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Algebra, STD2 A4 2011 HSC 20 MC

A function centre hosts events for up to 500 people. The cost , in dollars, for the centre
to host an event, where people attend, is given by:

The centre charges $100 per person. Its income , in dollars, is given by:


 

2UG 2011 20

How much greater is the income of the function centre when 500 people attend an event, than its income at the breakeven point?

  1.  
  3.  
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♦ Mean mark 50%
COMMENT: Students can read the income levels directly off the graph to save time and then check with the equations given.

 

 

Financial Maths, STD2 F1 2011 HSC 19 MC

Simon is a mechanic who receives a normal rate of pay of $22.35 per hour for a 40-hour
week.

When he is needed for emergency call-outs he is paid a special allowance of $150 for that
week. Additionally, every time he is called out to an emergency he is paid for a minimum
of 4 hours at double time.

In the week beginning 2 February, 2011 Simon worked 40 hours normal time and was
needed for emergency call-outs. His emergency call-out log book for the week is shown.

2011 19 mc

What was Simon’s total pay for that week?

  1.   $1189.28
  2.   $1296.30
  3.   $1334.55
  4.  $1446.30
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Show Worked Solution
♦ Mean mark 41%
COMMENT: To reduce errors, calculate each element of pay separately in your working and then add up the total, as per the Worked Solution.
 

 

 
 

 

 

 

Measurement, 2UG 2011 HSC 24c

A ship sails 6 km from to on a bearing of 121°. It then sails 9 km to .  The
size of angle is 114°.

2UG 2011 24c

Copy the diagram into your writing booklet and show all the information on it.

  1. What is the bearing of from ?   (1 mark)
  2. Find the distance . Give your answer correct to the nearest kilometre.   (2 marks)
  3. What is the bearing of from ? Give your answer correct to the nearest degree.   (3 marks)

 

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  1.  
  2.  
  3.  
Show Worked Solution
♦♦ Mean mark 24% 
STRATEGY: This deserves repeating again: Draw North-South parallel lines through major points to make the angle calculations easier.
(i)     2011 HSC 24c

 

 
 

  

 

(ii)   

♦ Mean mark 39%
 
 
 

 

(iii)    

♦♦♦ Mean mark 15%
MARKER’S COMMENT: The best responses clearly showed what steps were taken with working on the diagram. Note that all North/South lines are parallel.
 
 

 

 
 
 

Financial Maths, STD2 F5 2009 HSC 27a

The table shows the future value of a $1 annuity at different interest rates over different numbers of time periods. 
 

2UG-2009-27a

  1. What would be the future value of a $5000 per year annuity at 3% per annum for 6 years, with interest compounding yearly?   (1 mark)

  2. What is the value of an annuity that would provide a future value of  $407100  after 7 years at 5% per annum compound interest?   (1 mark)

  3. An annuity of $1000 per quarter is invested at 4% per annum, compounded quarterly for 2 years. What will be the amount of interest earned?    (3 marks)

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i. 

 


ii.   

♦ Mean mark 45%
MARKER’S COMMENT: A common error was to multiply $407 100 by 8.1420 rather than divide.

 
 

 

iii. 

♦♦ Mean mark 31%
MARKER’S COMMENT: When questions asked for the interest paid on annuities, remember to subtract the total principal amounts contributed.

 

 

 
 

 

Algebra, STD2 A4 2009 HSC 28c

The height above the ground, in metres, of a person’s eyes varies directly with the square of the distance, in kilometres, that the person can see to the horizon.

A person whose eyes are 1.6 m above the ground can see 4.5 km out to sea.

How high above the ground, in metres, would a person’s eyes need to be to see an island that is 15 km out to sea? Give your answer correct to one decimal place.   (3 marks)

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♦♦ Mean mark 22%
CRITICAL STEP: Reading the first line of the question carefully and establishing the relationship is the key part of solving this question.

 

 

 
 

Statistics, STD2 S4 2009 HSC 28b

The height and mass of a child are measured and recorded over its first two years. 

