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Financial Maths, SMB-013

The table shows the income tax rates for the 2022-23 financial year.

\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{    Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{    Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$45 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$45 001 – \$120 000}\rule[-1ex]{0pt}{0pt} & \text{\$5092 plus 32.5 cents for each \$1 over \$45 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$120 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$29 467 plus 37 cents for each \$1 over \$120 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$51 667 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}

Boonie is a professional cricketer and has a gross income of $145 000. During the financial year, he has allowable tax deductions of $1300 for cricket bats and pads.

What is Boonie's total amount of tax payable for the financial year?  (3 marks)

Show Answers Only

`$38\ 236`

Show Worked Solution

`text(Taxable income)\ =145\ 000-1300=$143\ 700`

`text{Tax payable}`

`= 29\ 467 + 0.37 (143\ 700-120\ 000)`

`= 29\ 467 + 8769`

`= $38\ 236`

Financial Maths, SMB-012 MC

\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{    Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{    Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$45 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$45 001 – \$120 000}\rule[-1ex]{0pt}{0pt} & \text{\$5092 plus 32.5 cents for each \$1 over \$45 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$120 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$29 467 plus 37 cents for each \$1 over \$120 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$51 667 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}

Using the tax table, what is the tax payable on $47 580?

  1. $5721.20
  2. $5930.50
  3. $15 463.50
  4. $20 555.50
Show Answers Only

`B`

Show Worked Solution

`text(Tax Payable)`

`= 5092 + 0.325 (47\ 580-45\ 000)`

`= 5092 + 838.50`

`= 5930.50`
 

`=>  B`

Financial Maths, SMB-011

George makes a single deposit of $9000 into an account that pays simple interest.

After 4 years, George's account has a balance of $10 350.

What simple interest rate did George receive on his investment?   (2 marks)

Show Answers Only

`3.75text(%)`

Show Worked Solution
`text(Interest earned)` `= 10\ 350-9000`
  `= $1350`

 

`text(Using)\ \ I = Prn,`

`1350` `= 9000 xx r xx 4`
`:. r` `= 1350/(4 xx 9000)`
  `= 0.0375`
  `= 3.75text(%)`

Financial Maths, SMB-009

Pamela deposits $2000 into a savings account which earns simple interest at the rate of 2.5% per annum.

No deposits or withdrawals are made from this account.

After 2 years, how much is in Pamela's savings account?  (2 marks)

Show Answers Only

`$2100`

Show Worked Solution
`text(Interest earned)` `= Prn`
  `= 2000 xx 2.5/100 xx 2`
  `= $100`

 

`:.\ text{Account balance}` `= 2000 + 100`
  `= $2100`

Financial Maths, SMB-008

Dante deposits $5000 into a savings account which earns simple interest.

No deposits or withdrawals are made from this account.

After 4 years, Dante notices there is $5600 in the account.

What is the annual rate of interest for the account?   (2 marks)

Show Answers Only

`3text(%)`

Show Worked Solution
`text(Interest per year)` `= (5600-5000)÷4`
  `= 600 ÷ 4`
  `= $150`

 

`:.\ text{Interest rate (p.a.)}` `= 150/5000 xx 100`
  `= 3 text(%)`

Financial Maths, SMB-007

Mr. Soros put $500 into a simple interest account for a year.

He did not take any money out or add any money to the account.

At the end of the year he had $530 in the account.

What was the annual percentage interest rate?   (2 marks)

Show Answers Only

`text(6%)`

Show Worked Solution
`text(Interest earned)` `= 530-500`
  `= $30`

 

`:.\ text{Interest rate (annual)}`

`= 30/500`

`= 0.06`

`= 6text(%)`

Financial Maths, SMB-006

Cassie opens a savings account and deposits $900 into it.

She makes no more deposits and earns simple interest on her original deposit at 3.5% each year.

How much interest will Cassie earn after 4 years?  (2 marks)

Show Answers Only

`$126`

Show Worked Solution
`text(Interest earned)` `=Prn`
  `= 900 xx 3.5/100 xx 4`
  `= $126`

Financial Maths, SMB-005

On a weekend, Abbey works 8 hours at her normal pay rate and 12 hours at time and a half of her normal pay.

Abbey was paid $707.20 in total for this work.

