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Statistics, STD2 S1 2009 HSC 24b

Tayvan is an international company that reports its profits in the USA, Belgium and India at the end of each quarter. The profits for 2008 are shown in the area chart.

2009 24b

  1. What was the total profit for Tayvan on June 30?   (1 mark)
  2. What was Tayvan’s profit in Belgium on March 31?     (1 mark)
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MARKER’S COMMENT: Area charts provide cumulative totals. Many students did not know this and incorrectly answered 14 million.
(i)    

 

(ii)    
   
 

Statistics, STD2 S1 2009 HSC 24a

The diagram below shows a stem-and-leaf plot for 22 scores. 
 

2UG-2009-24a
 

  1.  What is the mode for this data?   (1 mark)

  2.  What is the median for this data?     (1 mark)

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i.   

 

ii.   


 

 

Financial Maths, STD2 F1 2009 HSC 10 MC

Billy worked for 35 hours at the normal hourly rate of pay and for five hours at double time. He earned $561.60 in total for this work.

What was the normal hourly rate of pay?

  1. $7.02
  2. $12.48
  3. $14.04
  4. $16.05
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Financial Maths, STD2 F4 2009 HSC 6 MC

A house was purchased in 1984 for $35 000. Assume that the value of the house has increased by 3% per annum since then. 

Which expression gives the value of the house in 2009?  

  2.  
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Statistics, STD2 S3 2009 HSC 5 MC

Jamie wants to know how many songs were downloaded legally from the internet in the last 12 months by people aged 18–25 years. He has decided to conduct a statistical inquiry.

After he collects the data, which of the following shows the best order for the steps he should take with the data to complete his inquiry?

  1.    Display, organise, conclude, analyse
  2.    Organise, display, conclude, analyse
  3.    Display, organise, analyse, conclude
  4.    Organise, display, analyse, conclude
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Statistics, STD2 S1 2009 HSC 3 MC

The eye colours of a sample of children were recorded.

When analysing this data, which of the following could be found?

  1. Mean
  2. Median
  3. Mode
  4. Range
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Measurement, STD2 M6 2011 HSC 9 MC

Two trees on level ground, 12 metres apart, are joined by a cable. It is attached 2 metres above the ground to one tree and 11 metres above the ground to the other.

What is the length of the cable between the two trees, correct to the nearest metre? 

  1.  
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Probability, STD2 S2 2012 HSC 26e

The dot plot shows the number of push-ups that 13 members of a fitness class can do in one minute.

2012 26e

  1.  What is the probability that a member selected at random from the class can do more than 38 push-ups in one minute?   (1 mark)

  2.  A new member who can do 32 push-ups in one minute joins the class.

     

    Does the addition of this new member to the class change the probability calculated in part (i)? Justify your answer.    (1 mark)

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i. 
 

 
ii.   

 
MARKER’S COMMENT: The most successful candidates used the fraction in their part (ii) answer rather than relying solely on words.

Statistics, STD2 S1 2010 HSC 1 MC

The results of a survey are displayed in the dot plot.

What is the range of this data?
 

2010 1 MC 
 

  1.    7
  2.    8
  3.    9
  4.    10
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Probability, STD2 S2 2013 HSC 7 MC

In an experiment, a standard six-sided die was rolled 72 times. The results are shown in the table.
 

Which number on the die was obtained the expected number of times?

  1.    1
  2.    2
  3.    3
  4.    6
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Calculus, 2ADV C3 2009 HSC 9b

An oil rig,  ,  is 3 km offshore. A power station,  ,  is on the shore. A cable is to be laid from  to  .  It costs $1000 per kilometre to lay the cable along the shore and $2600 per kilometre to lay the cable underwater from the shore to  .

The point    is the point on the shore closest to  ,  and the distance    is 5 km.

The point    is on the shore, at a distance of    km from  ,  as shown in the diagram.


  

  1. Find the total cost of laying the cable in a straight line from    to    and then in a straight line from    to  .   (1 mark)

  2. Find the cost of laying the cable in a straight line from    to  .  (1 mark)

  3. Let    be the total cost of laying the cable in a straight line from    to  ,  and then in a straight line from  to  .
     
