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Complex Numbers, EXT2 N2 2023 HSC 12e

The complex number    is a zero of the polynomial

where and are real numbers.

  1. Explain why    is also a zero of the polynomial .  (1 marks)

  2. Find the remaining zeros of the polynomial .  (2 marks)

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i.   

ii.   

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i.   

 

ii.   

 
 
 

 

 
 
 

 

 
 
 
 

 

PHYSICS, M6 2023 HSC 2 MC

Which diagram best represents the transmission of energy from a power station to people's houses?
 

 
 

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→ Energy from power stations passes through a step-up transformer to increase the voltage and decrease the current.

→ This energy then passes through a step-down transformer to be used in homes.

 

Calculus, EXT1 C3 2023 HSC 13a

A hemispherical water tank has radius cm. The tank has a hole at the bottom which allows water to drain out.

Initially the tank is empty. Water is poured into the tank at a constant rate of  cm³ s, where is a positive constant.

After seconds, the height of the water in the tank is cm, as shown in the diagram, and the volume of water in the tank is cm³.
  

It is known that      (Do NOT prove this.)

While water flows into the tank and also drains out of the bottom, the rate of change of the volume of water in the tank is given by  .

  1. Show that  .  (2 marks)

  2. Show that the tank is full of water after  seconds.  (2 marks)

  3. The instant the tank is full, water stops flowing into the tank, but it continues to drain out of the hole at the bottom as before.
  4. Show that the tank takes 3 times as long to empty as it did to fill.  (3 marks)

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i.   

 
   
   

 
ii.   

 
   
   

 

 

♦ Mean mark (ii) 41%.

iii.   


 

♦♦♦ Mean mark (iii) 20%.
 

 
 
 

 

 
 

 

 
 
   

 

Networks, STD1 N1 2023 HSC 18

A network of running tracks connects the points , as shown. The number on each edge represents the time, in minutes, that a typical runner should take to run along each track.
 

 

  1. Which path could a typical runner take to run from point to point in the shortest time?  (2 marks)

  2. A spanning tree of the network above is shown.
     

  1. Is it a minimum spanning tree? Give a reason for your answer.  (2 marks)

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a.   

b.   

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a.   
 

 
   
   

 
b.  


 


 

♦♦ Mean mark (b) 21%.

Networks, STD1 N1 2023 HSC 15

The table shows some of the flight distances (rounded to the nearest 10 km between various Australian cities.

  1. Use the information in the table to complete the network diagram where the edges are labelled with distances.  (2 marks)
     


 

  1. Mahsa wants to travel from Hobart to Darwin. She wants to change planes only once.
  2. Using the network diagram, calculate how many kilometres she will travel by plane.  (1 mark)

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a.  

b.   

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a.

b.   

Measurement, STD1 M4 2023 HSC 14

The distance-time graph shows the first two stages of a car journey from home to a holiday house.
  

  1. At what speed, in kilometres per hour, did the car travel during stage of the journey?   (1 mark)

  2. For how long did the car stop during stage of the journey?   (1 mark)

  3. After stage , the car continues to travel towards the holiday house at a constant speed of for 2 hours. Graph this part of the journey on the grid above.   (2 marks)

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a.   

b.   

c.   

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a.   
 

b.   
 

c.   


♦ Mean mark (c) 50%.

Measurement, STD1 M5 2023 HSC 12

A floor plan is shown.
 

  1. What are the dimensions, in metres, of the living room?  (2 marks)

  2. The shaded area of the kitchen floor is to be tiled. Each tile is 40 cm by 40 cm. How many tiles are needed to cover the kitchen floor?  (2 marks)

  3. The tiles are supplied in boxes of 10 . Only full boxes can be purchased. How many boxes need to be purchased to tile the kitchen floor?  (1 mark)

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a.   

b.   

c.   

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a.   


♦ Mean mark (a) 51%.

b.   

 
   
 
   
 
   

  

 
 
 

♦ Mean mark (b) 43%.

c.   

Statistics, STD1 S1 2023 HSC 11

A company employs 50 people.

The annual income of the employees is shown in the grouped frequency distribution table.

  

  1. What are the values of and ?  (2 marks)

  2. Find the mean for this distribution.  (1 mark)

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a.    ,

b.   

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a.   

  

b.   

