SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

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)

Show Answers Only
Show Worked Solution

i. 

 
 

 

 

ii. 

­
­
 
 

 

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)

Show Answers Only
  4.  
Show Worked Solution

i.   
 

 

 

 

ii. 

 
 

 

 

 

 

 

iii. 

 


 

(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)

Show Answers Only
  2.   
  3. ²
Show Worked Solution

i.   

 

 

ii.   2UA HSC 2006 7b

 

iii. 
 
  ­­
 
 
  ²

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)

Show Answers Only
  3.  
Show Worked Solution
i.       
 

 

 

 

ii. 

 

 

 

iii. 

2UA HSC 2006 5a

 

(iv) 

 

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. 

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)

Show Answers Only
Show Worked Solution
i.   
   

 

ii. 

 

iii. 

 
 

 

 

 

iv.  

 

 


 

 

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. 

Measurement, STD2 M6 2005 HSC 27c

2UG-2005-27c
 

The bearing of from is 250° and the distance of from is 36 km.

  1. Explain why    is 110°.   (1 mark)

  2. If    is 15 km due north of  , calculate the distance of    from  , correct to the nearest kilometre.   (3 marks)

Show Answers Only
Show Worked Solution

i. 

 

ii.   
 
 
 

Measurement, 2UG 2005 HSC 27b

This diagram represents Earth. is at the centre, and and are points on the surface.

2UG-2005-27b

Calculate the distance from to along the great circle through and .
Give your answer in to the nearest km.   (2 marks)

Show Answers Only

Show Worked Solution

HSC 2005 27b

 

 

Data, 2UG 2005 HSC 27a

The area graph shows sales figures for Shoey’s shoe store.

2UG-2005-27a

  1. Approximately how many school shoes were sold in January?   (1 mark)
  2. For which month does the graph indicate that the same number of school shoes and business shoes was sold?   (1 mark)
  4. Identify ONE trend in this graph, and suggest a valid reason for this trend.   (2 marks)
Show Answers Only
  2.  


Show Worked Solution

(i)   

 

(ii)  

 

(iii) 

Statistics, STD2 S5 2005 HSC 26c

The weights of boxes of Brekky Bicks are normally distributed. The mean is 754 grams and the standard deviation is 2 grams.

  1. What is the -score of a box of Brekky Bicks with a weight of 754 g?   (1 mark)

  2. What is the weight of a box that has a -score of  –1?   (1 mark)

  3. Brekky Bicks boxes are labelled as having a weight of 750 g. What percentage of boxes will have a weight less than 750 g?   (2 marks)

Show Answers Only
  2.  
Show Worked Solution

i.   

 

ii.   
 
 

 

iii.
   

 

 
 

Financial Maths, STD2 F5 2005 HSC 26b

Rod is saving for a holiday. He deposits $3600 into an account at the end of every year for four years. The account pays 5% per annum interest, compounding annually.

The table shows future values of an annuity of $1.
 

2UG-2005-26b
 

  1. Use the table to find the value of Rod’s investment at the end of four years.   (2 marks)

  2. How much interest does Rod earn on his investment over the four years?   (2 marks)

Show Answers Only
Show Worked Solution

i.   

 

ii. 
 
 

Financial Maths, STD2 F4 2005 HSC 26a

A sports car worth $150 000 is bought in December 2005.

In December each year, beginning in 2006, the value of the sports car is depreciated by 10% using the declining balance method of depreciation.

In which year will the depreciated value first fall below $120 000?   (2 marks)

Show Answers Only

Show Worked Solution

 

 

 

 

Financial Maths, STD2 F4 2006* HSC 25b

In June, Ms Bigspender received a statement for her credit card account.

The account has no interest-free period. Compound interest is calculated daily and charged to her account on the statement date.
 

  1. For how many days is she charged interest on her purchase?  (1 mark)

  2. Calculate the interest charged to her account.  (2 marks)

Show Answers Only
  1.  
  2.  
Show Worked Solution

i. 

