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Geometry and Calculus, EXT1 2009 HSC 4b

Consider the function  . 

  1. Show that    is an even function.    (1 mark)
  2. What is the equation of the horizontal asymptote to the graph  ?    (1 mark)
  3. Find the  -coordinates of all stationary points for the graph  .   (3 marks)
  4. Sketch the graph  . You are not required to find any points of inflection.    (2 marks)
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  5.  
Show Worked Solution
(i)   
 
   
   

 

 

(ii)   
   
 

 

 

(iii)  
 
 
 
 

 

MARKER’S COMMENT: Many students did not realise the denominator of   could be ignored when equating  .

 

 

 

(iv)   
 

 

EXT1 2009 4b

Trig Calculus, EXT1 2009 HSC 3b

  1. On the same set of axes, sketch the graphs of  
    1.  and  , for  .    (2 marks)
  3. Use your graph to determine how many solutions there are to the equation    for  .     (1 mark)
  5. One solution of the equation    is close to  . Use one application of Newton’s method to find another approximation to this solution. Give your answer correct to three decimal places.   (3 marks)

 

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Show Worked Solution
(i)   

 

(ii)  

 

(iii)  

MARKER’S COMMENT: Better responses defined and and evaluated each for before calculating Newton’s formula, as done in the Worked Solution.

 

 
 

 

 
 
 
 

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%
 

 

 

 
 

Calculus, 2ADV C4 2014 HSC 12d

The parabola    and the line    intersect at the origin and at the point  .
  


  

  1. Find the  -coordinate of the point  .    (1 mark)

  2. Calculate the area enclosed by the parabola and the line.     (3 marks)

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  2. ²
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i.  

 

  

 

ii.
   
   
   
   
    ²

Probability, 2ADV S1 2014 HSC 12c

A packet of lollies contains 5 red lollies and 14 green lollies. Two lollies are selected at random without replacement.

  1. Draw a tree diagram to show the possible outcomes. Include the probability on each branch.   (2 marks)

  2. What is the probability that the two lollies are of different colours?     (1 mark)

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  1.  
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i.         2UA HSC 2014 12c

 

ii. 

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.   

 

 

 
 
 

 

 

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.
 

 

 
 
 
 

Linear Functions, 2UA 2014 HSC 12b

The points  ,    and    form a triangle, as shown in the diagram.
 

2014 12b
 

  1. Show that the equation of    is  .   (2 marks)
  2. Find the perpendicular distance from    to  .    (2 marks)
  3. Hence, or otherwise, find the area of  .   (2 marks)
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  3. ²
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(i) 

 
 

 

 

(ii) 

 
 
 
 

 

(iii) 

 
 
 
 

  ²

Statistics, STD2 S1 2014 HSC 29c

Terry and Kim each sat twenty class tests. Terry’s results on the tests are displayed in the box-and-whisker plot shown in part (i).
 

  1. Kim’s  5-number summary for the tests is  67,  69,  71,  73,  75.

     

    Draw a box-and-whisker plot to display Kim’s results below that of Terry’s results.   (1 mark)
     
         

  2. What percentage of Terry’s results were below 69?     (1 mark)

  3. Terry claims that his results were better than Kim’s. Is he correct?

     

    Justify your answer by referring to the summary statistics and the skewness of the distributions.    (4 marks)

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i.    

 

♦ Mean mark 39%

ii. 

 

iii.  

♦♦ Mean mark 29%
COMMENT: Examiners look favourably on using language of location in answers, particularly the areas they have specifically pointed students towards (skewness in this example).

Algebra, STD2 A2 2014 HSC 27b

Xuso is comparing the costs of two different ways of travelling to university.

Xuso’s motorcycle uses one litre of fuel for every 17 km travelled. The cost of fuel is $1.67/L and the distance from her home to the university car park is 34 km. The cost of travelling by bus is  $36.40 for 10 single trips.

Which way of travelling is cheaper and by how much? Support your answer with calculations.  (2 marks)

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Financial Maths, STD2 F1 2014 HSC 13 MC

Jane sells jewellery. Her commission is based on a sliding scale of 6% on the first $2000 of her sales, 3.5% on the next $1000, and 2% thereafter.

What is Jane’s commission when her total sales are $5670? 

  1. $188.40
  2. $208.40
  3. $321.85
  4. $652.05
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Financial Maths, STD2 F4 2014 HSC 9 MC

A car is bought for  $19 990. It will depreciate at 18% per annum. 

Using the declining balance method, what will be the salvage value of the car after 3 years, to the nearest dollar? 

  1. $8968
  2. $9195
  3. $11 022
  4. $16 392
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Show Worked Solution
 
 
 

 

Probability, STD2 S2 2014 HSC 8 MC

A group of 150 people was surveyed and the results recorded.
  

A person is selected at random from the surveyed group. 

What is the probability that the person selected is a male who does not own a mobile?

  2.  
  3.  
  4.  
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Measurement, STD2 M1 2014 HSC 2 MC

A measurement of  72 cm is increased by 20% and then the result is decreased by 20%. 

What is the new measurement, correct to the nearest centimetre?

  1. 46 cm
  2. 69 cm
  3. 72 cm
  4. 104 cm
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Show Worked Solution
STRATEGY: Students confident in this area could save time in the calculations as follows:

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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Show Worked Solution
(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)  

 

 

 

Quadratic, EXT1 2013 HSC 13b

The point    lies on the parabola  .  The tangent to the parabola at    meets the  -axis at  .  The normal to the tangent at    meets the  -axis at  .

2013 13b

The point    divides    externally in the ratio  . 

  1. Show that the coordinates of    are  .    (2 marks)
  2. Show that    lies on a parabola with the same directrix and focal length as the original parabola.    (2 marks)

 

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(i)   

 
 

 

♦♦ Mean mark 31%.
IMPORTANT: Remember that finding the locus involves eliminating the parameter. Expressing the locus in the form is critical to finding its directrix and focal length.

(ii) 

 

 

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!

 

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)   
 
 
 
 

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)  
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(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)  
   

 

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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Show Worked Solution

i. 

 

 

 

 
 
 

ii. 

 

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

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!

 

 

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 .

Functions, EXT1 F2 2011 HSC 2a

Let    be a polynomial, where    is a real number.

When    is divided by    the remainder is  .

Find the remainder when    is divided by  .    (3 marks)

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Show Worked Solution

 

 

 

Mechanics, EXT2* M1 2012 HSC 13c

A particle is moving in a straight line according to the equation

, 

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

  1. Prove that the particle is moving in simple harmonic motion by showing that  satisfies an equation of the form  .  (2 marks)

  2. When is the displacement of the particle zero for the first time?    (3 marks)

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i. 

 
 
 

 

ii. 

♦ Mean mark 42%.
IMPORTANT: The critical insight required to solve is to realise that the cosine of the difference between 2 angles, i.e. , applies.

 

 Calculus in the Physical World, EXT1 2012 HSC 13c Answer

 

 

 
 

Functions, EXT1 F1 2012 HSC 12b

Let   

  1.  Find the domain of  .   (1 mark)

  2.  Find an expression for the inverse function  .    (2 marks)

  3.  Find the points where the graphs    and    intersect.   (1 mark)

  4.  On the same set of axes, sketch the graphs    and    showing the information found in part (iii).   (2 marks)

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  4.   
    1. Inverse Functions, EXT1 2012 HSC 12b Answer
Show Worked Solution

i. 

 

ii. 

 

 

iii. 

 

 

 

iv.

Inverse Functions, EXT1 2012 HSC 12b Answer