What is her normal pay rate per hour?  (2 marks)

Show Answers Only

`$27.20\ text(per hour)`

Show Worked Solution

`text(Total normal hours)`

`= 8 + 1.5 xx 12`

`= 26`

 

`:.\ text(Abbey’s normal pay rate)`

`= 707.20/26`

`= $ 27.20\ text(per hour.)`

Financial Maths, SMB-004

John, Olivia and Louis are picking grapes to earn money.

Their pay is based on the number of tonnes of grapes they pick.
 

  Tonnes picked
John 4
Olivia 3
Louis 1

 

Their total pay is  $640.

How much does Olivia earn?  (2 marks)

Show Answers Only

`$240`

Show Worked Solution

`text(Payment per tonne)`

`= 640/8`

`= $80`
 

`:.\ text(Olivia earns)` `= 3 xx 80`
  `= $240`

Financial Maths, SMB-003 MC

David has a job selling mobile phone plans. His weekly salary, `W` dollars, is calculated using the rule below:

`W = 300 + 0.05 P`

where `P` is the total value in dollars of the mobile phone plans he sells that week.

David sold $32 000 worth of mobile phone plans in a given week.

What was David's salary in the week?

  1. `$640`
  2. `$1280`
  3. `$1900`
  4. `$3230`
Show Answers Only

`C`

Show Worked Solution
`W` `= 300 + 0.05 xx 32\ 000`
  `= 300 + 1600`
  `= 1900`

 
`=>C`

Financial Maths, SMB-002

Bella is an electrician.

She charges $150 to attend a job and then a fixed price for each minute she spends on the job.

The graph below shows Bella's charge based on the number of minutes she spends at the job.
 

How much will Bella charge for a job that takes 60 minutes?  (2 marks)

Show Answers Only

`$330`

Show Worked Solution

`text(Charge to attend = $150)`

`text(From the graph,)`

`=>\ text(50 minute job costs $300)`

`:.\ text(C)text(ost per minute)` `= (300-150)/50 = $3`

 

`:.\ text(C)text(ost of 60 minute job)`

`= 150 + 60 xx 3`

`= $330`

Financial Maths, SMB-001

Olivia earned $17.24 per hour working at a pizza store.

This week she worked for  `8 1/4` hours.

She used the money she earned this week to buy concert tickets for herself and her friends.

Each concert ticket cost $14.15.

Calculate the maximum number of concert tickets Olivia could have bought?  (2 marks)

Show Answers Only

`10`

Show Worked Solution
`text(Money earned)` `= 8 1/4 xx 17.24`
  `= $142.23`

 

`:.\ text(Maximum tickets)` `= 142.23/14.15`
  `= 10.05…`
  `= 10`

Algebra, STD1 A1 2019 HSC 34

Given the formula  `C = (A(y + 1))/24`, calculate the value of  `y`  when  `C = 120`  and  `A = 500`.  (3 marks)

Show Answers Only

`4.76`

Show Worked Solution

`text(Make)\ \ y\ \ text(the subject:)`

`C` `= (A(y + 1))/24`
`24C` `= A(y + 1)`
`y + 1` `= (24C)/A`
`y` `= (24C)/A-1`
  `= (24 xx 120)/500-1`
  `= 4.76`

Measurement, STD1 M5 2022 HSC 5 MC

Two similar figures are shown.
 

What is the value of `x` ?

  1. 6
  2. 8
  3. 18
  4. 27
Show Answers Only

`C`

Show Worked Solution

`text{Scale factor}\ =3/2 =1.5`

`:.\ x = 1.5 xx 12 = 18`
  

`text{Alternate solution}`

`text{Using sides of similar figures in the same ratio:}`

`x/12` `=3/2`  
`x` `=12 xx (3/2)`  
`x` `=18`  

   
`=> C`

Measurement, STD2 M1 2022 HSC 28

A dam is in the shape of a triangular prism which is 50 m long, as shown.

Both ends of the dam, `A B C` and `D E F`, are isosceles triangles with equal sides of length 25 metres. The included angles `B A C` and `E D F` are each `150^@`.