 

ii.   

 
 

 

 
   

Measurement, 2UG 2004 HSC 26b

The location of Sorong is and the location of Darwin is .

  1. What is the difference in the latitudes of Sorong and Darwin?  (1 mark)
  2. The radius of Earth is approximately

  3. Show that the great circle distance between Sorong and Darwin is approximately .  (2 marks)

 

Show Answers Only
Show Worked Solution

(i) 

 

(ii)   

Probability, STD2 S2 2006 HSC 25a

Three cards labelled , and can be arranged in any order.

  1. In how many different ways can the cards be arranged?  (1 mark)

  2. What is the probability that the second card in an arrangement is a ?  (1 mark)

  3. What is the probability that the last card in an arrangement is not a ?  (1 mark)

Show Answers Only
Show Worked Solution
i. 
 

 

ii. 

 

iii. 

Measurement, STD2 M6 2006 HSC 24b

A 130 cm long garden rake leans against a fence. The end of the rake is 44 cm from the base of the fence.

  1. If the fence is vertical, find the value of to the nearest degree.  (2 marks)
      
          2UG-2006-24b-i

     

  2. The fence develops a lean and the rake is now at an angle of 53° to the ground. Calculate the new distance ( cm) from the base of the fence to the head of the rake. Give your answer to the nearest centimetre.  (2 marks)
     
          2UG-2006-24b-ii

Show Answers Only
Show Worked Solution
i.   

2UG-2006-24b1 Answer
 
 

 

ii.   

2UG-2006-24b2 Answer

 
 

Data, 2UG 2006 HSC 23d

The graph shows the amounts charged by Company and Company to deliver parcels of various weights.

2UG-2006-23d

  1. How much does Company charge to deliver a kg parcel?  (1 mark)
  2. Give an example of the weight of a parcel for which both Company and Company charge the same amount.  (1 mark)
  4. For what weight(s) is it cheaper to use Company ?  (2 marks)
  5. What is the rate per kilogram charged by Company for parcels up to kg?  (1 mark)
Show Answers Only
Show Worked Solution

(i)  

 

(ii) 

 

(iii) 

 

(iv)  

 

Statistics, STD2 S1 2006 HSC 23c

Vicki wants to investigate the number of hours spent on homework by students at her high school.

  1. Briefly describe a valid method of randomly selecting 200 students for a sample.  (1 mark)

  2. Vicki chooses her sample and asks each student how many hours (to the nearest hour) they usually spend on homework during one week.

     

    The responses are shown in the frequency table.
     
         2UG-2006-23c

    What is the mean amount of time spent on homework?  (2 marks)

Show Answers Only

  1.  

     

Show Worked Solution

i.   

MARKER’S COMMENT: This “routine” exercise of finding a mean from grouped data was incorrectly answered by most students! The best responses copied the table and inserted a class-centre column (see solution).

 

ii.    2UG-2006-23c Answer

 

 
 

Statistics, STD2 S1 2006 HSC 24a

2UG-2006-24a

List TWO ways in which this graph is misleading.  (2 marks)

Show Answers Only

Show Worked Solution

Probability, 2UG 2005 HSC 25c

Robyn plays a game in which she randomly chooses one of these five cards. She plays the game times, replacing the card after each game.

2UG-2005-25c

  1. How many times would she expect to win ?   (1 mark)
  3. What is the financial expectation of the game?   (2 marks)
  4. Another card is added to the game with ‘Win nothing $0’ written on it.
    Robyn claims that the financial expectation will not change.
  6. Do you agree? Justify your answer with suitable calculations.   (2 marks)
Show Answers Only

 

Show Worked Solution

(i)   

 

(ii)  

 

(iii) 

 

Measurement, STD2 M6 2005 HSC 25b

2UG-2005-25b

  1. Use Pythagoras’ theorem to show that is a right-angled triangle.  (1 mark)

  2. Calculate the size of to the nearest minute.  (2 marks)

Show Answers Only
Show Worked Solution

i.   

