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Networks, STD2 EQ-Bank 21

Five friends \((A, B, C, D, E)\) live in walking distance to each other's houses. The weighted network diagram below shows the walking time between the houses.
 

The table below summarises the same information. Fill in the missing table values in the shaded cells.   (3 marks)

Show Answers Only

Show Worked Solution

\(\text{Note the symmetry about the table diagonal.}\)

Financial Maths, STD2 EQ-Bank 38

Priya works as a sales representative and earns a base wage plus commission. A spreadsheet is used to calculate her weekly earnings.

Total weekly earnings = Base wage + Total sales \(\times\) Commission rate

A spreadsheet showing Priya's weekly earnings is shown.
  

 

  1. Write down the formula used in cell B9, using appropriate grid references.   (1 mark)

  2. In the following week, Priya earned total weekly earnings of $1437.50. Her base wage and commission rate remained unchanged. Calculate Priya's total sales for that week.   (2 marks)

  3. Priya's employer increases her commission rate to 4.2% but keeps her base wage the same. If Priya makes $22 000 in sales, calculate her new total weekly earnings.   (2 marks)

Show Answers Only

a.   \(=\text{B4}+\text{B5}^*\text{B6}/100\)

b.    \($22\, 500\)

c.    \($1574\)

Show Worked Solution

a.   \(\text{Total weekly earnings} = \text{Base wage}+\text{Total sales} \times \text{Commission rate}\)

\(\therefore\ \text{Formula:}\ =\text{B4}+\text{B5}^*\text{B6}/100\)
 

b.   \(\text{Total weekly earnings} = \text{Base wage}+\text{Total sales} \times \text{Commission rate}\)

\(\text{Let the Total sales}=S\)

\(1437.50\) \(=650+S\times \dfrac{3.5}{100}\)
\(787.50\) \(=S\times \dfrac{3.5}{100}\)
\(S\) \(=\dfrac{787.50\times 100}{3.5}=$22\,500\)

 
\(\text{The amount of Priya’s total sales was \$22 500.}\)
 

c.    \(\text{Base wage}=650\)

\(\text{Total sales}=$22\,000\)

\(\text{New commission rate}=4.2\%\)

\(\text{Total weekly earnings}\) \(=650+22\,000\times \dfrac{4.2}{100}\)
  \(=650+924=$1574\)

Financial Maths, STD2 EQ-Bank 17 MC

A company uses a spreadsheet to calculate employees' monthly salaries from their weekly salaries. The spreadsheet is shown below.
 

Which formula has been used in cell B7 to calculate the monthly salary?
 
  1. \(=\text{B4}^*4\)
  2. \(=\text{B4}^*52/12\)
  3. \(=\text{B4}^*12\)
  4. \(=\text{B4}/52^*12\)
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Monthly salary} = \dfrac{\text{weekly salary} \times 52}{12}\)

\(\therefore\ \text{Formula:}\ =\text{B4}^*52/12\)

\(\Rightarrow B\)

Financial Maths, STD2 EQ-Bank 21

Liam works in a factory assembling electronic components and is paid on a piecework basis. A spreadsheet is used to calculate his weekly earnings.

Weekly earnings = Number of units completed \(\times\) Rate per unit

A spreadsheet showing Liam's earnings for one week is shown.
  

  1. Write down the formula used in cell B8, using appropriate grid references.   (1 mark)

  2. In the following week, Liam earned $2036.25. The rate per unit remained at $3.75. Calculate the number of units Liam completed that week.   (2 marks)

Show Answers Only

a.   \(=\text{B4}^*\text{B5}\)

b.    \(\text{Number of units completed}=543\)

Show Worked Solution

a.   \(\text{Weekly earnings} = \text{Number of units completed} \times \text{Rate per unit}\)

\(\text{Formula:}\ =\text{B4}^*\text{B5}\)
 

b.   \(\text{Weekly earnings} = \text{Number of units completed} \times \text{Rate per unit}\)

\(\text{Let the Weekly earnings}=E\)

\(2036.25\) \(=E\times 3.75 \)
\(E\) \(=\dfrac{2036.25}{3.75}=543\)

Financial Maths, STD2 EQ-Bank 19

Maria works as a freelance writer and earns income through royalties. A spreadsheet is used to calculate her monthly royalty earnings.

Royalties = Number of books sold \( \times \) Royalty rate per book

A spreadsheet showing Maria's royalty earnings is shown.
  

  1. Write down the formula used in cell B8, using appropriate grid references.   (1 mark)

  2. In April, Maria earned total royalty earnings of $8960. Her royalty rate remained at $2.85 per book. How many books did Maria sell in April?   (2 marks)

Show Answers Only

a.   \(=\text{B4}^*\text{B5}\)

b.    \(\text{Number of books}=3144\)

Show Worked Solution

a.   \(\text{Total royalty earnings} = \text{Number of books sold} \times \text{Royalty rate per book}\)

\(\text{Formula:}\ =\text{B4}^*\text{B5}\)
 

b.   \(\text{Total royalty earnings} = \text{Number of books sold} \times \text{Royalty rate per book}\)

\(\text{Let the Number of books sold}=N\)

\(8960.40\) \(=N\times 2.85 \)
\(N\) \(=\dfrac{8960.40}{2.85}=3144\)

Measurement, STD2 M1 2025 HSC 26*

A toy has a curved surface on the top which has been shaded as shown. The toy has a uniform cross-section and a rectangular base.
 

  1. Use two applications of the trapezoidal rule to find an approximate area of the cross-section of the toy.   (2 marks)

  2. The total surface area of the plastic toy is 1300 cm².
  3. What is the approximate area of the curved surface?   (2 marks)

Show Answers Only

a.   \(34.68 \ \text{cm}^2\)

b.   \(582.64 \ \text{cm}^2\)

Show Worked Solution

a.   \(\text{Solution 1}\)

\(A\) \(=\dfrac{5.1}{2}(6.0+3.8) + \dfrac{5.1}{2}(3.8+0) \)  
  \(=34.68\ \text{cm}^2\)  

 
\(\text{Solution 2}\)

\(\begin{array}{|c|c|c|c|}
\hline\rule{0pt}{2.5ex} \quad x \quad \rule[-1ex]{0pt}{0pt}& \quad 0 \quad & \quad 5.1 \quad & \quad 10.2 \quad\\
\hline \rule{0pt}{2.5ex}y \rule[-1ex]{0pt}{0pt}& 6 & 3.8 & 0 \\
\hline
\end{array}\)

\(A\) \(\approx \dfrac{h}{2}\left(y_0+2y_1+y_2\right)\)
  \(\approx \dfrac{5.1}{2}\left(6+2 \times 3.8+0\right)\)
  \(\approx 34.68 \ \text{cm}^2\)

 

b.    \(\text{Toy has 5 sides.}\)

\(\text{Area of base}=10.2 \times 40=408 \ \text{cm}^2\)

\(\text{Area of rectangle}=6.0 \times 40=240 \ \text{cm}^2\)

\(\text{Approximated areas}=2 \times 34.68=69.36 \ \text{cm}^2\)

\(\therefore \ \text{Area of curved surface}\) \(=1300-(408+240+69.36)=582.64 \ \text{cm}^2\)
♦ Mean mark (b) 48%.

Algebra, STD2 EQ-Bank 28

Fried's formula for determining the medicine dosage for children aged 1 - 2 years is:

\(\text{Dosage}=\dfrac{\text{Age of infant (months)}\  \times \ \text{adult dose}}{150}\)

The spreadsheet below is used as a calculator for determining an infant's medicine dosage according to Fried's formula.
 

Amber, a 12 month old child, is being discharged from hospital with two medications. Medicine A has an adult dosage of 325 milligrams and she is to take 26 milligrams. She must also take 9.6 milligrams of Medicine B but the equivalent adult dosage has been left off the spreadsheet.

  1. By using appropriate grid references, write down a formula that could appear in cell B10.   (2 marks)

  2. Calculate the equivalent adult dosage for Medicine B (cell B7) using the information in the spreadsheet.    (2 marks)

Show Answers Only

a.    \(=\text{B5}^*\text{B6}/150\)

b.    \(120 \ \text{milligrams}\)

Show Worked Solution

a.     \(=\text{B5}^*\text{B6}/150\)
 

b.    \(\text{Let} \ A= \text{Adult dose}\)

\(\text{Using given formula:}\)

\(9.6\) \(=\dfrac{12 \times A}{150}\)
\(A\) \(=\dfrac{9.6 \times 150}{12}=120 \ \text{milligrams}\)

Financial Maths, STD1 EQ-Bank 1

A company uses the spreadsheet below to calculate the fortnightly pay, after tax, of its employees.
 

  1. Write down the formula that was used in cell D5, using appropriate grid references.   (1 mark)
  2. Hence, calculate Kim's fortnightly pay.   (2 marks)
Show Answers Only

a.  \(=\text{B6}-\text{C6}\)

b.    \(\$3140.85\)

Show Worked Solution

a.    \(\text{Formula for Kim’s Salary after tax:}\)

\(=\text{B5}-\text{C5}\)
 

b.    \(\text{Kim’s salary after tax}\ = \$81\,662\)

\(\text{Kim’s fortnightly pay}\ = \dfrac{81\,662}{26}=\$3140.85\)

Financial Maths, STD2 EQ-Bank 17

The table shows the income tax rate for Australian residents for the 2024-2025 financial year.

\begin{array}{|l|l|}
\hline
\rule{0pt}{2.5ex}\textit{Taxable income} \rule[-1ex]{0pt}{0pt}& \textit{Tax on this income} \\
\hline
\rule{0pt}{2.5ex}0-\$18\,200 \rule[-1ex]{0pt}{0pt}& \text{Nil} \\
\hline
\rule{0pt}{2.5ex}\$18 \, 201-\$45\,000 \rule[-1ex]{0pt}{0pt}& \text{16 cents for each \$1 over \$18 200} \\
\hline
\rule{0pt}{2.5ex}\$45\,001-\$135\,000 \rule[-1ex]{0pt}{0pt}& \$4288 \text{ plus 30 cents for each \$1 over \$45 000} \\
\hline
\rule{0pt}{2.5ex}\$135\,001-\$190\,000 \rule[-1ex]{0pt}{0pt}& \$31 \, 288 \text{ plus 37 cents for each \$1 over \$135 000} \\
\hline
\rule{0pt}{2.5ex}\$190\,001 \text{ and over} \rule[-1ex]{0pt}{0pt}& \$51 \, 638 \text{ plus 45 cents for each \$1 over \$190 000} \\
\hline
\end{array}

A company's spreadsheet was created that calculates its employees' after tax fortnightly pay, based on the table and excluding the Medicare levy.

