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Geometry and Calculus, EXT1 2009 HSC 7b

A billboard of height    metres is mounted on the side of a building, with its bottom edge    metres above street level. The billboard subtends an angle    at the point  ,    metres from the building.
 

 
 

  1. Use the identity    to show that
  2.  
    1. .   (2 marks)
    2.  
  4. The maximum value of    occurs when    and    is positive.
  5.  
  6. Find the value of    for which    is a maximum.     (3 marks)

 

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Show Worked Solution

(i)   

♦♦ Mean mark data not available for parts of questions although Q7 as a whole scored <30%.
MARKER’S COMMENT: Answers that included a diagram and clearly labelled angles were generally successful.
 
 

 

 

(ii)  

 

 

 
 
 

 

Binomial, EXT1 2009 HSC 6b

  1. Sum the geometric series  
  3.  
  4. and hence show that
     
    1. .   (3 marks)
    2.  
  6. Consider a square grid with    rows and    columns of equally spaced points.
  7.  
    1. 2009 6b
  8. The diagram illustrates such a grid. Several intervals of gradient  , whose endpoints are a pair of points in the grid, are shown. 
  9.  
  10. (1)   Explain why the number of such intervals on the line    is equal to  .   (1 mark)
  11. (2)   Explain why the total number,  , of such intervals in the grid is given by
  12.  
      1. .   (1 mark)
    2.  
  14. Using the result in part (i), show that 
    1. .   (3 marks)

 

 

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  2. (1)
  3. (2)
  4.  
Show Worked Solution

(i)   

♦♦ Exact data unavailable although mean marks for all Q6 was < 25%.
MARKER’S COMMENT: A common mistake was to assume the number of terms in the series was  .

 

 
 
 

 

 

 

 

 

 

(ii)(1)  
 
 
   
     (2)  
 
 
 
 
 

 

 
 

 

♦♦♦ Exact data unavailable although very few completed part (iii) correctly. 
MARKER’S COMMENT: Note that even if part (i) wasn’t solved, the proof as stated in the question can be used to solve part (iii).
(iii)
   
 
 
 
 
 

Calculus, EXT1 C1 2009 HSC 5b

The cross-section of a 10 metre long tank is an isosceles triangle, as shown in the diagram. The top of the tank is horizontal.
 

 
 

When the tank is full, the depth of water is 3 m. The depth of water at time days is metres.   

  1. Find the volume, , of water in the tank when the depth of water is metres.     (1 mark)

  2. Show that the area, , of the top surface of the water is given by  .   (1 mark)

  3. The rate of evaporation of the water is given by  , where is a positive constant. 

     

    Find the rate at which the depth of water is changing at time .   (2 marks)

  4. It takes 100 days for the depth to fall from 3 m to 2 m. Find the time taken for the depth to fall from 2 m to 1 m.   (1 mark)

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  1. ³
Show Worked Solution
MARKER’S COMMENT: Students who drew a diagram and included their working calculations on it were the most successful.
(i) 
 
 
  ²

 

 
  ³

 

(ii)   
 
  ²

 

(iii)   
   

 

MARKER’S COMMENT: Half marks awarded for stating an appropriate chain rule, even if the following calculations were incorrect. Show your working!
 
 

 

 

♦♦♦ Exact data for part (iv) not available.
COMMENT: Interpreting a constant rate of change was very poorly understood!
(iv)   
 

 

Plane Geometry, 2UA 2014 HSC 15b

In  , a point    is chosen on the side  . The length of    is  , and the length of    is  . The line through    parallel to    meets    at  . The line through    parallel to    meets    at  .

The area of    is  . The areas of    and    are    and    respectively.

  1. Show that    is similar to  .    (2 marks)
  2. Explain why  .    (1 mark)
  3.  
  4. Show that  
    1. .  (2 marks)
    2.  
  5. Using the result from part (iii) and a similar expression for  
    1. , deduce that  .   (2 marks)

 

 

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  4.  
  7.  
Show Worked Solution

(i)   

 

(ii) 

 

♦♦ Mean mark 27%
COMMENT: The critical step in solving part (iii) is realising that you need the areas of 2 non-right angled triangles and therefore the formula  ½ is required.

(iii) 

 
 
 

 

(iv) 

 

 

♦♦ Mean mark 26%
 

 

 

 
 

Trigonometry, 2ADV T2 2014 HSC 7 MC

How many solutions of the equation    lie between    and  ?

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♦♦♦ Mean mark 25%, making it the toughest MC question in the 2014 exam.
COMMENT: Note that the “2 solutions” answer relies on the sum of an infinity of zeros not equalling zero. This concept created unintended difficulty in this question.


 


  

Statistics, STD2 S4 2014* HSC 30b

The scatterplot shows the relationship between expenditure per primary school student, as a percentage of a country’s Gross Domestic Product (GDP), and the life expectancy in years for 15 countries.
 

 
 

  1. For the given data, the correlation coefficient,  , is 0.83. What does this indicate about the relationship between expenditure per primary school student and life expectancy for the 15 countries?   (1 mark)

  2. For the data representing expenditure per primary school student,    is 8.4 and    is 22.5.

     

    What is the interquartile range?   (1 mark)

  3. Another country has an expenditure per primary school student of 47.6% of its GDP.

     

    Would this country be an outlier for this set of data? Justify your answer with calculations.   (2 marks)

  4. On the scatterplot, draw the least-squares line of best fit  .    (2 marks)

  5. Using this line, or otherwise, estimate the life expectancy in a country which has an expenditure per primary school student of 18% of its GDP.   (1 mark)

  6. Why is this line NOT useful for predicting life expectancy in a country which has expenditure per primary school student of 60% of its GDP?   (1 mark)

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  1.  

  4.  

  7.  

  8.  

Show Worked Solution
i.
 