    Show that  .    (2 marks)

  4. Find the minimum cost of laying the cable.    (4 marks)

  5. New technology means that the cost of laying the cable underwater can be reduced to $1100 per kilometre.

     

    Determine the path for laying the cable in order to minimise the cost in this case.    (2 marks)

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♦♦ Although specific data is unavailable for question parts, mean marks were 35% for Q9 in total.
i.   
 
 

 

 

ii.   

 
 

 

 
 

 

iii. 

 
 

 

iv.   

 

IMPORTANT: Tougher derivative questions often require students to deal with multiple algebraic constants. See Worked Solutions in part (iv).

 
MARKER’S COMMENT: Check the nature of the critical points in these type of questions. If using the first derivative test, make sure some actual values are substituted in.

 
   

 

 

v.   

 
 

 

 

 

MARKER’S COMMENT: Many students failed to interpret a correct calculation of    as providing no solution.

.

Mechanics, EXT2* M1 2011 HSC 6b

The diagram shows the trajectory of a ball thrown horizontally, at speed m/s, from the top of a tower metres above the ground level.
 

2011 6b
 

The ball strikes the ground at an angle of  45°, metres from the base of the tower, as shown in the diagram. The equations describing the trajectory of the ball are

  and   ,    (DO NOT prove this)

where is the acceleration due to gravity, and is time in seconds.

  1. Prove that the ball strikes the ground at time  
     
           seconds.    (1 mark)

  2. Hence, or otherwise, show that  .    (2 marks)

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  1.  
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i.  

 

ii.  

,

 

 

♦♦ Mean mark 33%
STRATEGY: The key to unlocking this proof is understanding that , where   is the angle of trajectory at impact of a projectile, equals  .

 

 
 

Calculus in the Physical World, 2UA 2008 HSC 6b

The graph shows the velocity of a particle,    metres per second, as a function of time,    seconds.
 


 

  1. What is the initial velocity of the particle?   (1 mark)
  2. When is the velocity of the particle equal to zero?    (1 mark)
  3. When is the acceleration of the particle equal to zero?    (1 mark)
  4. By using Simpson's Rule with five function values, estimate the distance travelled by the particle between    and  .   (3 marks)
  5.  
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(i)    

 

(ii)    

 

(iii) 

 

(iv)   

MARKER’S COMMENT: Less errors were made by students using a table and the given formula. Note however, that the formula produced the most errors.
 
 
 
 

 

 

Calculus, 2ADV C4 2009 HSC 7a

The acceleration of a particle is given by

,

where    is the displacement in metres and    is the time in seconds.

Initially its velocity is  and its displacement is 5 m.

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

  3. Find the time when the particle comes to rest.    (3 marks)

  4. Find the displacement  when the particle comes to rest.    (1 mark)

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i.   

 

 

 
 

 

 

 

ii.  

 
 

 


 


  

iii. 

ALGEBRA TIP: Helpful identity  . Easily provable as follows:



.

Calculus, EXT1* C1 2010 HSC 7a

The acceleration of a particle is given by

,

where    is the displacement in metres and    is the time in seconds. 

Initially the particle is at the origin with a velocity of  .

  1. Show that the velocity of the particle is given by

     

      .    (2 marks)

  2. Find the time when the particle first comes to rest.    (2 marks)

  3. Find the displacement,  ,  of the particle in terms of  .    (2 marks)

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i.   
 
 

 

 

 

ii.  

 

 

 

iii.   
 
 

 

 

Calculus, 2ADV C4 2012 HSC 15b

The velocity of a particle is given by

,

where    is the displacement in metres and    is the time in seconds. Initially the particle is 3 m to the right of the origin.

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

  2. Find the maximum velocity of the particle.    (1 mark)

  3. Find the displacement, ,  of the particle in terms of  .    (2 marks)

  4. Find the position of the particle when it is at rest for the first time.    (2 marks)

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i.    :

 
 

 

 

ii.  

♦♦ Mean mark 29%
MARKER’S COMMENT: Solution 2 is more efficient here. Using the -1 and +1 limits of trig functions can be very a effective way to calculate max/min values.


 

,  ,  , …

 

 

 

 

iii.   
 
 

 

 

 

iv.  