♦♦♦ Mean mark (b) 12%.

Statistics, EXT1 S1 2023 HSC 12c

A gym has 9 pieces of equipment: 5 treadmills and 4 rowing machines.

On average, each treadmill is used 65% of the time and each rowing machine is used 40% of the time.

  1. Find an expression for the probability that, at a particular time, exactly 3 of the 5 treadmills are in use.  (2 marks)

  2. Find an expression for the probability that, at a particular time, exactly 3 of the 5 treadmills are in use and no rowing machines are in use.  (1 mark)

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i.   


 

ii. 

Proof, EXT2 P1 2023 HSC 2 MC

Consider the following statement.

'If an animal is a herbivore, then it does not eat meat.'

Which of the following is the converse of this statement?

  1. If an animal is a herbivore, then it eats meat.
  2. If an animal is not a herbivore, then it eats meat.
  3. If an animal eats meat, then it is not a herbivore.
  4. If an animal does not eat meat, then it is a herbivore.
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Statistics, 2ADV S2 2023 HSC 18

A university uses gas to heat its buildings. Over a period of 10 weekdays during winter, the gas used each day was measured in megawatts (MW) and the average outside temperature each day was recorded in degrees Celsius (°C).

Using as the average daily outside temperature and as the total daily gas usage, the equation of the least-squares regression line was found.

The equation of the regression line predicts that when the temperature is 0°C, the daily gas usage is 236 MW.

The ten temperatures measured were: 0°, 0°, 0°, 2°, 5°, 7°, 8°, 9°, 9°, 10°,

The total gas usage for the ten weekdays was 1840 MW.

In any bivariate dataset, the least-squares regression line passes through the point , where is the sample mean of the -values and is the sample mean of the -values.

  1. Using the information provided, plot the point and the -intercept of the least-squares regression line on the grid.  (3 marks)
     

 

  1. What is the equation of the regression line?  (2 marks)

  2. In the context of the dataset, identify ONE problem with using the regression line to predict gas usage when the average outside temperature is 23°C.  (1 mark)

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a.    
       

b.   

c.   

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a.   


 

 

b.   

 
 
 

 
c.   

Financial Maths, 2ADV M1 2023 HSC 15

A table of future value interest factors for an annuity of $1 is shown.
 

  1. Micky wants to save $450 000 over the next 10 years.
  2. If the interest rate is 6% per annum compounding annually, how much should Micky contribute each year? Give your answer to the nearest dollar.  (2 marks)

  3. Instead, Micky decides to contribute  $8535 every three months for 10 years to an annuity paying 6% per annum, compounding quarterly.
  4. How much will Micky have at the end of 10 years?  (3 marks)

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Show Worked Solution

a.   

 
   

 
b.   

 
   

Probability, 2ADV S1 2023 HSC 12

The table shows the probability distribution of a discrete random variable.

  1. Show that the expected value .  (1 mark)

  2. Calculate the standard deviation, correct to one decimal place.  (2 marks)

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a.    
   
   

 

b.    
   
   
   

 

 
   
   

Algebra, STD2 A4 2023 HSC 21

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

  1. Complete the table showing Provider 's monthly charges for different levels of electricity usage.  (1 mark)
     
 
 
Provider charges 35 cents per kWh, with no fixed monthly charge. The graph shows how Provider 's charges vary with the amount of electricity used in a month.
 

 
  1. On the grid above, graph Provider '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 and Provider charge the same amount.   (1 mark)

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

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a. 

b.    
         

c.   

d.   

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a. 

b. 
          
 

c.   


 

d.    

Networks, STD2 N2 2023 HSC 19

A network of running tracks connects the points , as shown. The number on each edge represents the time, in minutes, that a typical runner should take to run along each track.


  1. Which path could a typical runner take to run from point to point in the shortest time?  (2 marks)

  2. A spanning tree of the network above is shown.
  3.  
  5. Is it a minimum spanning tree? Give a reason for your answer.  (2 marks)

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a.   

b.   

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a.   
 

 
   
   

 
b.   


 


 

♦ Mean mark (b) 47%.

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.   

Networks, STD2 N2 2023 HSC 17

The table shows some of the flight distances (rounded to the nearest 10 km) between various Australian cities.
 