 

ii.
   
   

 

♦ Mean mark 35% 

iii.   

 

iv.  

v. 

♦♦ Mean mark 39%

 

  

♦♦♦ Mean mark 0%. The toughest question on the 2014 paper.
COMMENT: Examiners regularly ask students to identify and comment on outliers where linear relationships break down.
vi.  
 
 
 

Algebra, STD2 A4 2014 HSC 29a

The cost of hiring an open space for a music festival is  $120 000. The cost will be shared equally by the people attending the festival, so that    (in dollars) is the cost per person when    people attend the festival.

  1. Complete the table below by filling in the THREE missing values.   (1 mark)
  2. Using the values from the table, draw the graph showing the relationship between    and  .   (2 marks)
     
  3. What equation represents the relationship between and ?   (1 mark)

  4. Give ONE limitation of this equation in relation to this context.   (1 mark)

  5. Is it possible for the cost per person to be $94? Support your answer with appropriate calculations.   (1 mark)

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i.   


 

ii. 

iii.   


 

iv.  

 

 
 

v.   

 

 

Show Worked Solution

i.   


 

ii. 

 

♦ Mean mark (iii) 48%

iii.   

 

♦♦♦ Mean mark (iv) 7%
COMMENT: When asked for limitations of an equation, look carefully at potential restrictions with respect to both the domain and range.

iv.  

 

 

 

v.   

 
♦ Mean mark (v) 38%

 

FS Driving, 2UG 2014 HSC 27a

Alex is buying a used car which has a sale price of  $13 380. In addition to the sale price there are the following costs:

2014 27a1

  1. Stamp Duty for this car is calculated at $3 for every $100, or part thereof, of the sale price.  
  2. Calculate the Stamp Duty payable.   (1 mark)
  3.  
  4. Alex borrows the total amount to be paid for the car including Stamp Duty and transfer of registration. Interest on the loan is charged at a flat rate of  7.5%  per annum. The loan is to be repaid in equal monthly instalments over 3 years.  
  5.  
    Calculate Alex’s monthly repayments.   (4 marks)
  6.  
  7. Alex wishes to take out comprehensive insurance for the car for 12 months. The cost of comprehensive insurance is calculated using the following: 
    1.  
    2. 2014 27a2
  8. Find the total amount that Alex will need to pay for comprehensive insurance.   (3 marks)
  9.  
  10. Alex has decided he will take out the comprehensive car insurance rather than the less expensive non-compulsory third-party car insurance.
  11. What extra cover is provided by the comprehensive car insurance?   (1 mark)

 

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  6.  
Show Worked Solution
♦♦♦ Mean mark 12%
IMPORTANT: “or part thereof ..” in the question requires students to round up to 134 to get the right multiple of $3 for their calculation.
(i)   
 

 

(ii)   
   

 

 
 

 

 

 

 
 

 

(iii)  

 
 
 
 

 

 

 

♦ Mean mark 34%.
(iv)  
 

Statistics, STD2 S1 2014 HSC 26e

The times taken for 160 music downloads were recorded, grouped into classes and then displayed using the cumulative frequency histogram shown.   
 

 

 
On the diagram, draw the lines that are needed to find the median download time.   (2 marks)

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Show Worked Solution
♦♦♦ Mean mark 17%.

HSC 2014 26ei

Probability, STD2 S2 2014 HSC 16 MC

In Mathsville, there are on average eight rainy days in October.

Which expression could be used to find a value for the probability that it will rain on two consecutive days in October in Mathsville?

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Show Worked Solution

♦♦♦ Mean mark 16%.
Lowest mark of any MC question in 2014!

Quadratic, EXT1 2014 HSC 13c

The point    lies on the parabola    with focus  .

The point    divides the interval    internally in the ratio  .

2014 13c 

  1. Show that the coordinates of    are  
  2.  and  .  (2 marks)
  4. Express the slope of    in terms of  .    (1 mark)
  5. Using the result from part (ii), or otherwise, show that    lies on a fixed circle of radius  .   (3 marks)

 

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Show Worked Solution

(i)  

 
 

 

(ii)   
   
   
   

 

(iii)  
 

 

 

♦♦ Mean mark 22%

 

 

 
 
 
 

 

Binomial, EXT1 2011 HSC 7b

The binomial theorem states that 
 

  for all integers  . 
 

  1. Show that    
    1. .   (2 marks)
    2.  
  2. By differentiating the result from part (i), or otherwise, show that
    1. .    (2 marks)
    2.  
  3. Assume now that    is even. Show that, for  , 
    1. .  
    2. (3 marks)

 

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Show Worked Solution
♦ Mean mark 36%. 
MARKER’S COMMENT: Better responses provided a narrative through the solution such as “Differentiate both sides” and “Mult. by ”, as seen in the solutions.

(i) 


 


 

 

 

(ii)  

♦♦ Mean mark 36%.

 


 


 

 

 

(iii)  

♦♦♦ Toughest question in the 2011 exam. Mean mark of 2%. The lowest mean of any Ext1 question since that data became available post-2009!

 

 

 

 

 

 

 

Polynomials, EXT1 2013 HSC 14c

The equation     has an approximate solution  

  1. Use one application of Newton’s method to show that    is another approximate solution of  .   (2 marks)
  3. Hence, or otherwise, find an approximation to the value of    for which the graphs    and    have a common tangent at their point of intersection.     (3 marks)
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Show Worked Solution
MARKER’S COMMENT: Many students had difficulty expressing a valid function while too many others carelessly used rather than 0.5.
(i)
 
 
 

 

 
 
 

 

(ii)
 
♦♦♦ 2nd toughest question on 2013 paper with mean mark 15%.
MARKER’S COMMENT: Few students used the common tangent information to equate the derivative functions.
 