 
     

Calculate the number of litres of water the dam will hold when full.  (4 marks)

Show Answers Only

`7\ 812\ 500\ text{L}`

Show Worked Solution

`V=Ah`

`text{Use sine rule to find}\ A:`

`A` `=1/2 ab\ sinC`  
  `=1/2 xx 25 xx 25 xx sin150^@`  
  `=156.25\ text{m}^2`  

 

`:.V` `=156.25 xx 50`  
  `=7812.5\ text{m}^3`  

 

`text{S}text{ince 1 m³ = 1000 litres:}`

`text{Dam capacity}` `=7812.5 xx 1000`  
  `=7\ 812\ 500\ text{L}`  

♦♦ Mean mark 33%.

Financial Maths, STD2 F1 2022 HSC 21

A real estate agent's commission for selling houses is 2% for the first $800 000 of the sale price and 1.5% for any amount over $800 000.

Calculate the commission earned in selling a house for $1 500 000.  (2 marks)

Show Answers Only

`$26\ 500`

Show Worked Solution
`text{Commission}` `=800\ 000 xx 2text{%} + (1\ 500\ 000-800\ 000) xx 1.5text{%}`  
  `=800\ 000 xx 0.02 + 700\ 000 xx 0.015`  
  `=16\ 000 + 10\ 500`  
  `=$26\ 500`  

Financial Maths, STD2 F1 2022 HSC 7 MC

Tian is paid $20.45 per hour, as well as a meal allowance of $16.20 per day.

What are Tian's total earnings if she works 9 hours per day for 5 days?

  1. `$329.85`
  2. `$936.45`
  3. `$1001.25`
  4. `$1649.25`
Show Answers Only

`C`

Show Worked Solution
`text{Earnings (5 days)}` `=5 xx [(9 xx 20.45) + 16.20]`  
  `=5 xx 200.25`  
  `=$1001.25`  

 
`=>C`

Statistics, STD1 S1 2020 HSC 24

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

\(38 \ \ 25 \ \ 38 \ \ 46 \ \ 55 \ \ 68 \ \ 72 \ \ 55 \ \ 36 \ \ 38\)

  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.     \(1\)

b.    \(\text{Standard deviation is a measure of how much the}\)

\(\text{ages of individuals differ from the mean age of the group.}\)
 

\(\Rightarrow\ \text{Standard deviation of Wednesday’s group would be}\)

\(\text{less as the mean is 70 and everyone’s age is 70.}\)

Show Worked Solution

a.     \(\text{Reorder ages in ascending order:}\)

    \(25, 36, 38, 38, 38, 46, 55 , 55, 68, 72\)

\(\text{Median} = \dfrac{\text{5th + 6th}}{2} = \dfrac{38 + 46}{2} = 42\)

\(\therefore\ \text{People with age between 42 − 47.1 = 1}\)

♦ Mean mark (a) 39%.

 
b.    \(\text{Standard deviation is a measure of how much the}\)

\(\text{ages of individuals differ from the mean age of the group.}\)
 

\(\Rightarrow\ \text{Standard deviation of Wednesday’s group would be}\)

\(\text{less as the mean is 70 and everyone’s age is 70.}\)

♦♦♦ Mean mark (b) 20%.

Measurement, STD2 M1 2020 HSC 25

A composite solid consists of a triangular prism which fits exactly on top of a cube, as shown.
 

Find the surface area of the composite solid.   (3 marks)

Show Answers Only

`424 \ text{cm}^2`

Show Worked Solution

`text{S.A. of 1 face of cube} = 8 xx 8 = 64 \ text{cm}^2`

`text{Height of triangle} = 11 – 8 = 3 \ text{cm}`

`therefore \ text{S.A. (triangular prism)}` `= 2 xx ( frac{1}{2} xx 8 xx 3 ) + 2 xx (5 xx 8)`
  `= 24 + 80`
  `= 104 \ text{cm}^2`

 

`therefore \ text{Total S.A.}` `= 5 xx 64 + 104`
  `= 424 \ text{cm}^2`

Financial Maths, STD2 F1 2020 HSC 20

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

\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{    Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{    Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$37 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$37 001 – \$90 000}\rule[-1ex]{0pt}{0pt} & \text{\$3572 plus 32.5 cents for each \$1 over \$37 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$90 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$20 797 plus 37 cents for each \$1 over \$90 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$54 097 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}

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

`$3111.40`

Show Worked Solution
`text(Tax paid)` `=12 xx 3000`
  `=$36\ 000`

 
`text(Tax payable on $122 680)`

`=20\ 797 + 0.37(122\ 680-90\ 000)`

`=20\ 797 + 0.37(32\ 680)`

`=$32\ 888.60`
 

`:.\ text(Tax refund)` `=36\ 000-32\ 888.60`  
  `=$3111.40`  

Measurement, STD2 M6 2020 HSC 16

Consider the triangle shown.
 


