SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, 2ADV C3 2015 HSC 16c

The diagram shows a cylinder of radius and height inscribed in a cone of radius and height , where and are constants.
  

The volume of a cone of radius and height is  

The volume of a cylinder of radius and height is  

  1. Show that the volume, , of the cylinder can be written as
     
            (3 marks)

  2. By considering the inscribed cylinder of maximum volume, show that the volume of any inscribed cylinder does not exceed of the volume of the cone.   (4 marks)

Show Answers Only
Show Worked Solution

i.   
 

♦♦ Mean mark (i) 16%.

 

 
 

 


 

♦♦ Mean mark (ii) 21%.

ii.   

 
 

 

 


 

 

 

 

 

Plane Geometry, 2UA 2015 HSC 15b

The diagram shows which has a right angle at . The point is the midpoint of the side . The point is chosen on such that . The line segment is produced to meet the line at .

Copy or trace the diagram into your writing booklet.

  1. Prove that is similar to   (2 marks)
  2. Explain why is isosceles.  (1 mark)
  3. Show that   (2 marks)
Show Answers Only
Show Worked Solution

(i) 

 

♦ Mean mark 45%.

(ii) 

 

(iii) 

♦♦ Mean mark 23%.
 
 
 
 

 

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)
Show Answers Only
  5.  
  6. ²
Show Worked Solution
(i)     

 

 
 
 

 

(ii) 

 

   
 
 

 

(iii) 
 
 

 

(iv) 

.

 

 
 
 
 

.

 

(v) 

 
 
 
 
 

  ²

 

  ²

 

²

Calculus, 2ADV C3 2006 HSC 9c

A cone is inscribed in a sphere of radius , centred at . The height of the cone is and the radius of the base is , as shown in the diagram.

  1. Show that the volume, , of the cone is given by
     
         .  (2 marks)

  2. Find the value of for which the volume of the cone is a maximum. You must give reasons why your value of gives the maximum volume.  (3 marks)

Show Answers Only
Show Worked Solution

i. 

 
 
 

 

ii. 

 

 

 

 

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)

Show Answers Only
Show Worked Solution

i. 

 

ii. 
 
 

 

 

iii. 


 

 

 

 

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)

Show Answers Only
Show Worked Solution
i.   Trig Ratios, 2UA 2005 HSC 9b Answer

 

 

 

 

 

ii.


 

 

 

 
 
 
 

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)

Show Answers Only
Show Worked Solution

i.   

 

ii. 

 

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

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)

Show Answers Only

  1.  

Show Worked Solution

i. 

 

ii. 

 

 

iii.   
   

 

iv. 


 

 

 

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)

Show Answers Only


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

Show Worked Solution

i. 

 

ii. 


 

 

iii. 

Statistics, STD2 S1 2007 HSC 22 MC

A set of examination results is displayed in a cumulative frequency histogram and polygon (ogive).
 

2007 22 mc
 
Sanath knows that his examination mark is in the 4th decile.

Which of the following could have been Sanath’s examination mark?

  1.    37
  2.    57
  3.    67
  4.    77
Show Answers Only

Show Worked Solution

Data, 2UG 2007 HSC 22 MC Answer

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)

Show Answers Only
Show Worked Solution

i. 

 

ii. 

 
 

 

iii. 

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)

Show Answers Only
Show Worked Solution
i.   
   
   
   

 

ii.  
    ²

 

 
 
 

 

 

 

²²

 

iii. 
   

 

 
 

Statistics, STD2 S1 2005 HSC 27d

Nine students were selected at random from a school, and their ages were recorded.

  1. What is the sample standard deviation, correct to two decimal places?   (2 marks)

  2. Briefly explain what is meant by the term standard deviation.   (1 mark)

Show Answers Only

  2.  

     

Show Worked Solution

i. 

 

ii. 

Algebra, 2UG 2004 HSC 28b

2004 28b

A set of garden gnomes is made so that the cost () varies directly with the cube of the base length ( centimetres). A gnome with a base length of has a cost of .

  1. Write an equation relating the variables and , and a constant .  (1 mark)
  2. Find the value of .  (1 mark)
  3. Felicity says, ‘If you double the base length, you double the cost.’ Is she correct? Justify your answer with mathematical calculations.  (2 marks)

 

Show Answers Only



 

Show Worked Solution
(i) 
 

 

(ii) 

 

 

(iii) 

 

 

Algebra, STD2 A4 2004 HSC 28a

A health rating, , is calculated by dividing a person’s weight, , in kilograms by the square of the person’s height, , in metres.