This information is displayed in a scatter graph. 
 

  1. Describe the correlation between the height and mass of this child, as shown in the graph.   (1 mark)

  2. A line of best fit has been drawn on the graph.

     

    Find the equation of this line.   (2 marks)

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  1.  

Show Worked Solution

i. 

♦ Mean mark 48%. 

 

ii. 

♦♦♦ Mean mark 18%. 
MARKER’S COMMENT: Many students had difficulty due to the fact the horizontal axis started at  and not the origin.
 
 
 

 

 

Algebra, STD2 A4 2009 HSC 28a

Anjali is investigating stopping distances for a car travelling at different speeds. To model this she uses the equation
 

,
 

where is the stopping distance in metres and is the car’s speed in km/h.

The graph of this equation is drawn below.

2009 28a

  1. Anjali knows that only part of this curve applies to her model for stopping distances.

     

    In your writing booklet, using a set of axes, sketch the part of this curve that applies for stopping distances.  (1 mark)

  2. What is the difference between the stopping distances in a school zone when travelling at a speed of 40 km/h and when travelling at a speed of 70 km/h?   (2 marks)

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  1.  
    2UG-2009-28a-Answer
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(i)

2UG-2009-28a-Answer

(ii)

♦ Mean mark 41%
COMMENT: Students could easily have used the graph for calculating at , although the formula was required when the speed increased to .
 
 

 

 
 

 

 

Probability, STD2 S2 2009 HSC 27c

In each of three raffles, 100 tickets are sold and one prize is awarded.

Mary buys two tickets in one raffle. Jane buys one ticket in each of the other two raffles.

Determine who has the better chance of winning at least one prize. Justify your response using probability calculations.   (4 marks)  

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♦♦ Mean mark 31%.
MARKER’S COMMENT: Very few students calculated Jane’s chance of winning correctly. Note the use of “at least” in the question. Finding (complement) is the best strategy here.

Measurement, STD2 M6 2009 HSC 27b

A yacht race follows the triangular course shown in the diagram. The course from    to    is 1.8 km on a true bearing of 058°.

At    the course changes direction. The course from    to    is 2.7 km and  .
 

 2009-2UG-27b
 

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

  2. What is the distance from    to  ?     (2 marks)

  3. The area inside this triangular course is set as a ‘no-go’ zone for other boats while the race is on.

     

    What is the area of this ‘no-go’ zone?   (1 mark)

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  3. ²
Show Worked Solution
i.    2UG-2009-27b-Answer

♦♦♦ Mean mark 18%.
TIP: Draw North-South parallel lines through relevant points to help calculate angles as shown in the Worked Solutions.

 

ii.   

♦ Mean mark 36%
 +  
   +  
 
 
 
 

 

iii.  

♦ Mean mark 44%
 
  ²

 

²

Measurement, STD2 M2 2009 HSC 26b

John lives in Denver and wants to ring a friend in Osaka.

  1. In Denver it is 9 pm Monday. Given Osaka has a UTC of +9 and Denver has a UTC of –7, what time and day is it in Osaka then?   (1 mark) 

  2. John’s friend in Osaka sent him a text message which happened to take 14 hours to reach him. It was sent at 10 am Thursday, Osaka time.

     

    What was the time and day in Denver when John received the text?    (2 marks)

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i.  

 

 

 

♦ Mean mark 34%
ii.  
   
   
   

Statistics, STD2 S1 2009 HSC 26a

In a school, boys and girls were surveyed about the time they usually spend on the internet over a weekend. These results were displayed in box-and-whisker plots, as shown below. 
 

2UG-2009-26a

  1. Find the interquartile range for boys.   (1 mark)

  2. What percentage of girls usually spend 5 or less hours on the internet over a weekend?  (1 mark)

  3. Jenny said that the graph shows that the same number of boys as girls usually spend between 5 and 6 hours on the internet over a weekend.

     

    Under what circumstances would this statement be true?    (1 mark)

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  5.  
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i.   
   

 

♦♦ Mean mark part ii: 31%
ii.   
 

