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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.  
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(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) 
 


 

Trig Ratios, EXT1 2010 HSC 5a

A boat is sailing due north from a point    towards a point    on the shore line.

The shore line runs from west to east.

In the diagram,    represents a tree on a cliff vertically above  , and    represents a landmark on the shore. The distance    is 1 km.

From    the point    is on a bearing of 020°, and the angle of elevation to    is 3°.

After sailing for some time the boat reaches a point  , from which the angle of elevation to    is 30°.
 

5a
 

  1. Show that 
     
    .   (3 marks) 
     
  2. Find the distance  . Give your answer to 1 decimal place.   (1 mark)
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(i)    

 

 

 

 

 

(ii)   
 
 
 
 

Quadratic, EXT1 2010 HSC 4c

The diagram shows the parabola  . The point  , where  , is on the parabola.

 

Quadratic, EXT1 2010 HSC 4c

The tangent to the parabola at  ,  , meets the  -axis at  .

The point    is on the directrix, such that    is perpendicular to the directrix.

Show that    is a rhombus.   (3 marks)

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Quadratic, EXT1 2010 HSC 4c Answer

 
 
 

 

 

 

 

 
 

 

Trigonometry, EXT1 T3 2010 HSC 4b

  1. Express    in the form  ,
     
    where    and  .    (3 marks)

  2. Hence, or otherwise, solve  ,   
  3. for  .    (2 marks)

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i.   
 
 
 
 

 

 
 

 

 

♦ Mean mark part (ii) 49%
MARKER’S COMMENT: Many students did not check their answers against the stated domain for .
ii.   
 
 
 

 

Mechanics, EXT2* M1 2010 HSC 4a

A particle is moving in simple harmonic motion along the -axis. 

Its velocity , at , is given by  . 

  1. Find all values of for which the particle is at rest.   (1 mark)

  2. Find an expression for the acceleration of the particle, in terms of .   (1 mark)

  3. Find the maximum speed of the particle.    (2 marks)

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Show Worked Solution
(i)   

 

 

(ii)   
   
   

 

(iii)  

MARKER’S COMMENT: While most students found correctly, too many made errors substituting this back in or failed to finish the answer by taking the square root. BE CAREFUL! .
 
 
 

 

 

Inverse Functions, EXT1 2010 HSC 3b

Let  .  The diagram shows the graph  .
 

 Inverse Functions, EXT1 2010 HSC 3b
 

  1. The graph has two points of inflection.  
  2. Find the    coordinates of these points.   (3 marks)
  3.  
  4. Explain why the domain of    must be restricted if    is to have an inverse function.    (1 mark)
  5. Find a formula for    if the domain of    is restricted to  .   (2 marks)
  6. State the domain of  .    (1 mark)
  7. Sketch the curve  .    (1 mark)
  8. (1)   Show that there is a solution to the equation    between    and  .   (1 mark)
  9. (2)   By halving the interval, find the solution correct to one decimal place.   (1 mark)

 

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  1.  
  2.  
  5.  

  6.  
  8.  
  9. Inverse Functions, EXT1 2010 HSC 3b Answer
  10. (1)  
  11. (2)  
Show Worked Solution
(i)   
 
 
   
   

 

 
   
 
   
COMMENT: It is also valid to show that is an even function and if a P.I. exists at , there must be another P.I. at .
 

 

 

(ii)  
 
 
 

 

(iii)  

 

 

 


 

♦ Parts (iv) and (v) were poorly answered with mean marks of 39% and 49% respectively.
(iv)  

 

(v) 

Inverse Functions, EXT1 2010 HSC 3b Answer

 

(vi)(1)  

 
 

 

♦ Mean mark 37%.
MARKER’S COMMENT: Better responses showed the change in sign between and as shown in the solution.

 

(vi)(2)  
   

 

Combinatorics, EXT1 A1 2010 HSC 3a

At the front of a building there are five garage doors. Two of the doors are to be painted red, one is to be painted green, one blue and one orange. 

  1. How many possible arrangements are there for the colours on the doors?   (1 mark)

  2. How many possible arrangements are there for the colours on the doors if the two red doors are next to each other?    (1 mark)

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Show Worked Solution
i.   
   

 

♦ Mean mark 50%
MARKER’S COMMENT: Drawing a diagram was a successful strategy for many students in this part.
ii.   
 