  1. Fred is 150 cm and weighs 72 kg. Calculate Fred’s health rating.  (1 mark)

  2. Over several years, Fred expects to grow 10 cm taller. By this time he wants his health rating to be 25. How much weight should he gain or lose to achieve his aim? Justify your answer with mathematical calculations.  (2 marks)

Show Answers Only
Show Worked Solution

i. 

 

 

ii. 

 

 

Financial Maths, STD2 F1 2006 HSC 22 MC

This income tax table is used to calculate Evelyn’s tax payable.
 

Evelyn’s taxable income increases from $50 000 to $80 000.

What percentage of her increase will she pay in additional tax?

Show Answers Only

Show Worked Solution
 
 
 
 
 

 


 

Statistics, STD2 S1 2005 HSC 22 MC

Two groups of people were surveyed about their weekly wages. The results are shown in the box-and-whisker plots.
 

Which of the following statements is true for the people surveyed?

  1. The same percentage of people in each group earned more than $325 per week.
  2. Approximately 75% of people under 21 years earned less than $350 per week.
  3. Approximately 75% of people 21 years and older earned more than $350 per week.
  4. Approximately 50% of people in each group earned between $325 and $350 per week.
Show Answers Only

Show Worked Solution


 


 

Probability, STD2 S2 2005 HSC 20 MC

Dave’s school has computer security codes made up of four digits (eg 0773). Juanita’s school has computer security codes made up of five digits (eg. 30 568).

How many more codes are available at Juanita’s school than at Dave’s school?

  1.    10
  2.    50
  3.    90 000
  4.    100 000
Show Answers Only

Show Worked Solution

 

Financial Maths, 2UG 2006 HSC 16 MC

Two families borrow different amounts of money on the same day.

The Wang family has a flat rate loan. The Salama family has a reducing balance loan and repays the loan earlier than the Wang family.

Which graph best represents this situation?

 

2UG-2006-16abMC

2UG-2006-16cdMC

Show Answers Only

Show Worked Solution

CORE*, FUR1 2008 VCAA 9 MC

An amount of $8000 is invested for a period of 4 years.

The interest rate for this investment is 7.2% per annum compounding quarterly.

The interest earned by the investment in the fourth year (in dollars) is given by

A.  

B.   

C.   

D.  

E.  

Show Answers Only

 

Show Worked Solution
 

♦♦ Mean mark 33%.
MARKERS’ COMMENT: The most common errors were not converting and applying the interest rate to a quarterly one.


 

 

 

 

 

CORE*, FUR1 2008 VCAA 7 MC

Ernie took out a reducing balance loan to buy a new family home.

He correctly graphed the amount paid off the principal of his loan each year for the first five years.

The shape of this graph (for the first five years of the loan) is best represented by
 

 

 

Show Answers Only

 

Show Worked Solution

♦♦ Mean mark 25%.

CORE*, FUR1 2007 VCAA 7 MC

At the start of each year Joe's salary increases to take inflation into account.

Inflation averaged 2% per annum last year and 3% per annum the year before that.

Joe's salary this year is $42 000.

Joe's salary two years ago, correct to the nearest dollar, would have been

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

♦♦ Mean mark 23%.
MARKERS’ COMMENT: Depreciating this year’s salary of $42 000 by 3% and then 2% to get ($42 000 x 0.97 x 0.98) is an incorrect strategy used by almost half of students. Make sure you understand why!
 

 

 

 

CORE*, FUR1 2006 VCAA 9 MC

Jenny borrowed $18 000. She will fully repay the loan in five years with equal monthly payments.

Interest is charged at the rate of 9.2% per annum, calculated monthly, on the reducing balance.

The amount Jenny has paid off the principal immediately following the tenth repayment is

A.   $1876.77

B.   $2457.60

C.   $3276.00

D.   $3600.44

E.   $3754.00

Show Answers Only

Show Worked Solution

♦♦ Mean mark 27%.

 

 

 

 

 

CORE*, FUR1 2006 VCAA 3 MC

Grandpa invested in an ordinary perpetuity from which he receives a monthly pension of $584.

The interest rate for the investment is 6.2% per annum.

The amount Grandpa has invested in the perpetuity is closest to

A.        $3600

B.        $9420

C.     $94 200

D.     $43 400

E.   $113 000

Show Answers Only

Show Worked Solution

♦♦ Mean mark 23%.
MARKERS’ COMMENT: Nearly half of students read the monthly pension amount as a yearly amount. Be careful!
 

