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Calculus, 2ADV C3 2014 HSC 16c

The diagram shows a window consisting of two sections. The top section is a semicircle of diameter    m. The bottom section is a rectangle of width    m and height    m.

The entire frame of the window, including the piece that separates the two sections, is made using 10 m of thin metal.

The semicircular section is made of coloured glass and the rectangular section is made of clear glass.

Under test conditions the amount of light coming through one square metre of the coloured glass is 1 unit and the amount of light coming through one square metre of the clear glass is 3 units.

The total amount of light coming through the window under test conditions is    units.

  1. Show that  .   (2 marks)

  2. Show that  .    (2 marks)

  3. Find the values of    and    that maximise the amount of light coming through the window under test conditions.   (3 marks)

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i.   

 

 

♦ Mean mark 35%
ii.   
 
   
   

 

 
 
 
 

 

♦ Mean mark 38%
COMMENT: A sanity check for your answer could be to compare your answers to the perimeter restriction of 10m.
iii.  
 
 

 

 

 

 
 

 

Calculus, 2ADV C3 2014 HSC 15c

The line    is a tangent to the curve    at a point  . 

  1. Sketch the line and the curve on one diagram.   (1 mark)

  2. Find the coordinates of  .     (3 marks)

  3. Find the value of  .   (1 mark)

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  1.   
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i. 

 

ii.
 

 

♦ Mean mark 40%
COMMENT: Given , it follows . Make sure you understand the arithmetic behind this (NB. Simply take the of both sides).

 

 
 

 

 

iii.   

♦♦ Mean mark 30%.

 

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.  
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(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%
 

 

 

 
 

Financial Maths, 2ADV M1 2014 HSC 14d

At the beginning of every 8-hour period, a patient is given 10 mL of a particular drug.

During each of these 8-hour periods, the patient’s body partially breaks down the drug. Only    of the total amount of the drug present in the patient’s body at the beginning of each 8-hour period remains at the end of that period.  

  1. How much of the drug is in the patient’s body immediately after the second dose is given?    (1 mark)

  2. Show that the total amount of the drug in the patient’s body never exceeds 15 mL.     (2 marks)

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i.   

 

 

ii.   

 

 

 
 
 

 

 

Trigonometry, 2ADV T1 2014 HSC 13d

Chris leaves island    in a boat and sails 142 km on a bearing of 078° to island  .  Chris then sails on a bearing of 191° for 220 km to island  , as shown in the diagram.
 

 

  1. Show that the distance from island    to island    is approximately 210 km.    (2 marks)

  2. Chris wants to sail from island    directly to island  . On what bearing should Chris sail? Give your answer correct to the nearest degree.    (3 marks)

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i.   

 


 

 

 

 
 
 

 

ii. 

 
 

 

 

 

Calculus, 2ADV C1 2014 HSC 13c

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

 ,

where    is the displacement from the origin in metres,    is the time in seconds, and  .

  1. Show that the acceleration of the particle is always negative.    (2 marks)

  2. What value does the velocity approach as    increases indefinitely?    (1 mark)

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i.   
   

 

 

 

 

 


 

 

ii.   

 

Calculus, EXT1* C1 2014 HSC 13b

A quantity of radioactive material decays according to the equation 

,

where    is the mass of the material in kg,    is the time in years and    is a constant.

  1. Show that    is a solution to the equation, where    is a constant.   (1 mark)

  2. The time for half of the material to decay is 300 years. If the initial amount of material is 20 kg, find the amount remaining after 1000 years.   (3 marks)

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i.
 
   

 

ii.    

TIP: Many students find it efficient to save the exact value of in the memory function of their calculator for these questions.
 

 

 
 
 
 

Calculus, 2ADV C1 2014 HSC 9 MC

The graph shows the displacement    of a particle moving along a straight line as a function of time  .

2014 9 mc

 Which statement describes the motion of the particle at the point  ? 

  1. The velocity is negative and the acceleration is positive.
  2. The velocity is negative and the acceleration is negative.
  3. The velocity is positive and the acceleration is positive.
  4. The velocity is positive and the acceleration is negative.
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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.  

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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 A1 2014 HSC 29b

Blood alcohol content of males can be calculated using the following formula

where    is the number of standard drinks consumed

is the number of hours drinking

is the person's mass in kilograms 

What is the maximum number of standard drinks that a male weighing 84 kg can consume over 4 hours in order to maintain a blood alcohol content (BAC) of less than 0.05?   (3 marks)

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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.   

 

 

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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 Resources, 2UG 2014 HSC 28d

An aerial diagram of a swimming pool is shown. 

The swimming pool is a standard length of 50 metres but is not in the shape of a rectangle.

(i)   Given  , determine the scale of the diagram such that

1 cm = m   (1 mark)

(ii)  If the length of a carpark next to the pool measured 5 cm (not shown), how long would it be in real life?   (1 mark)

(iii)  In the diagram of the swimming pool, the five widths are measured to be: 

 

 

The average depth of the pool is 1.2 m

Calculate the approximate volume of the swimming pool, in cubic metres. In your calculations, use TWO applications of Simpson’s Rule.   (3 marks)

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(i)  

(ii)  

(iii) ³

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(i)
 
   

 

(ii) 

 

 

(iii)

♦ Mean mark 50%. Be careful not to give away easy marks! 
 
 
 
  ²

 

 
 
  ³

Probability, STD2 S2 2014 HSC 28c

A fair coin is tossed three times. Using a tree diagram, or otherwise, calculate the probability of obtaining two heads and a tail in any order.   (2 marks)

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Measurement, STD2 M6 2014 HSC 28b

A radial compass survey of a sports centre is shown in the diagram. 
 

