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EXAMCOPY Algebra, STD2 A4 2019 HSC 33

The time taken for a car to travel between two towns at a constant speed varies inversely with its speed.

It takes 1.5 hours for the car to travel between the two towns at a constant speed of 80 km/h.

  1. Calculate the distance between the two towns.  (1 mark)

  2. By first plotting four points, draw the curve that shows the time taken to travel between the two towns at different constant speeds.  (3 marks)

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  2.  
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a.   
   
   

 
b.   

EXAMCOPY Algebra, STD2 A4 2014 HSC 29a

The cost of hiring an open space for a music festival is  $120 000. The cost will be shared equally by the people attending the festival, so that    (in dollars) is the cost per person when    people attend the festival.

  1. Complete the table below by filling in the THREE missing values.   (1 mark)
  2. Using the values from the table, draw the graph showing the relationship between    and  .   (2 marks)
     
  3. What equation represents the relationship between and ?   (1 mark)

  4. Give ONE limitation of this equation in relation to this context.   (1 mark)

  5. Is it possible for the cost per person to be $94? Support your answer with appropriate calculations.   (1 mark)

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i.   


 

ii. 

iii.   


 

iv.  

 

 
 

v.   

 

 

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i.   


 

ii. 

 

♦ Mean mark (iii) 48%

iii.   

 

♦♦♦ Mean mark (iv) 7%
COMMENT: When asked for limitations of an equation, look carefully at potential restrictions with respect to both the domain and range.

iv.  

 

 

 

v.   

 
♦ Mean mark (v) 38%

 

EXAMCOPY Algebra, STD2 A4 2022 HSC 24

A student believes that the time it takes for an ice cube to melt ( minutes) varies inversely with the room temperature . The student observes that at a room temperature of it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating and .   (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from to    (2 marks)

 
           

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a.   
 
 
 
   

 

b.   

 

     

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a.   
 
 
 
   

 


♦♦ Mean mark part (a) 29%.

b.   

 

     


♦ Mean mark 44%.

EXAMCOPY Algebra, STD2 A4 2011 HSC 28a

The air pressure, , in a bubble varies inversely with the volume, , of the bubble. 

  1. Write an equation relating , and , where is a constant.    (1 mark)

  2. It is known that when .

     

    By finding the value of the constant, , find the value of when .    (2 marks)

  3. Sketch a graph to show how varies for different values of .

     

    Use the horizontal axis to represent volume and the vertical axis to represent air pressure.   (2 marks)


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  3.  
     
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♦ Mean mark (i) 39%
COMMENT: Expressing the proportional relationship as the equation is a core skill here.
i.
   

 

ii.

 

  

♦ Mean mark (ii) 47%
 

  

♦♦ Mean mark (iii) 26%
COMMENT: An inverse relationship is reflected by a hyperbola on the graph.
iii.

EXAMCOPY Algebra, STD2 A4 2010 HSC 13 MC

The number of hours that it takes for a block of ice to melt varies inversely with the temperature. At 30°C it takes 8 hours for a block of ice to melt.

How long will it take the same size block of ice to melt at 12°C?  

  1. 3.2 hours
  2. 20 hours
  3. 26  hours
  4. 45 hours
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♦ Mean mark 50% 

  

 

 

Statistics, STD1 S3 2023 HSC 19

The scatterplot shows the number of ice-creams sold, , at a shop over a ten-day period, and the temperature recorded at 2 pm on each of these days.
 

  1. The data are modelled by the equation of the line of best fit given below.

, where is the temperature.

  1. Sam used a particular temperature with this equation and predicted that 23 ice-creams would be sold.
  2. What was the temperature used by Sam, to the nearest degree?  (2 marks)

  3. In using the equation to make the prediction in part (a), was Sam interpolating or extrapolating? Justify your answer.  (2 marks)

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a.   

b.   

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a.             
 
 

♦♦ Mean mark (a) 31%.

b.    


♦♦ Mean mark (b) 33%.

L&E, 2ADV E1 2008 HSC 7a

Solve     (3 marks)

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IMPORTANT: Students should recognise this equation as a quadratic, and the best responses substituted with a variable such as .
   
 
MARKER’S COMMENT: Many responses incorrectly stated that there is no solution to or could not find given .

 

Measurement, STD2 M1 2009 HSC 25b

The mass of a sample of microbes is 50 mg. There are approximately   microbes in the sample.

In scientific notation, what is the approximate mass in grams of one microbe?   (2 marks)

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♦♦♦ Mean mark 20%.
IMPORTANT: Can you solve: 8 apples weigh 1kg, what does 1 apple weigh? This is exactly the same concept.

 

 
 

Quadratics, SMB-015

The diagram shows the curve with equation  . The curve intersects the -axis at points . The point on the curve has the same -coordinate as the -intercept of the curve.
 

 

  1. Find the -coordinates of points   (2 marks)

  2. Write down the coordinates of   (2 marks)

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i.   
 

 

 

ii.   