 

♦♦♦ Mean mark part iii: 9%
iii.   
 
 
 
 

Statistics, STD2 S5 2009 HSC 25d

In Broken Hill, the maximum temperature for each day has been recorded. The mean of these maximum temperatures during spring is 25.8°C, and their standard deviation is 4.2° C. 

  1. What temperature has a -score of  –1?    (1 mark)

  2. What percentage of spring days in Broken Hill would have maximum temperatures between 21.6° C and 38.4°C?

     

    You may assume that these maximum temperatures are normally distributed and that

  3.  

    • 68% of maximum temperatures have -scores between –1 and 1
    • 95% of maximum temperatures have -scores between –2 and 2
    • 99.7% of maximum temperatures have -scores between –3 and 3.   (3 marks)

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i.   

 

 

 

ii.   

♦ Mean mark 41%
MARKER’S COMMENT: Many students failed to use the answer to (d)(i), costing them valuable exam time.

 

 

 

 

Measurement, 2UG 2009 HSC 25c

There is a lake inside the rectangular grass picnic area , as shown in the diagram.
 

2UG-2009-25c
 

  1. Use Simpson’s Rule to find the approximate area of the lake’s surface.   (3 marks)
  2. The lake is 60 cm deep. Bozo the clown thinks he can empty the lake using a four-litre bucket.
  3. How many times would he have to fill his bucket from the lake in order to empty the lake?  (Note that 1 m³ = 1000 L)`.    (2 marks)

 

 

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  1. ²
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(i)    
  ²

 

 
 
 
  ²

 

  ²

 

♦ Mean mark 44%
STRATEGY: Most students who did calculations in cm² and cm³ made errors. Keeping calculations in metres is much easier here.
(ii)   
   
    ³
    ³

 

 

 

Statistics, STD2 S5 2013 HSC 29b

Ali’s class sits two Geography tests. The results of her class on the first Geography test are shown.

The mean was 68.5 for the first test. 

  1. Calculate the standard deviation for the first test. Give your answer correct to one decimal place.    (1 mark)

  2. On the second Geography test, the mean for the class was 74.4 and the standard deviation was 12.4.

     

    Ali scored 62 on the first test. Calculate the mark that she needed to obtain in the second test to ensure that her performance relative to the class was maintained.   (3 marks)

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Show Worked Solution
♦ Mean mark 39%
COMMENT: Make sure you are confident with this function on your calculator!
i.
     

 

ii.
 

 

♦♦ Mean mark 21%.
MARKER’S COMMENT: When “performance relative to the class is maintained”, are the same in each test.

 

Algebra, STD2 A1 2013 HSC 29a

Sarah tried to solve this equation and made a mistake in Line 2. 

 
Copy the equation in Line 1 and continue your solution to solve this equation for .

Show all lines of working.   (2 marks)

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Show Worked Solution
♦♦ Mean mark 27%
STRATEGY: The RHS of the equation increases from 1 to 15 (from Line 1 to Line 2), indicating both sides must have been multiplied by 15.

Measurement, 2UG 2013 HSC 28c

A ship sails due South from Channel-Port-aux-Basques, Canada,    to Barbados, .

How far did the ship sail, to the nearest kilometre? Assume that the radius of Earth is 6400 km.  (2 marks)

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♦ Mean mark 50%
 
 
 
 

Statistics, STD2 S4 2013 HSC 28b

Ahmed collected data on the age () and height () of males aged 11 to 16 years.

He created a scatterplot of the data and constructed a line of best fit to model the relationship between the age and height of males.
 

  1. Determine the gradient of the line of best fit shown on the graph.   (1 mark)

  2. Explain the meaning of the gradient in the context of the data.   (1 mark)

  3. Determine the equation of the line of best fit shown on the graph.  (2 marks)

  4. Use the line of best fit to predict the height of a typical 17-year-old male.   (1 mark)

  5. Why would this model not be useful for predicting the height of a typical 45-year-old male?   (1 mark)

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  2.  

  6.  

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i.    
 

ii.   