 

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

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

Show Answers Only
  1. `39^@`
  2. `12.1 \ text{(to 1 d.p.)}`
Show Worked Solution
a.      `tan theta` `= frac{8}{10}`
  `theta` `= tan ^(-1) frac{8}{10}`
    `= 38.659…`
    `= 39^@ \ text{(nearest degree)}`

 

b.     `text{Using Pythagoras:}`

`x` `= sqrt{8^2 + 10^2}`
  `= 12.806…`
  `= 12.8 \ \ text{(to 1 d.p.)}`

Measurement, STD2 M1 2020 HSC 5 MC

A plant stem is measured to be 16.0 cm, correct to one decimal place.

What is the percentage error in this measurement?

  1.  0.3125%
  2.  0.625%
  3.  3.125%
  4.  6.25%
Show Answers Only

`A`

Show Worked Solution

♦ Mean mark 41%.

`text{Absolute error} = 1/2 xx \text{precision} = 1/2 xx 0.1 = 0.05\ text{cm}`

`%  text(error)` `= frac(0.05)(16.0) xx 100`
  `= 0.3125%`

 
`=> \ A`

Probability, STD2 S2 2019 HSC 25

A bowl of fruit contains 17 apples of which 9 are red and 8 are green.

Dennis takes one apple at random and eats it. Margaret also takes an apple at random and eats it.

By drawing a probability tree diagram, or otherwise, find the probability that Dennis and Margaret eat apples of the same colour.  (3 marks)

Show Answers Only

`8/17`

Show Worked Solution

`P(text(same colour))` `= P(R R) + P(GG)`
  `= 9/17 xx 8/16 + 8/17 xx 7/16`
  `= 72/272 + 56/272`
  `= 8/17`
♦♦ Mean mark 35%.

Financial Maths, STD2 F1 2019 HSC 9 MC

What is the interest earned, in dollars, if $800 is invested for `x` months at a simple interest rate of 3% per annum?

  1. `2x`
  2. `24x`
  3. `200x`
  4. `2400x`
Show Answers Only

`A`

Show Worked Solution

♦♦♦ Mean mark 20%!

`text(Interest)` `= 800 xx x/12 xx 3/100`
  `= 2x`

 
`=> A`

Measurement, STD2 M1 2019 HSC 8 MC

A person's weight is measured as 79.3 kg.

What is the absolute error of this measurement?

  1. 10 grams
  2. 50 grams
  3. 100 grams
  4. 500 grams
Show Answers Only

`B`

Show Worked Solution

♦ Mean mark 46%.

`text(A)text(bsolute error)` `= 1/2 xx\ text(precision)`
  `= 1/2 xx 0.1\ text(kg)`
  `= 1/2 xx 100\ text(grams)`
  `= 50\ text(grams)`

 
`=> B`

Financial Maths, STD2 F1 2019 HSC 7 MC

Julia earns $28 per hour. Her hourly pay rate increases by 2%.

How much will she earn for a 4-hour shift with this increase?

  1. $2.24
  2. $28.56
  3. $112
  4. $114.24
Show Answers Only

`D`

Show Worked Solution
`text(Hourly rate)` `= 28 xx 1.02`
  `= $28.56`

 

`:.\ text(Shift earnings)` `= 4 xx 28.56`
  `= $114.24`

`=> D`

Measurement, STD2 M6 2019 HSC 22

Two right-angled triangles, `ABC` and `ADC`, are shown.
 

Calculate the size of angle `theta`, correct to the nearest minute.  (3 marks)

Show Answers Only

`41°4^{′}\ \ text{(nearest minute)}`

Show Worked Solution

`text(Using Pythagoras in)\ DeltaACD:`

Mean mark 51%.

