\ text{kg/litre (to 2 d.p.)}`
♦ Mean mark part c. 38%.
c.   `text(A waist of 65 cm is outside the)`
  `text(range of the existing data set.)`

 
`:.\ text(Extrapolating)`

 

♦ Mean mark part d. 44%.
d.   `text(Body density decreases by 0.001512 kg/litre)`
  `text(for each increase in waist size of 1 cm.)`

 

e.   `text{Body density (predicted)}`

`= 1.195-0.001512 xx 122`

`~~ 1.0105\ text(kg/litre)`
 

`text(Residual)` `= text(Actual-predicted)`
  `~~ 0.995-1.0105`
  `~~ -0.0155`
  `~~ -0.02\ text{(to 2 d.p.)}`

 

♦♦ Mean mark part f. 25%.

f.   `r` `= -sqrt(0.6783)`
    `=-0.8235…`
    `= -0.824\ text{(to 3 d.p.)}`

 

g.  `text(The residual plot has no pattern and is centred around zero.)`

`:.\ text(It supports the assumption of linearity of the LSRL.)`

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)

Show Answers Only
  1.  i. `24`
  2. ii. `1.065\ text(kg/litre)`
  3. i. `text(Weight)`
  4. ii. `text(Slope) = -0.00112\ text{(by CAS)}`
  5. `29 text(%)`
Show Worked Solution
a.i.   `n = 12`  
  `text(Median)` `= (text{6th + 7th})/2`
    `= (24 + 24)/2`
    `= 24`

 

a.ii.   `text(Mean)` `= (∑\ text{body density})/12`
    `= 1.065\ text(kg/litre)`

 

b.i.   `text(Weight)`

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

b.ii.   `text(Slope) = -0.00112\ text{(by CAS)}`

 

c.   `r` `= -0.53847\ text{(by CAS)}`
  `r^2` `= 0.289…`

 
`:. 29 text(%)`

CORE, FUR2 2020 VCAA 2

The neck size, in centimetres, of 250 men was recorded and displayed in the dot plot below.
 

  1. Write down the modal neck size, in centimetres, for these 250 men.   (1 mark)

  2. Assume that this sample of 250 men has been drawn at random from a population of men whose neck size is normally distributed with a mean of 38 cm and a standard deviation of 2.3 cm.
  3.  i. How many of these 250 men are expected to have a neck size that is more than three standard deviations above or below the mean? Round your answer to the nearest whole number.   (1 mark)

  4. ii. How many of these 250 men actually have a neck size that is more than three standard deviations above or below the mean?   (1 mark)

  5. The five-number summary for this sample of neck sizes, in centimetres, is given below.
     

    Use the five-number summary to construct a boxplot, showing any outliers if appropriate, on the grid below.   (2 marks)

Show Answers Only

a.    `text(Mode) = 38\ text(cm)`

b.i.   `1`

b.ii. `1`

c.    `text(See Worked Solutions)`

Show Worked Solution

a.   `text(Mode) = 38\ text(cm)`

 

♦ Mean mark part b.i. 40%.
b.i.   `text(Expected number of men)` `= (1-0.997) xx 250`
    `= 0.75`
    `= 1\ text{(nearest whole)}`

 

♦ Mean mark part b.ii. 42%.
b.ii.   `text(When)\ \ z = +- 3`
  `text(Neck size limits)` `= 38 +- (2.3 xx 3)`
    `= 44.9 or 31.1`

 
`:.\ text(1 man has neck size outside 3 s.d.)`
 

c.   `IQR = 39-36=3`

`text(Upper fence)\ =Q_3 + 1.5 xx 3=39 + 4.5=43.5`

`text(Lower fence)\ =Q_1-1.5 xx 3=36-4.5=31.5`

 

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)

Show Answers Only
  1. `text(Positively skewed)`
  2. `24.55`
  3. `37.5 text(%)`
Show Worked Solution

a.   `text(Positively skewed)`
 

b.   `32\ text(data points)`

`text(Median)` `= text(16th + 17th)/2`
  `= (24.5 + 24.6)/2`
  `= 24.55`

 

c.    `text(Percentage)` `= 12/32 xx 100`
    `= 37.5%`

Statistics, NAPX-p167881v02

A menswear store recorded the number of items it sold in four different categories.

What category makes up 60% of the total number of items sold?
 


  

 
   Hoodies
 
   Shorts
 
   T-shirts
 
   Pants
Show Answers Only

`text{Pants}`

Show Worked Solution

`text{Pants represent 60% of the graph.}`

Number, NAPX-p167756v02

`8 xx` 
 
  `- 13 = 75`

What value would make the number sentence right?

 
   11
 
     9
 
   13
 
     7
Show Answers Only

`11`

Show Worked Solution

`text{Check each option (multiply then add):}`

`8 xx 11 -13 = 75 \ text{(Correct)}`

`8 3 xx 9 -13 = 59 \ text{(Incorrect)}`

`8 3 xx 9 -13 = 91 \ text{(Incorrect)}`

`8  xx 7 -13 = 43 \ text{(Incorrect)}`

Number, NAPX-p167756v01

`12 + 3 xx ` 
 
  `= 39`

 
For this number sentence to be true, what is the missing value?

 
   15
 
   11
 
     9
 
     7
Show Answers Only

`9`

Show Worked Solution

`text{Check each option (multiply then add):}`

`12 + 3 xx 15= 57 \ text{(Incorrect)}`

`12 + 3 xx 11= 57 \ text{(Incorrect)}`

`12 + 3 xx 9 = 57 \ text{(Correct)}`

`12 + 3 xx 7= 57 \ text{(Incorrect)}`