  1. Write down the formula that was used in cell D6, using appropriate grid references.   (1 mark)
  2. Write down the formula that was used in cell C5, using appropriate grid references.   (1 mark)
  3. Hence, calculate Greg's fortnightly pay.   (2 marks)
Show Answers Only

a.  \(=\text{B6}-\text{C6}\)

b.  \(=4288+(\text{B5}-45000)^*0.30 \)

c.    \(\$3140.85\)

Show Worked Solution

a.    \(\text{Formula for Ian’s Salary after tax:}\)

\(=\text{B6}-\text{C6}\)
 

b.    \(\text{Formula for Greg’s estimated tax:}\)

\(=4288+(\text{B5}-45000)^*0.30 \)
 

c.    \(\text{Greg’s estimated tax}\ =4288+(103\,500-45\,000) \times 0.30=\$21\,838\)

\(\text{Greg’s salary after tax}\ = 103\,500-21\,838=\$81\,662\)

\(\text{Greg’s fornightly pay}\ = \dfrac{81\,662}{26}=\$3140.85\)

Algebra, STD2 EQ-Bank 26

Fried's formula for determining the medicine dosage for children aged 1 - 2 years is:

\(\text{Dosage}=\dfrac{\text{Age of infant (months)}\  \times \ \text{adult dose}}{150}\)

The spreadsheet below is used as a calculator for determining an infant's medicine dosage according to Fried's formula.
 

  1. By using appropriate grid references, write down a formula that could appear in cell B9.   (2 marks)

  2. Another infant requiring the same medicine has been recommended a dosage of 2 millilitres. What is the age of the infant?   (2 marks)

Show Answers Only

a.    \(=\text{B5}^*\text{B6}/150\)

b.    \(15 \ \text{months}\)

Show Worked Solution

a.     \(=\text{B5}^*\text{B6}/150\)
 

b.    \(\text{Let} \ n= \text{age of infant}\)

\(\text{Using given formula:}\)

\(2\) \(=\dfrac{n \times 20}{150}\)
\(n\) \(=\dfrac{2 \times 150}{20}=15 \ \text{months}\)

Financial Maths, STD2 EQ-Bank 36

Nigel's weekly wages are calculated using the partially completed spreadsheet below.
 

  1. Calculate the total wages Nigel earned on Wednesday.   (2 marks)

  2. Determine the time Nigel finished work on Saturday, given he earned $196.35 on the day.   (2 marks)

Show Answers Only

a.    \(\text{Total wages}=119.00+35.70=\$ 154.70\)

b.    \(14:30\)

Show Worked Solution

a.    \(\text{Wednesday wages:}\)

\(\text{Regular hours:} \ 5 \times 23.80=\$ 119.00\)

\(\text{Time-and-a-half:} \ 1 \times 1.5 \times 23.80=\$35.70\)

\(\text{Total wages}=119.00+35.70=\$ 154.70\)
 

b.    \(\text{Let \(h=\) total hours worked:}\)

\(\text{Since Saturday wages are time-and-a-half rate:}\)

\(h \times 1.5 \times 23.80\) \(=196.35\)
\(h\) \(=\dfrac{196.35}{1.5 \times 23.80}=5.5\ \text{hours}\)

 

\(\text{Nigel’s shift started at 09:00 and lasted 5.5 hours.}\)

\(\therefore\ \text{Nigel finished work at 14:30.}\)

Financial Maths, STD2 EQ-Bank 37

Trust Us Realty has three salespeople, Ralph, Ritchie, and Fonzi.

Their June monthly wages include a base wage and commission earned, which is modelled in the spreadsheet below.

  1. Write down the formula that was used in cell C9, using appropriate grid references.   (1 mark)

  2. Calculate Fonzi's total pay for the month of June.   (2 marks)

  3. Ralph's total pay for June is $5850. Determine Ralph's total sales for the month.   (2 marks)

Show Answers Only

a.    \(=0.01^* \text{B9}\)

b.    \(\$ 20\,200\)

c.    \(\$ 385\,000\)

Show Worked Solution

a.    \(=0.01^* \text{B9}\)
 

b.    \(\text{Fonzi’s Commission:}\)

\(\text{Sales}\ \$0-\$500\,000=0.01 \times 500\,000=\$ 5000\)

\(\text{Sales over} \ \$500\, 000=(2\,150\,000-500\,000) \times 0.008=\$13\,200\)

\(\text{Total June wages}=5000+13\,200+2000=\$ 20\,200\)
 

c.    \(\text{Ralph’s sales commission}\ =5850-2000=\$3850\)

\(\text {Since Ralph earned} \ \$3850 \ \text{in commission:}\)

\(\text{Sales} \times 0.01\) \(=3850\)
\(\text{Sales}\) \(=\dfrac{3850}{0.01}=\$ 385\,000\)

Algebra, STD2 EQ-Bank 31

Clark's formula for determining the medicine dosage for children is:

\(\text{Dosage}=\dfrac{\text{weight in kilograms}\  \times \ \text{adult dosage}}{70}\)

The spreadsheet below is used as a calculator for determining a child's medicine dosage according to Clark's formula.
 

  1. By using appropriate grid references, write down a formula that could appear in cell E5.   (2 marks)

  2. Another child requiring the same medicine has been recommended a dosage of 62.5 milligrams. How much does the child weigh?   (2 marks)

Show Answers Only

a.    \(=\text{B6}^*\text{B5}/70\)

b.    \(17.5 \ \text{kilograms}\)

Show Worked Solution

a.    \(=\text{B6}^*\text{B5}/70\)
 

b.    \(\text{Let} \ w= \text{weight of the child.}\)

\(\text{Using given formula:}\)

\(62.5\) \(=\dfrac{w \times 250}{70}\)
\(w\) \(=\dfrac{62.5 \times 70}{250}=17.5 \ \text{kilograms}\)

Financial Maths, STD2 EQ-Bank 28

The spreadsheet shows a casual employee's partially completed timesheet.
 

Calculate Jane's total earnings for the week.   (2 marks)

Show Answers Only

\(\text{Earnings:}\)

\(\text{Sunday:}\ 4 \times 20.70 \times 2 = $165.60\)

\(\text{Thursday:}\ 3 \times 20.70 = $62.10\)

\(\text{Friday:}\ (2 \times 20.70) + (2 \times 20.70 \times 1.5) = $103.50\)

\(\text{Saturday:}\ 4 \times 20.70 \times 1.5 = $124.20\)
 

\(\text{Total earnings}\ =165.60+62.10+103.50+124.20=$455.40\)

Show Worked Solution

\(\text{Earnings:}\)

\(\text{Sunday:}\ 4 \times 20.70 \times 2 = $165.60\)

\(\text{Thursday:}\ 3 \times 20.70 = $62.10\)

\(\text{Friday:}\ (2 \times 20.70) + (2 \times 20.70 \times 1.5) = $103.50\)

\(\text{Saturday:}\ 4 \times 20.70 \times 1.5 = $124.20\)
 

\(\text{Total earnings}\ =165.60+62.10+103.50+124.20=$455.40\)

Algebra, STD2 EQ-Bank 24

QuickPrint Copy Centre charges for printing services using the formula below.

Total cost = Setup fee + Cost per page \( \times \) Number of pages

A spreadsheet used to calculate the total cost is shown.

  1. Write down the formula used in cell E3, using appropriate grid references.   (1 mark)

  2. QuickPrint increases their cost per page to \$0.42, but keeps the setup fee unchanged. Aisha needs to print 120 pages.

    How much more will Aisha pay compared to the original pricing shown in the spreadsheet?   (2 marks)

Show Answers Only

a.    \(=\text{B3}+ \text{B4} \ ^* \ \text{B5}\)

b.    \($14.40\)

Show Worked Solution

a.   \(=\text{B3}+ \text{B4} \ ^* \ \text{B5}\)
 

b.   \(\text{Original cost from spreadsheet}:\ $44.50\)

\(\text{At increased rate}:\)

\(\text{Total cost}\ =8.50+0.42\times 120=$58.90\)

\(\text{Additional amount}\ =58.90-44.50=$14.40\)

Algebra, STD2 EQ-Bank 23

Green Thumb Landscaping charges for their lawn mowing service based on the size of the lawn.

They use the formula below to calculate the cost of each service.

Total cost = Call-out fee + Cost per square metre \( \times \) Area of lawn

The spreadsheet they provide to their clients is included below.

  1. Write down the formula that has been used in cell E4, using appropriate grid references.   (1 mark)

  2. Miguel has a lawn with a different area. The call-out fee and cost per square metre remain the same. When Miguel's lawn area is entered into the spreadsheet, the total cost shown in cell E4 becomes \$153.00.

    What is the area of Miguel's lawn?   (2 marks)

Show Answers Only

a.    \(=\text{B4}+ \text{B5} \ ^* \ \text{B6}\)

b.    \(\text{60 m}^2\)

Show Worked Solution

a.    \(=\text{B4}+ \text{B5} \ ^* \ \text{B6}\)
 

b.     \(153\) \(=45+1.8A\)
  \(108\) \(=1.8A\)
  \(A\) \(=\dfrac{108}{1.8}=60\)

 
\(\therefore\ \text{The area of Miguel’s lawn is 60m}^2.\)

Algebra, STD2 EQ-Bank 17

Two quantities, \(M\) and \(t\), have a relationship such that \(M\) varies directly with \(t\) and  \(M=2.4\)  when  \(t=8\).

Find the value of \(t\) when  \(M=5.1\).   (2 marks)

Show Answers Only

\(t=17\)

Show Worked Solution

\(M \propto t \ \ \Rightarrow \ \ M=kt\)

\(\text{Given} \ \ M=2.4 \ \ \text{when} \ \ t=8:\)

\(2.4=8k \ \ \Rightarrow \ \ k=\dfrac{2.4}{8}=0.3\)

\(\text{When} \ \ M=5.1:\)

\(5.1\) \(=0.3 \times t\)  
\(t\) \(=\dfrac{5.1}{0.3}=17\)  

Algebra, STD2 EQ-Bank 18

\(F\) varies directly with \(m\) and \(F=14\) when \(m=3.5\).

Find the value of \(m\) when \(F=50\).   (2 marks)

Show Answers Only

\(m=12.5\)

Show Worked Solution

\(F \propto m \ \ \Rightarrow \ \ F=km\)

\(\text{Given} \ \ F=14 \ \ \text{when} \ \ m=3.5:\)

\(14=3.5k \ \ \Rightarrow \ \ k=4\)

\(\text{When} \ \ F=50:\)

\(50\) \(=4 \times m\)  
\(m\) \(=\dfrac{50}{4} = 12.5\)  

Algebra, STD2 EQ-Bank 4 MC

\(d\) varies directly with \(h\) and it is known that  \(d=1.2\)  when  \(h=5\).

The value of \(h\) when  \(d=48\)  is

  1. \(11.52\)
  2. \(57.6\)
  3. \(200\)
  4. \(240\)
Show Answers Only

\(C\)

Show Worked Solution

\(d \propto h \ \ \Rightarrow \ \ d=k \times h\)

\(\text{Find \(k\) given \(\ d=1.2\)  when \(\ h=5\):}\)

\(1.2\) \(=5 \times k\)  
\(k\) \(=\dfrac{1.2}{5}=0.24\)  

 
\(\text{Find} \ h \ \text{when} \ \ d=48:\)

\(48\) \(=0.24 \times h\)  
\(h\) \(=\dfrac{48}{0.24}=200\)  

 
\(\Rightarrow C\)

Algebra, STD2 EQ-Bank 1 MC

\(Q\) varies directly with \(r\) and  \(Q=2\) when  \(r=16\).