 

 
 
 

Binomial, EXT1 2013 HSC 14b

  1. Write down the coefficient of    in the binomial expansion of  .    (1 mark)
  2. Show that  
    1. .   (2 marks)
       
  3. It is known that  
     

    1.  
      1.     .   (Do NOT prove this.)
      2.  
  4. Show that
  5.  
    1. .   (3 marks)

 

 

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  2.  
Show Worked Solution

(i) 

 

(ii)  

♦♦ Mean mark 30%.

 
 

 

(iii)  

 

 

♦♦♦ Toughest question in the 2013 exam with Mean mark 12%!
MARKER’S COMMENT: Simply stating received 1 full mark, showing that even initial working can gain marks.
 

 

 

Combinatorics, EXT1 A1 2010 HSC 7c

  1. A box contains    identical red balls and    identical blue balls. A selection of    balls is made from the box, where  .

     

    Explain why the number of possible colour combinations is  .   (1 mark)

  2. Another box contains    white balls labelled consecutively from to  .  A selection of    balls is made from the box, where  .

     

    Explain why the number of different selections is  .   (1 mark)

  3. The    red balls, the    blue balls and the    white labelled balls are all placed into one box, and a selection of    balls is made.

     

    Using the identity,   or otherwise, show that the number of different selections is  .    (3 marks)

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Show Worked Solution
♦♦♦ Mean mark 4%.
COMMENT: Solving this part required high level logical reasoning which proved extremely challenging for the vast majority of candidates.
i.   
 
 

 


 

 

ii.   

♦♦♦ Mean mark 18%.
 

 
 

 

♦♦♦ Mean mark part (iii) 2%. Beast alert – equal lowest mean mark of any part of any question since this data has been available post-2009.

iii.   

   

   

 

 
 

 
 
 
 

Plane Geometry, EXT1 2010 HSC 5c

In the diagram,    is tangent to both the circles at  .

The points    and    are on the larger circle, and the line    is tangent to the smaller circle at  . The line    intersects the smaller circle at  .

 

Plane Geometry, EXT1 2010 HSC 5c

Copy or trace the diagram into your writing booklet 

  1. Explain why     (1 mark)
  2. Explain why     (1 mark)
  3. Hence show that    bisects  .    (2 marks)
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Show Worked Solution
(i)    Plane Geometry, EXT1 2010 HSC 5c Answer

 

♦ Mean mark 47%
(ii)   
   

 

(iii)  

 

♦♦ Mean mark 22%
IMPORTANT: Recognising that angles in the alternate segment are equal is an examiner favourite. LOOK FOR IT!

 

Calculus, EXT1 C1 2011 HSC 7a

The diagram shows two identical circular cones with a common vertical axis.  Each cone has height cm and semi-vertical angle 45°.
 

2011 7a

The lower cone is completely filled with water. The upper cone is lowered vertically into the water as shown in the diagram. The rate at which it is lowered  is given by  

,

where cm is the distance the upper cone has descended into the water after seconds.

As the upper cone is lowered, water spills from the lower cone. The volume of water remaining in the lower cone at time is  cm³.

  1. Show that  .   (1 mark)

  2. Find the rate at which is changing with respect to time when  .     (2 marks)

  3. Find the rate at which is changing with respect to time when the lower cone has lost    of its water. Give your answer in terms of .   (2 marks)

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  2. ³
  3. ³
Show Worked Solution

i.   

 ♦ Mean mark 42% 

 

 
 

 

 

ii. 

 

  

  ³

 

iii. 

♦♦♦ Mean mark 12%
MARKER’S COMMENT: Many unsuccessful answers attempted to find an alternate version of . Part (ii) directed students directly toward the correct strategy.

   

  ³

 
³

Statistics, EXT1 S1 2011 HSC 6c

A game is played by throwing darts at a target. A player can choose to throw two or three darts.

Darcy plays two games. In Game 1, he chooses to throw two darts, and wins if he hits the target at least once. In Game 2, he chooses to throw three darts, and wins if he hits the target at least twice.

The probability that Darcy hits the target on any throw is  , where  .

  1. Show that the probability that Darcy wins Game 1 is  .    (1 mark)

  2. Show that the probability that Darcy wins Game 2 is  .     (1 mark)

  3. Prove that Darcy is more likely to win Game 1 than Game 2.    (2 marks)

  4. Find the value of    for which Darcy is twice as likely to win Game 1 as he is to win Game 2.    (2 marks)

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Show Worked Solution

i. 

♦ Mean mark 47%.


 

 
 
 

 

ii. 

♦ Mean mark 40%.

 
 

 

♦♦♦ Mean mark 17%.
COMMENT: It is critical for students to utilise the limits of    to answer part (iii).

iii. 

 
   
   

 


 

iv. 

♦♦ Mean mark 21%.
MARKER’S COMMENT: BE CAREFUL in formulating your equation here. Many students multiplied the wrong side by 2.
 

  

 
 

 

Trig Ratios, EXT1 2011 HSC 5a

In the diagram,    is a point on the unit circle    at an angle    from the positive  -axis, where  . The line through    and    intersects the line    at  . The points    and   are on the  -axis.
 

 
 

  1. Use the fact that    and    are similar to show that
     
  2.      .  (2 marks)
  3.  
  4. Show that  .    (1 mark)
  5.  
  6. Show that  .   (1 mark)
  7.  
  8. Hence, or otherwise, show that  .   (1 mark)
  9.  
  10. Hence, or otherwise, solve  , where  .   (2 marks)

 

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Show Worked Solution
(i)

 

♦ Mean mark 41%
MARKER’S COMMENT: When questions direct you to use a certain fact for a proof, use that fact!

