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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 EQ-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 EQ-Bank 22

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

i.    \(6\)

ii.   \(\text{43%}\)

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)
     
    \begin{array} {|l|l|}
    \hline
    \rule{0pt}{2.5ex}\textit{    Taxable income}\rule[-1ex]{0pt}{0pt} & \textit{    Tax payable}\\
    \hline
    \rule{0pt}{2.5ex}\$0 - \$18 200\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
    \hline
    \rule{0pt}{2.5ex}\$18 201 - \$37 000\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
    \hline
    \rule{0pt}{2.5ex}\$37 001 - \$87 000\rule[-1ex]{0pt}{0pt} & \text{\$3572 plus 32.5 cents for each \$1 over \$37 000}\\
    \hline
    \rule{0pt}{2.5ex}\$87 001 - $180 000\rule[-1ex]{0pt}{0pt} & \text{\$19 822 plus 37 cents for each \$1 over \$87 000}\\
    \hline
    \rule{0pt}{2.5ex}\$180 001\text{ and over}\rule[-1ex]{0pt}{0pt} & \text{\$54 232 plus 45 cents for each \$1 over \$180 000}\\
    \hline
    \end{array}

  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

a.    `$26\ 038`

b.    `$10\ 584`

Show Worked Solution

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

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

 

b.    `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

a.    `200.875`

b.    `127.4\ \ text{(1 d.p.)}`

Show Worked Solution
a.   `text(Mean)` `= (220 + 105 + 101 + 450 + 37 + 338 + 151 + 205) ÷ 8`
    `= 200.875`
♦ Mean mark part (b) 47%.
IMPORTANT: The population standard deviation is required here.
  

b.   `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` `=13`
`x` `=-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

a.   `58.9\ text{cm}^3`

b.   `193\ text{cm}^3`

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

 

b.    2ug-2016-hsc-q28-answer1

`text(Volume of rectangle)`

♦ Mean mark part (b) 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}\$0 - \$18 200\rule[-1ex]{0pt}{0pt} & \text{Nil}\\
\hline
\rule{0pt}{2.5ex}\$18 201 - \$37 000\rule[-1ex]{0pt}{0pt} & \text{19 cents for each \$1 over \$18 200}\\
\hline
\rule{0pt}{2.5ex}\$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}\$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}\$180 001\text{ 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

a.    `$80\ 021`

b.    `$17\ 554.77`

c.    `$1369.81\ text(refund)`

Show Worked Solution

a.   `text(Taxable income)= 82\ 521 + 10\ 920-13\ 420= $80\ 021`

b.   `text(Tax payable)= 17\ 547 + (80\ 021-80\ 000) xx 0.37= $17\ 554.77`

c.   `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%.

Algebra, STD2 A2 2016 HSC 14 MC

The graph shows a line which has an equation in the form  `y = mx + c`.
 

Which of the following statements is true?

  1. `m` is positive and `c` is negative
  2. `m` is negative and `c` is positive
  3. `m` and `c` are both positive
  4. `m` and `c` are both negative
Show Answers Only

`=> A`

Show Worked Solution

`m` is the gradient and the line slopes to the right so `m` is positive.

`c` is the `y`-intercept which is negative.

`:.\ m` is positive and `c` is negative.

`=> A`

Measurement, STD2 M1 2016 HSC 12 MC

A container is in the shape of a triangular prism which has a capacity of 12 litres. The area of the base is 240 cm².
 

What is the distance, `h`, between the two triangular ends of the container?

  1. 5 cm
  2. 20 cm
  3. 25 cm
  4. 50 cm
Show Answers Only

`=> D`

Show Worked Solution
♦♦ Mean mark 35%.

`text{1 mL = 1 cm}^3\ \ =>\ \ text{1 L = 1000 cm}^3`

`text(Volume)` `= Ah`
`12\ 000` `= 240 xx h`
`h` `= (12\ 000)/240= 50\ text(cm)`

 
`=> D`

Algebra, STD2 A1 2016 HSC 5 MC

Which expression is equivalent to  `2(3x-4) + 2`?