 

CORE*, FUR1 2012 VCAA 9 MC

Peter took out a reducing balance loan where interest was calculated monthly. He planned to repay this loan fully, with eight equal monthly payments of $260.

Peter missed the fourth payment, but made a double payment of $520 in the fifth month. He then continued to make payments of $260 for the remaining three months.

Which graph could show the balance of the loan each month over the eight-month period?

 

Show Answers Only

Show Worked Solution

♦♦ Mean mark 20%.

CORE*, FUR1 2013 VCAA 8 MC

A car is purchased for $25 000. The value of the car is to be depreciated each year by 20% using the reducing balance method.

In the fourth year, the car will depreciate in value by

A.      $2048

B.      $2560

C.      $5000

D.   $10 240

E.   $14 760

Show Answers Only

Show Worked Solution

♦♦ Mean mark 29%.
MARKERS’ COMMENT: A common mistake was to not read the question carefully and give the depreciated value of the car after 4 years.


 


 

CORE*, FUR1 2014 VCAA 9 MC

Leslie borrowed $35 000 from a bank.

Interest is charged at the rate of 4.75% on the reducing monthly balance.

The loan is to be repaid with 47 monthly payments of $802.00 and a final payment that is to be adjusted so that the loan will be fully repaid after exactly 48 monthly payments.

Correct to the nearest cent, the amount of the final payment will be

A.       $0.39

B.       $3.57

C.   $802.00

D.   $802.39

E.   $805.57

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 18%.
MARKERS’ COMMENT: A majority of students found the amount still owed after the second last payment but failed to add interest during the last month.

 

 

GRAPHS, FUR1 2007 VCAA 6 MC

Russell is a wine producer. He makes both red and white wine.

Let represent the number of bottles of red wine he makes and represent the number of bottles of white wine he makes.

This year he plans to make at least twice as many bottles of red wine as white wine.

An inequality representing this situation is

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

♦♦ Mean mark 34%.

GRAPHS, FUR1 2008 VCAA 4 MC

When shopping, Betty can use either Easypark or Safepark to park her car.

At Easypark, cars can be parked for up to 8 hours per day.

The fee structure is as follows.

Safepark charges fees according to the formula

Betty wants to park her car for 5 hours on Monday and 3 hours on Tuesday.

The minimum total fee that she can pay for parking for the two days is

A.     

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

♦♦ Mean mark 29%.
MARKER’S COMMENT: Almost half of students failed to take into account that the same car park did not have to be used on each day, and incorrectly answered D.

GRAPHS, FUR1 2009 VCAA 9 MC

The graph below shows the cost (in dollars) of producing birthday cards.

GRAPHS, FUR1 2009 VCAA 9 MC

If the profit from the sale of 150 birthday cards is $175, the selling price of one card is

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

♦ Mean mark 37%.

 

 

 
 
 

GEOMETRY, FUR1 2006 VCAA 9 MC

Points and are the same distance from a third point .

The bearing of from is 038° and the bearing of from is 152°.

The bearing of from is

A.   between 000° and 090°

B.   between 090° and 180°

C.   exactly 180°

D.   between 180° and 270°

E.   between 270° and 360°

Show Answers Only

Show Worked Solution
♦♦ Mean mark 31%.

GEOMETRY, FUR1 2006 VCAA 9 MC Answer 

 
 

 

 

 

GEOMETRY, FUR1 2006 VCAA 8 MC

GEOMETRY, FUR1 2006 VCAA 8 MC

The cross-section of a water pipe is circular with a radius, , of 50 cm, as shown above.

The surface of the water has a width, , of 80 cm.

The depth of water in the pipe, , could be

A.   

B.    

C.   

D.   

E.    

Show Answers Only

Show Worked Solution
♦♦ Mean mark 22%.
MARKER’S COMMENT: 39% of students calculated in the solution, without then using it to find the depth.

 GEOMETRY, FUR1 2006 VCAA 8 MC Answer

 
   

Measurement, 2UG 2007 HSC 28c

A piece of plaster has a uniform cross-section, which has been shaded, and has dimensions as shown.
 