 
 

 

(ii)   

♦ Mean mark 35%
 
 
 
 
 

 

(iii)  

♦♦♦ Mean mark 11%

 
 
 

 

(iv)  

♦♦ Mean mark 28%

 

 
 
 

 

 

♦ Mean mark 45%
(v)   
 
 

Plane Geometry, EXT1 2011 HSC 4b

In the diagram, the vertices of    lie on the circle with centre  . The point    lies on    such that    is isosceles and  .

Copy or trace the diagram into your writing booklet. 

  1. Explain why  .    (1 mark)
  2. Prove that    is a cyclic quadrilateral.    (2 marks)
  3. Let    be the midpoint of    and    the centre of the circle through  and  . 
  4. Show that    and    are collinear.   (1 mark)
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  3.  
Show Worked Solution
(i)

 

♦ Mean mark 38%
STRATEGY: Proving part (ii) by showing opposite angles are supplementary also worked but was more time consuming.

(ii)  

 

 

(iii)

 

♦♦♦ Mean mark 7%. 2nd hardest question in the 2011 paper!

 

 

Mechanics, EXT2* M1 2011 HSC 3a

The equation of motion for a particle undergoing simple harmonic motion is 

 ,

where is the displacement of the particle from the origin at time , and is a positive constant.

  1. Verify that  , where and are constants, is a solution of the equation of motion.    (1 mark)

  2. The particle is initially at the origin and moving with velocity . 

     

    Find the values of and in the solution  .    (2 marks)

  3. When is the particle first at its greatest distance from the origin?   (1 mark)

  4. What is the total distance the particle travels between    and  ?   (1 mark)

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Show Worked Solution
i.  
 
 
   
   

 

ii.   

  
♦♦ Mean mark part (iii) 47%
 

iii.   

 

 

iv. 

♦♦ Mean mark 22%
MARKER’S COMMENT: Many students found the displacement at rather than the distance travelled while another common error found distance as 2 x amplitude.

 

 

Calculus, EXT1 C1 2012 HSC 14c

A plane takes off from a point . It flies due north at a constant angle to the horizontal. An observer is located at , 1 km from , at a bearing 060° from .

Let km be the distance from to the plane and let km be the distance from the observer to the plane. The point is on the ground directly below the plane.
 

2012 14c
 

  1. Show that  .   (3 marks)

  2. The plane is travelling at a constant speed of 360 km/h.
  3. At what rate, in terms of  , is the distance of the plane from the observer changing 5 minutes after take-off?    (2 marks)

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Show Worked Solution

i. 

♦ Mean mark 42%
IMPORTANT: Students should always be looking for opportunities to use the identity to clean up calculations with trig functions.

 

 

 

 
 

 

ii.  

♦♦♦ Mean mark 14%

 

 

 

 

Combinatorics, EXT1 A1 2012 HSC 11f

 

  1. Use the binomial theorem to find an expression for the constant term in the expansion of 
     
         .   (2 marks)

  2. For what values of    does    have a non-zero constant term?    (1 mark)

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Show Worked Solution

i. 

 


 

 

 
 

 

ii. 

♦♦♦ Mean mark 16%.

 


 

Calculus, 2ADV C3 2009 HSC 10

  1. Show that the graph of    has no turning points.   (2 marks)

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

  3. i. Show that  for  .   (1 mark)

     

    ii. Let  .

     

        Use the result in part c.i. to show that    for all  .   (2 marks)

  1. Sketch the graphs of    and    for  .    (2 marks)

  2. Show that  .   (2 marks)

  3. Find the area enclosed by the graphs of    and  , and the straight line  .   (2 marks)

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  3. i. 

     

    ii.  

  4.  
        Geometry and Calculus, 2UA 2009 HSC 10 Answer
  6. ²
Show Worked Solution
a.   
♦♦ Mean mark 28% for all of Q10 (note that data for each question part is not available).
 

 
 

 

 

b.   

 

 
 

 

c.i.     
 
 

 

c.ii.  
 
 

MARKER’S COMMENT: When 2 graphs are drawn on the same set of axes, you must label them. 
 

d. 

Geometry and Calculus, 2UA 2009 HSC 10 Answer
e.   
 
 
 
 

 

f.   
   
   
   
   
   
    ²

Probability, 2ADV S1 2009 HSC 9a

Each week Van and Marie take part in a raffle at their respective workplaces.
The probability that Van wins a prize in his raffle is  . The probability that Marie wins a prize in her raffle is  .  

What is the probability that, during the next three weeks, at least one of them wins a prize?   (2 marks)

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Calculus, 2ADV C3 2009 HSC 8a

Geometry and Calculus, 2UA 2009 HSC 8a

The diagram shows the graph of a function  . 

  1. For which values of    is the derivative,  , negative?    (1 mark)

  2. What happens to    for large values of  ?    (1 mark)

  3. Sketch the graph  .     (2 marks)

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  1.  


  2.  

  3.   
    Geometry and Calculus, 2UA 2009 HSC 8a Answer

Show Worked Solution

i.   

♦♦ Exact data not available.

 

ii.   

 

♦♦ Exact data not available.
MARKER’S COMMENT: Poorly drawn graphs with axes not labelled and inaccurate scales were common.

iii.

  Geometry and Calculus, 2UA 2009 HSC 8a Answer

Trigonometry, 2ADV T3 2009 HSC 7b

Between 5 am and 5 pm on 3 March 2009, the height, , of the tide in a harbour was given by

where    is in metres and    is in hours, with    at 5 am. 

  1. What is the period of the function  ?    (1 mark)

  2. What was the value of    at low tide, and at what time did low tide occur?     (2 marks)

  3. A ship is able to enter the harbour only if the height of the tide is at least 1.35 m.

     

    Find all times between 5 am and 5 pm on 3 March 2009 during which the ship was able to enter the harbour.    (3 marks)

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Show Worked Solution

i.   