`AC^2` `= 2.5^2 + 6^2`
  `= 42.25`
`:.AC` `= 6.5\ text(cm)`

 
`text(In)\ DeltaABC:`

`costheta` `= 4.9/6.5`
`theta` `= cos^(−1)\ 4.9/6.5`
  `= 41.075…`
  `= 41°4^{′}31^{″}`
  `= 41°5^{′}\ \ text{(nearest minute)}`

Statistics, STD2 S1 SM-Bank 2 MC

A dataset has the following five-number summary.

If the range of the dataset is 8, what is the minimum value of the dataset?

  1.  2
  2.  3
  3.  4
  4.  7
Show Answers Only

`D`

Show Worked Solution
`text(Range)` `=\ text{Max}-text{Min}`
`8` `= 15-text{Min Value}`
`:.\ text{Min}` `= 15-8`
  `=7`

 
`=> D`

Statistics, STD2 S1 SM-Bank 1

Write down the five-number summary for the dataset 

`3, \ 7, \ 8, \ 11, \ 13, \ 18.`  (2 marks)

Show Answers Only
`text(Minimum value:)`   `3`
`text(First quartile:)`   `7`
`text(Median:)`   `(11 + 8)/2 = 9.5`
`text(Third quartile:)`   `13`
`text(Maximum value:)`   `18`
Show Worked Solution
`text(Minimum value:)`   `3`
`text(First quartile:)`   `7`
`text(Median:)`   `(11 + 8)/2 = 9.5`
`text(Third quartile:)`   `13`
`text(Maximum value:)`   `18`

Statistics, STD2 S3 2017 HSC 29d*

All the students in a class of 30 did a test.

The marks, out of 10, are shown in the dot plot.
 

  1. Find the median test mark.  (1 mark)

  2. The mean test mark is 5.4. The standard deviation of the test marks is 4.22.
  3. Using the dot plot, calculate the percentage of the marks which lie within one standard deviation of the mean.  (2 marks)
Show Answers Only
  1. \(6\)
  2. \(\text{43%}\)
Show Worked Solution

i.    \(\text{30 data points}\)

\(\text{Median}\ = \dfrac{\text{15th + 16th}}{2} = \dfrac{4+8}{2} = 6\)
 

♦ Mean mark 50%.

ii.    \(\text{Lower limit}\ = 5.4-4.22 = 1.18\)

\(\text{Upper limit} = 5.4 + 4.22 = 9.62\)

\(\text{% between}\) \(= \dfrac{13}{30} \times 100\)  
  \(= 43.33… \%\)  
  \(=43\%\ \ \text{(nearest %)}\)  
♦♦ Mean mark 34%.

Financial Maths, STD2 F1 2018 HSC 30b

Last year, Luke’s taxable income was `$87\ 000` and the tax payable on this income was `$19\ 822`. This year, Luke’s taxable income has increased by `$16\ 800`.

  1. Use the table to calculate the tax payable by Luke this year.  (2 marks)
     

  2. How much extra money will Luke have this year, after paying tax, as a result of the increase in his taxable income? Ignore the Medicare levy.  (2 marks)

Show Answers Only
  1. `$26\ 038`
  2. `$10\ 584`
Show Worked Solution

i.   `text(Taxable income) = 87\ 000 + 16\ 800 = $103\ 800`

`:.\ text(Tax payable)` `= 19\ 822 + 0.37 (103\ 800-87\ 000)`
  `= $26\ 038`

 

ii.   `text(Net income from last year)`

`= 87\ 000-19\ 822`

`= $67\ 178`
 

`text(Net income in current year)`

`= 103\ 800-26\ 038`

`= $77\ 762`
 

`:.\ text(Extra money)` `= 77\ 762-67\ 178`
  `= $10\ 584`

Measurement, STD2 M1 2018 HSC 30a

A cylindrical water tank has a radius of 9 metres and a capacity of 1.26 megalitres.
 

What is the height of the water tank? Give your answer in metres, correct to two decimal places.  (3 marks)

Show Answers Only

`4.95\ text{m}`

Show Worked Solution

`text{Converting megalitres to m³  (using 1 m³ = 1000 L):}`

♦ Mean mark 48%.