The value of \(r\) when  \(Q=13\)  is

  1. \(1.625\)
  2. \(26\)
  3. \(76\)
  4. \(104\)
Show Answers Only

\(D\)

Show Worked Solution

\(Q \propto r \ \Rightarrow \ Q=kr\)

\(\text{Find \(k\) given \(\ Q=2\) given \(\ r=16\):}\)

\(2=16 \times k \ \Rightarrow \ k=\dfrac{1}{8}\)

\(\text{Find \(r\) when \(\ Q=13:\)}\)

\(13\) \(=\dfrac{1}{8} \times r\)  
\(r\) \(=13 \times 8=104\)  

 
\(\Rightarrow D\)

Algebra, STD2 EQ-Bank 27

Dial-A-Lift Luxury Transport offers ride services in the local area of Maitland.

They currently use the formula below to calculate the cost of each fare.

   Total fare \(=\) Booking fee \(+\) Cost per kilometre \(\times\) Number of kilometres travelled

A spreadsheet used to calculate the total fare is shown.
 

  1. By using appropriate grid references, write down a formula that could have been used in cell E4.   (2 marks)

  2. A trip with a different number of kilometres travelled is entered, but the booking fee and cost per kilometre remain unchanged. As a result, the value in cell E4 changes to $47.50.
  3. Calculate the value entered in cell B6.   (2 marks)

Show Answers Only

a.    \(=\text{B4}+ \text{B5} \ ^* \ \text{B6}\)

b.    \(\text{Cell B6 value = 13.}\)

Show Worked Solution

a.    \(=\text{B4}+ \text{B5} \ ^* \ \text{B6}\)
 

b.     \(47.5\) \(=15+2.5 n\)
  \(32.5\) \(=2.5n\)
  \(n\) \(=\dfrac{32.5}{2.5}=13\)

 
\(\text{Cell B6 value = 13.}\)

Algebra, STD2 EQ-Bank 25

The following formula can be used to calculate the recommended dosage of a medicine for a child.

   Recommended dosage \(=\) base dosage \(+\) adjustment factor \(\times\) weight of child,

where the recommended dosage and base dosage are in milligrams and the weight of the child is in kilograms.

A spreadsheet used to calculate the recommended dosage is shown.
 

  1. By using appropriate grid references, write down a formula that could have been used in cell E5.   (2 marks)

  2. The weight of a different child is entered, but the base dosage and adjustment factor remain unchanged. As a result, the value in cell E5 changes to 70.
  3. Calculate the value entered in cell B7.   (2 marks)

Show Answers Only

a.    \(=\text{B5}+ \text{B6} \ ^* \ \text{B7}\)

b.    \(\text{Cell B7 value = 25.}\)

Show Worked Solution

a.    \(=\text{B5}+ \text{B6} \ ^* \ \text{B7}\)
 

b.    \(\text{Let \(w\) = weight of the child}\)

\(70\) \(=50+0.8 w\)
\(20\) \(=0.8 w\)
\(w\) \(=\dfrac{20}{0.8}=25\)

 
\(\text{Cell B7 value = 25.}\)

Algebra, STD2 EQ-Bank 24

The area of a semicircle is given by  \(A=\dfrac{1}{2}\pi r^2\)  where \(r\) is the radius of the semicircle.

If the area of a semicircle is 250 cm², find the radius, to 1 decimal place.   (3 marks)

Show Answers Only

\(12.6\ \text{cm}\)

Show Worked Solution

\(A=\dfrac{1}{2}\pi r^2\ \ \Rightarrow \ \ r^2=\dfrac{2A}{\pi}\)

   
\(\text{When}\ \ A = 250:\)

\(r^2\) \(=\dfrac{2\times 250}{\pi}=\dfrac{500}{\pi}=159.154…\)
\( r\) \(=\sqrt{159.154…}=12.615…=12.6\ \text{cm (to 1 d.p.)}\)

Algebra, STD2 EQ-Bank 3 MC

If  \(w=4y^2-5\), what is the value of \(y\) when  \(w=43\)?

  1. \(3\)
  2. \(-3\)
  3. \(\sqrt{3}\)
  4. \(-2\sqrt{3}\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{When}\ w=43:\)

\(w\) \(=4y^2-5\)
\(43\) \(=4y^2-5\)
\(4y^2\) \(=48\)
\(y^2\) \(=\dfrac{48}{4}=12\)
\(y\) \(=\pm 2\sqrt{3}\)

 
\(y=-2\sqrt{3}\ \text{(only possibility given options)}\)

\(\Rightarrow D\)

Combinatorics, EXT1 EQ-Bank 11

Find the term independent of \(x\) in the expansion of  \(\left(2 x^3+\dfrac{1}{x^4}\right)^7\).   (2 marks)

Show Answers Only

\(560\)

Show Worked Solution

\(T_k=\ \text {General term of} \ \ \left(2 x^3+\dfrac{1}{x^4}\right)^7\)

\(T_k\) \(=\displaystyle \binom{7}{k}\left(2 x^3\right)^{7-k} \cdot\left(x^{-4}\right)^k\)
  \(=\displaystyle\binom{7}{k} \cdot 2^{7-k} \cdot x^{3(7-k)} \cdot x^{-4 k}\)
  \(=\displaystyle\binom{7}{k} \cdot 2^{7-k} \cdot x^{21-7 k}\)

 

\(\text{Independent term occurs when:}\)

\(x^{21-7 k}=x^0 \ \Rightarrow \ k=3\)

\(\therefore \text{Independent term}=\displaystyle \binom{7}{3} \cdot 2^4=560\)

Polynomials, EXT1 EQ-Bank 10

The polynomial  \(R(x)=2 x^4+a x^3+b x^2+c x+d\)  has a double zero at  \(x=1\), a zero at  \(x=-3\), and passes through the point \((0,-12)\).

Find the integer values of \(a, b, c, d\) and the fourth zero of the polynomial.   (4 marks)

Show Answers Only

\(a=-2, \ b=-14, \ c=26, \ d=-12\)

\(\text{Fourth zero:} \ \ x=2\)

Show Worked Solution

\(R(x)=2 x^4+a x^3+b x^2+c x+d\)

\(\text{Since leading coefficient is 2 with a double zero at 1 and a zero at }-3:\)

\(R(x)=2(x-1)^2(x+3)(x-k) \ \ \text{where} \ k \ \text{is the fourth zero.}\)

\(\text{The polynomial passes through}\ (0,-12):\)

\(R(0)=2(0-1)^2(0+3)(0-k)=-12\ \ \Rightarrow\ \ k=2\)
 

\(\text{Expanding}\ R(x):\)

\(R(x)\) \(=2(x-1)^2(x+3)(x-2)\)  
  \(=2\left(x^2-2 x+1\right)(x+3)(x-2) \)  
  \(=2(x^3+3 x^2-2 x^2-6 x+x+3)(x-2) \)  
  \(=2(x^3+x^2-5 x+3)(x-2) \)  
  \(=2(x^4+x^3-5 x^2+3 x-2 x^3-2 x^2+10 x-6) \)  
  \(=2 x^4-2 x^3-14 x^2+26 x-12\)  

 

\(\text{Equating coefficients:}\)

\(a=-2, \ b=-14, \ c=26, \ d=-12\)

\(\text{Fourth zero:} \ \ x=2\)

Trigonometry, 2ADV EQ-Bank 2 MC

A tower \(B T\) has height \(h\) metres.

From point \(A\), the angle of elevation to the top of the tower is 26° as shown.
 

Which of the following is the correct expression for the length of \(A B\) ?

  1. \(h\, \tan 26^{\circ}\)
  2. \(h\, \cot 26^{\circ}\)
  3. \(h\, \sin 26^{\circ}\)
  4. \(h\, \operatorname{cosec}\, 26^{\circ}\)
Show Answers Only

\(B\)

Show Worked Solution
\(\tan 26^{\circ}\) \(=\dfrac{h}{AB}\)  
\(AB\) \(=\dfrac{h}{\tan 26^{\circ}}\)  
\(AB\) \(=h\,\cot 26^{\circ}\)  

 
\(\Rightarrow B\)

Polynomials, EXT1 EQ-Bank 9

A polynomial has the equation

\(Q(x)=(x+1)^2(x-2)\left(x^2+2 x-8\right)\)

  1. Express  \(Q(x)\)  as a product of linear factors and determine the multiplicity of each of its roots.   (2 marks)

  2. Hence, without using calculus, draw a sketch of \(y=Q(x)\), showing all  \(x\)-intercepts.   (2 marks)

Show Answers Only

a.    \(Q(x) = (x+1)^2(x-2)\left(x^2+2 x-8\right) \)

\(\text{Roots:}\)

\(x=-1 \ \ \text{(multiplicity 2)}\)

\(x=2 \ \ \text{(multiplicity 2)}\)

\(x=4 \ \ \text{(multiplicity 1)}\)
 

b.

Show Worked Solution
a.     \(Q(x)\) \(=(x+1)^2(x-2)\left(x^2+2 x-8\right)\)
    \(=(x+1)^2(x-2)(x-2)(x+4)\)
    \(=(x+1)^2(x-2)^2(x-4)\)

 

\(\text{Roots:}\)

\(x=-1 \ \ \text{(multiplicity 2)}\)

\(x=2 \ \ \text{(multiplicity 2)}\)

\(x=4 \ \ \text{(multiplicity 1)}\)
 

b.    \(Q(x) \ \text{degree}=5, \ \text{Leading coefficient}=1\)

\(\text{As} \ \ x \rightarrow-\infty, y \rightarrow-\infty\)

\(\text{As} \ \ x \rightarrow \infty, y \rightarrow \infty\)

\(\text{At}\ \ x=0, \ y=1^2 \times (-2)^2 \times -4 = -16\)

Polynomials, EXT1 EQ-Bank 5

Consider the function  \(P(x)=(x-1)^2(x+2)\left(x^2+3 x-4\right)\)

  1. By expressing \(P(x)\) as a product of its linear factors, identify its zeroes and the multiplicity of each zero.   (2 marks)

  2. Without using calculus, draw a sketch of  \(y=P(x)\)  showing any \(x\)-intercepts.   (2 marks)

Show Answers Only

a.    \(P(x)=(x-1)^3(x+2)(x+4)\)

\(\text{Roots:}\)

\(x=-2 \ \ \text{(multiplicity 1)}\)

\(x=-4 \ \ \text{(multiplicity 1)}\)

\(x=1 \ \ \text{(multiplicity 3)}\)
 

b.  

