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Functions, EXT1 F1 2020 MET1 6

, where 

  1. Find  , and state its domain.  (2 marks)

The graph of  , where  , is shown on the axes below.
 

     
 

  1. On the axes above, sketch the graph of    over its domain. Label the endpoints and point(s) of intersection with  , giving their coordinates.  (2 marks)

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  1.  

  2.  
       
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a.   

 
 
 

 

 

b.     
  

Probability, 2ADV S1 2012 MET1 2

A car manufacturer is reviewing the performance of its car model X. It is known that at any given six-month service, the probability of model X requiring an oil change is , the probability of model X requiring an air filter change is and the probability of model X requiring both is .

  1. State the probability that at any given six-month service model X will require an air filter change without an oil change.  (1 mark)

  2. The car manufacturer is developing a new model. The production goals are that the probability of model Y requiring an oil change at any given six-month service will be , the probability of model Y requiring an air filter change will be and the probability of model Y requiring both will be , where .
     
    Determine in terms of if the probability of model Y requiring an air filter change without an oil change at any given six-month service is 0.05.  (2 marks)

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a.   
 
   
   

 

 

b.   

Probability, MET1 2020 VCAA 2

A car manufacturer is reviewing the performance of its car model X. It is known that at any given six-month service, the probability of model X requiring an oil change is , the probability of model X requiring an air filter change is and the probability of model X requiring both is .

  1. State the probability that at any given six-month service model X will require an air filter change without an oil change.  (1 mark)

  2. The car manufacturer is developing a new model. The production goals are that the probability of model Y requiring an oil change at any given six-month service will be , the probability of model Y requiring an air filter change will be and the probability of model Y requiring both will be , where .
  3. Determine in terms of if the probability of model Y requiring an air filter change without an oil change at any given six-month service is 0.05.  (2 marks)

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a.   
 
   
   

 

 

b.   

Complex Numbers, EXT2 N1 2004 HSC 2b

Let    and  .

  1. Find  , in the form  .   (1 mark)

  2. Express in modulus-argument form.   (3 marks)

  3. Given that has the modulus-argument form
     
         .
     
    find the modulus-argument form of  .   (1 mark)

  4. Hence find the exact value of     (1 mark)

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i.    
   
   
   

 

ii.  
 

 

iii.  
 
   

 

iv. 

 

Complex Numbers, EXT2 N2 EQ-Bank 1

  is a root of the equation  .

  1. Express    in exponential form.  (2 marks)

  2. Find the value of .  (1 mark)

  3. Find the other 4 roots of the equation in exponential form.  (3 marks)

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i.   

 

ii.   
 
   
   

 

 

iii.   


 

 

Calculus, EXT2 C1 2005 HSC 1e

Let 

  1. Show that     (1 mark)

  2. Show that     (2 marks)

  3. Use the substitution    to find     (2 marks)

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i.    
 
   
   

 

ii.   
 

 
 

 

iii.   

 
 
 

Complex Numbers, EXT2 N1 2005 HSC 2b

Let  .

  1. Express    in modulus-argument form.   (2 marks)

  2. Express    in modulus-argument form.   (2 marks)

  3. Hence express    in the form  .   (1 mark)

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i.   
 

 

 

ii.  
   
   

 

iii.  
   
   

Complex Numbers, EXT2 N1 2008 HSC 2b

  1. Write    in the form  , where and are real.  (2 marks)

  2. By expressing both    and    in  modulus-argument form, write    in modulus-argument form.   (3 marks)

  3. Hence find    in surd form.  (1 mark)

  4. By using the result of part (ii), or otherwise, calculate  .   (1 mark)

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i.     
   
   

 

ii.   


  


 


 

 
 

 

iii. 

 

 

iv.     
   
   

Vectors, EXT2 V1 2020 HSC 15b

The point divides the interval so that  . The position vectors of and are and respectively, as shown in the diagram.
 

  1. Show that  .  (2 marks)

     

  2. Prove that  .  (1 mark)

Let be a parallelogram with    and  . The point is the midpoint of and is the intersection of and , as shown in the diagram.
  
 
         
 

  1. Show that  .  (3 marks)

  2. Using parts (ii) and (iii), or otherwise, prove that is the point that divides the interval in the ratio 2 :1.   (1 mark)

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i.  

 

 

ii.   
   
   
   
   
   

 

iii. 

♦ Mean mark part (iii) 50%.
 

 


 

 
 
Mean mark part (iv) 51%.

 

iv.   

 

 

Proof, EXT2 P1 2020 HSC 15a

In the set of integers, let be the proposition:

'If    is divisible by 3, then    divisible by 3.'