`1.26\ text(ML)` `= (1.26 xx 10^6)/(10^3)`
  `= 1.26 xx 10^3\ text(m)^3`
  `= 1260\ text(m)^3`

 

`V` `= pir^2h`
`1260` `= pi xx 9^2 xx h`
`h` `= 1260/(pi xx 9^2)`
  `= 4.951…`
  `= 4.95\ text{m  (2 d.p.)}`

Measurement, STD2 M1 2018 HSC 27c

A shade shelter is to be constructed in the shape of half a cylinder with open ends. The diameter is 3.8 m and the length is 10 m.
 

 
The curved roof is to be made of plastic sheeting.

What area of plastic sheeting is required, to the nearest m²?  (2 marks)

Show Answers Only

`60\ text(m²  (nearest m²))`

Show Worked Solution

`text(Flatten out the half cylinder,)`

`text(Width)` `= 1/2 xx text(circumference)`
  `= 1/2 xx pi xx 3.8`
  `= 5.969…`

 

`:.\ text(Sheeting required)` `= 10 xx 5.969…`
  `= 59.69…`
  `= 60\ text(m²  (nearest m²))`

Measurement, STD2 M1 2018 HSC 18 MC

The length of a window is measured as 2.4 m.

Which calculation will give the percentage error for this measurement?

  1. `0.05/2.4 xx 100`
  2. `0.05/100 xx 2.4`
  3. `0.5/2.4 xx 100`
  4. `0.5/100 xx 2.4`
Show Answers Only

`A`

Show Worked Solution

`text{Absolute error}\ =1/2 xx text{precision}\ = 1/2 xx 0.1 = 0.05\ text{m}`

`text{% error}` `=\ frac{text{absolute error}}{text{measurement}} xx 100%`  
  `=0.05/2.4 xx 100%`  

 
`=>A`

Statistics, STD2 S1 2017 HSC 27a

Jamal surveyed eight households in his street. He asked them how many kilolitres (kL) of water they used in the last year. Here are the results.

`220, 105, 101, 450, 37, 338, 151, 205`

  1. Calculate the mean of this set of data.  (1 mark)

  2. What is the standard deviation of this set of data, correct to one decimal place?  (1 mark)

Show Answers Only
  1. `200.875`
  2. `127.4\ \ text{(1 d.p.)}`
Show Worked Solution
i.   `text(Mean)` `= (220 + 105 + 101 + 450 + 37 + 338 + 151 + 205) ÷ 8`
    `= 200.875`
♦ Mean mark part (ii) 47%.
IMPORTANT: The population standard deviation is required here.

 

ii.   `text(Std Dev)` `= 127.357…\ \ text{(by calc)}`
    `= 127.4\ \ text{(1 d.p.)}`

Measurement, STD2 M1 2017 HSC 25 MC

In the circle, centre `O`, the area of the quadrant is 100 cm².
 


 

What is the arc length `l`, correct to one decimal place?

  1. 8.9 cm
  2. 11.3 cm
  3. 17.7 cm
  4. 25.1 cm
Show Answers Only

`C`

Show Worked Solution

`text(Find)\ r:`

♦ Mean mark 44%.
`text(Area)` `= 1/4 pir^2`
`100` `= 1/4 pir^2`
`r^2` `= 400/pi`
`:. r` `= 11.283…\ text(cm)`

 

`text(Arc length)` `= theta/360 xx 2pir`
  `= 90/360 xx 2pi xx 11.283…`
  `= 17.724…`
  `= 17.7\ text(cm)`

 
`=> C`

Algebra, STD2 A1 2017 HSC 9 MC

What is the value of  `x`  in the equation  `(5-x)/3 = 6`?

  1. `-13`
  2. `-3`
  3. `3`
  4. `13`
Show Answers Only

`A`

Show Worked Solution
`(5-x)/3` `= 6`
`5-x` `= 18`
`x` `= 5-18`
  `= -13`

`=>A`

Financial Maths, STD2 F1 2017 HSC 6 MC

Tom earns a weekly wage of $1025. He also receives an additional allowance of $87.50 per day when handling toxic substances.

What is Tom’s income in a fortnight in which he handles toxic substances on 5 separate days?