  1. `6x-2`
  2. `6x-4`
  3. `6x-6`
  4. `6x-10`
Show Answers Only

`C`

Show Worked Solution

`2(3x-4) + 2`

`= 6x-8 + 2= 6x-6` 

`=> C`

Measurement, STD2 M1 2016 HSC 1 MC

What is  208.345  correct to two significant figures?

  1. 208
  2. 210
  3. 208.34
  4. 208.35
Show Answers Only

`B`

Show Worked Solution

`208.345 = 210\ (2\ text(sig. fig.))`

♦♦ Mean mark 36%!!

`=> B`

Probability, STD2 S2 2015 HSC 16 MC

The probability of winning a game is `7/10`.

Which expression represents the probability of winning two consecutive games?

  1. `7/10 xx 6/9`
  2. `7/10 xx 6/10`
  3. `7/10 xx 7/9`
  4. `7/10 xx 7/10`
Show Answers Only

`D`

Show Worked Solution

`text{Since the two events are independent:}`

`P text{(W)}` `= 7/10`
`P text{(WW)}` `= 7/10 xx 7/10`

 
`=>D`

Measurement, STD2 M1 2015 HSC 12 MC

The length of a fish was measured to be 49 cm, correct to the nearest cm.

What is the percentage error in this measurement, correct to one significant figure?

  1. 0.01%
  2. 0.5%
  3. 1%
  4. 2%
Show Answers Only

`C`

Show Worked Solution
♦ Mean mark 41%.

`text{Absolute error}\ =1/2 xx text{precision}\ = 1/2 xx 1 = 0.5\ text{cm}`

`text{% error}` `=\ frac{text{absolute error}}{text{measurement}} xx 100%`  
  `=0.5/49 xx 100%`  
  `=1.020… %`  
  `=1%\ \ text{(to 1 sig fig)}`  

 
`=>C`

Statistics, STD2 S1 2015 HSC 6 MC

The times, in minutes, that a large group of students spend on exercise per day are presented in the box‑and‑whisker plot.
 

What percentage of these students spend between 40 minutes and 60 minutes per day on exercise?

  1. 17%
  2. 20%
  3. 25%
  4. 50%
Show Answers Only

`C`

Show Worked Solution

`text{Q}_1 = 40, \ text(Median) = 60`

`:.\ text(% Students between 40 and 60)`

`= 50text{%}-25text{%}`

`=25 text{%}`
 

`=>C`

Algebra, STD2 A1 2015 HSC 2 MC

Which of the following is  `4x+3y-x-5y`   in its simplest form?

  1. `3x-2y`
  2. `3x+8y`
  3. `5x-2y`
  4. `5x+8y`
Show Answers Only

`A`

Show Worked Solution

`4x+3y-x-5y= 3x-2y`

`⇒ A`

Measurement, STD2 M1 2015 HSC 1 MC

What is  1 560 200 km written in standard form correct to two significant figures?

  1. `1.56 × 10^4 \ text(km)`
  2. `1.6 × 10^5 \ text(km)`
  3. `1.56 × 10^6 \ text(km)`
  4. `1.6 × 10^6 \ text(km)`
Show Answers Only

`D`

Show Worked Solution

♦♦ Mean mark 30%.
COMMENT: Incredibly, the first MC question in 2015 had the lowest mean mark of all MC questions in the exam!