 
 

  1. Use two applications of Simpson’s rule to approximate the area of the cross-section.    (3 marks)
  2. The total surface area of the piece of plaster is  ².
  3. Calculate the area of the curved surface as shown on the diagram.   (2 marks)

 

Show Answers Only
  1. ²
  2. ²
Show Worked Solution
(i)   
 
 
  ²

 

(ii)  ²

  ²
  ²
  ²

 

²

Algebra, STD2 A1 2007 HSC 28b

This shape is made up of a right-angled triangle and a regular hexagon.
 

 

The area of a regular hexagon can be estimated using the formula    where    is the side-length.

Calculate the total area of the shape using this formula.   (3 marks)

Show Answers Only

²

Show Worked Solution

 
  ²
 
  ²

 

  ²

Probability, 2UG 2007 HSC 28a

Two unbiased dice,    and  , with faces numbered  ,  ,  ,  ,    and    are rolled.

The numbers on the uppermost faces are noted. This table shows all the possible outcomes.

 

A game is played where the difference between the highest number showing and the lowest number showing on the uppermost faces is calculated.

  1. What is the probability that the difference between the numbers showing on the uppermost faces of the two dice is one?   (1 mark)
  2. In the game, the following applies. 
  3. What is the financial expectation of the game?   (3 marks)
  4. If Jack pays    to play the game, does he expect a gain or a loss, and how much will it be?  (1 mark)

 

Show Answers Only
Show Worked Solution

(i)   

 

(ii) 

 
 
 

 

(iii) 

GEOMETRY, FUR1 2007 VCAA 6 MC

A solid cylinder has a height of 30 cm and a diameter of 40 cm.

A hemisphere is cut out of the top of the cylinder as shown below.

GEOMETRY, FUR1 2007 VCAA 6 MC

In square centimetres, the total surface area of the remaining solid (including its base) is closest to

A.     

B.     

C.     

D.     

E. 

Show Answers Only

Show Worked Solution

♦♦ Mean mark 28%.
 
 
 
 
 
 
  ²

GEOMETRY, FUR1 2007 VCAA 5 MC

A block of land has an area of 4000 m².

When represented on a map, this block of land has an area of 10 cm².

On the map 1 cm would represent an actual distance of

A.       

B.      

C.       

D.    

E.   

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 18%.
MARKER’S COMMENT: A majority of students obtained the the “area” scale factor (k² = 400) but failed to convert this to the corresponding linear scale factor.
 
 

 

GEOMETRY, FUR1 2011 VCAA 6 MC

In parallelogram , 

In this parallelogram,  and

The length of the longer diagonal of this parallelogram is closest to

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution
♦♦ Mean mark 34%.

geometry 2011 VCAA 6mci

 

 
 

GEOMETRY, FUR1 2009 VCAA 9 MC

GEOMETRY, FUR1 2009 VCAA 9 MC

 

A vertical pole, , is  4 metres tall and stands on level ground near a vertical wall.

The wall is 6 metres long and 4 metres high.

The base of the pole, , is  5 metres from one end of the wall at and 4 metres from the other end of the wall at .

The pole falls and hits the wall.

The maximum height above ground level at which the pole could hit the wall is closest to

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution
♦♦ Mean mark 27%.
MARKER’S COMMENT: The key concept to finding the solution is to realise that max height occurs when the pole falls at right angles to the wall.

GEOMETRY, FUR1 2009 VCAA 9 MC Answer1 

 

 

 

 

GEOMETRY, FUR1 2009 VCAA 9 MC Answer2  

 

PATTERNS, FUR1 2009 VCAA 9 MC

There are 10 checkpoints in a 4500 metre orienteering course.

Checkpoint 1 is the start and checkpoint 10 is the finish.

The distance between successive checkpoints increases by 50 metres as each checkpoint is passed.

The distance, in metres, between checkpoint 2 and checkpoint 3 is

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

♦♦ Mean mark 26%.

GEOMETRY, FUR1 2014 VCAA 9 MC

The middle section of a cone is shaded, as shown in the diagram below.

The surface area of the unshaded top section of the cone is 180 cm².

The surface area of the middle section of the cone, in square centimetres, is

A.     

B.   

C.   

D.   

E.    

Show Answers Only

Show Worked Solution

♦♦ Mean mark 33%.

 

 

 
  ²

 

²

GEOMETRY, FUR1 2012 VCAA 9 MC

The solid  , as shown below, is one-eighth of a sphere of radius 15 cm.

The point is the centre of the sphere and the points  and   are on the surface of the sphere.


 

 The total surface area of the solid  , in cm², is closest to

A.    619

B.    648

C.    706

D.    884

E.  1767

Show Answers Only

Show Worked Solution

♦♦ Mean mark 31%.