  1. $1112.50
  2. $1462.50
  3. $2225.00
  4. $2487.50
Show Answers Only

`D`

Show Worked Solution
`text(Fortnightly wage)` `= 2 xx 1025`
  `= $2050`
`text(Allowances)` `= 5 xx 87.50`
  `= $437.50`

 

`:.\ text(Income)` `= 2050 + 437.50`
  `= $2487.50`

 
`=>D`

Statistics, STD2 S1 2017 HSC 1 MC

The box-and-whisker plot for a set of data is shown.
 

What is the median of this set of data?

  1. 15
  2. 20
  3. 30
  4. 35
Show Answers Only

`C`

Show Worked Solution

`text(Median = 30)`

`=> C`

Area, SMB-028

 

What is the area of the shaded part of the figure?   (2 marks)

Show Answers Only

`8.5\ text(cm)^2`

Show Worked Solution

`text(Area of large triangle)`

`= 1/2 xx 5 xx 5`

`= 12.5\ text(cm)^2`

`text(Area of smaller triangle)`

`= 1/2 xx 2 xx 2`

`= 2\ text(cm)^2`

 

`:.\ text(Shaded area)`

`= 12.5-(2 xx 2)`

`= 8.5\ text(cm)^2`

Measurement, STD2 M1 2016 HSC 28e

A company makes large marshmallows. They are in the shape of a cylinder with diameter 5 cm and height 3 cm, as shown in the diagram.

2ug-2016-hsc-q28_4

  1. Find the volume of one of these large marshmallows, correct to one decimal place.  (2 marks)

A cake is to be made by stacking 24 of these large marshmallows and filling the gaps between them with chocolate. The diagrams show the cake and its top view. The shading shows the gaps to be filled with chocolate.
 

2ug-2016-hsc-q28_5

  1. What volume of chocolate will be required? Give your answer correct to the nearest whole number.  (3 marks)

Show Answers Only
  1. `58.9\ text{cm}^3`
  2. `193\ text{cm}^3`
Show Worked Solution
i.    `V` `= pir^2h`
    `= pi xx 2.5^2 xx 3`
    `= 58.904…`
    `= 58.9\ text{cm³  (1 d.p.)}`

 

ii.    2ug-2016-hsc-q28-answer1

`text(Volume of rectangle)`

♦ Mean mark part (ii) 35%.

`= 15 xx 10 xx 6`

`= 900\ text(cm)^3`
 

`text(Volume of marshmallows in rectangle)`

`= 6 xx 2 xx 58.9`

`= 706.8\ text(cm)^3`

 

`:.\ text(Volume of chocolate)`

`= 900-706.8`

`= 193.2`

`= 193\ text{cm}^3 \ text{(nearest cm}^3 text{)}`

Probability, STD2 S2 2016 HSC 28c

A cricket team is about to play two matches. The probability of the team having a win, a loss or a draw is 0.7, 0.1 and 0.2 respectively in each match. The possible results in the two matches are displayed in the probability tree diagram.
  

2ug-2016-hsc-q28_2

  1. What is the probability of the team having a win and a draw, in any order?  (2 marks)

  2. Paul claims that 1.4 is the probability of the team winning both matches.

     

    Give one reason why this is NOT correct.  (1 mark)

Show Answers Only
  1. `0.28`
  2. `text(Probabilities cannot exceed 1.)`
Show Worked Solution

i.   `P(W\ text(and)\ D)`

`= P(W,D) + P(D,W)`

`= 0.7 xx 0.2 + 0.2 xx 0.7`

`= 0.28`

♦ Mean mark (i) 45%.
♦ Mean mark (ii) 49%.

 
ii.   `text(Probabilities cannot exceed 1.)`

Financial Maths, STD2 F1 2016 HSC 26f

Theo is completing his tax return. He has a gross salary of $82 521 and income from a rental property totalling $10 920. He is claiming $13 420 in allowable deductions.