  1. Use the information in the table to complete the network diagram where the edges are labelled with distances.  (2 marks)
     

  1. Mahsa wants to travel from Hobart to Darwin. She wants to change planes only once.
  2. Using the network diagram, calculate how many kilometres she will travel by plane.  (1 mark)

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a.

 

b.   

Show Worked Solution

a. 

       

 

b.   

 
   
   

Measurement, STD2 M7 2023 HSC 16

The graph shows Peta's heart rate, in beats per minute, during the first 60 minutes of a marathon.
 

  1. What was Peta's heart rate 20 minutes after she started her marathon?   (1 mark)

  2. Peta started the marathon at 10 am. At what time would her heart rate first reach 140 beats/minute?   (1 mark)

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a.   

b.   

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a.   

b.   

Financial Maths, STD2 F1 2023 HSC 4 MC

A delivery truck was valued at $65 000 when new. The value of the truck depreciates at a rate of 22 cents per kilometre travelled.

What is the value of the truck after it has travelled a total distance of 132 600 km?

  1. $35 828
  2. $29 172
  3. $14 872
  4. $14 300
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Show Worked Solution
 
   

 

 
   

 

BIOLOGY, M3 2013 HSC 16 MC

The theory of evolution has been supported by studying the structures of vertebrate forelimbs from the fossil record.

This type of study is best described as

  1. biogeography.
  2. comparative biochemistry.
  3. comparative embryology.
  4. palaeontology.
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→ Palaeontology is simply, the study of fossils.

BIOLOGY, M2 2014 HSC 35a

  1. Name a scientist from the 17th or 18th century who contributed to the development of ideas on the structure and function of plants.   (1 mark)

  2. Outline how the findings from a 17th or 18th century experiment informed scientists about plant structure and/or function.   (3 marks)

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i.    Scientists could include one of:

→ Van Helmont, Hales, Priestley, Ingen-Housz, Senebier, Saussure
 

ii.   Priestly’s Experiment

→ In 1771, Joseph Priestly ignited a candle in a jar with some mint, in which he ignited the candle and then it went out. After a few trials, the candle wouldn’t ignite as all the oxygen in the air had been used (note that oxygen was not yet discovered). However after 27 days, by igniting the candle with a mirror and sunlight as to not open the jar, it reignited.

→ Similarly, by putting mice into a jar with and without a sprig of mint, the jar with the mint allowed the mouse to survive much longer.

→ These observations lead to questions about plant structure, and how they were somehow able to ‘restore air’ that is used by animals breathing and combustion reactions. 

Show Worked Solution

i.    Scientists could include one of:

→ Van Helmont, Hales, Priestley, Ingen-Housz, Senebier, Saussure
 

ii.   Priestly’s Experiment

→ In 1771, Joseph Priestly ignited a candle in a jar with some mint, in which he ignited the candle and then it went out. After a few trials, the candle wouldn’t ignite as all the oxygen in the air had been used (note that oxygen was not yet discovered). However after 27 days, by igniting the candle with a mirror and sunlight as to not open the jar, it reignited.

→ Similarly, by putting mice into a jar with and without a sprig of mint, the jar with the mint allowed the mouse to survive much longer.

→ These observations lead to questions about plant structure, and how they were somehow able to ‘restore air’ that is used by animals breathing and combustion reactions. 

BIOLOGY, M3 2015 HSC 1 MC

What is the name of the theory which describes evolution as patterns of rapid first appearances or extinctions followed by periods of little or no change?

  1. Gradualism
  2. Convergence
  3. Adaptive radiation
  4. Punctuated equilibrium
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→ Punctuated equilibrium describes when species adapt until they reach a stable or equilibrium state.

BIOLOGY, M3 2017 HSC 5 MC

Four vertebrate forelimbs are shown.
 

In which area of study do these forelimbs support the theory of evolution?

  1. Biogeography
  2. Comparative anatomy
  3. Comparative embryology
  4. Palaeontology
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→ Comparative anatomy is the practise of comparing body structures of different species.

→ In this example we see how across vertebrate species there are similar forelimb structures, suggesting divergence from a common ancestor. 

BIOLOGY, M3 2018 HSC 5 MC

Darwin and Wallace proposed the theory of evolution by natural selection.

