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Algebra, STD2 A4 2025 HSC 30

It costs $465 to register a passenger car and $350 to register a motorcycle.

Let    \(P\) \(\ =\ \text{the number of passenger cars, and}\)
  \(B\) \(\ =\ \text{the number of motorcycles}\)

 
Write TWO linear equations that represent the relationship below.

  • There are 11 times as many passenger cars as motorcycles.
  • The total registration fees for passenger cars and motorcycles is $494 million.   (2 marks)
Show Answers Only

\(\text{Equation 1:}\ P=11B\)

\(\text{Equation 2:}\ 465P+350B=494\ 000\ 000\)

Show Worked Solution

\(\text{Equation 1:}\ P=11B\)

\(\text{Equation 2:}\ 465P+350B=494\ 000\ 000\)

Algebra, STD2 A4 2023 HSC 21

Electricity provider \(A\) charges 25 cents per kilowatt hour (kWh) for electricity, plus a fixed monthly charge of $40.

  1. Complete the table showing Provider \(A\)'s monthly charges for different levels of electricity usage.  (1 mark)
     
\(\textit{Electricity used in a
 month (kWh)}\)		 \(\ \ 0\ \ \) \(400\) \(1000\)
\(\textit{Monthly charge (\$)}\) \(40\)   \(290\)
 
Provider \(B\) charges 35 cents per kWh, with no fixed monthly charge. The graph shows how Provider \(B\)'s charges vary with the amount of electricity used in a month.
 

 
  1. On the grid above, graph Provider \(A\)'s charges from the table in part (a).   (1 mark)
  2. Use the two graphs to determine the number of kilowatt hours per month for which Provider \(A\) and Provider \(B\) charge the same amount.   (1 mark)

  3. A customer uses an average of 800 kWh per month.
  4. Which provider, \(A\) or \(B\), would be the cheaper option and by how much?   (2 marks)

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a. 

\(\textit{Electricity used in a
 month (kWh)}\)		 \(\ \ 0\ \ \) \(400\) \(1000\)
\(\textit{Monthly charge (\$)}\) \(40\) \(140\) \(290\)

b.    
         

c.    \(\text{400 kWh}\)

d.    \(\text{Provider}\ A\ \text{is cheaper by \$40.}\)

Show Worked Solution

a. 

\(\textit{Electricity used in a
 month (kWh)}\)		 \(\ \ 0\ \ \) \(400\) \(1000\)
\(\textit{Monthly charge (\$)}\) \(40\) \(140\) \(290\)

b. 
          
 

c.    \(A_{\text{charge}} = B_{\text{charge}}\ \text{at intersection.}\)

\(\therefore\ \text{Same charge at 400 kWh}\)
 

d.    \(\text{Cost at 800 kWh:}\)

\(\text{Provider}\ A: \ 40 + 0.25 \times 800 = $240\)

\(\text{Provider}\ B: \ 0.35 \times 800 = $280\)

\(\therefore \text{Provider}\ A\ \text{is cheaper by \$40.}\)

Algebra, STD2 A4 2021 HSC 34

In a park the only animals are goannas and emus. Let `x` be the number of goannas and let `y` be the number of emus.

The number of goannas plus the number of emus in the park is 31. Hence  `x + y = 31`.

Each goanna has four legs and each emu has two legs. In total the emus and goannas have 76 legs.

By writing another relevant equation and graphing both equations on the grid on the following page, find the number of goannas and the number of emus in the park.  (4 marks)

   
 

Number of goannas = _______________

Number of emus = _________________

Show Answers Only

`text{7 goannas, 24 emus}`

Show Worked Solution

`text{Total goanna legs} = 4x`

♦♦♦ Mean mark 29%.

`text{Total emu legs} = 2y`

`4x + 2y` `= 76`  
`2x + y` `= 38 \ …\ (1)`  
`x + y` `= 31 \ …\ (2)`  

 

`text{Number of goannas} = 7`

`text{Number of emus} = 24`

Algebra, STD2 A4 2020 HSC 24

There are two tanks on a property, Tank A and Tank B. Initially, Tank A holds 1000 litres of water and Tank B is empty.

