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Algebra, STD2 A4 2024 HSC 26

A sheet of metal is folded to make a gutter, as shown. The cross-section of the gutter is a rectangle of width cm and height cm.
 

 

The area, cm, of the cross-section can be modelled by the quadratic formula

A graph of this model is shown.
 

Find the width and height of the rectangle which will give the greatest possible area of the cross-section.   (3 marks)

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♦ Mean mark 43%.

Algebra, STD2 A4 2023 HSC 22

The braking distance of a car, in metres, is directly proportional to the square of its speed in km/h, and can be represented by the equation

where is the constant of variation.

The braking distance for a car travelling at 50 km/h is 20 m.

  1. Find the value of .  (2 marks)

  2. What is the braking distance when the speed of the car is 90 km/h?  (1 mark)

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a.   

b.   

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a. 

 
 
   

 
b.   

 
   

Algebra, STD2 A4 2023 HSC 20

On another planet, a ball is launched vertically into the air from the ground.

The height above the ground, metres, can be modelled using the function  , where is measured in seconds. The graph of the function is shown.

  1. Based on the graph, what is the maximum height reached by the ball?  (1 mark)

  2. Based on the graph, at what TWO times is the ball at of its maximum height?  (2 marks)

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a.   

b.   

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a.   

b.   

Algebra, STD2 A4 2021 HSC 35

A publisher sells a book for $10. At this price, 5000 copies of the book will be sold and the revenue raised will be  .

The publisher is considering increasing the price of the book. For every dollar the price of the book is increased, the publisher will sell 50 fewer copies of the book.

If the publisher charges dollars for each book, a quadratic model for the revenue raised, , from selling books is

 


 

  1. By first finding a suitable value of , find the price the publisher should charge for each book to maximise the revenues raised from sales of the book.  (2 marks)

  2. Find the value of the intercept of the parabola with the vertical axis.  (1 mark) 

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a.   

♦♦♦ Mean mark part (a) 16%.


 

♦♦♦ Mean mark part (b) 21%.

b.  

 

Algebra, STD2 A4 2020 HSC 19

A fence is to be built around the outside of a rectangular paddock. An internal fence is also to be built.

The side lengths of the paddock are metres and metres, as shown in the diagram.
 

 
A total of 900 metres of fencing is to be used. Therefore  .
 
The area, , in square metres, of the rectangular paddock is given by  .

The graph of this equation is shown.
  

  1. If the area of the paddock is , what is the largest possible value of ?   (1 mark)

  2. Find the values of and so that the area of the paddock is as large as possible.   (2 marks)

  3. Using your value from part (b), find the largest possible area of the paddock.   (1 mark)

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a.     

♦ Mean mark part (a) 39%.

 


 

b.   

♦♦ Mean mark part (b) 34%.

 

♦ Mean mark part (c) 40%.
c.   
   
   

Algebra, STD2 A4 2019 HSC 31

A rectangle has width centimetres. The area of the rectangle, , in square centimetres, is  .

The graph    is shown.
 


 

  1. Explain why, in this context, the model    only makes sense for the bold section of the graph.  (1 mark)

  2. The area of the rectangle is 18 cm². Calculate the perimeter of the rectangle.  (2 marks)

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a.   

♦ Mean mark 44%.

 

b.   

♦ Mean mark 42%.

 

 

Algebra, STD2 A4 SM-Bank 2

Moses finds that for a Froghead eel, its mass is directly proportional to the square of its length.

An eel of this species has a length of 72 cm and a mass of 8250 grams.

What is the expected length of a Froghead eel with a mass of 10.2 kg? Give your answer to one decimal place.  (3 marks)

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Algebra, STD2 A4 2017 HSC 28e

A movie theatre has 200 seats. Each ticket currently costs $8.

The theatre owners are currently selling all 200 tickets for each session. They decide to increase the price of tickets to see if they can increase the income earned from each movie session.

It is assumed that for each one dollar increase in ticket price, there will be 10 fewer tickets sold.

A graph showing the relationship between an increase in ticket price and the income is shown below.
 


 

  1. What ticket price should be charged to maximise the income from a movie session?  (1 mark)

  2. What is the number of tickets sold when the income is maximised?  (1 mark)

  3. The cost to the theatre owners of running each session is $500 plus $2 per ticket sold.

     

    Calculate the profit earned by the theatre owners when the income earned from a session is maximised.  (2 marks)

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i.   

♦ Mean mark 50%.
 

 

ii.   

♦ Mean mark 45%.
 