♦ Mean mark 50%
MARKER’S COMMENT: Many students found    but failed to gain full marks by omitting to find  . Remember that for calculus, angles are measured in radians, NOT degrees!

 

 

 
 

Calculus, EXT1 C1 2010 HSC 2b

The mass of a whale is modelled by

 

where    is measured in tonnes,    is the age of the whale in years and    is a positive constant.

  1. Show that the rate of growth of the mass of the whale is given by the differential equation
     
       (1 mark)

  2. When the whale is 10 years old its mass is 20 tonnes.

     

    Find the value of  ,  correct to three decimal places.    (2 marks)

  3. According to this model, what is the limiting mass of the whale?    (1 mark)

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i. 

IMPORTANT: Know this standard proof well and be able to produce it quickly.

 
 

 

ii.  

 
 

 

iii. 

Calculus, EXT1 C1 2013 HSC 12c

A cup of coffee with an initial temperature of 80°C is placed in a room with a constant temperature of 22°C.

The temperature,  °C,  of the coffee after minutes is given by

,

where  ,    and    are positive constants. The temperature of the coffee drops to 60°C after 10 minutes.

How long does it take for the temperature of the coffee to drop to 40°C?  Give your answer to the nearest minute.    (3 marks)

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IMPORTANT: Students should either keep   in exact form (using their calculator’s memory button), or ensure they take  to a sufficient number of decimal places to reach an accurate answer.


 

 

ALGEBRA TIP: Keep  in the equation right until the last calculation and then substitute in (see Worked Solution).
 
 
 

 

Calculus, EXT1* C1 2008 HSC 5c

Light intensity is measured in lux. The light intensity at the surface of a lake is 6000 lux. The light intensity,    lux, a distance    metres below the surface of the lake is given by

where  ,  and    are constants.

  1. Write down the value of  .   (2 marks)

  2. The light intensity 6 metres below the surface of the lake is 1000 lux. Find the value of  .    (2 marks)

  3. At what rate, in lux per metre, is the light intensity decreasing 6 metres below the surface of the lake?    (2 marks)

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i.   

 

ii.   

MARKER’S COMMENT: Many students used the “log” function on their calculator rather than the function. BE CAREFUL!
 
 

 

iii. 

ALGEBRA TIP: Tidy your working by calculating using and then only substituting for in part (ii) at the final stage.

 

 
 

 

Calculus, EXT1* C1 2009 HSC 6b

Radium decays at a rate proportional to the amount of radium present. That is, if    is the amount of radium present at time  ,  then  ,  where    is a positive constant and    is the amount present at  . It takes 1600 years for an amount of radium to reduce by half.

  1. Find the value of  .   (2 marks)

  2. A factory site is contaminated with radium. The amount of radium on site is currently three times the safe level.

     

    How many years will it be before the amount of radium reaches the safe level.    (2 marks)

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i.   

MARKER’S COMMENT: Students must be familiar with “half-life” and the algebra required. i.e. using within their calculations.

 

 

ii.   

IMPORTANT: Know how to use the memory function on your calculator to store the exact value of found in part (i) to save time.
 
 

 

Calculus, EXT1* C1 2012 HSC 14c

Professor Smith has a colony of bacteria. Initially there are 1000 bacteria. The number of bacteria,  ,  after    minutes is given by

. 

  1. After 20 minutes there are 2000 bacteria.

     

    Show that  correct to four decimal places.   (1 mark)

  2. How many bacteria are there when  ?    (1 mark)

  3. What is the rate of change of the number of bacteria per minute, when  ?     (1 mark)

  4. How long does it take for the number of bacteria to increase from 1000 to 100 000?    (2 marks)

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  4.  
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i.   

 

 

 ii.  

NOTE: Students could have used the exact value of    in parts (ii), (iii) and (iv), which would yield the answers (ii) 64,000, (iii) 2,218, and (iv) 133 minutes.
 
 

 

 

iii.  

MARKER’S COMMENT: This part proved challenging for many students. Differentiation is required to find the rate of change in these type of questions.
 

 

 
 

 

 

iv. 