 
 

  1. Show that the size of angle    is  114°.   (1 mark)

  2. Calculate the length of the boundary , to the nearest metre.     (2 marks)

  3. Find the area of triangle in hectares, correct to two significant figures.   (3 marks)

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i.   

 

 

ii. 

 
 
 
 
♦ Mean marks of 42% and 41% for parts (ii) and (iii) respectively.
 

iii. 

  ²
  ²
 

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.  
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♦♦♦ 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)  
 

Algebra, STD2 A2 2014 HSC 26f

The weight of an object on the moon varies directly with its weight on Earth.  An astronaut who weighs 84 kg on Earth weighs only 14 kg on the moon.

A lunar landing craft weighs 2449 kg when on the moon. Calculate the weight of this landing craft when on Earth.   (2 marks)

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Algebra, STD2 A2 2014 HSC 22 MC

Heather’s car uses fuel at the rate of 6.6 L per 100 km for long-distance driving and  8.9 L per 100 km for short-distance driving.

She used the car to make a journey of 560 km, which included 65 km of short-distance driving.  

Approximately how much fuel did Heather’s car use on the journey?

  1.  37 L
  2. 38 L
  3. 48 L
  4. 50 L
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Probability, STD2 S2 2014 HSC 19 MC

Jaz has 2 bags of apples.

Bag A contains 4 red apples and 3 green apples. 

Bag B contains 3 red apples and 1 green apple.

Jaz chooses an apple from one of the bags. 

Which tree diagram could be used to determine the probability that Jaz chooses a red apple?
 

2014 19 mc1

2014 19 mc2

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Measurement, STD2 M7 2014 HSC 17 MC

A child who weighs 14 kg needs to be given 15 mg of paracetamol for every 2 kg of body weight.

Every 10 mL of a particular medicine contains 120 mg of paracetamol.

What is the correct dosage of this medicine for the child?

  1. 5.6 mL
  2. 8.75 mL
  3. 11.43 mL
  4. 17.5 mL
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Measurement, STD2 M1 2014 HSC 12 MC

A path 1.5  metres wide surrounds a circular lawn of radius 3 metres. 

What is the approximate area of the path?

  1. 7.1 m²
  2. 21.2 m²
  3. 35.3 m²
  4. 56.5 m²
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²
 

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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(i)  

 
 

 

(ii)   
   
   
   

 

(iii)  
 

 

 

♦♦ Mean mark 22%

 

 

 
 
 
 

 

Quadratic, EXT1 2009 HSC 2c

The diagram shows points    and    which move along the parabola  . The tangents to the parabola at    and    meet at  . 

2009 2c

  1. Show that the equation of the tangent at    is     (2 marks)
  2. Write down the equation of the tangent at  , and find the coordinates of the point    in terms of  .     (2 marks)
  3. Find the Cartesian equation of the locus of  .   (1 mark)
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(i)   
 
 
IMPORTANT: Deriving this equation was specifically asked in 2009 and 2011, as well as being required in 2014. KNOW IT! Completing this in 60 seconds gains 2 full minutes on allocated time.

 

 

 

 

(ii)   

 

 

 

 

 

 

 

(iii)  

 

 

 

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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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.
 

 

 

Calculus, EXT1 C1 2013 HSC 13a

A spherical raindrop of radius metres loses water through evaporation at a rate that depends on its surface area.  The rate of change of the volume of the raindrop is given by

, 

where is time in seconds and is the surface area of the raindrop. The surface area and the volume of the raindrop are given by    and    respectively.

  1. Show that    is constant.   (1 mark)

  2. How long does it take for a raindrop of volume   m3  to completely evaporate?     (2 marks)

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i.   

 

 

 

 

ii.    ³
 
 
 

 

♦♦ Mean mark 31%

 

Mechanics, EXT2* M1 2013 HSC 12e

A particle moves along a straight line. The displacement of the particle from the origin is , and its velocity is . The particle is moving so that  , where is a constant.

Show that the particle moves in simple harmonic motion with period  .   (2 marks)

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Geometry and Calculus, EXT1 2013 HSC 12d

The point    lies on the curve  . The line    has equation  . The perpendicular distance from    to the line    is  .

2013 12d

  1. Show that  
    1. .   (2 marks)
  3. Find the value of    when    is closest to  .     (1 mark)
  4. Show that, when    is closest to  , the tangent to the curve at    is parallel to  .   (1 mark)
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(i)   
 
 
 
Mean mark 40%
MARKER’S COMMENT: The biggest difficulty in part (i) was to explain the removal of the absolute value sign.

 

 

(ii)   

 

(iii)  
 

 

Calculus, EXT1 C2 2013 HSC 11f

Use the substitution    to evaluate  .   (3 marks)

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 MARKER’S COMMENT: Many students did not calculate in radians and incorrectly got an answer of 8.2. BE CAREFUL!
Note that converting your answer to 0.14 is also correct but not required.

 

 

 

Geometry and Calculus, EXT1 2013 HSC 11d

Consider the function  .

  1. Show that    for all    in the domain of  .     (2 marks)
  2. Sketch the graph  , showing all asymptotes.     (2 marks)
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(i)   

 
 
 

 

 

 

(ii)  EXT1 2013 11d

COMMENT: Note that is a short way of writing as approaches 2 from the negative (or left-hand) side. This notation can save time when required.
 
 
 

Binomial, EXT1 2010 HSC 7b

The binomial theorem states that 
 

 
  

  1. Show that  
    1. .   (1 mark)
  2.  
  3. Hence, or otherwise, find the value of
     

    1. .   (1 mark)
    2.  
  4. Show that
    1.  .   (2 marks)
    2.  
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(i)   

 

 

(ii)   

♦ Mean mark 37%
 

(iii)