 

♦♦ Mean marks of 38%, 26% and 25% respectively for parts (i)-(iii).
MARKER’S COMMENT: Interpreting gradients has been consistently examined in recent history and almost always poorly answered. 

iii.   

 

 

iv.   


  

v.   
 

Measurement, STD2 M1 2013 HSC 27d

A rectangular wooden chopping board is advertised as being 17 cm by 25 cm, with each side measured to the nearest centimetre.

  1. Calculate the percentage error in the measurement of the longer side.   (1 mark)

  2. Between what lower and upper limits does the actual area of the top of the chopping board lie?     (2 marks)

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i.   

♦♦ Mean mark 23%
MARKER’S COMMENT: Be aware that measurements accurate to the nearest cm have an absolute error for calculation purposes of 0.5 cm.

 
   
   

 

ii.   

♦ Mean mark 35%
 

 

 

 

Financial Maths, STD2 F1 2013 HSC 27b

The table shows the tax payable to the Australian Taxation Office for different taxable incomes.

2013 27b

Peta has a gross annual salary of  $84 000. She has tax deductions of $1000 for work-related travel and $500 for stationery. The Medicare levy that she pays is calculated at 1.5% of her taxable income.

Peta has already paid $18 500 in tax.

Will Peta receive a tax refund or will she owe money to the Australian Taxation Office? Justify your answer by calculating the refund or amount owed.   (4 marks)

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Show Worked Solution
 
 
 

 

♦ Mean mark 44%
IMPORTANT: Note that ‘Tax’ and the ‘Medicare Levy’ are calculated separately using the ‘Taxable Income’ figure and added together to find the amount owed to the ATO.
 
 

 

 

 

 
   

 

 

 

Statistics, STD2 S1 2013 HSC 26f

Jason travels to work by car on all five days of his working week, leaving home at 7 am each day. He compares his travel times using roads without tolls and roads with tolls over a period of 12 working weeks.

He records his travel times (in minutes) in a back-to-back stem-and-leaf plot.
 

2013 26f
 

  1. What is the modal travel time when he uses roads without tolls?  (1 mark)

  2. What is the median travel time when he uses roads without tolls?   (1 mark)

  3. Describe how the two data sets differ in terms of the spread and skewness of their distributions.   (2 marks)

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  6.  

Show Worked Solution

i. 

♦ Mean mark 36%
MARKER’S COMMENT: Finding a median proved challenging for many students. Take note!

 

ii. 

 
 

 

♦ Mean mark 39%

 

 iii. 

Financial Maths, STD2 F4 2013 HSC 26e

Kimberley has invested $3500.  

Interest is compounded half-yearly at a rate of 2% per half-year.

2013 26e

Use the table to calculate the value of her investment at the end of 4 years.  (2 marks)

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Show Worked Solution
♦ Mean mark 44%
COMMENT: Structure your answer: 1-Find the interest rate per compounding period (same in this case). 2-Find the number of compounding periods.

 

 

Measurement, STD2 M1 2013 HSC 26d

A section of Jim’s electricity bill is shown.

2013 26d

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

  2. How much will Jim save if he uses 154 kWh of energy at the Off-peak rate rather than at the Peak rate?    (2 marks)

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i.   
   
   
♦ Mean mark part (i) 43% and part  (ii) 46%.

 

ii.   
   
 

 

 
     

Probability, STD2 S2 2013 HSC 26c

The probability that Michael will score more than 100 points in a game of bowling is . 

  1. A commentator states that the probability that Michael will score less than 100 points in a game of bowling is  .

     

    Is the commentator correct? Give a reason for your answer.   (1 mark)

  2. Michael plays two games of bowling. What is the probability that he scores more than 100 points in the first game and then again in the second game?   (1 mark)

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♦♦♦ Mean mark 11%

i.  

 

♦ Mean mark 34%
ii.  
   

Measurement, STD2 M6 2010 HSC 24d

The base of a lighthouse, , is at the top of a cliff 168 metres above sea level. The angle of depression from to a boat at is 28°. The boat heads towards the base of the cliff, , and stops at . The distance is 126 metres.
 