 
 
 

MARKER’S COMMENT: Know your calculator process for producing an angle in minutes/seconds. Note >30 “seconds” rounds up to the higher “minute”.

 
ii. 

 
 

Statistics, STD2 S1 2005 HSC 24d

The sector graph shows the proportion of people, as a percentage, living in each region of Sumcity. There are 24 000 people living in the Eastern Suburbs.
 

2UG-2005-24d1
 

  1. Show that the total number of people living in Sumcity is  160 000.  (1 mark)

Jake used the information above to draw a column graph.
 

2UG-2005-24d2

  1. The column graph height is incorrect for one region.

     

    Identify this region and justify your answer.   (2 marks)

Show Answers Only
Show Worked Solution

i.   

   
   

 

ii. 

Algebra, STD2 A1 2005 HSC 24b

The formula    is used to calculate the dosage of Hackalot cough medicine to be given to a child.

    • D is the dosage of Hackalot cough medicine in millilitres (mL).
    • A is the age of the child in months.
  1. If George is nine months old, what dosage of Hackalot cough medicine should he be given?  (1 mark)

The correct dosage of Hackalot cough medicine for Sam is 4 mL.

  1. What is the difference in the ages of Sam and George, in months?  (3 marks)

Show Answers Only
Show Worked Solution
i.
   
   

 

 

ii.   

 
   

 

Statistics, STD2 S1 2005 HSC 24a

  1. Draw a stem-and-leaf plot for the following set of scores.

  2.  

     

     (2 marks)

  3. What is the median of the set of scores?   (1 mark)

  4. Comment on the skewness of the set of scores.   (1 mark)

Show Answers Only
  1.  

  4.  

Show Worked Solution
i.    HSC 2005 24a

 

ii. 

 
 

 

iii.  

Probability, STD2 S2 2005 HSC 23c

Moheb owns five red and seven blue ties. He chooses a tie at random for himself and puts it on. He then chooses another tie at random, from the remaining ties, and gives it to his brother.

  1. What is the probability that Moheb chooses a red tie for himself?  (1 mark)

Copy the tree diagram into your writing booklet.
 

2UG-2005-23c
 

  1. Complete your tree diagram by writing the correct probability on each branch.  (2 marks)
  2. Calculate the probability that both of the ties are the same colour.  (2 marks)

Show Answers Only
  2.  
Show Worked Solution
i.
   

 

ii.  

 

iii. 

Financial Maths, STD2 F1 2004 HSC 27b

David is paid at these rates:
  

 

His time sheet for last week is:
  

  1. Calculate David’s gross pay for last week.  (3 marks)

  2. David decides not to work on Saturdays. He wants to keep his weekly gross pay the same. How many extra hours at the weekday rate must he work?  (1 mark)

Show Answers Only
Show Worked Solution
i.   
 
 
 

 

 

 

 ii. 

 

Financial Maths, STD2 F4 2004 HSC 27a

Aaron decides to borrow  $150 000  over a period of 20 years at a rate of 7.0% per annum.

2004 27a

  1. Using the Monthly Repayment Table, calculate Aaron’s monthly repayment.  (2 marks)

  2. How much interest does he pay over the 20 years?  (2 marks)

  3. Aaron calculates that if he repays the loan over 15 years, his total repayments would be .

     

    How much interest would he save by repaying the loan over 15 years instead of 20 years?  (2 marks)

Show Answers Only
Show Worked Solution

i.   


 

 

ii. 


 

 

iii. 
 
 

Algebra, STD2 A4 2004 HSC 26a

  1. The number of bacteria in a culture grows from 100 to 114 in one hour.

     

    What is the percentage increase in the number of bacteria?   (1 mark)

  2. The bacteria continue to grow according to the formula  , where    is the number of bacteria after    hours.

     

    What is the number of bacteria after 15 hours?   (1 mark)

  1. Use the values of from    to    to draw a graph of  .

     

    Use about half a page for your graph and mark a scale on each axis.   (4 marks)

  2. Using your graph or otherwise, estimate the time in hours for the number of bacteria to reach 300.   (1 mark)

Show Answers Only
Show Worked Solution

i.   