`1\ 560\ 200`

`= 1.5602 xx 10^6`

`= 1.6 xx 10^6\ text(km)\ \ \ text{(2 sig fig)}`

 `=> D`

Financial Maths, STD2 F1 2004 HSC 27b

David is paid at these rates:
  

 

His time sheet for last week is:
  

  1. Calculate David’s gross pay for last week.   (3 marks)

  2. David decides not to work on Saturdays. He wants to keep his weekly gross pay the same. How many extra hours at the weekday rate must he work?   (1 mark)

Show Answers Only

a.    `$414.00`

b.    `text(9 extra hours)`

Show Worked Solution

a.    `text{Pay (Fri)}= text(4 hours) xx 18.00= $72.00`

`text{Pay (Sat)}= 6\ text(hours) xx 1.5 xx 18.00=$162.00`

`text{Pay (Sun)}= 5\ text(hours) xx 2 xx 18.00=$180.00`

`:.\ text(Gross pay)= 72 + 162 + 180= $414.00`

b.    `text(Pay on Sat) = $162.00`

`text(Weekly equivalent hours)= 162/18= 9\ text(hours)`

`:.\ text(He will have to work 9 extra hours on)`

`text(a weekday for the same gross pay)`

 

Financial Maths, STD2 F1 2006 HSC 22 MC

This income tax table is used to calculate Evelyn’s tax payable.
 

Evelyn’s taxable income increases from `$50\ 000` to `$80\ 000`.

What percentage of her increase will she pay in additional tax?

  1. `15.25%`
  2. `40.7%`
  3. `43.5%`
  4. `52%`
Show Answers Only

`B`

Show Worked Solution
`text(Tax on $50 000)` `= 2500 + 0.35 xx (50\ 000-45\ 000)`
  `= 2500 + 1750= $4250`
`text(Tax on $80 000)` `= 11\ 250 + 0.52 xx (80\ 000-70\ 000)`
  `= 11\ 250 + 5200= $16\ 450`

  
`:.\ text(Extra tax)= 16\ 450-4250= $12\ 200`

`:.\ text(% Increase paid in tax)= (12\ 200) / (30\ 000) xx 100=\ text(40.66… %)`

 `=>  B`

Measurement, STD2 M1 2005 HSC 23b

A clay brick is made in the shape of a rectangular prism with dimensions as shown.
 

  1. Calculate the volume of the clay brick.   (1 mark)

Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.  
 

  1. What is the volume of clay remaining in the brick after the holes have been made? (Give your answer to the nearest cubic centimetre.)   (3 marks)

  2. What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.)   (1 mark)

Show Answers Only

a.   `text(1512 cm)^3`

b.   `text{1364 cm}^3`

c.   `text{9.8%}`

Show Worked Solution

a.    `V= l × b × h= 21 × 8 × 9= 1512\ text(cm)^3`

b.    `text(Volume of each hole)`

`= pir^2h= pi × 1.4^2 × 8= 49.260…\ text(cm)^3`

 `:.\ text(Volume of clay still in brick)`

`= 1512 − (3 × 49.260…)`

`= 1364.219…= 1364\ text{cm}^3\ text{(nearest whole)}`

c.     `text(Percentage of clay removed)`

`= ((3 × 49.260…))/1512 × 100`

`= 9.773…= 9.8 text{%   (1 d.p.)}`

Statistics, STD2 S1 2005 HSC 22 MC

Two groups of people were surveyed about their weekly wages. The results are shown in the box-and-whisker plots.
 

Which of the following statements is true for the people surveyed?

  1. The same percentage of people in each group earned more than $325 per week.
  2. Approximately 75% of people under 21 years earned less than $350 per week.
  3. Approximately 75% of people 21 years and older earned more than $350 per week.
  4. Approximately 50% of people in each group earned between $325 and $350 per week.
Show Answers Only

`B`

Show Worked Solution

`text{Option A: 50% of Under 21 group earned over $325 and 75%}`

`text{of Over 21 group did. NOT TRUE.}`
 

`text{Option B: 75% of Under 21 group earned below $350 is TRUE.}`
 

`text{Options C and D: can both be proven to be untrue using their}`

`text{median and quartile values.}`

`=>  B`

Financial Maths, STD2 F1 2006 HSC 5 MC

A salesman earns $200 per week plus $40 commission for each item he sells.