  1. Determine Theo’s taxable income.  (1 mark)

  2. Using the tax table below, calculate Theo’s tax payable.  (2 marks)

\begin{array} {|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{    Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{    Tax payable}\\
\hline
\rule{0pt}{2.5ex}\text{\$0 – \$18 200}\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\text{\$18 201 – \$37 000}\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\text{\$37 001 – \$80 000}\rule[-1ex]{0pt}{0pt} & \text{\$3572 plus 32.5 cents for each \$1 over \$37 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$80 001 – \$180 000}\rule[-1ex]{0pt}{0pt} & \text{\$17 547 plus 37 cents for each \$1 over \$80 000}\\
\hline
\rule{0pt}{2.5ex}\text{\$180 001 and over}\rule[-1ex]{0pt}{0pt} & \text{\$54 547 plus 45 cents for each \$1 over \$180 000}\\
\hline
\end{array}

  1. In addition to the above tax, Theo must also pay a Medicare levy of $1600.42
  2. Theo has already paid $20 525 as Pay As You Go (PAYG) tax.
  3. Should Theo receive a tax refund or will he owe more tax? Justify your answer with calculations.  (2 marks)

Show Answers Only
  1. `$80\ 021`
  2. `$17\ 554.77`
  3. `$1369.81\ text(refund)`
Show Worked Solution

i.   `text(Taxable income)`

`= 82\ 521 + 10\ 920-13\ 420`

`= $80\ 021`

 

ii.   `text(Tax payable)`

`= 17\ 547 + (80\ 021-80\ 000) xx 0.37`

`= $17\ 554.77`

 

iii.   `text(Total tax payable)`

`= 17\ 554.77 + 1600.42`

`= $19\ 155.19`
 

`text(Tax paid > tax payable)`

`:.\ text(Refund)` `= 20\ 525-19\ 155.19`
  `= $1369.81`

Financial Maths, STD2 F1 2016 HSC 26e

Jenny earns a yearly salary of  $63 752. Her annual leave loading is 17.5% of four weeks pay.

Calculate her total pay for her four weeks of annual leave.  (3 marks)

Show Answers Only

`$5762.20`

Show Worked Solution

`text(4 weeks’ normal pay)`

`= 4/52 xx 63\ 752`

`= $4904`

 

`:.\ text(Annual leave pay)`

`= 4904(1 + 17.5/100)`

`= $5762.20`

Statistics, STD2 S1 2016 HSC 22 MC

The box-and-whisker plots show the results of a History test and a Geography test.
 

In History, 112 students completed the test. The number of students who scored above 30 marks was the same for the History test and the Geography test.

How many students completed the Geography test?

  1. 8
  2. 50
  3. 56
  4. 112
Show Answers Only

`=> C`

Show Worked Solution

`text{In History} \ => \  text{Q}_3 = 30\ \text{marks}`

`:.\ text{Scoring over 30}\ = 25text(%) xx 112 = 28\ \text{students}`
 

`text{In Geography} \ => \ text{Median}\ = 30\ \text{marks}`

`:.\ text{Students completing Geography}\ =2 xx 28 = 56\ \text{students}`

`=> C`

Financial Maths, STD2 F1 2016 HSC 20 MC

Isabella works a 35-hour week and is paid at an hourly rate of $18. Any overtime hours worked are paid at time-and-a-half. In a particular week, she earned $1008.

How many hours in total did Isabella work in this week to earn this amount?

  1. 37.3
  2. 42
  3. 49
  4. 56
Show Answers Only

`=> C`

Show Worked Solution

`text(Let)\ \ X=\ text(number of extra hours worked.)`

`text(Total wage)` `= 35 xx 18 + X xx 27`
 `1008` `= 630 + 27X`
`27X` `=378`
`X` `=14`

 

`:.\ text(Total hours worked)\ =35+14=49`

`=> C`

Statistics, STD2 S1 2016 HSC 19 MC

A soccer referee wrote down the number of goals scored in 9 different games during the season.

`2,  \ 3,  \ 3,  \ 3,  \ 5,  \ 5,  \ 8,  \ 9,  \ ...`

The last number has been omitted. The range of the data is 10.

What is the five-number summary for this data set?

  1. `2, 3, 5, 8.5, 12`
  2. `2, 3, 5, 8.5, 10`
  3. `2, 3, 5, 8, 12`
  4. `2, 3, 5, 8, 10`
Show Answers Only

`=> A`

Show Worked Solution

`text{Since range is 10} \ => \ text{Last data point = 12}`

`text{Q}_1 = 3`

`text{Q}_3 = (8 + 9)/2 = 8.5`

`text(Median = 5)`

`=> A`

♦ Mean mark 46%.