  1. What is the angle of depression from to , correct to the nearest degree?   (3 marks)

  2. How far did the boat travel from to , correct to the nearest metre?   (2 marks)

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Show Worked Solution
♦♦ Mean mark 31%
i.   
 
     

 

 
 

 

ii.     

♦♦ Mean mark 31%
MARKER’S COMMENT: Solve efficiently by using right-angled trigonometry. Many students used non-right angled trig, adding to the calculations and the difficulty.
 
 

 

 
 
 

Statistics, STD2 S5 2010 HSC 24c

The marks in a class test are normally distributed. The mean is 100 and the standard deviation is 10.

  1. Jason's mark is 115. What is his  -score?  (1 mark)

  2. Mary has a -score of 0. What mark did she achieve in the test?   (1 mark)

  3. What percentage of marks lie between 80 and 110?

     

    You may assume the following:

     

    • 68% of marks have a -score between –1 and 1

     

    • 95% of marks have a -score between  –2 and 2

     

    • 99.7% of marks have a -score between –3 and 3.   (2 marks) 

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i.     

MARKER’S COMMENT: Too may students had calculator errors in this question, giving away easy marks. BE CAREFUL!

 
 

 

ii.     

 

♦ Mean mark 42%
iii.   

 

 

 

 

Algebra, STD2 A4 2010 HSC 24b

Ashley makes picture frames as part of her business. To calculate the cost,  , in dollars, of making    frames, she uses the equation  .

She sells the frames for $20 each and determines her income,  , in dollars, using the equation  .
 

Use the graph to solve the two equations simultaneously for    and explain the significance of this solution for Ashley's business.   (2 marks)

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Show Worked Solution

♦ Mean mark 36%.
MARKER’S COMMENT: The intersection on the graph is the same point at which the two simultaneous equations are solved for the given value of .

 

Probability, STD2 S2 2010 HSC 20 MC

Lou and Ali are on a fitness program for one month. The probability that Lou will finish the program successfully is 0.7 while the probability that Ali will finish successfully is 0.6. The probability tree shows this information

 

What is the probability that only one of them will be successful ?

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♦ Mean mark 48%.

Statistics, STD2 S1 2010 HSC 16 MC

This back-to-back stem-and-leaf plot displays the test results for a class of 26 students.
 

2010 Q16 MC  
 

What is the median test result for the class?

  1.    
  2.    
  3.    
  4.    
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♦♦ Mean mark 35%

 

 

Financial Maths, STD2 F4 2009 HSC 24e

Jay bought a computer for $3600. His friend Julie said that all computers are worth nothing (i.e. the value is $0) after 3 years.

  1. Find the amount that the computer would depreciate each year to be worth nothing after 3 years, if the straight line method of depreciation is used.   (1 mark)

  2. Explain why the computer would never be worth nothing if the declining balance method of depreciation is used, with 30% per annum rate of depreciation. Use suitable calculations to support your answer.    (2 marks)

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i.   
 
 
 
   

 

 

♦ Mean mark 45%
ii  
 

 

 
   

 

Algebra, STD2 A2 2009 HSC 24d

A factory makes boots and sandals. In any week

• the total number of pairs of boots and sandals that are made is 200
• the maximum number of pairs of boots made is 120
• the maximum number of pairs of sandals made is 150.

The factory manager has drawn a graph to show the numbers of pairs of boots () and sandals () that can be made.
 

 

  1. Find the equation of the line .   (1 mark)

  2. Explain why this line is only relevant between and for this factory.     (1 mark)

  3. The profit per week, , can be found by using the equation  .

     

    Compare the profits at and .     (2 marks)

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  2.  

     

     

     

Show Worked Solution

i.    

♦♦♦ Mean mark part (i) 14%. 
Using is a less efficient but equally valid method, using    and  (-intercept).

 

ii. 

♦ Mean mark 49%

 

iii. 

♦ Mean mark 40%.
 