How many items does he need to sell to earn a total of $2640 in two weeks?

  1. 33
  2. 56
  3. 61
  4. 66
Show Answers Only

`B`

Show Worked Solution

`text(Let items sold) = n`

`text{Wages over 2 weeks}\ (w)`

`= (2 xx 200) + 40n=400 + 40n` 

`text(Find)\ n\ text(when)\ w = 2640:`

`2640` `= 400 + 40n`
`40n` `= 2240`
`n` `= 56`

 
`=>  B`

Probability, STD2 S2 2005 HSC 23a

There are 100 tickets sold in a raffle. Justine sold all 100 tickets to five of her friends. The number of tickets she sold to each friend is shown in the table.
 

  1. Justine claims that each of her friends is equally likely to win first prize.

     

    Give a reason why Justine’s statement is NOT correct.   (1 mark)

  2. What is the probability that first prize is NOT won by Khalid or Herman?   (2 marks)

Show Answers Only
  1. `text(The claim is incorrect because each of her friends)`
    `text(bought a different number of tickets and therefore)`
    `text(their chances of winning are different.)`
  2. `69/100`
Show Worked Solution

i.    `text(The claim is incorrect because each of her friends bought)`

`text(a different number of tickets and therefore their chances of)`

`text(winning are different.)`

 

ii.  `text(Number of tickets not sold to K or H)`

`= 45 + 10 + 14`

`= 69`
 

`:.\ text(Probability 1st prize NOT won by K or H)`

`= 69/100`

Algebra, STD2 A1 2005 HSC 2 MC

What is the value of  `(a-b)/4`, if  `a = 240`  and  `b = 56`?

  1. `4`
  2. `46`
  3. `226`
  4. `736`
Show Answers Only

`B`

Show Worked Solution

`(a-b)/4= (240-56)/4= 46`

`=> B`

Probability, STD2 S2 2007 HSC 25c

In a stack of 10 DVDs, there are 5 rated PG, 3 rated G and 2 rated M.

  1. A DVD is selected at random. What is the probability that it is rated M?   (1 mark)

Grant chooses two DVDs at random from the stack. Copy or trace the tree diagram into your writing booklet.
 

  1. Complete the tree diagram by writing the correct probability on each branch.   (2 marks)

  2. Calculate the probability that Grant chooses two DVDs with the same rating.   (2 marks)

Show Answers Only
  1. `1/5`
  2.  
  3. `14/45`
Show Worked Solution

i.    `text(5 PG, 3 G, 2 M)`

`P text{(M)} = 2/10 = 1/5` 

 

ii.   

 

iii.  `P text{(same rating)}`

`= P text{(PG, PG)} + P text{(G, G)} + P text{(M, M)}`

`= (1/2 xx 4/9) + (3/10 xx 2/9) + (1/5 xx 1/9)`

`= 2/9 + 1/15 + 1/45`

`= 14/45`

Algebra, STD2 A1 2007 HSC 24b

The distance in kilometres (`D`) of an observer from the centre of a thunderstorm can be estimated by counting the number of seconds (`t`) between seeing the lightning and first hearing the thunder.

Use the formula  `D = t/3`  to estimate the number of seconds between seeing the lightning and hearing the thunder if the storm is 1.2 km away.   (1 mark)

Show Answers Only

`3.6\ text(seconds)`

Show Worked Solution

`D = t/3`

`text(When)\ \ D = 1.2,`

`t/3` `= 1.2`
`t` `= 3.6\ text(seconds)`

Measurement, STD2 M7 2007 HSC 4 MC

What scale factor has been used to transform Triangle `A` to Triangle `B`?
  