 
 

 

 

ii. 

IMPORTANT: Using for a minimum here is very effective and time efficient. This property of trig functions is often very useful in harder questions.

 

 

 

 


iii. 

   

 

Trig Calculus, 2UA 2009 HSC 7b Answer

 

Calculus, 2ADV C1 2009 HSC 6c

The diagram illustrates the design for part of a roller-coaster track. The section    is a straight line with slope 1.2, and the section    is a straight line with slope  – 1.8. The section    is a parabola  . The horizontal distance from the  -axis to    is 30 m.
 

2009 6c
 

In order that the ride is smooth, the straight line sections must be tangent to the parabola at    and at  .  

  1. Find the values of    and    so that the ride is smooth.   (3 marks)

  2. Find the distance  , from the vertex of the parabola to the horizontal line through  , as shown on the diagram.     (2 marks)

Show Answers Only
Show Worked Solution
i.   
 
 
 
♦♦♦ Exact data unavailable.
MARKER’S COMMENT: Successful students equated the curve gradient to the straight section as a requirement for a smooth ride.
 


 

 

 

 

 

 

ii.   
 

 

 

 

 

 
 

 

 
 

 

 

Plane Geometry, 2UA 2009 HSC 4c

In the diagram,    is a right-angled triangle, with the right angle at  . The midpoint of    is  , and  .

Plane Geometry, 2UA 2009 HSC 4c_1

Copy or trace the diagram into your writing booklet. 

  1. Prove that    is similar to  .   (2 marks)
  2. What is the ratio of    to  ?    (1 mark)
  3. Prove that    is isosceles.   (2 marks)
  4. Show that    can be divided into two isosceles triangles.   (1 mark)
  5. Plane Geometry, 2UA 2009 HSC 4c_2
  6. Copy or trace this triangle into your writing booklet and show how to divide it into four isosceles triangles.   (1 mark)
  7.  
Show Answers Only
  5. Plane Geometry, 2UA 2009 HSC 4c_2 Answer1
Show Worked Solution
(i)   
 
 
 

 

(ii)  
   

 

 

(iii)

Plane Geometry, 2UA 2009 HSC 4c_1 Answer
 

 

(iv)  
   
   

 

(v)

Plane Geometry, 2UA 2009 HSC 4c_2 Answer1
♦♦ Mean mark 25%.
MARKER’S COMMENT: Use a ruler, draw large diagrams and always pay careful attention to previous parts of the question!

Calculus, EXT1* C3 2010 HSC 10b

The circle    has radius and centre . The circle meets the positive -axis at . The point is on the interval . A vertical line through meets the circle at . Let  .
  

2010 10b1

  1. The shaded region bounded by the arc and the intervals and is rotated about the -axis. Show that the volume, , formed is given by
  2.   (3 marks)

  3. A container is in the shape of a hemisphere of radius metres. The container is initially horizontal and full of water. The container is then tilted at an angle of to the horizontal so that some water spills out. 

  1. (1) Find so that the depth of water remaining is one half of the original depth.   (1 mark)

  2. (2) What fraction of the original volume is left in the container?   (2 marks)

Show Answers Only
  2. (1)
  3. (2)
Show Worked Solution

i.   

♦♦♦ Mean mark (i) 16%.
MARKER’S COMMENT: A common error was to integrate to instead of (note that is a constant).  

 

 
 
 
 
 

 

ii. (1)

♦♦♦ Part (ii) mean marks 3% and 2% for (ii)(1) and (ii)(2) respectively.

 

 MARKER’S COMMENT: Previous parts of a question should always be at the front and centre of a student’s mind and direct their strategy.
ii. (2)   
   

 

 
 
 
 

 

Plane Geometry, 2UA 2010 HSC 10a

In the diagram    is an isosceles triangle with  . The point    on the interval    is chosen so that  . Let  ,    and  .
 

 
 

  1. Show that    is similar to  .    (2 marks)
  2. Show that  .     (1 mark)
  3. Show that  .   (2 marks)
  4. Deduce that  .   (1 mark) 
Show Answers Only
Show Worked Solution
MARKER’S COMMENT: When dealing with multiple triangles, the best performing students label all angles clearly and unambiguously.
(i)


 

 

♦♦ Mean mark 25%.
MARKER’S COMMENT: The similarity proof in part (i) should have immediately alerted students to using similar ratios of sides to solve part (ii). Many did not follow this obvious hint.

(ii) 


 

 

 

(iii) 

 
 
 

 

(iv) 

♦♦♦ Parts (iii) and (iv) mean marks – 18% and just 4% respectively.
IMPORTANT: Limits of trig functions are often extremely useful in proving inequalities (see Worked Solutions).

 

Calculus, 2ADV C3 2010 HSC 9b

Let    be a function defined for  , with  . 

The diagram shows the graph of the derivative of  ,  . 

The shaded region    has area    square units. The shaded region    has area    square units. 

  1. For which values of    is    increasing?  (1 mark)

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

  3. Find the value of  .   (1 mark)

  4. Draw a graph of    for  .   (2 marks)

Show Answers Only
  4.  
  5.  
Show Worked Solution
i.   
 
 

 

ii.  
 
 
 
 
 

 

♦♦♦ Parts (ii) and (iii) proved particularly difficult for students with mean marks of 12% and 11% respectively.

 

 

iii.  
   

 
 
 

 

 

  

♦♦ Mean mark 28%
EXAM TIP: Clearly identify THE EXTREMES when given a defined domain. In this case, the origin is obvious graphically, and the other extreme at , is CLEARLY LABELLED! 
iv.  2UA HSC 2010 9bi

Financial Maths, 2ADV M1 2011 HSC 9d

 

  1. Rationalise the denominator in the expression    where    is an integer and  .   (1 mark)

  2. Using your result from part (i), or otherwise, find the value of the sum
     
            (2 marks)

Show Answers Only
Show Worked Solution

i. 

