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Calculus, 2ADV C3 2015 HSC 13c

Consider the curve  .

  1. Find the stationary points and determine their nature.   (4 marks)

  2. Given that the point    lies on the curve, prove that there is a point of inflection at  .  (2 marks)

  3. Sketch the curve, labelling the stationary points, point of inflection and -intercept.  (2 marks)

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Quadratic, 2UA 2015 HSC 12e

The diagram shows the parabola with focus A tangent to the parabola is drawn at

  1. Find the equation of the tangent at the point .   (2 marks)
  2. What is the equation of the directrix of the parabola?   (1 mark)
  3. The tangent and directrix intersect at .
    Show that lies on the -axis.   (1 mark)

  4. Show that is isosceles.   (1 mark)
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Plane Geometry, 2UA 2006 HSC 10b

A rectangular piece of paper has sides cm and cm. The point is the midpoint of . The points and are to be chosen on and respectively, so that when the paper is folded along , the corner that was at lands on the edge at . Let cm and cm.

Copy or trace the diagram into your writing booklet.

  1. Show that .  (1 mark)
  2. Let be the point on for which is perpendicular to .
  3. By showing that , deduce that .  (3 marks)
  4. Show that the area, , of is given by
    1.   (1 mark)
    2.  
  5. Use the fact that to find the possible values of .  (2 marks)
  6. Find the minimum possible area of .  (3 marks)
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Calculus, 2ADV C4 2006 HSC 9b

During a storm, water flows into a 7000-litre tank at a rate of litres per minute, where and is the time in minutes since the storm began.

  1. At what times is the tank filling at twice the initial rate?  (2 marks)

  2. Find the volume of water that has flowed into the tank since the start of the storm as a function of .  (1 mark)

  3. Initially, the tank contains 1500 litres of water. When the storm finishes, 30 minutes after it began, the tank is overflowing.

     

    How many litres of water have been lost?  (2 marks)

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Quadratic, 2UA 2006 HSC 7c

  1. Write down the discriminant of    where    is a constant.  (1 mark)
  2. Hence, or otherwise, find the values of    for which the parabola   does not intersect the line  .  (2 marks)

 

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HSC quadratic

Calculus, EXT1* C1 2006 HSC 6b

A rare species of bird lives only on a remote island. A mathematical model predicts that the bird population, , is given by

where is the number of years after observations began.

  1. According to the model, how many birds were there when observations began?  (1 mark)

  2. According to the model, what will be the rate of change in the bird population ten years after observations began?  (2 marks)

  3. What does the model predict will be the limiting value of the bird population?  (1 mark)

  4. The species will become eligible for inclusion in the endangered species list when the population falls below . When does the model predict that this will occur?  (2 marks)

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Calculus, 2ADV C4 2006 HSC 5b

  1. Show that   (1 mark)


  2.   
     
    2006 5b
     
    The shaded region in the diagram is bounded by the curve    and the lines    and 

     

    Using the result of part (i), or otherwise, find the area of the shaded region.  (3 marks)

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Calculus, 2ADV C3 2004 HSC 4b

Consider the function  .

  1. Find the coordinates of the stationary points of the curve    and determine their nature.   (3 marks)

  2. Sketch the curve showing where it meets the axes.   (2 marks)

  3. Find the values of    for which the curve    is concave up.   (2 marks)

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    1. Geometry and Calculus, 2UA 2004 HSC 4b Answer
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 Geometry and Calculus, 2UA 2004 HSC 4b Answer

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Trigonometry, 2ADV T1 2004 HSC 3c

Trig Ratios, 2UA 2004 HSC 3c
 

The diagram shows a point    which is  30 km due west of the point  .

The point    is 12 km from    and has a bearing from    of  070°. 

  1. Find the distance of    from  .   (2 marks)

  2. Find the bearing of    from  .   (2 marks)

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Trig Ratios, 2UA 2004 HSC 3c Answer

 
 
 

 

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Probability, 2ADV S1 2006 HSC 4c

A chessboard has 32 black squares and 32 white squares. Tanya chooses three different squares at random.

  1. What is the probability that Tanya chooses three white squares?  (2 marks)

  2. What is the probability that the three squares Tanya chooses are the same colour?.  (1 mark)

  3. What is the probability that the three squares Tanya chooses are not the same colour?  (1 mark)

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Trigonometry, 2ADV T1 2006 HSC 4a

In the diagram, represents a garden. The sector    has centre and 

The points and lie on a straight line and  metres.

Copy or trace the diagram into your writing booklet.

  1. Show that    (1 mark)

  2. Find the length of  .  (2 marks)

  3. Find the area of the garden  .  (2 marks)

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Linear Functions, 2UA 2006 HSC 3a

In the diagram, are the points    respectively. The line    meets the y-axis at .