  1. `1/2`
  2. `3/4`
  3. `2`
  4. `3`
Show Answers Only

`A`

Show Worked Solution

`text(Take two corresponding sides)`

`text(In)\ Delta A:\ 3\ text(cm)`

`text(In)\ Delta B:\ 1 \frac{1}{2}\ text(cm)`

`:.\ text(Scale factor converting)\ Delta A\ text(to)\ Delta B = frac{1}{2}`

`=>  A`

Financial Maths, STD2 F1 2007 HSC 3 MC

Joe is about to go on holidays for four weeks. His weekly salary is $280 and his holiday loading is 17.5% of four weeks pay.

What is Joe’s total pay for the four weeks holiday?

  1. $196
  2. $329
  3. $1169 
  4. $1316 
Show Answers Only

`D`

Show Worked Solution

`text(Salary)\ text{(4 weeks)}= 4 xx 280= $1120`

`text(Holiday loading)= 1120 xx 17.5%= $196`

`:.\ text(Total pay)= 1120 + 196= $1316`

`=>  D`

Financial Maths, STD2 F1 2008 HSC 24a

Bob is employed as a salesman. He is offered two methods of calculating his income.

\begin{array} {|l|}
\hline
\rule{0pt}{2.5ex}\text{Method 1: Commission only of 13% on all sales}\rule[-1ex]{0pt}{0pt} \\
\hline
\rule{0pt}{2.5ex}\text{Method 2: \$350 per week plus a commission of 4.5% on all sales}\rule[-1ex]{0pt}{0pt} \\
\hline
\end{array}

Bob’s research determines that the average sales total per employee per month is $15 670. 

  1. Based on his research, how much could Bob expect to earn in a year if he were to choose Method 1?   (2 marks)

  2. If Bob were to choose a method of payment based on the average sales figures, state which method he should choose in order to earn the greater income. Justify your answer with appropriate calculations.   (3 marks)

Show Answers Only
  1. `$24\ 445.20`
  2. `text(Proof)\ \ text{(See Worked Solutions)}`
Show Worked Solution

a.    `text(Method 1)`

`text(Yearly sales)= 12 xx 15\ 670= 188\ 040`

`:.\ text(Earnings)= text(13%) xx 188\ 040= $24\ 445.20`

b.    `text(Method 2)`

`text(In 1 Year, Weekly Wage)= 350 xx 52= 18\ 200`

`text(Commission)= text(4.5%) xx 188\ 040= 8461.80`

`text(Total earnings)= 18\ 200 + 8461.80= $26\ 661.80`

`:.\ text(Bob should choose Method 2.)`

Statistics, STD2 S1 2008 HSC 10 MC

The marks for a Science test and a Mathematics test are presented in box-and-whisker plots.
 

 Which measure must be the same for both tests?

  1. Mean
  2. Range
  3. Median
  4. Interquartile range
Show Answers Only

`D`

Show Worked Solution

`text(IQR)=text(Upper Quartile)-text(Lower Quartile)`

`text{In both box plots, IQR = 3 intervals (against bottom scale)}`

`=>  D`

Financial Maths, STD2 F1 2008 HSC 7 MC

Luke’s normal rate of pay is $15 per hour. Last week he was paid for 12 hours, at time-and-a-half.

How many hours would Luke need to work this week, at double time, to earn the same amount?

  1. 4
  2. 6
  3. 8
  4. 9
Show Answers Only

`D`

Show Worked Solution

`text(Amount earned last week)= 270/30= 9\ text(hrs)`

`text(Double time rate)= 2 xx 15= $30 text(/hr)`

`:.\ text(# Hours at double time)= 270/30= 9\ text(hrs)`

`=>  D`

Measurement, STD2 M1 2014 HSC 27c

The base of a water tank is in the shape of a rectangle with a semicircle at each end, as shown.

The tank is 1400 mm long, 560 mm wide, and has a height of 810 mm.  
  

What is the capacity of the tank, to the nearest litre?   (4 marks) 

Show Answers Only

`581\ text(L)`

Show Worked Solution

`V = Ah` 

♦ Mean mark 41%
STRATEGY: Adjusting measurements to metres makes the final conversion to litres simple.