Show Worked Solution
a.     \(P(x)\) \(=(x-1)^2(x+2)\left(x^2+3 x-4\right)\)
    \(=(x-1)^2(x+2)(x-1)(x+4)\)
    \(=(x-1)^3(x+2)(x+4)\)

 
\(\text{Roots:}\)

\(x=-2 \ \ \text{(multiplicity 1)}\)

\(x=-4 \ \ \text{(multiplicity 1)}\)

\(x=1 \ \ \text{(multiplicity 3)}\)
 

b.    \(\text{Degree} \ P(x)=5, \ \ \text {Leading coefficient }=1\)

\(\text{As} \ \ x \rightarrow \infty, \ y \rightarrow \infty\)

\(\text{As} \ \ x \rightarrow -\infty, \ y \rightarrow -\infty\)

\(\text{At} \ \ x=0, y=-8\)
 

HMS, HAG 2025 HSC 32b

To what extent do THREE factors that create health inequities affect ONE population group in Australia?   ( 12 marks)

Show Answers Only

Population group: The aged (65+ years)

Judgement Statement

  • Three factors—access to services, socioeconomic disadvantage and social isolation— create significant health inequities for aged Australians, with compounding effects that substantially reduce health outcomes and quality of life.

Access to Services and Transport

  • Limited access to healthcare services creates major health inequities for older Australians.
  • Rural and remote aged populations face substantial barriers reaching specialist care, diagnostic services and preventative health programs.
  • Transport difficulties compound these issues as declining mobility reduces medical appointment attendance.
  • This results in delayed diagnosis of cardiovascular disease, cancer and dementia related illnesses, increasing mortality rates among the elderly.
  • Fewer preventative visits mean chronic conditions progress undetected whilst inadequate allied health access reduces rehabilitation opportunities.
  • Geographic isolation intensifies these barriers, creating cycles of declining health status compared to urban counterparts.

Socioeconomic Factors

  • Financial constraints significantly affect aged Australians relying solely on age pensions.
  • Lower incomes restrict access to private healthcare, specialists and medications not fully covered by Medicare.
  • Gap payments and medication costs force delays in seeking care, resulting in untreated chronic conditions.
  • This leads to preventable hospitalisations and poorer disease management outcomes.
  • Socioeconomically disadvantaged older adults experience higher rates of inadequate nutrition and poor housing quality, further compromising health.

Social Isolation

  • Social isolation profoundly impacts mental and physical health for older adults living alone following bereavement or family relocation.
  • Reduced social connections correlate strongly with increased depression, anxiety and cognitive decline including dementia progression.
  • Isolation decreases motivation for self-care and physical activity whilst poor housing modifications increase fall risk and injury rates.
  • Limited support networks mean delayed help-seeking when unwell, worsening health outcomes and recovery times.

Reaffirmation

  • These three factors interact substantially to create significant health inequities for aged Australians.
  • The combined effect exceeds individual factors as limited access, financial barriers and isolation reinforce each other.
  • Aged populations experiencing multiple disadvantages demonstrate markedly worse health outcomes including higher chronic disease rates and premature mortality.
Show Worked Solution

Population group: The aged (65+ years)

Judgement Statement

  • Three factors—access to services, socioeconomic disadvantage and social isolation— create significant health inequities for aged Australians, with compounding effects that substantially reduce health outcomes and quality of life.

Access to Services and Transport

  • Limited access to healthcare services creates major health inequities for older Australians.
  • Rural and remote aged populations face substantial barriers reaching specialist care, diagnostic services and preventative health programs.
  • Transport difficulties compound these issues as declining mobility reduces medical appointment attendance.
  • This results in delayed diagnosis of cardiovascular disease, cancer and dementia related illnesses, increasing mortality rates among the elderly.
  • Fewer preventative visits mean chronic conditions progress undetected whilst inadequate allied health access reduces rehabilitation opportunities.
  • Geographic isolation intensifies these barriers, creating cycles of declining health status compared to urban counterparts.

Socioeconomic Factors

  • Financial constraints significantly affect aged Australians relying solely on age pensions.
  • Lower incomes restrict access to private healthcare, specialists and medications not fully covered by Medicare.
  • Gap payments and medication costs force delays in seeking care, resulting in untreated chronic conditions.
  • This leads to preventable hospitalisations and poorer disease management outcomes.
  • Socioeconomically disadvantaged older adults experience higher rates of inadequate nutrition and poor housing quality, further compromising health.

Social Isolation

  • Social isolation profoundly impacts mental and physical health for older adults living alone following bereavement or family relocation.
  • Reduced social connections correlate strongly with increased depression, anxiety and cognitive decline including dementia progression.
  • Isolation decreases motivation for self-care and physical activity whilst poor housing modifications increase fall risk and injury rates.
  • Limited support networks mean delayed help-seeking when unwell, worsening health outcomes and recovery times.

Reaffirmation

  • These three factors interact substantially to create significant health inequities for aged Australians.
  • The combined effect exceeds individual factors as limited access, financial barriers and isolation reinforce each other.
  • Aged populations experiencing multiple disadvantages demonstrate markedly worse health outcomes including higher chronic disease rates and premature mortality.

♦♦ Mean mark 45%.

HMS, TIP 2025 HSC 31b

Justify THREE elements a coach needs to consider when designing a training session for a sport.   ( 12 marks)

Show Answers Only

Sport: Netball

Position Statement:

  • Warm-up, skill instruction and conditioning are essential elements when designing effective training sessions.
  • These elements prepare athletes physically, develop technical competence and build sport-specific fitness for optimal performance.

Warm-Up Preparation:

  • Structured warm-ups prepare athletes physiologically and psychologically for training. Research confirms gradual cardiovascular activation and dynamic stretching increase muscle temperature whilst reducing injury risk.
  • For example, netball players performing 10 minutes jogging at 70% maximum heart rate followed by side-stepping and pivoting experience improved muscle elasticity. This justifies warm-up inclusion because prepared muscles respond effectively to training whilst preventing strains.
  • Athletes completing structured warm-ups demonstrate noticeably better performance quality during skill work, proving warm-up effectiveness in optimising session outcomes.

Skill Instruction and Practice:

  • Deliberate skill instruction develops technical proficiency essential for competitive success. Structured practice with immediate feedback accelerates skill acquisition compared to unguided repetition.
  • Netball coaches implementing progression models—basic chest pass drills advancing to contested passing under defensive pressure—enable athletes to master fundamentals whilst introducing pressure gradually. This justifies systematic instruction because controlled complexity builds confidence and competence simultaneously.
  • Athletes receiving structured instruction show significantly faster technique improvement than those practising without guidance, demonstrating critical importance for training efficiency.

Conditioning for Sport-Specific Fitness:

  • Sport-specific conditioning builds physiological capacities for competition demands. Targeted fitness development enhances performance whilst reducing fatigue-related errors.
  • Combining agility ladder drills with defensive shadowing in netball develops quick directional changes and sustained movement capacity players need. This justifies conditioning because it transfers directly to game performance.
  • Athletes following sport-specific conditioning reportedly maintain higher intensity for longer during competition, supporting conditioning as essential for advantage.
Show Worked Solution

Sport: Netball

Position Statement:

  • Warm-up, skill instruction and conditioning are essential elements when designing effective training sessions.
  • These elements prepare athletes physically, develop technical competence and build sport-specific fitness for optimal performance.

Warm-Up Preparation:

  • Structured warm-ups prepare athletes physiologically and psychologically for training. Research confirms gradual cardiovascular activation and dynamic stretching increase muscle temperature whilst reducing injury risk.
  • For example, netball players performing 10 minutes jogging at 70% maximum heart rate followed by side-stepping and pivoting experience improved muscle elasticity. This justifies warm-up inclusion because prepared muscles respond effectively to training whilst preventing strains.
  • Athletes completing structured warm-ups demonstrate noticeably better performance quality during skill work, proving warm-up effectiveness in optimising session outcomes.

Skill Instruction and Practice:

  • Deliberate skill instruction develops technical proficiency essential for competitive success. Structured practice with immediate feedback accelerates skill acquisition compared to unguided repetition.
  • Netball coaches implementing progression models—basic chest pass drills advancing to contested passing under defensive pressure—enable athletes to master fundamentals whilst introducing pressure gradually. This justifies systematic instruction because controlled complexity builds confidence and competence simultaneously.
  • Athletes receiving structured instruction show significantly faster technique improvement than those practising without guidance, demonstrating critical importance for training efficiency.

Conditioning for Sport-Specific Fitness:

  • Sport-specific conditioning builds physiological capacities for competition demands. Targeted fitness development enhances performance whilst reducing fatigue-related errors.
  • Combining agility ladder drills with defensive shadowing in netball develops quick directional changes and sustained movement capacity players need. This justifies conditioning because it transfers directly to game performance.
  • Athletes following sport-specific conditioning reportedly maintain higher intensity for longer during competition, supporting conditioning as essential for advantage.

♦♦ Mean mark 37%.

HMS, TIP 2025 HSC 31aii

How can an athlete ensure their training procedures are safe? In your answer, refer to ONE training method.  ( 5 marks)

Show Answers Only

Training Method: Plyometrics

  • Athletes must complete comprehensive warm-ups before plyometric sessions. This occurs because muscles and connective tissues require increased temperature and blood flow before explosive movements to prevent strain injuries.
  • For example, performing 10 minutes of dynamic stretching and light jogging prepares joints for impact forces. This reduces injury risk during box jumps and depth jumps.
  • Proper landing technique ensures safety during plyometric exercises. This happens when athletes maintain correct body positioning with bent knees and controlled movements, which prevents excessive joint stress.
  • Progressive overload principles must be applied carefully. This means gradually increasing jump height and repetitions over weeks, allowing tissues to adapt without overuse injuries.
  • Training on appropriate surfaces like gym mats or grass protects joints from impact forces. Consequently, athletes avoid stress fractures and tendon damage during high-intensity explosive training.
Show Worked Solution

Training Method: Plyometrics

  • Athletes must complete comprehensive warm-ups before plyometric sessions. This occurs because muscles and connective tissues require increased temperature and blood flow before explosive movements to prevent strain injuries.
  • For example, performing 10 minutes of dynamic stretching and light jogging prepares joints for impact forces. This reduces injury risk during box jumps and depth jumps.
  • Proper landing technique ensures safety during plyometric exercises. This happens when athletes maintain correct body positioning with bent knees and controlled movements, which prevents excessive joint stress.
  • Progressive overload principles must be applied carefully. This means gradually increasing jump height and repetitions over weeks, allowing tissues to adapt without overuse injuries.
  • Training on appropriate surfaces like gym mats or grass protects joints from impact forces. Consequently, athletes avoid stress fractures and tendon damage during high-intensity explosive training.

HMS, TIP 2025 HSC 31ai

Outline the suitability of ONE training method for an athlete in a sport of your choice.   ( 3 marks)

Show Answers Only

Sport: Basketball

Training Method: Plyometrics

  • Plyometric training suits basketball players who require explosive vertical power for jumping and rebounding. Box jumps and depth jumps develop fast-twitch muscle fibres essential for contested rebounds.
  • The training method matches basketball’s movement demands of rapid acceleration and deceleration during gameplay. Exercises like lateral bounds replicate the quick directional changes required when defending opponents.
  • Plyometrics improves reaction time and explosive leg strength needed for successful lay-ups and blocks. This training enhances performance during critical high-intensity moments throughout matches.
Show Worked Solution

Sport: Basketball

Training Method: Plyometrics

  • Plyometric training suits basketball players who require explosive vertical power for jumping and rebounding. Box jumps and depth jumps develop fast-twitch muscle fibres essential for contested rebounds.
  • The training method matches basketball’s movement demands of rapid acceleration and deceleration during gameplay. Exercises like lateral bounds replicate the quick directional changes required when defending opponents.
  • Plyometrics improves reaction time and explosive leg strength needed for successful lay-ups and blocks. This training enhances performance during critical high-intensity moments throughout matches.