  1. Show that the equation of the line    is  .  (2 marks)
  2. Find the coordinates of the point .  (1 mark)
  3. Find the perpendicular distance of the point from the line  .  (1 mark)
  4. Hence, or otherwise, find the area of the triangle  .  (2 marks)
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Trigonometry, 2ADV T1 2005 HSC 9b

Trig Ratios, 2UA 2005 HSC 9b
 

The triangle    has a right angle at and  . The line is drawn perpendicular to . The line is then drawn perpendicular to . This process continues indefinitely as shown in the diagram.

  1. Find the length of the interval , and hence show that the length of the interval    is  .  (2 marks)

  2. Show that the limiting sum 
     

     
    is given by  .  (3 marks)

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i.   Trig Ratios, 2UA 2005 HSC 9b Answer

 

 

 

 

 

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Calculus, EXT1* C1 2005 HSC 9a

A particle is initially at rest at the origin. Its acceleration as a function of time, , is given by

  1. Show that the velocity of the particle is given by  .  (2 marks)

  2. Sketch the graph of the velocity for  π  AND determine the time at which the particle first comes to rest after  .  (3 marks)

  3. Find the distance travelled by the particle between    and the time at which the particle first comes to rest after  .  (2 marks)

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Calculus in the Physical World, 2UA 2005 HSC 9a Answer

 

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Financial Maths, 2ADV M1 2005 HSC 8c

Weelabarrabak Shire Council borrowed $3 000 000 at the beginning of 2005. The annual interest rate is 12%. Each year, interest is calculated on the balance at the beginning of the year and added to the balance owing. The debt is to be repaid by equal annual repayments of $480 000, with the first repayment being made at the end of 2005.

Let    be the balance owing after the -th repayment.

  1. Show that  .  (1 mark)

  2. Show that  .  (2 marks)

  3. In which year will Weelabarrabak Shire Council make the final repayment?  (2 marks)

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Calculus, 2ADV C3 2005 HSC 8a

2005 8a

A cylinder of radius    and height    is to be inscribed in a sphere of radius    centred at    as shown.

  1. Show that the volume of the cylinder is given by
     
          (1 mark)

  2. Hence, or otherwise, show that the cylinder has a maximum volume when   (3 marks)

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Calculus, EXT1* C1 2005 HSC 7b

Calculus in the Physical World, 2UA 2005 HSC 7b
 

The graph shows the velocity, , of a particle as a function of time. Initially the particle is at the origin. 

  1. At what time is the displacement, , from the origin a maximum?  (1 mark)

  2. At what time does the particle return to the origin? Justify your answer.  (2 marks)

  3. Draw a sketch of the acceleration,  , as a function of time for  .  (2 marks)

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iii.   Calculus in the Physical World, 2UA 2005 HSC 7b Answer

Financial Maths, 2ADV M1 2005 HSC 7a

Anne and Kay are employed by an accounting firm.

Anne accepts employment with an initial annual salary of  $50 000. In each of the following years her annual salary is increased by $2500.

Kay accepts employment with an initial annual salary of  $50 000. In each of the following years her annual salary is increased by 4%.

  1. What is Anne’s annual salary in her thirteenth year?  (2 marks)

  2. What is Kay’s annual salary in her thirteenth year?  (2 marks)

  3. By what amount does the total amount paid to Kay in her first twenty years exceed that paid to Anne in her first twenty years?  (3 marks)

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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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Financial Maths, STD2 F4 2006 HSC 27c

Kai purchased a new car for $30 000. It depreciated in value by $2000 per year for the first three years.

After the end of the third year, Kai changed the method of depreciation to the declining balance method at the rate of 25% per annum.

  1. Calculate the value of the car at the end of the third year.  (1 mark)

  2. Calculate the value of the car seven years after it was purchased.  (2 marks)

  3. Without further calculations, sketch a graph to show the value of the car over the seven years.

     

    Use the horizontal axis to represent time and the vertical axis to represent the value of the car.  (3 marks)

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Statistics, STD2 S4 2006 HSC 27b

Each member of a group of males had his height and foot length measured and recorded. The results were graphed and a line of fit drawn.
 

  1. Why does the value of the -intercept have no meaning in this situation?  (1 mark)

  2. George is 10 cm taller than his brother Harry. Use the line of fit to estimate the difference in their foot lengths.  (1 mark)

  3. Sam calculated a correlation coefficient of  −1.2  for the data. Give TWO reasons why Sam must be incorrect.  (2 marks)

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  3. `text(A correlation co-efficient must be between –1 and 1.)`

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Financial Maths, STD2 F4 2006 HSC 27a

Liliana wants to borrow money to buy a house. The bank sent her an email with the following table attached.