  1. Tank A begins to lose water at a constant rate of 20 litres per minute.

     

    The volume of water in Tank A is modelled by  `V = 1000 - 20t`  where `V` is the volume in litres and  `t`  is the time in minutes from when the tank begins to lose water.

     

    On the grid below, draw the graph of this model and label it as Tank A.   (1 mark)
     

     

  2. Tank B remains empty until  `t=15`  when water is added to it at a constant rate of 30 litres per minute.
     By drawing a line on the grid (above), or otherwise, find the value of  `t`  when the two tanks contain the same volume of water.  (2 marks)

  3. Using the graphs drawn, or otherwise, find the value of  `t`  (where  `t > 0`) when the total volume of water in the two tanks is 1000 litres.  (1 mark)

Show Answers Only
  1.  `text{T} text{ank} \ A \ text{will pass trough (0, 1000) and (50, 0)}` 
      
  2. `29 \ text{minutes}`
  3. `45 \ text{minutes}`
Show Worked Solution

a.     `text{T} text{ank} \ A \ text{will pass trough (0, 1000) and (50, 0)}` 
 

 

b.   `text{T} text{ank} \ B \ text{will pass through (15, 0) and (45, 900)}`  
 

   

`text{By inspection, the two graphs intersect at} \ \ t = 29 \ text{minutes}`

 
c.   `text{Strategy 1}`

♦♦ Mean mark part (c) 22%.

`text{By inspection of the graph, consider} \ \ t = 45`

`text{T} text{ank A} = 100 \ text{L} , \ text{T} text{ank B} =900 \ text{L} `

`:.\ text(Total volume = 1000 L when  t = 45)`
  

`text{Strategy 2}`

`text{Total Volume}` `=text{T} text{ank A} + text{T} text{ank B}`
`1000` `= 1000 – 20t + (t – 15) xx 30`
`1000` `= 1000 – 20t + 30t – 450 `
`10t` `= 450`
`t` `= 45 \ text{minutes}`

Algebra, STD2 A4 EQ-Bank 8

Two friends, Sequoia and Raven, sold organic chapsticks at the the local market.

Sequoia sold her chapsticks for $4 and Raven sold hers for $3 each. In the first hour, their total combined sales were $20.

If Sequoia sold `x` chapsticks and Raven sold `y` chapsticks, then the following equation can be formed:

`4x + 3y = 20`

In the first hour, the friends sold a total of 6 chapsticks between them.

Find the number of chapsticks each of the friends sold during this time by forming a second equation and solving the simultaneous equations graphically.  (5 marks)

Show Answers Only

`text(Sequoia sold 2 and Raven sold 4.)`

Show Worked Solution
`text(Graphing)\ \ 4x + 3y` `= 20`
`y` `= −4/3x + 20/3`

 
`ytext(-intercept) = (0, 20/3)`

`xtext(-intercept)= (5, 0)`

`text(Gradient)\ = -4/3`
 

`text(S)text(econd equation:)`

`x + y = 6`

`text(From the graph,)`

`text(Sequoia sold 2 and Raven sold 4.)`

Algebra, STD2 A4 SM-Bank 3

Temperature can be measured in degrees Celsius (`C`) or degrees Fahrenheit (`F`).

The two temperature scales are related by the equation  `F = (9C)/5 + 32`.

  1. Calculate the temperature in degrees Fahrenheit when it is  −20 degrees Celsius.  (1 mark)

  2. The following two graphs are drawn on the axes below:
     
           `F = (9C)/5 + 32`  and  `F = C`
     

         

    Explain what happens at the point where the two graphs intersect.  (1 mark)

Show Answers Only
  1. `−4^@F`
  2. `text(The two graphs intersect at a temperature where)`

     

    `text(Celcius and Farenheit are the same.)`

Show Worked Solution
i.   `F` `= (9(−20))/5 + 32`
    `= −4^@F`

 

ii.   `text(The two graphs intersect at a temperature where)`

`text(Celcius and Farenheit are the same.)`