  1. Prove that the proposition is true.  (2 marks)

  2. Write down the contrapositive of the proposition .  (1 mark)

  3. Write down the converse of the proposition and state, with reasons, whether this converse is true or false.  (3 marks)

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i.     

 
 
 
 

 

 

ii.    


 

♦♦ Mean mark part (iii) 36%.

iii.  


 

 

 
 
 

 

 
 
 
 

 

Complex Numbers, EXT2 N2 2020 HSC 14a

Let be a complex number and let 

The diagram shows points and which represent and , respectively, in the Argand plane.
 


 

  1. Explain why triangle    is an equilateral triangle.   (2 marks)

  2. Prove that  .  (3 marks)

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i.

 

    
 
   

 

 

ii.    
 
 
 
 
 

COMMENT: The identity to prove suggests the addition of 2 cubes is a possible strategy.

Complex Numbers, EXT2 N1 2020 HSC 13d

  1. Show that for any integer ,  .   (1 mark)

  2. By expanding    show that
     
       .   (3 marks)

  3. Hence, or otherwise, find  .   (2 marks)

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i.     
   
   

 

ii.   
   

 


 

 

iii.   
   
   
   
   

Mechanics, EXT2 M1 2020 HSC 12b

A particle is projected from the origin with initial velocity m/s at an angle    to the horizontal. The particle lands at    on the -axis. The acceleration vector is given by  , where is the acceleration due to gravity. (Do NOT prove this.)
 

  1. Show that the position vector    of the particle is given by
     
         .   (3 marks)

  2. Show that the Cartesian equation of the path of flights is given by
     
         .   (3 marks)

  3. Given  , prove that there are 2 distinct values of    for which the particle will land at  .   (2 marks)

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i.   


 

 


 


 


 

ii.   


 

 
 
 
 

 

iii.   

Mean mark part (iii) 52%.
   
 
  
 

 

 
 
 
 
 
 

 

Mechanics, EXT2 M1 2020 HSC 12a

A 50-kilogram box is initially at rest. The box is pulled along the ground with a force of 200 newtons at an angle of 30° to the horizontal. The box experiences a resistive force of    newtons, where is the normal force, as shown in the diagram.
 
Take the acceleration due to gravity to be 10m/s2.
 

  1. By resolving the forces vertically, show that  .   (2 marks)

  2. Show that the net force horizontally is approximately 53.2 newtons.  (2 marks)

  3. Find the velocity of the box after the first three seconds.  (2 marks)

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i.   

 

ii.   

 
 
 

 

iii.   
 
 

 

 
 

Proof, EXT2 P1 2020 HSC 7 MC

Consider the proposition:

'If    is not prime, then is not prime'. 

Given that each of the following statements is true, which statement disproves the proposition?

  1. is prime
  2. is divisible by 9
  3. is prime
  4. is divisible by 23
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Algebra, STD1 A3 2020 HSC 19

Each year the number of fish in a pond is three times that of the year before.

  1. The table shows the number of fish in the pond for four years.

    Complete the table above showing the number of fish in 2021 and 2022.   (2 marks)
     

  2. Plot the points from the  table in part (a) on the grid.   (2 marks)
     
  3. Which model is more suitable for this dataset: linear or exponential? Briefly explain your answer.   (2 marks)

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a.   

b.   
       

c.     The more suitable model is exponential.

A linear dataset would graph a straight line which is not the case here.

An exponential curve can be used to graph populations that grow at an increasing rate, such as this example.

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a.        

b.  

c.     The more suitable model is exponential.

A linear dataset would graph a straight line which is not the case here.

An exponential curve can be used to graph populations that grow at an increasing rate, such as this example.

♦ Mean mark (c) 31%.

Statistics, STD1 S1 2020 HSC 6 MC

When blood pressure is measured, two numbers are recorded: systolic pressure and diastolic pressure. If the measurements recorded are 140 systolic and 90 diastolic, then the blood pressure is written as 140/90 mmHg.
 
Blood pressure measurements are categorized as shown in the diagram.
 


 

Amy has a blood pressure of 97/76 mmHg and Betty has a blood pressure of 125/75 mmHg.

Which row of the table describes the blood pressure for Amy and Betty?
  

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Functions, EXT1 F2 2020 HSC 11a

Let  .

  1. Show that  .  (1 mark)

  2. Hence, factor the polynomial    as  , where    is a quadratic polynomial.  (2 marks)

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i.   
   

 

ii.   

Measurement, STD2 M6 2020 HSC 31

Mr Ali, Ms Brown and a group of students were camping at the site located at . Mr Ali walked with some of the students on a bearing of 035° for 7 km to location . Ms Brown, with the rest of the students, walked on a bearing of 100° for 9 km to location .
 