2006 27a

  1. Liliana decides that she can afford $1000 per month on repayments.

     

    What is the maximum amount she can borrow, and how many years will she have to repay the loan?  (1 mark)

  2. Zali intends to borrow  $160 000  over 15 years from the same bank.

     

    If she chooses to borrow  $160 000  over 20 years instead, how much more interest will she pay?  (2 marks)

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Measurement, STD2 M2 2006 HSC 26d

Cassie flew from London to Manila. Manila is 8 hours ahead of London.

Her plane left London at 9.30 am Monday (London time), stopped for 5 hours in Singapore and arrived in Manila at 4.00 pm Tuesday (Manila time).

What was the total flying time? (Ignore time zones.)  (2 marks)

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Probability, STD2 S2 2006 HSC 26c

A new test has been developed for determining whether or not people are carriers of the Gaussian virus.

Two hundred people are tested. A two-way table is being used to record the results.
 

  1.  What is the value of  ?  (1 mark)

  2.  A person selected from the tested group is a carrier of the virus.

     

     What is the probability that the test results would show this?  (2 marks) 

  3.  For how many of the people tested were their test results inaccurate?  (1 mark)

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Measurement, STD2 M6 2006 HSC 26a

Daniel conducts an offset survey to sketch a diagram, , of a block of land.

Daniel walks from to , a distance of 62 m.

When he is 15 m from , he notes that point is 25 m to his right.

When he is 57 m from , he notes that point is 20 m to his left.

This is his notebook entry.

  1. Draw a neat sketch of the block of land. Label and on your diagram.  (1 mark)

  2. Calculate the distance from to . (Give your answer to the nearest metre.)  (2 marks)

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Probability, STD2 S2 2006 HSC 25c

Sonia buys three raffle tickets.

HSC 2006 25c

  1. What is the probability that Sonia wins first prize?  (1 mark)

  2. What is the probability that she wins both prizes?  (2 marks)

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Calculus, EXT1* C3 2005 HSC 6c

2005 6c
 

The graphs of the curves    and    are shown in the diagram.

  1. Find the points of intersection of the two curves.  (1 mark)

  2. The shaded region between the curves and the -axis is rotated about the -axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed.  (3 marks)

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Calculus, 2ADV C3 2005 HSC 6b

A tank initially holds 3600 litres of water. The water drains from the bottom of the tank. The tank takes 60 minutes to empty.

A mathematical model predicts that the volume,  litres, of water that will remain in the tank after    minutes is given by
  

.
 

  1. What volume does the model predict will remain after ten minutes?  (1 mark)

  2. At what rate does the model predict that the water will drain from the tank after twenty minutes?  (2 marks)

  3. At what time does the model predict that the water will drain from the tank at its fastest rate?  (2 marks)

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Probability, 2ADV S1 2005 HSC 5d

A total of 300 tickets are sold in a raffle which has three prizes. There are 100 red, 100 green and 100 blue tickets.

At the drawing of the raffle, winning tickets are NOT replaced before the next draw.

  1. What is the probability that each of the three winning tickets is red?  (2 marks)

  2. What is the probability that at least one of the winning tickets is not red?  (1 mark)

  3. What is the probability that there is one winning ticket of each colour?  (2 marks)

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Calculus, 2ADV C3 2005 HSC 4b

A function    is defined by  .

  1. Find all solutions of   (2 marks)

  2. Find the coordinates of the turning points of the graph of  , and determine their nature.  (3 marks)

  3. Hence sketch the graph of  , showing the turning points and the points where the curve meets the -axis.  (2 marks)

  4. For what values of is the graph of    concave down?  (1 mark)

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iii.   Geometry and Calculus, 2UA 2005 HSC 4b Answer

 

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Trigonometry, 2ADV T1 2005 HSC 4a

 Trig Calculus, 2UA 2005 HSC 4a
 

A pendulum is 90 cm long and swings through an angle of 0.6 radians. The extreme positions of the pendulum are indicated by the points and in the diagram.

  1. Find the length of the arc  .  (1 mark)

  2. Find the straight-line distance between the extreme positions of the pendulum.  (2 marks)

  3. Find the area of the sector swept out by the pendulum.  (1 mark)

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 Trig Calculus, 2UA 2005 HSC 4a Answer

 
 

 

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Plane Geometry, 2UA 2005 HSC 3c

2005 3c

In the diagram, ,  and    are the points    and   respectively. The equation of the line    is  . The point    on    is chosen so that    is parallel to  . The point    on    is chosen so that    is parallel to the -axis.