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii. 

 
 

 

iii. 

 

iv. 

Probability, 2UG 2004 HSC 25b

Joe sells three different flavours of ice-cream from three different tubs in a cabinet. The flavours are chocolate, strawberry and vanilla.

2004 25b

  1. In how many different ways can he arrange the tubs in a row? Show working to justify your answer.  (2 marks)
  3. Paul buys an ice-cream from Joe on two different days. He chooses the flavour at random. What is the probability that he chooses chocolate on both days?  (1 mark)
  5. Mei-Ling buys an ice-cream from Joe and chooses any two different flavours at random. What is the probability that she chooses chocolate first and then strawberry?  (2 marks)

 

Show Answers Only
Show Worked Solution

(i) 

 

(ii) 
 

 

(iii) 
 

Data, 2UG 2004 HSC 24a

The following graphs have been constructed from data taken from the Bureau of Meteorology website. The information relates to a town in New South Wales.

The graphs show the mean 3 pm wind speed (in kilometres per hour) for each month of the year and the mean number of days of rain for each month (raindays).

2004 24a

  1. What is the mean 3 pm wind speed for September?  (1 mark)
  2. Which month has the lowest mean 3 pm wind speed?  (1 mark)
  3. In which three-month period does the town have the highest number of raindays?  (1 mark)
  4. Briefly describe the pattern relating wind speed with the number of raindays for this town. Refer to specific months.  (2 marks)

 

Show Answers Only





 

 

Show Worked Solution

(i)   

(ii)  

(iii) 

(iv) 

Measurement, STD2 M1 2005 HSC 23b

A clay brick is made in the shape of a rectangular prism with dimensions as shown.
 

  1. Calculate the volume of the clay brick.  (1 mark)

Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.  
 

  1. What is the volume of clay remaining in the brick after the holes have been made? (Give your answer to the nearest cubic centimetre.)  (3 marks)

  2. What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.)  (1 mark)

Show Answers Only
Show Worked Solution
i.   
   
   

 

ii. 

 

 

iii. 

Probability, STD2 S2 2005 HSC 16 MC

On a television game show, viewers voted for their favourite contestant. The results were recorded in the two-way table.

One male viewer was selected at random from all of the male viewers.

What is the probability that he voted for Contestant 1?

Show Answers Only

Show Worked Solution

  


 

Probability, STD2 S2 2004 HSC 25c

Lie detector tests are not always accurate. A lie detector test was administered to 200 people.

The results were:

• 50 people lied. Of these, the test indicated that 40 had lied;
• 150 people did NOT lie. Of these, the test indicated that 20 had lied.

  1. Complete the table using the information above   (2 marks)
      
        

  2. For how many of the people tested was the lie detector test accurate?   (1 mark)

  3. For what percentage of the people tested was the test accurate?   (1 mark)

  4. What is the probability that the test indicated a lie for a person who did NOT lie?   (1 mark)

Show Answers Only
Show Worked Solution

i.

ii. 


 

iii. 


 

iv. 

Financial Maths, STD2 F4 2004 HSC 25a

Tai uses the declining balance method of depreciation to calculate tax deductions for her business. Tai’s computer is valued at $6500 at the start of the 2003 financial year. The rate of depreciation is 40% per annum.

  1. Calculate the value of her tax deduction for the 2003 financial year.  (1 mark)

  2. What is the value of her computer at the start of the 2006 financial year?  (2 marks)

Show Answers Only
Show Worked Solution
i. 
 

 

ii. 

Statistics, STD2 S5 2004 HSC 24c

The normal distribution shown has a mean of 170 and a standard deviation of 10.
 