`text(Finding Area of base)`

`text(Semi-circles have radius 280 mm) = 0.28\ text(m)`

`:.\ text(Area of 2 semicircles)`

`=2 xx 1/2 xx pi r^2= pi xx (0.28)^2`

`= 0.2463…\ text(m)^2`
 

`text(Area of rectangle)`

`= l xx b`

`= (1.4-2 xx 0.28) xx 0.56`

`= 0.4704\ text(m)^2`

 

`:.\ text(Volume)` `= Ah= (0.2463… + 0.4704) xx 0.810`
  `= 0.580527…\ text(m)^3`
  `= 580.527…\ text(L)\ \ text{(using 1m³} = 1000\ text{L)}`
  `= 581\ text(L)\ text{(nearest L)}`

Algebra, STD2 A1 2014 HSC 26c

Solve the equation  `(5x + 1)/3-4 = 5-7x`.   (3 marks)

Show Answers Only

 `x = 1`

Show Worked Solution
`(5x + 1)/3-4` `= 5-7x`
`5x + 1-3(4)` `= 3(5-7x)`
`5x + 1-12` `= 15-21x`
`26x` `= 26`
`:. x` `= 1`

Probability, STD2 S2 2014 HSC 16 MC

In Mathsville, there are on average eight rainy days in October.

Which expression could be used to find a value for the probability that it will rain on two consecutive days in October in Mathsville?

  1. `8/31 xx 7/30`
  2. `8/31 xx 7/31`
  3. `8/31 xx 8/30`
  4. `8/31 xx 8/31`
Show Answers Only

`D`

Show Worked Solution

`P text{(rains)} = 8/31\ \ \text{(independent event for each day)}`

`text{Since each day has same probability:}`

`P(R_1 R_2) = 8/31 xx 8/31`

`=>  D`

♦♦♦ Mean mark 16%.
Lowest mark of any MC question in 2014!

Financial Maths, STD2 F1 2014 HSC 13 MC

Jane sells jewellery. Her commission is based on a sliding scale of 6% on the first $2000 of her sales, 3.5% on the next $1000, and 2% thereafter.

What is Jane’s commission when her total sales are $5670? 

  1. $188.40
  2. $208.40
  3. $321.85
  4. $652.05
Show Answers Only

`B`

Show Worked Solution

`text(Commission)`

`= (2000 xx text(6%)) + (1000 xx text(3.5%)) + (5670-3000) xx text(2%)`

`= (2000 xx 0.06) + (1000 xx 0.035) + (2670 xx 0.02)`

`= 120 + 35 + 53.40= $208.40` 

`=>  B`

Measurement, STD2 M1 2014 HSC 10 MC

The top of the Sydney Harbour Bridge is measured to be 138.4 m above sea level. 

What is the percentage error in this measurement?

  1. 0.036%
  2. 0.050%
  3. 0.072%
  4. 0.289%
Show Answers Only

`A`

Show Worked Solution
 
♦ Mean mark 48%

`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%`  
  `=0.05/138.4 xx 100%`  
  `=0.036%`  

 
`=>  A`

Statistics, STD2 S1 2010 HSC 27b

The graphs show the distribution of the ages of children in Numbertown in 2000 and 2010.
  

  1. In 2000 there were 1750 children aged 0–18 years.

     

    How many children were aged 12–18 years in 2000?   (1 mark)

  2. The number of children aged 12–18 years is the same in both 2000 and 2010.

     

    How many children aged 0–18 years are there in 2010?    (1 mark)

  3. Identify TWO changes in the distribution of ages between 2000 and 2010. In your answer, refer to measures of location or spread or the shape of the distributions.   (2 marks)

  4. What would be ONE possible implication for government planning, as a consequence of this change in the distribution of ages?   (1 mark)