 

 
 

 

Probability, STD2 S2 2009 HSC 23b

A personal identification number (PIN) is made up of four digits. An example of a PIN is 

2009 23b 

  1. When all ten digits are available for use, how many different PINs are possible?    (1 mark)

  2. Rhys has forgotten his four-digit PIN, but knows that the first digit is either 5 or 6.   
  3. What is the probability that Rhys will correctly guess his PIN in one attempt?   (1 mark)

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Show Worked Solution
♦ Mean mark 43%
i. 
   

 

♦♦♦ Mean mark 18%
MARKER’S COMMENT: A common error is finding the number of possible combinations but not then calculating the probability.
 ii.
   
     
 
   

Measurement, STD2 M6 2009 HSC 23a

The point is 25 m from the base of a building. The angle of elevation from  to the top of the building is 38°.
 

  1. Show that the height of the building is approximately 19.5 m.   (1 mark)

  2. A car is parked 62 m from the base of the building.

     

    What is the angle of depression from the top of the building to the car?

     

    Give your answer to the nearest minute.   (2 marks)

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i.   

ii.   

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i. 

 
 

 

ii.  

♦♦ Mean mark 33%
MARKER’S COMMENT: If >30 “seconds”, round to the higher “minute”.
 
 
 
 

 

Financial Maths, STD2 F1 2009 HSC 20 MC

Lou bought a plasma TV which was priced at $3499. He paid $1000 deposit and got a loan for the balance that was paid off by 24 monthly instalments of $135.36.

What simple interest rate per annum, to the nearest percent, was charged on his loan?

  1.   11%
  2.   15%
  3.   30%
  4.   46%
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Show Worked Solution
♦♦ Mean mark 34%
COMMENT: A multi-step question targeting higher bands that can be a time-trap for many students.
 
 

 

   

 

 

 

 
 

Algebra, STD2 A1 2009 HSC 16 MC

The time for a car to travel a certain distance varies inversely with its speed.

Which of the following graphs shows this relationship?
 

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Show Worked Solution

 

♦ Mean mark 38%

Algebra, STD2 A2 2009 HSC 14 MC

If   , and    is increased by  2, what will be the corresponding increase in ?

  1.  
  2.  
  3.  
  4.  
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Show Worked Solution
♦ Mean mark 50%.
STRATEGY: Substituting real numbers into the equation can work well in these type of questions. eg. If and when .

Algebra, STD2 A2 2009 HSC 13 MC

The volume of water in a tank changes over six months, as shown in the graph.
 

 2UG-2010-13MC

 
Consider the overall decrease in the volume of water.

What is the average percentage decrease in the volume of water per month over this time, to the nearest percent?

  1.  6%
  2. 11%
  3. 32%
  4. 64%
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Show Worked Solution
♦ Mean mark 48%
COMMENT: Remember that % decrease requires the decrease in volume to be divided by the original volume (50,000L).
 

 

 
 

 

Probability, 2UG 2009 HSC 7 MC

Two people are to be selected from a group of four people to form a committee.

How many different committees can be formed?

(A)    

(B)    

(C)    

(D)    

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♦ Mean mark 38%
 

Algebra, STD2 A1 2011 HSC 21 MC

A train departs from Town A at 3.00 pm to travel to Town B. Its average speed for the
journey is 90 km/h, and it arrives at 5.00 pm. A second train departs from Town A at
3.10 pm and arrives at Town B at 4.30 pm.

What is the average speed of the second train?

  1.   135 km/h
  2. 150 km/h
  3.   216 km/h
  4.   240 km/h
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Show Worked Solution

 

♦ Mean mark 49%

 
 

 

 
 
 

Measurement, STD2 M1 2010 HSC 17 MC

During a flood 1.5 hectares of land was covered by water to a depth of  17 cm.