 

  1. Roberto has a raw score in the shaded region. What could his -score be?  (1 mark)

  2. What percentage of the data lies in the shaded region?  (2 marks)

Show Answers Only
Show Worked Solution
i. 
 
 

 

 

 

 

ii.   

Measurement, STD2 M6 2004 HSC 24b

The diagram shows a radial survey of a piece of land.
 


 

  1. is south-east of . What is the size of angle ?  (1 mark)

  2. What is the bearing of from ?  (1 mark)

  3. Find the size of angle to the nearest degree.  (3 marks)

Show Answers Only
Show Worked Solution
i.   

 

 

ii.  

 

iii. 

 
 

Algebra, 2UG 2004 HSC 23b

Kirbee is shopping for computer software. Novirus costs more than
Funmaths. Let dollars be the cost of Funmaths.

  1. Write an expression involving for the cost of Novirus.  (1 mark)
  2. Novirus and Funmaths together cost . Write an equation involving
  3. and solve it to find the cost of Funmaths.  (2 marks)
Show Answers Only
Show Worked Solution

(i)  

 

(ii) 

Linear Functions, 2UA 2004 HSC 2a

The diagram shows the points    and  .

The line    is drawn perpendicular to the  -axis through the point  .
 

Linear Functions, 2UA 2004 HSC 2a 
 

  1. Calculate the length of the interval  .   (1 mark)
  2. Find the gradient of the line  .   (1 mark)
  3. What is the size of the acute angle between the line    and the line  ?   (1 mark)
  4. Show that the equation of the line    is  .    (1 mark)
  5. Copy the diagram into your writing booklet and shade the region defined by  .   (1 mark)
  6. Write down the equation of the line  .   (1 mark)
  7. The point    is on the line    such that    is perpendicular to  . Find the coordinates of  .   (2 marks)

 

Show Answers Only
  5. Linear Functions, 2UA 2004 HSC 2a Answer
Show Worked Solution
(i)   
 
 
 
 

 

(ii)  
   
   

 

(iii)  

 

(iv)  

 

 

(v)

Linear Functions, 2UA 2004 HSC 2a Answer

 

(vi)  
 

 

(vii)  
 
 

 

 

Statistics, STD2 S1 2006 HSC 12 MC

The mean of a set of 5 scores is 62.

What is the new mean of the set of scores after a score of 14 is added?

  1.   38
  2.   54
  3.   62
  4.   76
Show Answers Only

Show Worked Solution

 

Probability, STD2 S2 2006 HSC 10 MC

Kay randomly selected a marble from a bag of marbles, recorded its colour and returned it to the bag. She repeated this process a number of times.
  


  

Based on these results, what is the best estimate of the probability that Kay will choose a green marble on her next selection?

  1.  
  2.  
  3.  
  4.  
Show Answers Only

Show Worked Solution
 
 

Probability, STD2 S2 2006 HSC 6 MC

Marcella is planning to roll a standard six-sided die 60 times.

How many times would she expect to roll the number 4?

  1.   6
  2.   10
  3.   15
  4.   20
Show Answers Only

Show Worked Solution

Statistics, STD2 S1 2006 HSC 4 MC

A set of scores is displayed in a stem-and-leaf plot.
 

 2UG-2006-4MC

 
What is the median of this set of scores?

  1.   28
  2.   30
  3.   33
  4.   47
Show Answers Only

Show Worked Solution

 
 

Probability, STD2 S2 2005 HSC 23a

There are 100 tickets sold in a raffle. Justine sold all 100 tickets to five of her friends. The number of tickets she sold to each friend is shown in the table.
 

  1. Justine claims that each of her friends is equally likely to win first prize.

     

    Give a reason why Justine’s statement is NOT correct.   (1 mark)

  2. What is the probability that first prize is NOT won by Khalid or Herman?   (2 marks)

Show Answers Only


Show Worked Solution

i.   

 

ii.