Show Answers Only

a.    `875`

b.    `3500`

c.    `text{Changes in distribution (include 2 of the following):}`

  • `text(the lower quartile age is lower in 2010)`
  • `text(the median is lower in 2010)`
  • `text(the upper quartile age is lower in 2010)`
  • `text(the interquartile range is greater in 2010)`
  • `text(2010 is positively skewed while 2000 is negatively)`

d.  `text(Implication for government planning:)`

`text(Since the children are getting younger in 2010,)`

  • `text(Approve and build more childcare facilities)`
  • `text(Build more school and public playgrounds)`
Show Worked Solution

a.    `text{Since the median = 12 years}`

♦ Mean mark (a) 45%

`=>\ text{50% of children are aged 12–18 years}`

`:.\ text{Children aged 12–18}\ = 50\text{%}\ xx 1750 = 875`

 

♦♦ Mean mark (b) 25%

b.   `text{Upper quartile (2010) = 12 years}`

`text{Children in upper quartile = 875 (from part (i))}`

`:.\ text{Children aged 0–18}\ =4 xx 875= 3500`
 

c.  `text{Changes in distribution (include 2 of the following):}`

♦ Mean mark (c) 35%
MARKER’S COMMENT: A number of students incorrectly identified “positive” skew as “negative” skew here.
  • `text(the lower quartile age is lower in 2010)`
  • `text(the median is lower in 2010)`
  • `text(the upper quartile age is lower in 2010)`
  • `text(the interquartile range is greater in 2010)`
  • `text(2010 is positively skewed while 2000 is negatively)`

d.  `text(Implication for government planning:)`

♦ Mean mark (d) 46%
MARKER’S COMMENT: Answers should reflect the 1 mark allocation.

`text(Since the children are getting younger in 2010,)`

  • `text(Approve and build more childcare facilities)`
  • `text(Build more school and public playgrounds)`

Probability, STD2 S2 2013 HSC 30b

In a class there are 15 girls (G) and 7 boys (B). Two students are chosen at random to be class representatives.

  1. Complete the tree diagram below.    (2 marks)
     
     

  2. What is the probability that the two students chosen are of the same gender?    (2 marks)

Show Answers Only
  1.  
  2. `6/11`
Show Worked Solution

i.  

ii.    `Ptext{(same gender)}` `=P(G,G) + P(B,B)`
    `=(15/22 xx 14/21) + (7/22 xx 6/21)`
    `=210/462 + 42/462`
    `=252/462`
    `=6/11`
♦ Mean mark (ii) 40%.

Probability, STD2 S2 2011 HSC 15 MC

An unbiased coin is tossed 10 times.

A tail is obtained on each of the first 9 tosses.

What is the probability that a tail is obtained on the 10th toss?

  1. `1/2^10`
  2. `1/2`
  3. `1/10`
  4. `9/10`
Show Answers Only

`B`

Show Worked Solution

`text(Each toss is an independent event and has an even chance)`

`text(of being a head or tail.)`

`=> B`

Algebra, STD2 A1 2009 HSC 25a

Simplify  `5-2(x + 7)`.   (2 marks)

Show Answers Only

 `-2x-9`

Show Worked Solution
♦ Mean mark 47%
`5-2(x + 7)` `= 5-2x-14`
  `= -2x-9`

Measurement, STD2 M1 2013 HSC 27d

A rectangular wooden chopping board is advertised as being 17 cm by 25 cm, with each side measured to the nearest centimetre.

  1. Calculate the percentage error in the measurement of the longer side.   (1 mark)

  2. Between what lower and upper limits does the actual area of the top of the chopping board lie?     (2 marks)

Show Answers Only
  1. `text(2%)`
  2. `404.25\ text{cm}^2 and 446.25\ text{cm}^2`
Show Worked Solution

i.    `text(Longer side) = 25\ text(cm)`

♦♦ Mean mark 23%
MARKER’S COMMENT: Be aware that measurements accurate to the nearest cm have an absolute error for calculation purposes of 0.5 cm.