 
 

 

Calculus, EXT1* C1 2013 HSC 16b

Trout and carp are types of fish. A lake contains a number of trout. At a certain time, 10 carp are introduced into the lake and start eating the trout. As a consequence, the number of trout,  ,  decreases according to

,

where is the time in months after the carp are introduced.

The population of carp,  ,  increases according to  .

  1. How many trout were in the lake when the carp were introduced?    (1 mark)

  2. When will the population of trout be zero?    (1 mark)

  3. Sketch the number of trout as a function of time.     (1 marks)

  4. When is the rate of increase of carp equal to the rate of decrease of trout?    (3 marks)

  5. When is the number of carp equal to the number of trout?    (2 marks)

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  3.  
     
    2UA 2013 HSC 16b Answer
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i.   

MARKER’S COMMENT: A number of students did not equate in this part.

 

ii.   

NOTE: The last line of the solution isn’t necessary but is included as good practice as a check that the answer matches the exact question asked.
 
 

 

 

iii.   2UA 2013 HSC 16b Answer

♦♦ Mean mark 33% for part (iii)

 

iv.   


  

♦♦♦ Mean mark 20%
COMMENT: Students who progressed to received 2 full marks in this part. Show your working!

 
 

 

 
 

 

v.   

♦♦♦ Mean mark 4%!
MARKER’S COMMENT: Correctly applying substitution to exponentials to form a solvable quadratic proved very difficult for almost all students.

 

 
 

Calculus, 2ADV C3 2011 HSC 7b

The velocity of a particle moving along the -axis is given by

,

where is the time in seconds and is the displacement in metres.

  1. Show that the particle is initially at rest.     (1 mark)

  2. Show that the acceleration of the particle is always positive.     (1 mark)

  3. Explain why the particle is moving in the positive direction for all  .     (2 marks)

  4. As  , the velocity of the particle approaches a constant.

     

    Find the value of this constant.     (1 mark) 

  5. Sketch the graph of the particle's velocity as a function of time.     (2 marks)

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i.   

 
 

 

 

MARKER’S COMMENT: Students whose working showed as , tended to score highly in this question.

ii.   

.

.
 

.
 

iii. 

♦♦♦ Mean mark 22%
COMMENT: Students found part (iii) the most challenging part of this question by far.


 

.
 

iv.   ,  


 

 

IMPORTANT: Use previous parts to inform this diagram. i.e. clearly show velocity was zero at    and the asymptote at  . 
v.   

Calculus in the Physical World, 2UA 2011 HSC 7b Answer

Financial Maths, 2ADV M1 2008 HSC 4b

The zoom function in a software package multiplies the dimensions of an image by 1.2.  In an image, the height of a building is 50 mm. After the zoom function is applied once, the height of the building in the image is 60 mm. After the second application, it is 72 mm.

  1. Calculate the height of the building in the image after the zoom function has been applied eight times. Give your answer to the nearest mm.     (2 marks)

  2. The height of the building in the image is required to be more than 400 mm. Starting from the original image, what is the least number of times the zoom function must be applied?     (2 marks)

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i.   
 
 

 

 

MARKER’S COMMENT: Within this GP, note that is the term where the zoom has been applied 8 times.
 

 

 

ii.   
 
 
 
 
 

 

Calculus, EXT1* C1 2013 HSC 14a

The velocity of a particle moving along the -axis is given by  , where is the displacement from the origin in metres and is the time in seconds. Initially the particle is 5 metres to the right of the origin.

  1. Show that the acceleration of the particle is constant.  (1 mark)

  2. Find the time when the particle is at rest.  (1 mark)

  3. Show that the position of the particle after 7 seconds is 26 metres to the right of the origin.  (2 marks)

  4. Find the distance travelled by the particle during the first 7 seconds.   (2 marks)

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  1.  
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i.   

 

 

 ii.  

 

 

 iii.

 
 

 

 

 

 
 

 

 

 iv.  

♦ Mean mark 37%.
IMPORTANT: Students can also draw a number line that shows where a particle starts from, changes direction and finishes in order to reduce errors.

,

 
 

 


 

Financial Maths, 2ADV M1 2009 HSC 8b

One year ago Daniel borrowed $350 000 to buy a house. The interest rate was 9% per annum, compounded monthly. He agreed to repay the loan in 25 years with equal monthly repayments of $2937.