²

 
 
²
 
 
²

 

CORE, FUR1 2014 VCAA 12 MC

The seasonal index for heaters in winter is 1.25.

To correct for seasonality, the actual heater sales in winter should be

A.   reduced by 20%. 

B.   increased by 20%. 

C.   reduced by 25%.

D.   increased by 25%.

E.   reduced by 75%.

Show Answers Only

Show Worked Solution

♦♦ Mean mark 26%.
MARKER’S COMMENT: A majority of students appeared to answer this question by inspection only which was not possible. Don’t be tricked!

CORE, FUR1 2011 VCAA 12 MC

The seasonal index for headache tablet sales in summer is 0.80.

To correct for seasonality, the headache tablet sales figures for summer should be

A.   reduced by 80%

B.   reduced by 25%

C.   reduced by 20%

D.   increased by 20%

E.   increased by 25%

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 10%!
MARKERS’ COMMENT: The key to solving this problem is to rearrange the “Seasonal Index” formula in the formula sheet.

 

PATTERNS, FUR1 2010 VCAA 8 MC

The th term in a geometric sequence is  .

The common ratio is greater than one.

A graph that could be used to display the terms of this sequence is

 

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 16%!

 

CORE, FUR1 2014 VCAA 8 MC

A single back-to-back stem plot would be an appropriate graphical tool to investigate the association between a car’s speed, in kilometres per hour, and the

A.   driver’s age, in years. 

B.   car’s colour (white, red, grey, other). 

C.   car’s fuel consumption, in kilometres per litre.

D.   average distance travelled, in kilometres.

E.   driver’s sex (female, male).

Show Answers Only

Show Worked Solution

♦♦ Mean mark 28%.

CORE, FUR1 2014 VCAA 3-5 MC

The following table shows the data collected from a sample of seven drivers who entered a supermarket car park. The variables in the table are:

distance – the distance that each driver travelled to the supermarket from their home
 

    • sex – the sex of the driver (female, male)
    • number of children – the number of children in the car
    • type of car – the type of car (sedan, wagon, other)
    • postcode – the postcode of the driver’s home.
       

Part 1

The mean,  , and the standard deviation, , of the variable, distance, are closest to

A. 

B. 

C. 

D. 

E. 

 

Part 2

The number of categorical variables in this data set is

A. 

B.  

C.  

D. 

E.  

 

Part 3

The number of female drivers with three children in the car is

A. 

B. 

C. 

D. 

E.  

 

Show Answers Only

Show Worked Solution

 

 

♦♦ Mean mark 29%.
MARKER’S NOTE: Postcodes here are categorical variables. Ask yourself “Does it make sense to calculate the mean of this variable?” If the answer is “No”, the variable is categorical.

 

Measurement, 2UG 2008 HSC 28b

A tunnel is excavated with a cross-section as shown.
 

 
 

  1. Find an expression for the area of the cross-section using TWO applications of Simpson’s rule.  (2 marks)
  2. The area of the cross-section must be 600 m2. The tunnel is 80 m wide.   
  3. If the value of increases by 2 metres, by how much will change?   (2 marks)

 

Show Answers Only
  2.  
Show Worked Solution
(i)   
   
   

 

(ii)    ²
 

 

 

Statistics, STD2 S5 2008 HSC 28a

The following graph indicates  -scores of ‘height-for-age’ for girls aged  5 – 19 years.
 

 
 

  1. What is the  -score for a six year old girl of height 120 cm? (1 mark)

  2. Rachel is 10 ½  years of age. 

     

    (1)  If  2.5% of girls of the same age are taller than Rachel, how tall is she?   (1 mark)

     

    (2)  Rachel does not grow any taller. At age 15 ½, what percentage of girls of the same age will be taller than Rachel?   (2 marks)

  3. What is the average height of an 18 year old girl?   (1 mark)

For adults (18 years and older), the Body Mass Index is given by

 where    in kilograms and    in metres.

The medically accepted healthy range for    is  .

  1. What is the minimum weight for an 18 year old girl of average height to be considered healthy? (2 marks)

  2. The average height, , in centimetres, of a girl between the ages of 6 years and 11 years can be represented by a line with equation
     
               where is the age in years. 
     
    (1)  For this line, the gradient is 6. What does this indicate about the heights of girls aged 6 to 11?   (1 mark)

     

    (2)  Give ONE reason why this equation is not suitable for predicting heights of girls older than 12.   (1 mark)

Show Answers Only
  2. (1)

     

    (2)

  5. (1)

     

         

     

    (2)

     

         

Show Worked Solution
i.   