MARKER’S COMMENT: Students connecting the information between parts (i) and (ii) were easily the most successful. AGAIN, the information from earlier parts of multi-part questions is gold plated information for directing your answer.

 

ii. 

♦♦♦ Mean mark 15%.

 

Calculus, 2ADV C4 2011 HSC 9b

A tap releases liquid    into a tank at the rate of    litres per minute, where    is time in minutes. A second tap releases liquid    into the same tank at the rate of    litres per minute. The taps are opened at the same time and release the liquids into an empty tank. 

  1. Show that the rate of flow of liquid    is greater than the rate of flow of liquid    by    litres per minute.    (1 mark)

  2. The taps are closed after 4 minutes. By how many litres is the volume of liquid    greater than the volume of liquid    in the tank when the taps are closed?    (2 marks)

Show Answers Only
Show Worked Solution
♦♦♦ Mean mark 16%
MARKER’S COMMENT: Many students incorrectly differentiated in part (i). The 1 mark allocation indicates the answer will not require an involved multi-step process.

i.   

 
 
 
 
 

 

ii.   

♦♦♦ Mean mark 15%
MARKER’S COMMENT: Few students were able to answer this part. Previous parts of any question should be front and centre of your thinking when working out strategies.


 

Plane Geometry, 2UA 2011 HSC 9a

The diagram shows  , where    is the midpoint of    and    is the midpoint of  . The intervals    and    meet at  .

Plane Geometry, 2UA 2011 HSC 9a 

 

  1. Explain why    is similar to  .    (1 marks)
  2. Hence, or otherwise, prove that the ratio  .     (2 marks)
Show Answers Only
Show Worked Solution

(i) 

 

♦♦ Mean mark 22%.
TIP: Proving similarity via (AAA) test only requires proving 2 angles are equal because the remaining angle then must be equal – this will help save time in proofs.
(ii)

Plane Geometry, 2UA 2011 HSC 9a Answer
 

 

 

Calculus, 2ADV C3 2012 HSC 14a

A function is given by  . 

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

  2. Hence, sketch the graph     showing the stationary points.   (2 marks)

  3. For what values of    is the function increasing?   (1 mark)

  4. For what values of    will    have no solution?   (1 mark)

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  2.  
        
        2UA HSC 2012 14ai
     
Show Worked Solution
i.
 
 

 

 

 

 

 

 

 
 
 

 

ii.  2UA HSC 2012 14ai
♦ Mean mark 42%
MARKER’S COMMENT: Be careful to use the correct inequality signs, and not carelessly include ≥ or ≤ by mistake.

 

iii.
 

 

iv.   

♦♦♦ Mean mark 12%.

Calculus, EXT1* C3 2011 HSC 8b

The diagram shows the region enclosed by the parabola  , the  -axis and the line  , where  . This region is rotated about the  -axis to form a solid called a paraboloid. The point    is the intersection of   and  .

The point    has coordinates  .
 

2011 8b

  1. Find the exact volume of the paraboloid in terms of  .    (2 marks)

  2. A cylinder has radius    and height  .    

     

    What is the ratio of the volume of the paraboloid to the volume of the cylinder?   (1 mark)

Show Answers Only
  1. ³
Show Worked Solution
IMPORTANT: Most common errors: 1-use the correct axis, and 2-check the limits!
i.   
   
   
   
    ³

 
³
 

ii.   

♦♦ Mean mark of 24%.

 

 
 

 

Plane Geometry, 2UA 2013 HSC 16c

The diagram shows triangles    and    with    parallel to  . The sides    and    intersect at  . The point    lies on    such that    is parallel to    and  .

2UA 2013 HSC 16c

  1. Prove that    is similar to  .   (2 marks)
  2. Hence, or otherwise, prove that  .   (2 marks)
Show Answers Only
Show Worked Solution
(i)   2UA 2013 HSC 16c Answer

COMMENT: A great example of easy marks that appear in the early stages of the final questions.

 

(ii)  

♦♦♦ Mean mark 13%.
COMMENT: The presence of in the required proof should alert students to the strategy of finding another set of congruent triangles where this side can be utilised.

   
   

Functions, 2ADV F2 2013 HSC 15c

  1. Sketch the graph  .   (1 mark)

  2. Using the graph from part (i), or otherwise, find all values of    for which the equation    has exactly one solution.   (2 marks)

Show Answers Only
  1.  
    2UA 2013 HSC 15c Answer
Show Worked Solution

i. 

♦ Mean mark 49%
MARKER’S COMMENT: Many students drew diagrams that were “too small”, didn’t use rulers or didn’t use a consistent scale on the axes!

2UA 2013 HSC 15c Answer

 

ii.

   2UA 2013 HSC 15c1 Answer

 

♦♦ Mean mark 25%.
COMMENT: Students need a clear graphical understanding of what they are finding to solve this very challenging, Band 6 question.

 
 

  

 


 

Integration, 2UA 2013 HSC 15a

The diagram shows the front of a tent supported by three vertical poles. The poles are 1.2 m apart. The height of each outer pole is 1.5 m, and the height of the middle pole is 1.8 m. The roof hangs between the poles.

2013 15a

The front of the tent has area ². 

  1. Use the trapezoidal rule to estimate  .    (1 mark)
  2. Use Simpson’s rule to estimate  .     (1 mark)
  3. Explain why the trapezoidal rule gives the better estimate of  .    (1 mark)
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  1. ²
  2. ²
  3. 2UA 2013 15a ans
Show Worked Solution
(i)   
   
   
    ²

 

(ii)   
   
   
    ²

 

♦♦♦ This proved the hardest mark to get in the whole 2013 HSC paper with a mean mark of just 3%!
MARKER’S COMMENT: Ensure you understand the different lines that result from using Simpson’s Rule vs the Trapezoidal Rule. Drawing a sketch is often the clearest way to explain (see Worked Solutions).