  1.  Show that the equation of the line is .  (2 marks)
  2. Find the coordinates of the point .  (2 marks)
  3. Find the coordinates of the point .  (1 marks)
  4. Prove that  .  (2 marks)
  5. Hence, or otherwise, find the ratio of the lengths and .  (1 marks)
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Trigonometry, 2ADV T1 2005 HSC 3b

The lengths of the sides of a triangle are 7 cm, 8 cm and 13 cm.

  1. Find the size of the angle opposite the longest side.  (2 marks)

  2. Find the area of the triangle.  (1 marks)

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 Trig Ratios, 2UA 2005 HSC 3b Answer1 

 

 

 

 

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Financial Maths, 2ADV M1 2006 HSC 8b

Joe borrows $200 000 which is to be repaid in equal monthly instalments. The interest rate is 7.2% per annum reducible, calculated monthly.

It can be shown that the amount, , owing after the th repayment is given by the formula:

,

where    and    is the monthly repayment.  (Do NOT show this.)

  1. The minimum monthly repayment is the amount required to repay the loan in 300 instalments.

     

    Find the minimum monthly repayment.  (3 marks)

  2. Joe decides to make repayments of $2800 each month from the start of the loan.

     

    How many months will it take for Joe to repay the loan?  (2 marks)

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Calculus, 2ADV C1 2006 HSC 8a

A particle is moving in a straight line. Its displacement, metres, from the origin, , at time seconds, where  , is given by  .

  1. Find the initial displacement of the particle.  (1 mark)

  2. Find the velocity of the particle as it passes through the origin.  (3 marks)

  3. Show that the acceleration of the particle is always negative.  (1 mark)

  4. Sketch the graph of the displacement of the particle as a function of time.  (2 marks)

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(iv)  2UA HSC 2006 8a

Trigonometry, 2ADV T3 2006 HSC 7b

A function    is defined by  .

  1. Show that the graph of    cuts the -axis at  .   (1 mark)

  2. Sketch the graph of   for    showing where the graph cuts each of the axes.   (3 marks)

  3. Find the area under the curve   between   and  .   (3 marks)

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ii.   2UA HSC 2006 7b

 

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Calculus, 2ADV C3 2006 HSC 5a

A function    is defined by  .

  1. Find the coordinates of the turning points of    and determine their nature.  ( 3 marks)

  2. Find the coordinates of the point of inflection.  (1 mark)

  3. Hence sketch the graph of  , showing the turning points, the point of inflection and the points where the curve meets the -axis.  (3 marks)

  4. What is the minimum value of    for  ?  (1 mark)

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2UA HSC 2006 5a

 

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Probability, STD2 S2 2006 HSC 28a

On a bridge, the toll of $2.50 is paid in coins collected by a machine. The machine only accepts two-dollar coins, one-dollar coins and fifty-cent coins.

  1. List the different combinations of coins that could be used to pay the $2.50 toll.  (1 mark)

  2. Jill has three two-dollar coins, six one-dollar coins and two fifty-cent coins. She selects two coins at random.

     

    What is the probability that she selects exactly $2.50?  (3 marks)

  3. At the end of a day, the machine contains two-dollar coins, one-dollar coins and fifty-cent coins.

     

    Write an expression for the total value of coins in dollars in the machine.  (1 mark)

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Algebra, STD2 A4 2005 HSC 28b

Sue and Mikey are planning a fund-raising dance. They can hire a hall for $400 and a band for $300. Refreshments will cost them $12 per person.

  1. Write a formula for the cost ($C) of running the dance for people. (1 mark)

The graph shows planned income and costs when the ticket price is $20 

2005 28b

  1. Estimate the minimum number of people needed at the dance to cover the costs.  (1 mark)

  2. How much profit will be made if 150 people attend the dance? (1 mark)

Sue and Mikey plan to sell 200 tickets. They want to make a profit of $1500.

  1. What should be the price of a ticket, assuming all 200 tickets will be sold?  (3 marks)

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Measurement, STD2 M1 2005 HSC 28a

The Mitchell family has moved to a new house which has an empty swimming pool. The base of the pool is in the shape of a rectangle, with a semicircle on each end.

2005 28a1

  1. Explain why the expression for the area of the base of the pool is  π.   (1 mark) 

      
            2005 28a2
      

The pool is 1.1 metres deep.

  1. The sides and base of the pool are covered in tiles. If    and  , find the total area covered by tiles. (Give your answer correct to the nearest square metre.)   (4 marks)

Before filling the pool, the Mitchells need to install a new shower head, which saves 6 litres of water per minute.

The shower is used 5 times every day, for 3 minutes each time.

  1. If the charge for water is $1.013 per kilolitre, how much money would be saved in one year by using this shower head? (Assume there are 365 days in a year.)   (2 marks)

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