 

ii. (1)   ½
   
   
     
   (2)   ½
   
   
   
     
   
   
   

 
½

 

iii.  
 

 

iv.  
 

 

 
 

 

v. (1)  
   
  (2)
   

Measurement, STD2 M6 2008 HSC 25c

Pieces of cheese are cut from cylindrical blocks with dimensions as shown.

 

Twelve pieces are packed in a rectangular box. There are three rows with four pieces of cheese in each row. The curved surface is face down with the pieces touching as shown.
  

  1. What are the dimensions of the rectangular box?  (4 marks)

     

    To save packing space, the curved section is removed.
     
             
     

  2. What is the volume of the remaining triangular prism of cheese? Answer to the nearest cubic centimetre.    (2 marks)

Show Answers Only
  2. ³
Show Worked Solution

i. 

♦ Mean mark 45%.

 

 
 

 

 

 

 

ii. 

♦♦♦ Mean mark 22%.

 
 

 

 
  ³

Probability, STD2 S2 2008 HSC 24b

Three-digit numbers are formed from five cards labelled  1,  2,  3,  4  and  5.

  1. How many different three-digit numbers can be formed?  (1 mark)

  2. If one of these numbers is selected at random, what is the probability that it is odd?   (1 mark)

  3. How many of these three-digit numbers are even?   (1 mark)

  4. What is the probability of randomly selecting a three-digit number less than 500 with its digits arranged in descending order?   (2 marks)

Show Answers Only
Show Worked Solution

i. 

♦ Mean mark 45%.

 

ii. 

 

iii. 

♦ Mean mark 48%.

 

iv. 

♦♦♦ Mean mark 10%.

Algebra, 2UG 2008 HSC 19 MC

The height of a particular termite mound is directly proportional to the square root of the number of termites.

The height of this mound is 35 cm when the number of termites is 2000.

What is the height of this mound, in centimetres, when there are  10 000  termites? 

(A)   

(B)   

(C)   

(D)   

Show Answers Only

Show Worked Solution

 

 

 

Calculus, 2ADV C4 2008 HSC 9c

A beam is supported at    and    as shown in the diagram.
 

2008 9c

 
It is known that the shape formed by the beam has equation  , where    satisfies

  ( is a positive constant) 
and        .

 

  1. Show that  .   (2 marks)

  2. How far is the beam below the  -axis at  ?   (2 marks)

Show Answers Only
Show Worked Solution
i.   
 
 

 

 

 

ii.  
   
   

 

 
 

 

Combinatorics, EXT1 A1 2014 HSC 14b

Two players    and    play a game that consists of taking turns until a winner is determined. Each turn consists of spinning the arrow on a spinner once. The spinner has three sectors  ,    and  . The probabilities that the arrow stops in sectors  ,   and    are  ,    and    respectively.
 

2014 14b
 

The rules of the game are as follows:

• If the arrow stops in sector  , then the player having the turn wins.

• If the arrow stops in sector  , then the player having the turn loses and the other player wins.

• If the arrow stops in sector  , then the other player takes a turn.

Player    takes the first turn.

  1. Show that the probability of player    winning on the first or the second turn of the game is  .   (2 marks)

  2. Show that the probability that player    eventually wins the game is  .   (3 marks)

Show Answers Only
Show Worked Solution

i.  

 

♦♦ Mean mark 25%.
HINT: Expand out the solution  to get a better idea of what you need to prove. 


 


 

ii. 

 


 


 

♦♦♦ Mean mark 11%. Lowest in the 2014 HSC exam!
HINT: The use of ‘eventually’ in the question should flag the possibility of solving by using  in a GP.
 
 
 

Plane Geometry, EXT1 2009 HSC 7c

Consider the billboard below. There is a unique circle that passes through the top and bottom of the billboard (points    and    respectively) and is tangent to the street at  .

Let    be the angle subtended by the billboard at  , the point where    intersects the circle.

Copy the diagram into your writing booklet. 

  1. Show that    when    and    are different points, and hence show that    is a maximum when    and    are the same point.   (3 marks)
  2. Using circle properties, find the distance of    from the building.     (1 mark)
Show Answers Only
Show Worked Solution
(i) 
♦♦ Both parts proved extremely challenging for students.
MARKER’S COMMENT: Very few students recognised that is the external angle of .

 

 

(ii)