Calculus, EXT1 C1 2010 HSC 2d

A radio transmitter is situated 6 km from a straight road. The closest point on the road to the transmitter is .

A car is travelling away from along the road at a speed of −1.  The distance from the car to is    and from the car to is  .
 

2010 2d
 

Find an expression in terms of    for  , where is time in hours.    (3 marks) 

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Functions, EXT1 F2 2010 HSC 2c

Let  , 

where    is a polynomial and    and    are real numbers.

The polynomial    has a factor of  .

When    is divided by    the remainder is  . 

  1. Find the values of    and  .  (2 marks)

  2. Find the remainder when    is divided by  .     (1 mark)

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i. 

 

 

 

 
 
 

ii. 

 

COMMENT: This question requires a fundamental understanding of the remainder theorem.

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
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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

Geometry and Calculus, EXT1 2011 HSC 4a

Consider the function  . 

  1. Find  .   (1 mark)
  2. The graph    has one maximum turning point. 
  3. Find the coordinates of the maximum turning point.   (2 marks)
  4.  
  5. Evaluate  .   (1 mark)
  6. Describe the behaviour of    as  .   (1 mark)
  7. Find the  -intercept of the graph  .   (1 mark)
  8. Sketch the graph    showing the features from parts  (ii) - (v).
  9. You are not required to find any points of inflection.    (2 marks)
  10.  
  11.  
  12.  
  13.  
  14.  
  15.  
  16.  
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  7.  
  8.  
  9.  
  10.  
  11.  
Show Worked Solution
(i)   
 

 

(ii)  

 

 

ALGEBRA TIP: Note that can often help calculations. This is easily proven by simply taking the of both sides of the equation.
 
 
 

 

(iii)  
   
   
   

 

(iv)  
   
 

 

(v)   
 

 

(vi)

EXT1 2011 4a

Quadratic, EXT1 2011 HSC 3b

The diagram shows two distinct points    and    on the parabola  .  The point    is the intersection of the tangents to the parabola at    and  . 

 

  1. Show that the equation of the tangent to the parabola at    is  .   (2 marks)
  2. Using part  , write down the equation of the tangent to the parabola at  .     (1 mark)
  3. Show that the tangents at    and    intersect at
    .   (2 marks)
  5. Describe the locus of    as    varies, stating any restriction on the  -coordinate.   (2 marks)

 

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Show Worked Solution
(i)

 

 

(ii) 

MARKER’S COMMENT: Many students derived this equation rather than substituting the new parameter, costing them valuable time. This is a benefit of using the parametric approach.

 

(iii) 

 

 

(iv) 

♦♦ Mean mark of 22%. 
MARKER’S COMMENT: Many students stated the locus as rather than realising it had to be a straight line since ½, and that simply restricted the values of .

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.

 

 

Combinatorics, EXT1 A1 2011 HSC 2e

Alex’s playlist consists of 40 different songs that can be arranged in any order.  

  1. How many arrangements are there for the 40 songs?    (1 mark)

  2. Alex decides that she wants to play her three favourite songs first, in any order.
  3. How many arrangements of the 40 songs are now possible?   (1 mark)

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i.   

 

ii.   
   

Polynomials, EXT1 2011 HSC 2b

The function    has a zero near  .

Use one application of Newton’s method to obtain another approximation to this zero. Give your answer correct to two decimal places.    (3 marks)

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Show Worked Solution
 
 

 

 
 
 
 

Mechanics, EXT2* M1 2012 HSC 14b

A firework is fired from , on level ground, with velocity metres per second at an angle of inclination  . The equations of motion of the firework are

 and. (Do NOT prove this.) 

The firework explodes when it reaches its maximum height.
 

2012 14b
 

  1. Show that the firework explodes at a height of    metres.   (2 marks)

  2. Show that the firework explodes at a horizontal distance of    metres from  .    (1 mark)

  3. For best viewing, the firework must explode at a horizontal distance between 125 m and 180 m from , and at least 150 m above the ground.

     

    For what values of    will this occur?   (3 mark)

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Show Worked Solution

i.   


 

 

 
 

 

ii. 

 
 

 

iii. 

♦♦ Mean mark 28%
MARKER’S COMMENT: Many students ignored the findings in earlier parts and tried unsuccessfully to solve inequalities that still contained .

 

 

 

 

 

 

 

.