HMS, TIP 2025 HSC 30b

Justify the use of heat and cold and progressive mobilisation as rehabilitation procedures for a shoulder dislocation.   ( 12 marks)

Show Answers Only

Position Statement:

  • Heat and cold therapy combined with progressive mobilisation provides optimal rehabilitation for shoulder dislocations.
  • These procedures effectively manage inflammation whilst restoring range of motion and facilitating safe return to function.

Cold Therapy in Acute Phase:

  • Immediate cold application reduces inflammatory response and controls swelling after dislocation. Evidence confirms cold constricts blood vessels, limiting fluid accumulation around damaged tissue.
  • Research shows applying ice for 15-20 minutes every two hours during first 48-72 hours significantly decreases pain and tissue damage. This demonstrates cold therapy’s effectiveness in protecting injured structures during the acute inflammatory phase.
  • Cold application enables earlier mobilisation by controlling pain and swelling. Athletes experience reduced discomfort, allowing gentle movement exercises to begin sooner whilst preventing excessive inflammation that delays healing. This justifies cold as essential for initial injury management.

Heat Application in Later Stages:

  • After initial inflammation subsides (typically 72 hours post-injury), heat therapy increases blood flow to promote tissue repair. Studies indicate heat dilates vessels, delivering oxygen and nutrients essential for healing damaged ligaments and capsule tissue.
  • Heat application before mobilisation exercises improves muscle elasticity and joint flexibility. Evidence shows warming tissues reduces stiffness, allowing greater range of movement during rehabilitation exercises without causing re-injury or excessive discomfort.

Progressive Mobilisation Throughout Recovery:

  • Progressive mobilisation systematically restores shoulder function through graduated exercises matched to healing stages. This approach begins with passive pendulum movements, advancing to active-assisted exercises, then resistance training as tissue strength develops.
  • Evidence demonstrates controlled movement prevents joint stiffness, maintains neuromuscular patterns and reduces muscle atrophy that prolonged immobilisation causes.
  • Athletes following progressive protocols can achieve full range of motion faster than those using rest alone.

Reinforcement:

  • Some argue rest alone suffices for recovery. However, research consistently demonstrates controlled movement combined with appropriate thermal therapy optimises healing timeframes whilst minimising complications.
  • This evidence-based protocol remains valid because it addresses both tissue healing requirements and functional restoration needs.
  • The combined approach facilitates complete recovery whilst significantly reducing re-dislocation risk during return to sport.
Show Worked Solution

Position Statement:

  • Heat and cold therapy combined with progressive mobilisation provides optimal rehabilitation for shoulder dislocations.
  • These procedures effectively manage inflammation whilst restoring range of motion and facilitating safe return to function.

Cold Therapy in Acute Phase:

  • Immediate cold application reduces inflammatory response and controls swelling after dislocation. Evidence confirms cold constricts blood vessels, limiting fluid accumulation around damaged tissue.
  • Research shows applying ice for 15-20 minutes every two hours during first 48-72 hours significantly decreases pain and tissue damage. This demonstrates cold therapy’s effectiveness in protecting injured structures during the acute inflammatory phase.
  • Cold application enables earlier mobilisation by controlling pain and swelling. Athletes experience reduced discomfort, allowing gentle movement exercises to begin sooner whilst preventing excessive inflammation that delays healing. This justifies cold as essential for initial injury management.

Heat Application in Later Stages:

  • After initial inflammation subsides (typically 72 hours post-injury), heat therapy increases blood flow to promote tissue repair. Studies indicate heat dilates vessels, delivering oxygen and nutrients essential for healing damaged ligaments and capsule tissue.
  • Heat application before mobilisation exercises improves muscle elasticity and joint flexibility. Evidence shows warming tissues reduces stiffness, allowing greater range of movement during rehabilitation exercises without causing re-injury or excessive discomfort.

Progressive Mobilisation Throughout Recovery:

  • Progressive mobilisation systematically restores shoulder function through graduated exercises matched to healing stages. This approach begins with passive pendulum movements, advancing to active-assisted exercises, then resistance training as tissue strength develops.
  • Evidence demonstrates controlled movement prevents joint stiffness, maintains neuromuscular patterns and reduces muscle atrophy that prolonged immobilisation causes.
  • Athletes following progressive protocols can achieve full range of motion faster than those using rest alone.

Reinforcement:

  • Some argue rest alone suffices for recovery. However, research consistently demonstrates controlled movement combined with appropriate thermal therapy optimises healing timeframes whilst minimising complications.
  • This evidence-based protocol remains valid because it addresses both tissue healing requirements and functional restoration needs.
  • The combined approach facilitates complete recovery whilst significantly reducing re-dislocation risk during return to sport.

♦♦ Mean mark 52%.

HMS, TIP 2025 HSC 30aii

How does fluid intake support evaporation to regulate the body's temperature?   ( 5 marks)

Show Answers Only
  • Adequate fluid intake maintains blood plasma volume for sweat production. This occurs because the body requires sufficient water stores to transport heat from core to skin surface.
  • When athletes consume fluids regularly, sweat glands produce moisture continuously on skin. This allows evaporation to occur, which removes heat energy from the body surface.
  • For example, a cyclist drinking 200ml every 15 minutes during summer training sustains perspiration rates. As a result, evaporative cooling prevents dangerous core temperature elevation during prolonged effort.
  • Insufficient hydration reduces sweat production capacity. Consequently, the body loses its primary cooling mechanism, leading to heat exhaustion or heatstroke risk during physical activity.
  • The reason for this is that without adequate fluid replacement, blood volume decreases, limiting the body’s ability to deliver water to sweat glands for effective thermoregulation.
Show Worked Solution
  • Adequate fluid intake maintains blood plasma volume for sweat production. This occurs because the body requires sufficient water stores to transport heat from core to skin surface.
  • When athletes consume fluids regularly, sweat glands produce moisture continuously on skin. This allows evaporation to occur, which removes heat energy from the body surface.
  • For example, a cyclist drinking 200ml every 15 minutes during summer training sustains perspiration rates. As a result, evaporative cooling prevents dangerous core temperature elevation during prolonged effort.
  • Insufficient hydration reduces sweat production capacity. Consequently, the body loses its primary cooling mechanism, leading to heat exhaustion or heatstroke risk during physical activity.
  • The reason for this is that without adequate fluid replacement, blood volume decreases, limiting the body’s ability to deliver water to sweat glands for effective thermoregulation.

Polynomials, EXT1 EQ-Bank 5 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`

Polynomials, EXT1 EQ-Bank 4

The polynomial  \(p(x) = x^3 + ax^2 + b\)  has a zero at \(r\) and a double zero at 4.

Find the values of \(a, b\) and \(r\).   (3 marks)

Show Answers Only

\(a =-6, b = 32, r = -2\)

Show Worked Solution

\(p(x) = x^3 + ax^2 + b\)

\(\text{Zero at \(r\) and double zero at 4:}\)

\(p(x)\) \(=(x-4)^2(x-r) \)  
  \(=(x^2-8x+16)(x-r)\)  
  \(=x^3-8x^2+16x-rx^2+8rx-16r\)  
  \(=x^3+(-8-r)x^2+(16+8r)x-16r\)  

 

\(\text{Equating coefficients:}\)

\(16+8r=0\ \ \Rightarrow \ \ r=-2\)

\(a=-8-(-2)=-6\)

\(b=-16 \times -2=32\)

Polynomials, EXT1 EQ-Bank 1

A polynomial has the equation

\(P(x)=(x-1)(x-3)(x+2)^2\left(x^2-x-6\right)\).

  1. By expressing \(P(x)\) as a product of its linear factors, determine the multiplicity of each of the roots of  \(P(x)=0\).   (2 marks)

  2. Hence, without using calculus, draw a sketch of  \(y=P(x)\)  showing all \(x\)-intercepts.   (2 marks)

Show Answers Only
a.     \(P(x)\) \(=(x-1)(x-3)(x+2)^2\left(x^2-x-6\right)\)
    \(=(x-1)(x-3)(x+2)^2(x-3)(x+2)\)
    \(=(x-1)(x-3)^2(x+2)^3\)

 

\(\text{Roots:}\)

\(x=1\ \text{(multiplicity 1)}\)

\(x=-2\ \text{(multiplicity 3)}\)

\(x=3\ \text{(multiplicity 2)}\)
 

b.
     

Show Worked Solution
a.     \(P(x)\) \(=(x-1)(x-3)(x+2)^2\left(x^2-x-6\right)\)
    \(=(x-1)(x-3)(x+2)^2(x-3)(x+2)\)
    \(=(x-1)(x-3)^2(x+2)^3\)

 

\(\text{Roots:}\)

\(x=1\ \text{(multiplicity 1)}\)

\(x=-2\ \text{(multiplicity 3)}\)

\(x=3\ \text{(multiplicity 2)}\)
 

b.
     

Functions, EXT1 EQ-Bank 10

Consider the function  \(f(\theta)=\operatorname{cosec}\left(\frac{\pi}{2}-\theta\right)\)  for  \(0 \leqslant \theta \leqslant 2 \pi\).

  1. Sketch the graph of  \(y=\operatorname{cosec}\left(\frac{\pi}{2}-\theta\right)\),  showing all key features.   (2 marks)

     
     
  2. In set notation, state the range of \(\theta\).   (1 mark)

Show Answers Only

a.     


 

b.   \(\text{Range} \ \ f(\theta):\ y \in(-\infty,-1] \cup[1, \infty)\)

Show Worked Solution

a.    \(y=\operatorname{cosec}\left(\frac{\pi}{2}-\theta\right)=\dfrac{1}{\sin \left(\frac{\pi}{2}-\theta\right)}=\dfrac{1}{\cos\, \theta}\)
 


 

b.   \(\text{Range} \ \ f(\theta):\ y \in(-\infty,-1] \cup[1, \infty)\)

Functions, EXT1 EQ-Bank 9

Consider the functions  \(f(x)=\tan x\)  and  \(g(x)=\cot x\).

  1. Explain why  \(\cot x \neq \dfrac{1}{\tan x}\)  for all values of \(x\).   (2 marks)

  2. On the same set of axes below, sketch  \(y=\tan x\)  and  \(y=\cot x\)  for  \(0<x<\pi\), identifying any points where the graphs intersect.   (2 marks)
     
     
Show Answers Only

a.    \(\text{At}\ \  x=\dfrac{\pi}{2}:\)

\(\cot \dfrac{\pi}{2}=\dfrac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}}=\dfrac{0}{1}=0 \ \Rightarrow \ \text{defined}\).

\(\tan \dfrac{\pi}{2}=\dfrac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}}=\dfrac{1}{0} \Rightarrow \ \text{undefined}\).