Punctuated equilibrium differs from this theory in that punctuated equilibrium

  1. occurs in relatively short bursts of rapid change.
  2. requires genetic variation to exist in a population.
  3. is driven by selection pressures in the environment.
  4. discounts the idea that living organisms share a common ancestor.
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→ Darwin and Wallace’s theory explains how species adapt overtime to their environment.

→ Punctuated equilibrium is a seperate theory which explains how species adapt until they reach a stable stage but this can be effected by rapid evolutionary change to their environment.

CHEMISTRY, M2 2005 HSC 25a

A student collected a 500 mL sample of water from a local creek for analysis. It was filtered and the filtrate evaporated to dryness. The following data were collected.

Calculate the percentage of total dissolved solids in the creek sample.  (2 marks)

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→ The dissolved solids in the creek sample are contained in the evaporating basin and the undissolved solids are contained in the filter paper.

→ Mass of undissolved solids in filter paper =

→ Mass of dissolved solids in evaporating basin =

→ Total Mass of solids =

→ Percentage of dissolved solids = of total solids in the creek sample

CHEMISTRY, M2 2006 HSC 18

A student studying the mass change that occurs during fermentation added glucose, water and yeast to a flask and stoppered the flask with some cotton wool.

The student measured the mass of the flask daily for seven days. The table shows the data collected.

  1. Calculate the moles of released between days 1 and 7.  (1 mark)

  2. Calculate the mass of glucose that underwent fermentation between days 1 and 7. Include a balanced chemical equation in your answer.  (3 marks)

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a.    59.37 moles

b.    5395.92 g

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a.   

 

b. 

→ The moles of released and the reaction’s molar ratio can be used to calculate the mass of glucose that underwent fermentation.

→ Molar ratio between glucose and carbon dioxide =

CHEMISTRY, M2 2009 HSC 22

The nitrogen content of bread was determined using the following procedure:

    • A sample of bread weighing 2.80 g was analysed.
    • The nitrogen in the sample was converted into ammonia.
    • The ammonia was collected in 50.0 mL of 0.125 mol L hydrochloric acid. All of the ammonia was neutralised, leaving an excess of hydrochloric acid.
    • The excess hydrochloric acid was titrated with 23.30 mL of 0.116 mol L sodium hydroxide solution.
  1. Write balanced equations for the TWO reactions involving hydrochloric acid.  (2 marks)

  2. Calculate the moles of excess hydrochloric acid.  (1 mark)

  3. Calculate the moles of ammonia.   (2 marks)

  4. Calculate the percentage by mass of nitrogen in the bread.  (2 marks)

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a.   

b.   

c.   

d.   

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a.   


 

b.   
 

c.   


 

d.   

CHEMISTRY, M2 2010 HSC 31ai

A student collected a 250 mL sample of water from a local dam for analysis. The data collected are shown in the table.

The water was filtered and the filtrate evaporated to dryness.

Calculate the percentage of the total dissolved solids in the dam sample.  (2 marks)

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0.46%

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CHEMISTRY, M1 2008 HSC 3 MC

Which term describes the relationship between oxygen and ozone?

  1. Allotropes
  2. Conjugates
  3. Isomers
  4. Isotopes
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→ Allotropes are different forms of the same element as a result of different structures and bonding. 

→ As both oxygen and ozone are made up of only oxygen atoms, they are called allotropes.

CHEMISTRY, M1 2009 HSC 1 MC

Which of the following is an important factor in predicting the nuclear stability of an isotope?

  1. Atomic radius
  2. Nuclear radius
  3. The ratio of neutrons to protons
  4. The ratio of electrons to protons
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→ Nuclear stability refers to the stability of the nucleus within the atom. 

→ As the nucleus is made up of protons and neutrons, it is the ratio of these particles which are an important factor in predicting nuclear stability.

CHEMISTRY, M1 2013 HSC 6 MC

A representation of the Periodic Table is shown. The positions of six different elements, and are given.
 

Which row of the following table shows the correct acid and base reactivities of the oxides of these elements?
 

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→ Acidic Oxides: are often the oxides of non-metals and form acidic solutions

→ Basic Oxides: are usually formed by reacting oxygen with metals, they participate with acids in neutralisation rections

→ Neutral Oxides: react with neither acids or bases and do not lead to either acidic or basic solutions

→ Amphoteric Oxides: exhibit both acidic and basic properties, they can chemically react as either an acid or base