`text{Absolute error}\ =1/2 xx text{precision}\ = 1/2 xx 1 = 0.5\ text{cm}`

`text{% error}` `=\ frac{text{absolute error}}{text{measurement}} xx 100%`  
  `=0.5/25 xx 100%`  
  `=2%`  

 

ii.   `text(Area) = l xx b`

♦ Mean mark 35%
`text{Area (upper)}` `=25.5 xx 17.5`
  `=446.25\ text{cm}^2`

 

`text{Area (lower)}` `=24.5 xx 16.5`
  `=404.25\ text{cm}^2`

 
`:.\ text{Area is between 404.25 cm}^2\ text{and 446.25 cm}^2.`

Probability, STD2 S2 2013 HSC 26c

The probability that Michael will score more than 100 points in a game of bowling is `31/40`. 

  1. A commentator states that the probability that Michael will score less than 100 points in a game of bowling is  `9/40`.

     

    Is the commentator correct? Give a reason for your answer.   (1 mark)

  2. Michael plays two games of bowling. What is the probability that he scores more than 100 points in the first game and then again in the second game?   (1 mark)

Show Answers Only
  1. `text{Incorrect. Less than “or equal to 100” is correct.}`
  2. `961/1600`
Show Worked Solution
♦♦♦ Mean mark 11%

i.   `text(The commentator is incorrect. The correct)`

`text(statement is)\ Ptext{(score} <=100 text{)} =9/40`

`text{(i.e. less than “or equal to 100” is the correct statement)}`

 

♦ Mean mark 34%
ii. `\ \ \ P(text{score >100 in both})` `= 31/40 xx 31/40` 
    `= 961/1600`

Algebra, STD2 A1 2010 HSC 24a

Fred tried to solve this equation and made a mistake in Line 2. 

\begin{array}{rl}
4(y+2)-3(y+1)= -3\ & \ \ \ \text{Line 1} \\
4y+8-3y+3= -3\ &\ \ \ \text{Line 2} \\
y+11 =-3\ &\ \ \ \text{Line 3} \\
y =-14& \ \ \ \text{Line 3}
\end{array}

Copy the equation in Line 1.

  1. Rewrite Line 2 correcting his mistake.   (1 mark)

  2. Continue your solution showing the correct working for Lines 3 and 4 to solve this equation for `y`.   (1 mark)

Show Answers Only

a.   `4y+8+3y-3=-3`

b.   `y+5=-3`

`y=-8`

Show Worked Solution
a.     `4(y+2)-3(y+1)` `=-3\ \ \ \ \ \ text(Line)\ 1`
  `4y+8-3y-3` `=-3\ \ \ \ \  text(Line)\ 2`

 

b.     `y+5` `=-3\ \ \ \ \ \ text(Line)\ 3`
  `y` `=-8\ \ \ \ \ \ text(Line)\ 4`

Probability, STD2 S2 2010 HSC 20 MC

Lou and Ali are on a fitness program for one month. The probability that Lou will finish the program successfully is 0.7 while the probability that Ali will finish successfully is 0.6. The probability tree shows this information

 

What is the probability that only one of them will be successful ?

  1. `0.18`
  2. `0.28`
  3. `0.42`
  4. `0.46`
Show Answers Only

`D`

Show Worked Solution

`text(Let)\ \ Ptext{(Lou successful)}=P(L) = 0.7, \ P(\text{not}\ L) = 0.3`

`text(Let)\ \ Ptext{(Ali successful)}=P(A) = 0.6, \ P(\text{not}\ A) = 0.4`

`P text{(only 1 successful)}` `=P(L)xxP(text(not)\ A)+P(text(not)\ L)xxP(A)`
  `=(0.7xx0.4)+(0.3xx0.6)`
  `=0.28+0.18`
  `=0.46`

 
`=>  D`

♦ Mean mark 48%.