\(\dfrac{1}{\tan \frac{\pi}{2}}\ \ \text{is therefore undefined}\).

\(\therefore \cot x \neq \dfrac{1}{\tan x} \ \ \text{for all values of }\ x\).
 

b.
       

Show Worked Solution

a.    \(\text{At}\ \  x=\dfrac{\pi}{2}:\)

\(\cot \dfrac{\pi}{2}=\dfrac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}}=\dfrac{0}{1}=0 \ \Rightarrow \ \text{defined}\).

\(\tan \dfrac{\pi}{2}=\dfrac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}}=\dfrac{1}{0} \Rightarrow \ \text{undefined}\).

\(\dfrac{1}{\tan \frac{\pi}{2}}\ \ \text{is therefore undefined}\).

\(\therefore \cot x \neq \dfrac{1}{\tan x} \ \ \text{for all values of }\ x\).
 

b.
       

Functions, EXT1 EQ-Bank 8

Consider the function  \(y=\operatorname{cosec}\,x\)  for  \(-\pi \leqslant x \leqslant \pi\).

  1. State the equations of all vertical asymptotes in the given domain.   (1 mark)

  2. Sketch the graph of  \(y=\operatorname{cosec} x\), showing all key features.   (2 marks)

     
     
Show Answers Only

a.    \(y=\operatorname{cosec}\,x=\dfrac{1}{\sin x}\)

\(\text{Asymptotes when \(\ \sin x=0 \ \) in given domain.}\)

\(\therefore \ \text{Asymptotes at} \ \ x=-\pi, 0, \pi\)
 

b.
       

Show Worked Solution

a.    \(y=\operatorname{cosec}\,x=\dfrac{1}{\sin x}\)

\(\text{Asymptotes when \(\ \sin x=0 \ \) in given domain.}\)

\(\therefore \ \text{Asymptotes at} \ \ x=-\pi, 0, \pi\)
 

b.
       

Functions, EXT1 EQ-Bank 7

  1. Sketch the graph of  \(y=\sec x\)  for  \(0 \leqslant x \leqslant 2 \pi\).
  2. In your answer, identify all asymptotes and the coordinates of any maximum and minimum turning points.   (2 marks)

     

  3. Using set notation, state the domain and range of  \(y=\sec x\).   (1 mark)

Show Answers Only

a.
   

b.    \(\text{Domain:} \ x \in\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right]\)

\(\text{Range:} \ y \in(-\infty,-1] \cup[1, \infty)\)

Show Worked Solution

a.    \(\text{Draw}\ \ y=\cos\,x\ \ \text{to inform graph:}\)

 
   

\(\text{Minimum TPs:}\ (0,1), (2\pi, 1) \)

\(\text{Maximum TP:}\ (\pi, -1)\)
 

b.    \(\text{Domain:} \ x \in\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right]\)

\(\text{Range:} \ y \in(-\infty,-1] \cup[1, \infty)\)

Probability, 2ADV EQ-Bank 8

A survey of 50 students found that:

  • 28 students study Mathematics (set \(M\))
  • 22 students study Physics (set \(P\) )
  • 12 students study both Mathematics and Physics.
  1. How many students study Mathematics or Physics, or both?   (1 mark)

  2. If two students are chosen at random, what is the probability that both DO NOT study either Mathematics or Physics?   (2 marks)

Show Answers Only

a.
     

\(n(M \cup P)=38 \ \text {students}\)
 

b.    \(\dfrac{66}{1225}\)

Show Worked Solution

a.
     

\(n(M \cup P)=38 \ \text {students}\)
 

b.    \(P(M \cup P)=\dfrac{38}{50}\)

\(\text{Student 1:}\ P(\overline{M \cup P})=1-\dfrac{38}{50}=\dfrac{12}{50}\)

\(\text{Student 2:} \ P(\overline{M \cup P})=\dfrac{11}{49}\)

\(\therefore P\left(\text{Both study neither }\right)=\dfrac{12}{50} \times \dfrac{11}{49}=\dfrac{66}{1225}\)

Probability, 2ADV EQ-Bank 6

A survey of 85 households asked if they subscribed to the streaming services provided by Netflix (set \(N\)), Apple TV (set \(A\)), and Stan (set \(S\)).

The survey found that 17 households had no subscription and that \(n(N)=42, n(A)=35, n(S)=28\).

The survey also found

\(n(N \cap A)=18, n(N \cap S)=15, n(A \cap S)=12\) and \(n(N \cap A \cap S)=8\)

  1. Complete the Venn diagram below to accurately describe the information given.   (2 marks)


     

  2. A household is chosen randomly and is found to subscribe to Apple TV. What is the probability the household also subscribes to Stan?   (2 marks)

Show Answers Only

a.
           
 

b.    \(P(S \mid A)=\dfrac{12}{35}\)

Show Worked Solution

a.
           
 

b.    \(n(A)=35, n(A \cap S)=12\)

\(P(S \mid A)=\dfrac{n(A \cap S)}{n(A)}=\dfrac{12}{35}\)

HMS, HIC 2025 HSC 28aii

Explain how becoming involved in community service can assist young people in attaining better health.  ( 5 marks)

Show Answers Only
  • Community service creates opportunities for social connection with like-minded peers. This occurs when young people work alongside others towards shared goals in volunteer settings.
  • For example, a young person volunteering at environmental clean-up events develops teamwork and communication abilities. As a result, their social health improves through new friendships and increased confidence in group interactions.
  • Volunteering enhances mental health by providing sense of purpose and direction. This happens because contributing meaningfully to community needs generates feelings of value and belonging.
  • For instance, a young person supporting elderly residents through hospital visits experiences improved emotional wellbeing. The reason for this is that helping others creates perspective on personal challenges whilst building resilience through meaningful relationships.
  • Community service leads to increased physical activity in many volunteer roles. Consequently, young people gain health benefits from active engagement rather than sedentary screen time.
Show Worked Solution
  • Community service creates opportunities for social connection with like-minded peers. This occurs when young people work alongside others towards shared goals in volunteer settings.
  • For example, a young person volunteering at environmental clean-up events develops teamwork and communication abilities. As a result, their social health improves through new friendships and increased confidence in group interactions.
  • Volunteering enhances mental health by providing sense of purpose and direction. This happens because contributing meaningfully to community needs generates feelings of value and belonging.
  • For instance, a young person supporting elderly residents through hospital visits experiences improved emotional wellbeing. The reason for this is that helping others creates perspective on personal challenges whilst building resilience through meaningful relationships.
  • Community service leads to increased physical activity in many volunteer roles. Consequently, young people gain health benefits from active engagement rather than sedentary screen time.

♦♦ Mean mark 51%.

HMS, HIC 2025 HSC 27

Explain the responsibilities of individuals, communities and governments in creating supportive environments to promote health. Support your answer with examples.  ( 8 marks)

Show Answers Only

Individual Responsibilities:

  • Individuals must adopt health-promoting behaviours that create safe environments for themselves and others. This occurs when people apply health literacy to make informed decisions.
  • For example, a parent choosing active transport to school reduces vehicle emissions and models healthy physical activity patterns. This leads to improved air quality and encourages children to adopt active lifestyles.

Community Responsibilities:

  • Communities must advocate for and support their members through accessible programs and resources. This enables individuals to access culturally appropriate health services.
  • For instance, neighbourhood walking groups organised by local councils provide social connection and physical activity opportunities. Consequently, participants experience improved mental and physical health through regular engagement and peer support networks.

Government Responsibilities:

  • Governments must develop and enforce policies that facilitate health-promoting environments. This works through legislation creating safe public spaces and restricting harmful exposures.
  • For example, mandatory bicycle helmet laws and dedicated cycling infrastructure protect cyclists from injury whilst encouraging active transport adoption. As a result, communities experience reduced traffic congestion, improved air quality and increased population physical activity levels.

Collective Impact:

  • These responsibilities work together to create comprehensive supportive environments. The significance is that sustained health improvements require coordinated action across all three levels rather than isolated individual efforts.
Show Worked Solution

Individual Responsibilities:

  • Individuals must adopt health-promoting behaviours that create safe environments for themselves and others. This occurs when people apply health literacy to make informed decisions.
  • For example, a parent choosing active transport to school reduces vehicle emissions and models healthy physical activity patterns. This leads to improved air quality and encourages children to adopt active lifestyles.

Community Responsibilities:

  • Communities must advocate for and support their members through accessible programs and resources. This enables individuals to access culturally appropriate health services.
  • For instance, neighbourhood walking groups organised by local councils provide social connection and physical activity opportunities. Consequently, participants experience improved mental and physical health through regular engagement and peer support networks.

Government Responsibilities:

  • Governments must develop and enforce policies that facilitate health-promoting environments. This works through legislation creating safe public spaces and restricting harmful exposures.
  • For example, mandatory bicycle helmet laws and dedicated cycling infrastructure protect cyclists from injury whilst encouraging active transport adoption. As a result, communities experience reduced traffic congestion, improved air quality and increased population physical activity levels.

Collective Impact:

  • These responsibilities work together to create comprehensive supportive environments. The significance is that sustained health improvements require coordinated action across all three levels rather than isolated individual efforts.

HMS, TIP 2025 HSC 26

Analyse the relationship between training thresholds and TWO physiological adaptations. In your answer, provide examples of both aerobic and resistance training.  (8 marks)

Show Answers Only

Overview Statement:

  • Training thresholds represent critical intensity levels that trigger specific physiological adaptations.
  • Understanding how aerobic and resistance thresholds connect to metabolic and muscular changes can open up pathways to enhanced athletic performance.

Aerobic Threshold and Cardiovascular Adaptations:

  • The aerobic training threshold occurs at approximately 70% of maximum heart rate. Training at this intensity influences fuel utilisation and cardiovascular function.
  • This causes the body to shift from primarily using fat to using carbohydrates for energy. For example, a marathon runner training at 70% max heart rate stimulates this metabolic adaptation.
  • The threshold also triggers increased stroke volume through enhanced left ventricle filling capacity. This relationship results in improved cardiac output and oxygen delivery to working muscles.
  • Consequently, athletes sustain effort over extended periods with greater efficiency. This shows that aerobic threshold training enables both metabolic and cardiovascular improvements for endurance performance.

Resistance Threshold and Muscular Adaptations:

  • Resistance training thresholds involve working at 70-85% of one-rep maximum with 6-12 repetitions. This intensity creates sufficient mechanical stress to stimulate muscle hypertrophy.
  • For instance, a weightlifter performing squats at 80% of 1RM bmicroscopic muscle fibre damage. This initiates increased protein synthesis and muscle repair processes.
  • The threshold works through progressive overload that leads to enlarged muscle fibres with increased actin and myosin filaments. As a result, greater force production capacity develops.
  • The significance is that athletes gain strength and power output essential for explosive movements.