How many kilolitres of water covered the land?   (1 hectare = 10 000 m²)  

  1.    2.55 kL
  2.    2550 kL
  3.    255 000 kL
  4.    2 550 000 kL
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Show Worked Solution
♦♦ Mean mark 34%
NOTE: The unit conversion 1 m³ = 1000 L  is contained in the Formulae and Data sheet given out in the exam.
  ²
 
  ³

 

³

Algebra, STD2 A4 2010 HSC 13 MC

The number of hours that it takes for a block of ice to melt varies inversely with the temperature. At 30°C it takes 8 hours for a block of ice to melt.

How long will it take the same size block of ice to melt at 12°C?  

  1. 3.2 hours
  2. 20 hours
  3. 26  hours
  4. 45 hours
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Show Worked Solution
 
♦ Mean mark 50% 

  

 

 

Algebra, STD2 A4 2012 HSC 30c

In 2010, the city of Thagoras modelled the predicted population of the city using the equation

.

That year, the city introduced a policy to slow its population growth. The new predicted population was modelled using the equation

.

In both equations, is the predicted population and is the number of years after 2010.  

The graph shows the two predicted populations.
 

  1. Use the graph to find the predicted population of Thagoras in 2030 if the population policy had NOT been introduced.   (1 mark)

  2. In each of the two equations given, the value of is 3 000 000.

     

    What does represent?   (1 mark)

  3. The guess-and-check method is to be used to find the value of , in  .

     

    (1) Explain, with or without calculations, why 1.05 is not a suitable first estimate for .  (1 mark)

     

    (2) With    and  , use the guess-and-check method and the equation    to estimate the value of to two decimal places. Show at least TWO estimate values for , including calculations and conclusions.  (2 marks)

  4. The city of Thagoras was aiming to have a population under 7 000 000 in 2050. Does the model indicate that the city will achieve this aim?

     

    Justify your answer with suitable calculations.  (2 marks)

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  1.  
  2.  
  3.  

     

Show Worked Solution

i.   

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii.    


 

♦ Mean mark part (ii) 48%

iii. (1) 

   
 

iii. (2) 

 

 

 

 

 

iv. 

 

 

Measurement, STD2 M6 2012 HSC 29c

Raj cycles around a course. The course starts at , passes through , and  and finishes at . The distances and are equal.
  

2012 29c

  1. What is the length of , to the nearest kilometre?  (2 marks)

  2. What is the total distance that Raj cycles, to the nearest kilometre?   (3 marks)

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Show Worked Solution

i.   

♦ Mean mark 48%.
 
 

 

 
 

 

ii. 

♦ Mean mark 37%.
MARKER’S COMMENT: Students could also have used Pythagoras or the Sine rule to calculate .
 

 

 
 
 

Statistics, STD2 S5 2012 HSC 29b

A machine produces nails. When the machine is set correctly, the lengths of the nails are normally distributed with a mean of  6.000 cm  and a standard deviation of  0.040 cm.

To confirm the setting of the machine, three nails are randomly selected. In one sample the lengths are  5.950,  5.983 and  6.140.

The setting of the machine needs to be checked when the lengths of two or more nails in a sample lie more than 1 standard deviation from the mean.

Does the setting on the machine need to be checked? Justify your answer with suitable calculations.   (2 marks)

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Show Worked Solution
♦ Mean mark 44%

σ

 

σ

Statistics, STD2 S3 2012 HSC 29a*

Tourists visit a park where steam erupts from a particular geyser.

The brochure for the park has a graph of the data collected for this geyser over a period of time.

The graph shows the duration of an eruption and the time until the next eruption, timed from the end of one eruption to the beginning of the next.
 

  

  1. Tony sees an eruption that lasts 4 minutes. Based on the data in the graph, what is the minimum time that he can expect to wait for the next eruption? (1 mark)

  2. Julia saw two consecutive eruptions, one hour apart. Based on the data in the graph, what was the longest possible duration of the first eruption that she saw?    (1 mark)

  3. What does the graph suggest about the association between the duration of an eruption and the time to the next eruption? (1 mark)

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  4.  

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i. 

 

ii. 

♦ Mean mark (b) 47%

 

iii. 