 

 

 

Algebra, STD1 A3 2019 HSC 9 MC

The container shown is initially full of water.
 

Water leaks out of the bottom of the container at a constant rate.

Which graph best shows the depth of water in the container as time varies?
 

A. B.
C. D.
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Statistics, STD1 S3 2022 HSC 23

A teacher surveyed the students in her Year 8 class to investigate the relationship between the average number of hours of phone use per day and the average number of hours of sleep per day.

The results are shown on the scatterplot below.
 

  1. The data for two new students, Alinta and Birrani, are shown in the table below. Plot their results on the scatterplot.  (2 marks)

  1. By first fitting the line of best fit by eye on the scatterplot, estimate the average number of hours of sleep per day for a student who uses the phone for an average of 2 hours per day.  (2 marks)

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  1.  
  2. 9 hours (see LOBF in diagram above)
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a.     
 

b.   

Probability, STD1 S2 2022 HSC 17

Each number from 1 to 30 is written on a separate card. The 30 cards are shuffled. A game is played where one of these cards is selected at random. Each card is equally likely to be selected.

Ezra is playing the game, and wins if the card selected shows an odd number between 20 and 30.

  1. List the numbers which would result in Ezra winning the game.  (1 mark)

  2. What is the probability that Ezra does NOT win the game?  (2 marks)

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a.   
 

b.   
   
   
   


♦ Mean mark 51%.

Functions, EXT1 F2 2022 HSC 3 MC

Let be a polynomial of degree 5. When is divided by the polynomial , the remainder is .

Which of the following is true about the degree of ?

  1. The degree must be 1.
  2. The degree could be 1.
  3. The degree must be 2.
  4. The degree could be 2.
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♦ Mean mark 51%.

Functions, 2ADV F1 2022 HSC 12

A student believes that the time it takes for an ice cube to melt ( minutes) varies inversely with the room temperature . The student observes that at a room temperature of it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating and .    (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from to .   (2 marks)
     

 
                   

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a.   

 b.    

       

 

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a.   
 
 
 
   

 

 


♦ Mean mark (a) 49%.

b.   

Algebra, STD2 A4 2022 HSC 24

A student believes that the time it takes for an ice cube to melt ( minutes) varies inversely with the room temperature . The student observes that at a room temperature of it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating and .   (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from to    (2 marks)

 
           

Show Answers Only
a.   
 
 
 
   

 

b.   

 

     

Show Worked Solution
a.   
 
 
 
   

 


♦♦ Mean mark part (a) 29%.

b.   

 

     


♦ Mean mark 44%.

Measurement, STD2 M1 2022 HSC 34

A composite solid is shown. The top section is a cylinder with a height of 3 cm and a diameter of 4 cm. The bottom section is a hemisphere with a diameter of 6 cm. The cylinder is centred on the flat surface of the hemisphere.
 


 

Find the total surface area of the composite solid in cm², correct to 1 decimal place.  (4 marks)

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♦ Mean mark 50%.

Measurement, STD2 M1 2022 HSC 28

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

Both ends of the dam, and , are isosceles triangles with equal sides of length 25 metres. The included angles and are each .

 
     

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

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³

 
   

♦♦ Mean mark 33%.

Measurement, STD2 M6 2022 HSC 26

The diagram shows two right-angled triangles, and ,

where and .
 
     

Calculate the size of angle , to the nearest minute.  (4 marks)

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♦ Mean mark 50%.

Algebra, STD2 A4 2022 HSC 22

The formula    is used to calculate the cost of producing laptops, where is the cost in dollars, is the number of laptops produced and is the fixed cost in dollars.

  1. Find the cost when 1943 laptops are produced and the fixed cost is $20 180.  (1 mark)

  2. Some laptops have some extra features added. The formula to calculate the production cost for these is
  3.      
  4. where is the additional cost in dollars per laptop produced.
  5. Find the number of laptops produced if the additional cost is $26 per laptop and the total production cost is $97 040.  (2 marks)

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a.   

 
   

 

b.   

 
 
 
 
   

Measurement, STD1 M5 2021 HSC 26

The diagrams show two similar shapes. The dimensions of the small shape are enlarged by a scale factor of 1.5 to produce the large shape.
 


 

Calculate the area of the large shape.  (3 marks)

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♦♦ Mean mark 32%.

 

Probability, STD1 S2 2021 HSC 20

In a bag, there are six playing cards, 2, 4, 6, 8, Queen and King. The Queen and King are known as picture cards.

Two of these cards are chosen randomly. All the possible outcomes are shown.
 

   
 

  1. What is the probability that the two cards chosen include one or both picture cards?  (1 mark)

  2. What is the probability that the two cards chosen do NOT include any picture cards?  (1 mark)

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a.   

 

b.   
   

Statistics, STD1 S1 2021 HSC 2 MC

A survey of which of the following would provide data that are both categorical and
nominal?

  1. Hair colour
  2. Height in centimetres
  3. Number of people present at a concert
  4. Size of coffee cup classified as small, medium or large
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♦♦ Mean mark 25%.