 

 
 

 

iii.  
   

 

Algebra, STD2 A4 2015 HSC 29e

A diver springs upwards from a diving board, then plunges into the water. The diver’s height above the water as it varies with time is modelled by a quadratic function. Graphing software is used to produce the graph of this function.
 

Explain how the graph could be used to determine how high above the height of the diving board the diver was when he reached the maximum height.  (2 marks)

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♦♦ Mean mark 19%.


 

Algebra, STD2 A4 2006 HSC 28b

A new tunnel is built. When there is no toll to use the tunnel, 6000 vehicles use it each day. For each dollar increase in the toll, 500 fewer vehicles use the tunnel.

  1. Find the lowest toll for which no vehicles will use the tunnel.  (1 mark)

  2. For a toll of $5.00, how many vehicles use the tunnel each day and what is the total daily income from tolls?  (2 marks)

  3. If (dollars) represents the value of the toll, find an equation for the number of vehicles using the tunnel each day in terms of .  (2 marks)

  4. Anne says ‘A higher toll always means a higher total daily income’.

     

    Show that Anne is incorrect and find the maximum daily income from tolls. (Use a table of values, or a graph, or suitable calculations.)  (3 marks)

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  1.  

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i. 

 

ii. 

 

 

iii.   
   

 

iv. 


 

 

 

Algebra, STD2 A4 2009 HSC 28c

The height above the ground, in metres, of a person’s eyes varies directly with the square of the distance, in kilometres, that the person can see to the horizon.

A person whose eyes are 1.6 m above the ground can see 4.5 km out to sea.

How high above the ground, in metres, would a person’s eyes need to be to see an island that is 15 km out to sea? Give your answer correct to one decimal place.   (3 marks)

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♦♦ Mean mark 22%
CRITICAL STEP: Reading the first line of the question carefully and establishing the relationship is the key part of solving this question.

 

 

 
 

Algebra, STD2 A4 2009 HSC 28a

Anjali is investigating stopping distances for a car travelling at different speeds. To model this she uses the equation
 

,
 

where is the stopping distance in metres and is the car’s speed in km/h.

The graph of this equation is drawn below.

2009 28a

  1. Anjali knows that only part of this curve applies to her model for stopping distances.

     

    In your writing booklet, using a set of axes, sketch the part of this curve that applies for stopping distances.  (1 mark)

  2. What is the difference between the stopping distances in a school zone when travelling at a speed of 40 km/h and when travelling at a speed of 70 km/h?   (2 marks)

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  1.  
    2UG-2009-28a-Answer
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(i)

2UG-2009-28a-Answer

(ii)

♦ Mean mark 41%
COMMENT: Students could easily have used the graph for calculating at , although the formula was required when the speed increased to .
 
 

 

 
 

 

 

Algebra, STD2 A4 2012 HSC 30b

A golf ball is hit from point to point , which is on the ground as shown. Point is 30 metres above the ground and the horizontal distance from point to point is  300 m.
 

The path of the golf ball is modelled using the equation 

 

where 

is the height of the golf ball above the ground in metres, and 

is the horizontal distance of the golf ball from point in metres.

The graph of this equation is drawn below.

  

  1. What is the maximum height the ball reaches above the ground?    (1 mark)

  2. There are two occasions when the golf ball is at a height of 35 metres.

     

    What horizontal distance does the ball travel in the period between these two occasions?   (1 mark)

  3. What is the height of the ball above the ground when it still has to travel a horizontal distance of 50 metres to hit the ground at point ?   (1 mark)

  4. Only part of the graph applies to this model.

     

    Find all values of that are not suitable to use with this model, and explain why these values are not suitable.   (2 marks)

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i.   

COMMENT: With a mean mark of 92% in (i), a classic example of low hanging fruit in later questions.

 

ii.  

 

 

iii.   

MARKER’S COMMENT: Responses for (iii) in the range    were deemed acceptable estimates read off the graph.

 

iv.   

♦♦♦ Mean mark (iv) 12%
MARKER’S COMMENT: Many students did not refer to the domain as unsuitable to the model.

Algebra, STD2 A4 2013 HSC 22 MC

Leanne wants to build a rectangular vegetable garden in her backyard. She has 20 metres of fencing and will use a wall as one side of the garden. The plan for her garden is shown, where    metres is the width of her garden.
 

 

Which equation gives the area,  ,  of the vegetable garden?

  1.    
  2.    
  3.    
  4.    
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♦♦♦ Mean mark 24% (lowest mean of any MC question in 2013 exam)