Implications and Synthesis:

  • These thresholds work together as intensity markers that determine adaptation type. Aerobic thresholds influence metabolic and cardiovascular systems whilst resistance thresholds affect muscular structure.
  • Therefore, coaches must apply appropriate threshold intensities to achieve specific performance goals. This reveals that training success depends on understanding the precise relationship between intensity levels and resulting physiological changes.
Show Worked Solution

Overview Statement:

  • Training thresholds represent critical intensity levels that trigger specific physiological adaptations.
  • Understanding how aerobic and resistance thresholds connect to metabolic and muscular changes can open up pathways to enhanced athletic performance.

Aerobic Threshold and Cardiovascular Adaptations:

  • The aerobic training threshold occurs at approximately 70% of maximum heart rate. Training at this intensity influences fuel utilisation and cardiovascular function.
  • This causes the body to shift from primarily using fat to using carbohydrates for energy. For example, a marathon runner training at 70% max heart rate stimulates this metabolic adaptation.
  • The threshold also triggers increased stroke volume through enhanced left ventricle filling capacity. This relationship results in improved cardiac output and oxygen delivery to working muscles.
  • Consequently, athletes sustain effort over extended periods with greater efficiency. This shows that aerobic threshold training enables both metabolic and cardiovascular improvements for endurance performance.

Resistance Threshold and Muscular Adaptations:

  • Resistance training thresholds involve working at 70-85% of one-rep maximum with 6-12 repetitions. This intensity creates sufficient mechanical stress to stimulate muscle hypertrophy.
  • For instance, a weightlifter performing squats at 80% of 1RM bmicroscopic muscle fibre damage. This initiates increased protein synthesis and muscle repair processes.
  • The threshold works through progressive overload that leads to enlarged muscle fibres with increased actin and myosin filaments. As a result, greater force production capacity develops.
  • The significance is that athletes gain strength and power output essential for explosive movements.

Implications and Synthesis:

  • These thresholds work together as intensity markers that determine adaptation type. Aerobic thresholds influence metabolic and cardiovascular systems whilst resistance thresholds affect muscular structure.
  • Therefore, coaches must apply appropriate threshold intensities to achieve specific performance goals. This reveals that training success depends on understanding the precise relationship between intensity levels and resulting physiological changes.

♦♦ Mean mark 36%.

Probability, 2ADV EQ-Bank 4

In Year 11 there are 80 students. The students may choose to study Spanish (S), Japanese (J) and Mandarin (M).

The Venn diagram shows their choices.
 

 

Two of the students are selected at random.

  1. What is the probability that both students study only Spanish?   (2 marks)

  2. What is the probability that at least one of the students studies two languages.   (2 marks)

Show Answers Only

a.    \(\dfrac{93}{632}\)

b.   \(\dfrac{81}{158} \)

Show Worked Solution

a.    \(\text{Students only studying Spanish = 31}\)

\(P(\text{both study only Spanish}\ =\dfrac{31}{80} \times \dfrac{30}{79} = \dfrac{93}{632}\)
 

b.   \(\text{1st student chosen:}\)

\(P(2L) =\dfrac{6+4+14}{80} = \dfrac{24}{80}\ \ \Rightarrow\ \ P(\overline{2L})=\dfrac{56}{80} \)

\(\text{2nd student chosen:}\)

\(P(\overline{2L})=\dfrac{55}{79} \)
 

\(P(\text{at least one studies two languages})\)

\(= 1- P(\text{both don’t study two languages)}\)

\(=1-\dfrac{56}{80} \times \dfrac{55}{79} \)

\(=\dfrac{81}{158} \)

Calculus, 2ADV C1 EQ-Bank 5

A drone travels vertically from its launch pad.

It's height above ground, \(h\) metres, at time \(t\) minutes is modelled by

\(h(t)=-0.2 t^3+3 t^2+5 t\)  for  \(0 \leq t \leq 12\)

  1. Find the velocity of the drone at time \(t\) minutes.   (1 mark)

  2. Determine the exact time interval during which the drone is descending.   (2 marks)

Show Answers Only

a.    \(\dfrac{dh}{dt}=-0.6 t^2+6 t+5\)
 

b.    \(\dfrac{15+10 \sqrt{3}}{3}<t \leqslant 12\)

Show Worked Solution

a.    \(h=-0.2 t^3+3 t^2+5 t\)

\(\text{Velocity of the drone}=\dfrac{d h}{d t}.\)

\(\dfrac{dh}{dt}=-0.6 t^2+6 t+5\)
 

b.    \(\text{Drone is descending when} \ \ \dfrac{dh}{dt}<0:\)

\(-0.6 t^2+6 t+5\) \(<0\)  
\(0.6 t^2-6 t-5\) \(>0\)  
\(6 t^2-60 t-50\) \(>0\)  

 
\(\text{Solve}\ \ 6 t^2-60 t-50=0:\)

\(t=\dfrac{60 \pm \sqrt{(-60)^2+4 \times 6 \times 50}}{2 \times 6}=\dfrac{60 \pm \sqrt{4800}}{12}=\dfrac{15 \pm 10 \sqrt{3}}{3}\)

 
\(\text{Since parabola is concave up:}\)

\(6 t^2-60 t-50>0\ \ \text{when}\ \ t>\dfrac{15+10 \sqrt{3}}{3} \quad\left( t=\dfrac{15-10 \sqrt{3}}{3}<0\right)\)

\(\therefore \text{Drone is descending for} \ \ \dfrac{15+10 \sqrt{3}}{3}<t \leqslant 12\)

HMS, TIP 2025 HSC 25b

Explain how ONE relaxation technique could be used to manage the anxiety of an athlete. Support your answer with an example.   (5 marks)

Show Answers Only
  • Controlled breathing reduces physiological anxiety symptoms in athletes. This occurs because slow, deep breathing activates the parasympathetic nervous system, which decreases heart rate and muscle tension throughout the body.
  • Athletes practise rhythmic breathing patterns before high-pressure situations. For example, a penalty shooter in football uses four-second inhale, four-second exhale cycles to calm pre-kick nerves and reduce performance anxiety.
  • This technique allows the athlete to shift focus from anxious thoughts to breath control. As a result, mental clarity improves and intrusive worries diminish during critical competitive moments.
  • The breathing pattern creates a sense of control over physical responses. This leads to reduced trembling, steadier hands and improved concentration on technical execution requirements.
  • Regular practice strengthens the relaxation response over time through conditioning. Consequently, athletes can activate calmness quickly when needed in competition, enabling optimal arousal levels for peak performance delivery.
Show Worked Solution
  • Controlled breathing reduces physiological anxiety symptoms in athletes. This occurs because slow, deep breathing activates the parasympathetic nervous system, which decreases heart rate and muscle tension throughout the body.
  • Athletes practise rhythmic breathing patterns before high-pressure situations. For example, a penalty shooter in football uses four-second inhale, four-second exhale cycles to calm pre-kick nerves and reduce performance anxiety.
  • This technique allows the athlete to shift focus from anxious thoughts to breath control. As a result, mental clarity improves and intrusive worries diminish during critical competitive moments.
  • The breathing pattern creates a sense of control over physical responses. This leads to reduced trembling, steadier hands and improved concentration on technical execution requirements.
  • Regular practice strengthens the relaxation response over time through conditioning. Consequently, athletes can activate calmness quickly when needed in competition, enabling optimal arousal levels for peak performance delivery.

HMS, BM 2025 HSC 25a

How can goal setting be used by an athlete to enhance motivation?   (3 marks)

Show Answers Only
  • Goal setting creates clear targets for achievement. This increases motivation because athletes gain sense of purpose and direction in their training efforts.
  • Measurable goals allow athletes to track progress consistently. For example, a cyclist aiming to improve 5km time trial by 30 seconds experiences increased motivation through visible improvements.
  • Achievable goals generate confidence through successful completion. As a result, athletes remain motivated to pursue increasingly challenging objectives in their sport.
Show Worked Solution
  • Goal setting creates clear targets for achievement. This increases motivation because athletes gain sense of purpose and direction in their training efforts.
  • Measurable goals allow athletes to track progress consistently. For example, a cyclist aiming to improve 5km time trial by 30 seconds experiences increased motivation through visible improvements.
  • Achievable goals generate confidence through successful completion. As a result, athletes remain motivated to pursue increasingly challenging objectives in their sport.

Calculus, 2ADV EQ-Bank 4

The volume of water in a tank, \(V\) litres, at time \(t\) minutes is given by:

\(V(t)=2 t^3-15 t^2+24 t+50\)  for  \(0 \leqslant t \leqslant 6\)

  1. Find an expression for the rate at which water is flowing at time \(t\).   (1 mark)

  2. Calculate the rate of flow at  \(t=4\)  minutes and interpret this value.   (1 mark)

  3. Deduce when the water level in the tank is increasing.   (2 marks)

Show Answers Only

a.    \(\dfrac{dV}{d t}=6 t^2-30 t+24\)
 

b.    \(\text{At} \ \ t=4:\ \ \dfrac{dV}{d t}=0 \ \text{litres/minute}\)

\(\text{Interpretation: At \(\ t=4 \ \), water has stopped flowing either into or}\)

\(\text{out of the tank.}\)
 

c.    \(\text{If water level is increasing} \ \Rightarrow \ \dfrac{dV}{d t}>0:\)

\(\text {Find \(t\) when}\ \dfrac{dV}{d t}>0:\)

\(6 t^2-30t+24\) \(\gt 0\)  
\(6\left(t^2-5 t+4\right)\) \(\gt 0\)  
\((t-4)(t-1)\) \(\gt 0\)  

 

\(\therefore \ \text{Water level increases for}\ \ t \in [0,1) \cup (4, 6]\)

Show Worked Solution

a.    \(V=2 t^3-15 t^2+24 t+50\)

\(\dfrac{dV}{d t}=6 t^2-30 t+24\)
 

b.    \(\text{At} \ \ t=4:\)

\(\dfrac{dV}{d t}=6 \times 4^2-30 \times 4+24=0 \ \text{litres/minute}\)

\(\text{Interpretation: At \(\ t=4 \ \), water has stopped flowing either into or}\)

\(\text{out of the tank.}\)
 

c.    \(\text{If water level is increasing} \ \Rightarrow \ \dfrac{dV}{d t}>0:\)

\(\text {Find \(t\) when}\ \dfrac{dV}{d t}>0:\)

\(6 t^2-30t+24\) \(\gt 0\)  
\(6\left(t^2-5 t+4\right)\) \(\gt 0\)  
\((t-4)(t-1)\) \(\gt 0\)  

 

\(\therefore \ \text{Water level increases for}\ \ t \in [0,1) \cup (4, 6]\)

Functions, 2ADV EQ-Bank 5

The cost per person \((C)\), in dollars, for hiring a function room varies inversely with the number of people \((n)\) attending, since the total venue cost is fixed at $2400.

  1. Write an equation relating \(C\) and \(n\).   (1 mark)

  2. Sketch a graph showing the relationship between the number of people and cost per person for  \(0<n \leqslant 100\). Label your axes.   (2 marks)
     

           

Show Answers Only

a.    \(C=\dfrac{2400}{n}\)

b.    
         