 

  1. Show that the angle is 65°.  (1 mark)

  2. Find the distance .  (2 marks)

  3. Find the bearing of Ms Brown's group from Mr Ali's group. Give your answer correct to the nearest degree.  (2 marks)

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a.   
   

 

b.   

Mean mark 53%.
 
 
 

 
c.

 

♦♦ Mean mark 22%.

 
   
 
   

 

 

Statistics, EXT1 S1 2020 HSC 12b

When a particular biased coin is tossed, the probability of obtaining a head is .

This coin is tossed 100 times.

Let be the random variable representing the number of heads obtained. This random variable will have a binomial distribution.

  1. Find the expected value, .  (1 mark)

  2. By finding the variance, , show that the standard deviation of is approximately 5.  (1 mark)

  3. By using a normal approximation, find the approximate probability that is between 55 and 65.  (1 mark)

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i.   

 
 

 

ii.   
   
   

 

 

 

iii.   
   

Functions, 2ADV F1 2020 HSC 11

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

  1.  Tank begins to lose water at a constant rate of 20 litres per minute. The volume of water in Tank is modelled by    where    is the volume in litres and    is the time in minutes from when the tank begins to lose water.   (1 mark)
     
    On the grid below, draw the graph of this model and label it as Tank .
     
       

  2. Tank remains empty until    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    when the two tanks contain the same volume of water.  (2 marks)

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

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  1.   
      
Show Worked Solution

a.      
 


 

b.     
 

   

 
c.  


  

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    where is the volume in litres and    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    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    when the two tanks contain the same volume of water.  (2 marks)

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

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  1.   
      
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a.      
 

 

b.     
 

   

 
c.  

♦♦ Mean mark part (c) 22%.


  

Financial Maths, 2ADV M1 2020 HSC 26

Tina inherits $60 000 and invests it in an account earning interest at a rate of 0.5% per month. Each month, immediately after the interest has been paid, Tina withdraws $800.

The amount in the account immediately after the th withdrawal can be determined using the recurrence relation

,

where   and 

  1. Use the recurrence relation to find the amount of money in the account immediately after the third withdrawal.  (2 marks)

  2. Calculate the amount of interest earned in the first three months.  (2 marks)

  3. Calculate the amount of money in the account immediately after the 94th withdrawal.  (3 marks) 

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a.   
 
 

 

b.   


 


 

c.   

 
 
 
 
 

Probability, 2ADV S1 2020 HSC 14

History and Geography are two of the subjects students may decide to study. For a group of 40 students, the following is known.

    • 7 students study neither History nor Geography
    • 20 students study History
    • 18 students study Geography
  1. A student is chosen at random. By a using a Venn diagram, or otherwise, find the probability that the student studies both History and Geography.  (2 marks)

  2. A students is chosen at random. Given that the student studies Geography, what is the probability that the student does NOT study History?  (1 mark)

  3. Two different students are chosen at random, one after the other. What is the probability that the first student studies History and the second student does NOT study History?  (2 marks)

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  1.  
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a.   
 

 

♦ Mean mark (b) 49%.
b.   
   

 

c.   
   

Calculus, 2ADV C3 2020 HSC 21

Hot tea is poured into a cup. The temperature of tea can be modelled by  , where is the temperature of the tea, in degrees Celsius, minutes after it is poured.

  1. What is the temperature of the tea 4 minutes after it has been poured?  (1 mark)

  2. At what rate is the tea cooling 4 minutes after it has been poured?  (2 marks)

  3. How long after the tea is poured will it take for its temperature to reach 55°C?  (3 marks)

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a.   
   
   

 

b.   
   

 

 
 

 

c.   

♦ Mean mark part (c) 44%.
 
 

Trigonometry, 2ADV T1 2020 HSC 15

Mr Ali, Ms Brown and a group of students were camping at the site located at . Mr Ali walked with some of the students on a bearing of 035° for 7 km to location . Ms Brown, with the rest of the students, walked on a bearing of 100° for 9 km to location .
 


 

  1. Show that the angle is 65°.  (1 mark)

  2. Find the distance .  (2 marks)

  3. Find the bearing of Ms Brown's group from Mr Ali's group. Give your answer correct to the nearest degree.  (2 marks)

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a.   
   

 

b.   

 
 
 

 
c.

 
   
 
   

 

 

Measurement, STD2 M6 2020 HSC 16

Consider the triangle shown.
 


 

  1. Find the value of , correct to the nearest degree.   (2 marks)

  2. Find the value of , correct to one decimal place.   (2 marks)

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a.     
 
   
   

 

b.