 

(iii)  
    2UA 2013 15a ans

Calculus, 2ADV C4 2013 HSC 14d

The diagram shows the graph  .
 

2013 14d
 

What is the value of  , where  , so that  ?   (1 mark)  

Show Answers Only

 

Show Worked Solution

 ♦♦♦ A devilish 1-mark question mid-paper that had a mean mark of just 12%. 
MARKER’S COMMENT: The fact that this question was worth only 1 mark means that it is not necessary for students to show any detailed working.

 
   

 

Calculus, EXT1* C1 2013 HSC 10 MC

A particle is moving along the  -axis. The displacement of the particle at time    seconds is    metres.

At a certain time,    and  .

Which statement describes the motion of the particle at that time?

  1. The particle is moving to the right with increasing speed.
  2. The particle is moving to the left with increasing speed.
  3. The particle is moving to the right with decreasing speed.
  4. The particle is moving to the left with decreasing speed.
Show Answers Only

Show Worked Solution
♦♦ Mean mark 26%.

-1

-2

Financial Maths, STD2 F4 2011 HSC 28b

Norman and Pat each bought the same type of tractor for $60 000 at the same time. The value of their tractors depreciated over time.

The salvage value , in dollars, of each tractor, is its depreciated value after years.

Norman drew a graph to represent the salvage value of his tractor.
 

 2011 28b

  1. Find the gradient of the line shown in the graph.   (1 mark)

  2. What does the value of the gradient represent in this situation?   (1 mark)

  3. Write down the equation of the line shown in the graph.   (1 mark)

  4. Find all the values of that are not suitable for Norman to use when calculating the salvage value of his tractor. Explain why these values are not suitable.   (2 marks)

Pat used the declining balance formula for calculating the salvage value of her tractor. The depreciation rate that she used was 20% per annum.

  1. What did Pat calculate the salvage value of her tractor to be after 14 years?   (2 marks)

  2. Using Pat’s method for depreciation, describe what happens to the salvage value of her tractor for all values of greater than 15.   (1 mark)

Show Answers Only



  7.  
Show Worked Solution
♦♦♦ Mean mark 14%
COMMENT: The intercepts of both axes provide points where the gradient can be quickly found.
i.   
   
   

 

♦ Mean mark 37%

ii.   

 

♦♦ Mean mark 28%
COMMENT: Using the general form is quick here because you have the gradient (from part (i)) and the -intercept is obviously .
iii.  
 
 

 

iv.    

♦♦♦ Mean mark 20%

 

v.   
 
 

 

 

♦ Mean mark 37%
vi.  
 

Statistics, STD2 S1 2011 HSC 25a

A study on the mobile phone usage of NSW high school students is to be conducted.

Data is to be gathered using a questionnaire.

The questionnaire begins with the three questions shown.

2UG 2011 25a

  1. Classify the type of data that will be collected in Q2 of the questionnaire.  (1 mark)

  2. Write a suitable question for this questionnaire that would provide discrete ordinal data.   (1 mark)

  3. An initial study is to be conducted using a stratified sample.

     

    Describe a method that could be used to obtain a representative stratified sample.  (1 mark)

  4. Who should be surveyed if it is decided to use a census for the study?  (1 mark)

Show Answers Only
Show Worked Solution

i. 

 

ii. 

 

iii. 

♦♦♦ Mean mark 7%. Toughest mark to get in the 2011 exam!
COMMENT: Know and be able to describe random, systematic and stratified sampling!

 

♦♦♦ Mean mark 18%.
MARKER’S COMMENT: A specific population needed (i.e. high school students).

iv. 

Algebra, STD2 A2 2010 HSC 27c

The graph shows tax payable against taxable income, in thousands of dollars.

2010 27c

  1. Use the graph to find the tax payable on a taxable income of $21 000.  (1 mark)

  2. Use suitable points from the graph to show that the gradient of the section of the graph marked    is  .    (1 mark)

  3. How much of each dollar earned between  $21 000  and  $39 000  is payable in tax?   (1 mark)

  4. Write an equation that could be used to calculate the tax payable, , in terms of the taxable income, , for taxable incomes between  $21 000  and  $39 000.   (2 marks)

Show Answers Only
Show Worked Solution
i.

 

 

ii. 

♦♦ Mean mark 25%
 
 
 

 

iii. 

♦♦♦ Mean mark 12%!
MARKER’S COMMENT: Interpreting gradients is an examiner favourite, so make sure you are confident in this area.
 

 

iv. 

♦♦♦ Mean mark 15%.
STRATEGY: The earlier parts of this question direct students to the most efficient way to solve this question. Make sure earlier parts of a question are front and centre of your mind when devising strategy.

 
 

Statistics, STD2 S1 2010 HSC 26b

A new shopping centre has opened near a primary school. A survey is conducted to determine the number of motor vehicles that pass the school each afternoon between 2.30 pm and 4.00 pm.

The results for 60 days have been recorded in the table and are displayed in the cumulative frequency histogram.
 

2010 26b

  1. Find the value of  Χ  in the table.   (1 mark)

  2. On the cumulative frequency histogram above, draw a cumulative frequency polygon (ogive) for this data.   (1 mark)
  3. Use your graph to determine the median. Show, by drawing lines on your graph, how you arrived at your answer.   (1 mark)
  4. Prior to the opening of the new shopping centre, the median number of motor vehicles passing the school between  2.30 pm  and  4.00 pm  was 57 vehicles per day.

     

    What problem could arise from the change in the median number of motor vehicles passing the school before and after the opening of the new shopping centre?