  1. Calculate how much Daniel owed after his first monthly repayment.    (1 mark)

Daniel has just made his 12th monthly repayment. He now owes $346 095. The interest rate now decreases to 6% per annum, compounded monthly.

 

The amount  , owing on the loan after the th monthly repayment is now calculated using the formula  
 

  
where is the monthly repayment, and .   (DO NOT prove this formula.)

  1. Calculate the monthly repayment if the loan is to be repaid over the remaining 24 years (288 months).    (3 marks)

  2. Daniel chooses to keep his monthly repayments at $2937. Use the formula in part (ii) to calculate how long it will take him to repay the $346 095.   (3 marks)

  3. How much will Daniel save over the term of the loan by keeping his monthly repayments at $2937, rather than reducing his repayments to the amount calculated in part (ii)?   (1 mark)

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i.   

 
 

 

 

ii.   

 

MARKER’S COMMENT: Careless setting out and poor handwriting, especially where indexes were involved, was a major contributor to errors in this question.

 

 

 

 

iii. 

♦♦ A poorly answered question.
MARKER’S COMMENT: Many students struggled to handle the exponential and logarithm calculations in this question.
ALGEBRA TIP: Dividing by in part (iii) is equivalent to multiplying by 200, and cleans up working calculations (see Worked Solutions).
 

 

 

 

iv.  

 

Financial Maths, 2ADV M1 2010 HSC 4a

Susanna is training for a fun run by running every week for 26 weeks. She runs 1 km  in the first week and each week after that she runs 750 m more than the previous week, until she reaches 10 km in a week. She then continues to run 10 km each week.

  1. How far does Susannah run in the 9th week?     (1 mark)

  2. In which week does she first run 10 km?     (1 mark)

  3. What is the total distance that Susannah runs in 26 weeks?     (2 marks)

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i.    

 
 

 

 

ii. 

MARKER’S COMMENT: Better responses wrote the formula for the term before clearly substituting in known values and .
 

 

 

iii. 

MARKER’S COMMENT: Many students incorrectly calculated , not taking into account the AP stopped at the 13th term.
 
 
 
 

 

Financial Maths, 2ADV M1 2011 HSC 3a

A skyscraper of 110 floors is to be built. The first floor to be built will cost $3 million. The cost of building each subsequent floor will be $0.5 million more than the floor immediately below.

  1. What will be the cost of building the 25th floor?     (2 marks)

  2. What will be the cost of building all 110 floors of the skyscraper?     (2 marks)

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i.     

 
 

MARKER’S COMMENT: Better responses listed the sequence of terms in the series as illustrated.
 
 

 
 

MARKER’S COMMENT: Better responses stated the formula BEFORE any calculations were performed. This allows markers to allocate part marks to students who had errors in calculation.

 

ii.     
   
 
 
 

 

Financial Maths, 2ADV M1 2012 HSC 15a

Rectangles of the same height are cut from a strip and arranged in a row. The first rectangle has width 10cm. The width of each subsequent rectangle is 96% of the width of the previous rectangle.
 

2012 15a
 

  1. Find the length of the strip required to make the first ten rectangles.     (2 marks)

  2. Explain why a strip of 3m is sufficient to make any number of rectangles.     (1 mark)

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i.   

 

MARKER’S COMMENT: A common error was to find instead of
 
 
 

 

ii.  

 
 

 

Probability, 2ADV S1 2013 HSC 15d

Pat and Chandra are playing a game. They take turns throwing two dice. The game is won by the first player to throw a double six. Pat starts the game.

  1. Find the probability that Pat wins the game on the first throw.     (1 mark)

  2. What is the probability that Pat wins the game on the first or on the second throw?     (2 marks)

  3. Find the probability that Pat eventually wins the game.     (2 marks)

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i.   

 
 

 

ii.  

 

♦♦ Mean mark 33%
MARKER’S COMMENT: Many students did not account for Chandra having to lose when Pat wins on the 2nd attempt.

 

iii. 

 

♦♦♦ Mean mark 8%!
 COMMENT: Be aware that diminishing probabilities and within the Series and Applications are a natural cross-topic combination.

 

 
 
 

 

.