Show Worked Solution

a.    \(C=\dfrac{2400}{n}\)

b.   \(\text{Table of values:}\)

\(\begin{array}{|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex}\ \ n \ \ \rule[-1ex]{0pt}{0pt}& 20 & \ 40 \ & \ 60 \ & 100 \\
\hline \rule{0pt}{2.5ex}\ \ C \ \ \rule[-1ex]{0pt}{0pt}& 120 & 60 & 40 & 24 \\
\hline
\end{array}\)
 

Functions, 2ADV EQ-Bank 4

Consider the function  \(g(x)=-\dfrac{12}{x}\)

  1. Is \(g(x)\) an odd or even function? Give reason(s) for your answer.   (2 marks)

  2. Sketch the graph \(y=g(x)\) in the domain \(-6 \leq x \leq 6\). Label the endpoints and two other points on the curve.   (2 marks)

Show Answers Only

a.    \(\text{If}\ g(x)\ \text{is odd}\ \ \Rightarrow \ \ g(-x)=-g(x)\)

\(-g(x)= \dfrac{12}{x}\)

\(g(-x) = -\dfrac{12}{(-x)} = \dfrac{12}{x} = -g(x)\)

\(\therefore g(x)\ \text{is odd (and cannot be even).}\)
 

b.    \(\text{Table of values:}\)

\begin{array}{|c|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex}\ \ x \ \ \rule[-1ex]{0pt}{0pt}& -4 & -2 & \ \ 0 \ \ & 2 & 4 \\
\hline \rule{0pt}{2.5ex}y \rule[-1ex]{0pt}{0pt}& 3 & 6 & \infty & -6 & -3 \\
\hline
\end{array}

\(\text{Endpoints:}\ (-6,2), (6,-2) \)
 

Show Worked Solution

a.    \(\text{If}\ g(x)\ \text{is odd}\ \ \Rightarrow \ \ g(-x)=-g(x)\)

\(-g(x)= \dfrac{12}{x}\)

\(g(-x) = -\dfrac{12}{(-x)} = \dfrac{12}{x} = -g(x)\)

\(\therefore g(x)\ \text{is odd.}\)
 

b.    \(\text{Table of values:}\)

\begin{array}{|c|c|c|c|c|c|}
\hline \rule{0pt}{2.5ex}\ \ x \ \ \rule[-1ex]{0pt}{0pt}& -4 & -2 & \ \ 0 \ \ & 2 & 4 \\
\hline \rule{0pt}{2.5ex}y \rule[-1ex]{0pt}{0pt}& 3 & 6 & \infty & -6 & -3 \\
\hline
\end{array}

\(\text{Endpoints:}\ (-6,2), (6,-2) \)
 

HMS, HAG 2025 HSC 24

Explain the impact of an ageing population on the health service workforce in Australia.   (5 marks)

Show Answers Only
  • Increased demand for aged care services creates workforce shortages. This occurs because more Australians require assistance with daily living activities and chronic condition management.
  • Health professionals experience elevated workloads due to complex care requirements. As a result, staff manage multiple patients with conditions like dementia requiring specialised expertise and extended treatment periods.
  • Workforce pressure leads to burnout and high staff turnover rates. This happens when healthcare workers face sustained stress from increased patient numbers and inadequate staffing levels.
  • The reason for this is that chronic disease prevalence rises with age. Consequently, healthcare systems require additional specialists in geriatrics, palliative care and rehabilitation services.
  • Recruitment and training needs intensify significantly. This creates challenges in maintaining service quality whilst expanding workforce capacity through education programs and upskilling initiatives.
Show Worked Solution
  • Increased demand for aged care services creates workforce shortages. This occurs because more Australians require assistance with daily living activities and chronic condition management.
  • Health professionals experience elevated workloads due to complex care requirements. As a result, staff manage multiple patients with conditions like dementia requiring specialised expertise and extended treatment periods.
  • Workforce pressure leads to burnout and high staff turnover rates. This happens when healthcare workers face sustained stress from increased patient numbers and inadequate staffing levels.
  • The reason for this is that chronic disease prevalence rises with age. Consequently, healthcare systems require additional specialists in geriatrics, palliative care and rehabilitation services.
  • Recruitment and training needs intensify significantly. This creates challenges in maintaining service quality whilst expanding workforce capacity through education programs and upskilling initiatives.

HMS, HIC 2025 HSC 22

Explain how infant mortality can be used to measure the health status of a population.   (4 marks)

Show Answers Only
  • Infant mortality reflects a population’s overall wellbeing and survival prospects. This occurs because it measures deaths before age one, which indicates underlying health conditions.
  • Low infant death rates demonstrate effective healthcare access and quality medical services. This shows that preventative measures like immunisation programs and maternal support are working successfully.
  • For instance, when Australian infant mortality declined from 2010 to 2020, this revealed improvements in prenatal screening and neonatal care. As a result, more infants survived their vulnerable first year.
  • Infant mortality data enables health authorities to identify at-risk communities. This allows targeted interventions where mortality rates remain elevated.
Show Worked Solution
  • Infant mortality reflects a population’s overall wellbeing and survival prospects. This occurs because it measures deaths before age one, which indicates underlying health conditions.
  • Low infant death rates demonstrate effective healthcare access and quality medical services. This shows that preventative measures like immunisation programs and maternal support are working successfully.
  • For instance, when Australian infant mortality declined from 2010 to 2020, this revealed improvements in prenatal screening and neonatal care. As a result, more infants survived their vulnerable first year.
  • Infant mortality data enables health authorities to identify at-risk communities. This allows targeted interventions where mortality rates remain elevated.

HMS, HAG 2025 HSC 21

Outline TWO risk factors for ONE of the conditions listed below:   (3 marks)

  • diabetes
  • respiratory disease
  • injury
  • mental health problems and illnesses.
Condition selected:..........................................................
Show Answers Only

Any ONE of the following conditions:

Condition Selected: Diabetes

Risk Factor 1: Overweight and Obesity

  • Excess body weight increases insulin resistance. This prevents cells from responding effectively to insulin.
  • High body mass index is strongly associated with Type 2 diabetes development.

Risk Factor 2: Physical Inactivity

  • Sedentary behaviour reduces glucose uptake by muscles. This impairs blood sugar regulation.
  • Insufficient physical activity contributes to weight gain and metabolic dysfunction.

Condition Selected: Respiratory Disease 

Risk Factor 1: Tobacco Smoking

  • Smoking damages airways and lung tissue. This causes inflammation and reduces lung function.
  • Tobacco smoke exposure is the primary cause of chronic obstructive pulmonary disease.

Risk Factor 2: Air Pollution

  • Pollutants irritate respiratory passages and worsen chronic conditions. This increases symptoms and exacerbations.
  • Poor air quality exposure contributes to asthma development and progression.

Condition selected: Injury

Risk Factor 1: Alcohol Consumption

  • Alcohol impairs judgement and coordination. This increases accident and injury risk.
  • Excessive drinking is associated with higher rates of falls, assaults and transport accidents.

Risk Factor 2: Risky Behaviours

  • Not wearing safety equipment during activity increases injury severity. This includes seatbelts and helmets.
  • Participation in high-risk activities without proper training elevates injury likelihood.

Condition selected: Mental Health Problems and Illness

Risk Factor 1: Social Isolation

  • Limited social connections reduce support networks. This increases vulnerability to mental health issues.
  • Loneliness and disconnection are strongly linked to depression and anxiety development.

Risk Factor 2: Chronic Stress

  • Prolonged stress exposure affects brain chemistry and functioning. This contributes to anxiety and depression.
  • Ongoing stressful life circumstances without adequate coping mechanisms increase mental illness risk.
Show Worked Solution

Any ONE of the following conditions:

Condition Selected: Diabetes

Risk Factor 1: Overweight and Obesity

  • Excess body weight increases insulin resistance. This prevents cells from responding effectively to insulin.
  • High body mass index is strongly associated with Type 2 diabetes development.

Risk Factor 2: Physical Inactivity

  • Sedentary behaviour reduces glucose uptake by muscles. This impairs blood sugar regulation.
  • Insufficient physical activity contributes to weight gain and metabolic dysfunction.

Condition Selected: Respiratory Disease 

Risk Factor 1: Tobacco Smoking

  • Smoking damages airways and lung tissue. This causes inflammation and reduces lung function.
  • Tobacco smoke exposure is the primary cause of chronic obstructive pulmonary disease.

Risk Factor 2: Air Pollution

  • Pollutants irritate respiratory passages and worsen chronic conditions. This increases symptoms and exacerbations.
  • Poor air quality exposure contributes to asthma development and progression.

Condition selected: Injury

Risk Factor 1: Alcohol Consumption

  • Alcohol impairs judgement and coordination. This increases accident and injury risk.
  • Excessive drinking is associated with higher rates of falls, assaults and transport accidents.

Risk Factor 2: Risky Behaviours

  • Not wearing safety equipment during activity increases injury severity. This includes seatbelts and helmets.
  • Participation in high-risk activities without proper training elevates injury likelihood.

Condition selected: Mental Health Problems and Illness

Risk Factor 1: Social Isolation

  • Limited social connections reduce support networks. This increases vulnerability to mental health issues.
  • Loneliness and disconnection are strongly linked to depression and anxiety development.

Risk Factor 2: Chronic Stress

  • Prolonged stress exposure affects brain chemistry and functioning. This contributes to anxiety and depression.
  • Ongoing stressful life circumstances without adequate coping mechanisms increase mental illness risk.

HMS, HIC 2025 HSC 20 MC

Which row of the table accurately reflects epidemiological data in Australia?

Show Answers Only

\(A\)

Show Worked Solution

  • A is correct: Dementia leads female deaths with increasing rates; coronary heart disease leads male deaths with decreasing rates

Other Options:

  • B is incorrect: Coronary heart disease is not the leading female cause; dementia is
  • C is incorrect: Dementia rates increased, not decreased, for females
  • D is incorrect: Coronary heart disease leads for males, not dementia; rates decreased, not increased

HMS, BM 2025 HSC 19 MC

During a training session, a runner completes 40 m high intensity sprints with a 2 minute recovery period between each.

Which of the following explains the runner's rate of recovery and the efficiency of ATP production during the recovery periods between sprints?

  1. The lactic acid system is responsible for recovery, leading to the production of lactate and slower resynthesis of ATP.
  2. The ATP-PCr system is quickly replenishing phosphocreatine stores, allowing for a fast recovery and efficient ATP production.
  3. The lactic acid system is primarily used during recovery, allowing for efficient ATP production, but it takes several minutes to recover.
  4. The ATP-PCr system is slower to replenish phosphocreatine stores, leading to inefficient ATP production and more time is needed to recover.
Show Answers Only

\(B\)

Show Worked Solution

  • B is correct: Alactacid system rapidly restores phosphocreatine stores within 2-3 minutes, enabling quick ATP resynthesis between sprints

Other Options:

  • A is incorrect: Lactic acid system produces lactate during work, not recovery; recovery removes lactate, doesn’t produce it
  • C is incorrect: Lactic acid system creates ATP during exercise, not recovery; recovery period removes accumulated lactate
  • D is incorrect: Alactacid system replenishes quickly (50% in 30 seconds, 100% in 2-3 minutes), not slowly