     

    Briefly recommend a solution to this problem.   (2 marks)

Show Answers Only
  2.  
  3.  

  7.  

Show Worked Solution
i.
   

 

♦♦♦ Mean mark 18%
MARKER’S COMMENT: The ogive was poorly drawn with many students incorrectly joining the middle of each column rather than from corner to corner.
ii.
♦♦ Mean mark 25%
MARKER’S COMMENT: Many students did not “show by drawing lines on the graph” as the question asked.

iii. 

♦ Mean mark 47%
MARKER’S COMMENT: Short answers were often the best. Be concise when you can.
iv.
 
 
   
 
 
 

Measurement, STD2 M7 2013 HSC 30c

Joel mixes petrol and oil in the ratio  40 : 1  to make fuel for his leaf blower. 

  1. Joel pours 5 litres of petrol into an empty container to make fuel for his leaf blower.

     

    How much oil should he add to the petrol to ensure that the fuel is in the correct ratio?   (1 mark)

  2. Joel has 4.1 litres of fuel left in his container after filling his leaf blower.

     

    He wishes to use this fuel in his lawnmower. However, his lawnmower requires the petrol and oil to be mixed in the ratio  25 : 1.

     

    How much oil should he add to the container so that the fuel is in the correct ratio for his lawnmower?   (3 marks)

Show Answers Only
Show Worked Solution
♦ Mean mark 50% 
i.   
 

 

 

ii. 

♦♦♦ Mean mark 8% 
STRATEGY: The critical information required is how much petrol is in the container. Once known, the oil required to satisfy a 25:1 ration can be easily found.

 

 

 

 

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)

 

Show Answers Only
  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.
 
 

 

 
 
 

Probability, STD2 S2 2009 HSC 28d

In an experiment, two unbiased dice, with faces numbered  1, 2, 3, 4, 5, 6  are rolled 18 times.

The difference between the numbers on their uppermost faces is recorded each time. Juan performs this experiment twice and his results are shown in the tables.

 2009 28d

Juan states that Experiment 2 has given results that are closer to what he expected than the results given by Experiment 1.

Is he correct? Explain your answer by finding the sample space for the dice differences and using theoretical probability.    (4 marks)

 

Show Answers Only

 

Show Worked Solution
♦♦♦ Mean mark 7%. Toughest question in the 2009 exam.
MARKER’S COMMENT: This question guides students by asking for an explanation using the sample space for the dice differences. This step alone received 2 full marks. Note that instructions to explain your answer requires mathematical calculations to support an argument.

2UG-2009-28d1

2UG-2009-28d2_1

2UG-2009-28d3_1

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)

Show Answers Only

  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.
 
 
 

 

 

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)

Show Answers Only
  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

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)

Show Answers Only

  5.  
Show Worked Solution
i.   
   

 

♦♦ Mean mark part ii: 31%
ii.   
 

 

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

Measurement, STD2 M1 2009 HSC 25b

The mass of a sample of microbes is 50 mg. There are approximately   microbes in the sample.

In standard form, what is the approximate mass in grams of one microbe?   (2 marks)

Show Answers Only

Show Worked Solution
♦♦♦ Mean mark 20%.
IMPORTANT: Can you solve: 8 apples weigh 1kg, what does 1 apple weigh? This is exactly the same concept.

 

 
 

Financial Maths, STD2 F4 2013 HSC 28d

Adhele has 2000 shares. The current share price is  $1.50  per share. Adhele is paid a dividend of  $0.30  per share. 

  1. What is the current value of her shares?   (1 mark)
  2. Calculate the dividend yield.    (1 mark)
Show Answers Only
Show Worked Solution
i.   
 

 

 
♦♦♦ Mean mark 13%
MARKER’S COMMENT: A large majority of students had a poor understanding of the term dividend yield.
  

ii.   
   
   

Statistics, STD2 S1 2013 HSC 27c

A retailer has collected data on the number of televisions that he sold each week in 2012.

He grouped the data into classes and displayed the data using a cumulative frequency histogram and polygon (ogive).
 

2013 27c

  1. Use the cumulative frequency polygon to determine the interquartile range.  (2 marks)

  2. Oscar said that the retailer sold 300 televisions in 6 of the weeks in 2012.

     

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

Show Answers Only

  3.  

  4.  

Show Worked Solution

i. 

♦♦♦ Mean mark 13%
COMMENT: Finding median and quartile values from cumulative frequency charts is an important skill that is often examined.

 
 

 

ii. 

♦♦♦ This question proved a beast, with a mean mark of 1%. Toughest question in the 2013 exam!

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)

Show Answers Only
Show Worked Solution
♦♦♦ Mean mark 11%

i.  

 

♦ Mean mark 34%
ii.  
   

Financial Maths, STD2 F4 2010* HSC 22 MC

In July, Ms Alott 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.
 

2010 22 mc 
 

What is the minimum payment due on this account? 

  1.    
  2.    
  3.    
  4.    
Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 20%. Lowest scoring MC question in the 2010 exam.

 
 

 

 

 

Statistics, STD2 S1 2010 HSC 21 MC

The area graph shows the cost and profits for a business over a period of time.

2010 21-1

The information in the area graph is then displayed as a line graph.

Which of the following line graphs best displays the data from the area graph?

2010 21-2

2010 21-3

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♦♦♦ Mean mark 24%
COMMENT: Area graph questions have proven very challenging over recent years. Review this area.

 

Probability, 2UG 2010 HSC 14 MC

A restaurant serves three scoops of different flavoured ice-cream in a bowl. There are five flavours to choose from.

How many different combinations of ice-cream could be chosen?

(A)   

(B)   

(C)   

(D)   

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♦♦♦ Mean mark 21%
COMMENT: Here, we divide by (3x2x1) because the 3 ice-cream scoops are unordered.