Measurement, STD2 M7 2012 HSC 28c

Jacques and a flagpole both cast shadows on the ground. The difference between the lengths of their shadows is 3 metres.
 

What is the value of , the length of Jacques’ shadow?     (3 marks)

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♦♦ Mean mark 24%


 

 
 

Probability, STD2 S2 2012 HSC 27e

A box contains 33 scarves made from two different fabrics. There are 14 scarves made from silk (S) and 19 made from wool (W).
Two girls each select, at random, a scarf to wear from the box.

  1. Complete the probability tree diagram below.   (2 marks) 
      
       

  2. Calculate the probability that the two scarves selected are made from silk.    (1 mark)

  3. Calculate the probability that the two scarves selected are made from different fabrics.   (2 marks)

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  1.  
       
     
Show Worked Solution

i. 

♦ Mean mark (i) 43%.
ii. 
 
 

 

iii. 
 
 
 
♦ Mean mark (iii) 41%.
MARKER’S COMMENT: In better responses, students multiplied along the branches and then added these two results together

Measurement, STD2 M6 2012 HSC 27d

A disability ramp is to be constructed to replace steps, as shown in the diagram.

The angle of inclination for the ramp is to be 5°.   
  

Calculate the extra distance, , that the ramp will extend beyond the bottom step.

Give your answer to the nearest centimetre.   (3 marks)

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Show Worked Solution

♦♦ Mean mark 35%
MARKER’S COMMENT:  The better responses used a diagram of a simplified version of the ramp as per the Worked Solution.
 
   
 
 

Measurement, STD2 M7 2012 HSC 27c

A map has a scale of  1 : 500 000.

  1. Two mountain peaks are 2 cm apart on the map.

     

    What is the actual distance between the two mountain peaks, in kilometres?  (1 mark)

  2. Two cities are 75 km apart. How far apart are the two cities on the map, in centimetres?   (1 mark)

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Show Worked Solution
♦ Mean mark 37%
MARKER’S COMMENT: Better responses realised that 1 unit on the map represented 1 unit x 500,000 in real life.
i. 
 
 
 

 

 

♦ Mean mark 44%

ii.  

 

Measurement, STD2 M1 2012 HSC 27b

The sector shown has a radius of 13 cm and an angle of 230°. 
 

 2012 27b
 

 What is the perimeter of the sector to the nearest centimetre?    (2 marks) 

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Show Worked Solution
♦ Mean mark 39%
MARKER’S COMMENT: The formula of the length of an arc is given in the formula sheet.
 
 
   
 
 

Statistics, STD2 S1 2012 HSC 28d

The test results in English and Mathematics for a class were recorded and displayed in the box-and-whisker plots.
 

  1. What is the interquartile range for English?  (1 mark)

  2. Compare and contrast the two data sets by referring to the skewness of the distributions and the measures of location and spread.   (3 marks)

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i.   

ii. 

Show Worked Solution

i.   

♦ Mean mark (ii) 35%
MARKER’S COMMENT: Markers are looking for students to use the correct language of location and spread such as mean, median, interquartile range, standard deviation and skewness.

ii. 

Financial Maths, STD2 F4 2011* HSC 10 MC

A television was purchased for $2100 on 12 April 2011 using a credit card. Compound interest was charged daily at a rate basis 19.71% per annum for purchases on this credit card. There were no other purchases on this credit card account.

There was no interest-free period. The period for which interest was charged included the date of purchase and the date of payment.

What amount was paid when the account was paid in full on 20 May 2011?

  1.   $2143.09
  2.   $2143.53
  3.   $2144.23
  4.   $2144.68
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Show Worked Solution
♦♦ Mean mark 32%
COMMENT: Make sure you know how to adjust an annual rate to a daily rate, as shown in the Worked Solutions.

 
 

Statistics, STD2 S1 2011 HSC 7 MC

A set of data is displayed in this box-and-whisker plot.
 

Which of the following best describes this set of data?

  1. Symmetrical
  2. Positively skewed
  3. Negatively skewed
  4. Normally distributed
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Show Worked Solution

♦ Mean mark 47%.