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Statistics, STD2 S1 2008 HSC 3 MC

The stem-and-leaf plot represents the daily sales of soft drink from a vending machine.

If the range of sales is 43, what is the value of  2008 3 mc  ?

 
 

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Probability, 2ADV S1 2008 HSC 9a

It is estimated that 85% of students in Australia own a mobile phone.

  1. Two students are selected at random. What is the probability that neither of them owns a mobile phone?  (2 marks)

  2. Based on a recent survey, 20% of the students who own a mobile phone have used their mobile phone during class time. A student is selected at random. What is the probability that the student owns a mobile phone and has used it during class time?   (1 mark)

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i.     

COMMENT: is syllabus notation for the complement of event .

 
 

 

ii. 

Plane Geometry, 2UA 2008 HSC 8b

In the diagram,    is a parallelogram and    and    are both squares.

Copy or trace the diagram into your writing booklet.

  1. Prove that  .  (1 mark)
  2. Prove that  .   (3 marks)
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(i) 

 

(ii)

   
 
 

 

Calculus, 2ADV C3 2008 HSC 8a

Let  .

  1. Find the coordinates of the points where the graph of    crosses the axes.  (2 marks)

  2. Show that    is an even function.   (1 mark)

  3. Find the coordinates of the stationary points of    and determine their nature.   (4 marks)

  4. Sketch the graph of  .   (1 mark)

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  4.  
  5. 2UA HSC 2008 8ai
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i.   

 

 

ii.   
 
   
   

 

 

iii.  
 
 

 
   
 

 

   
 

 

iv. 

Probability, 2ADV S1 2008 HSC 7c

Xena and Gabrielle compete in a series of games. The series finishes when one player has won two games. In any game, the probability that Xena wins is    and the probability that Gabrielle wins is  .

Part of the tree diagram for this series of games is shown.
 

 
 

  1. Complete the tree diagram showing the possible outcomes.  (1 mark)
  2. What is the probability that Gabrielle wins the series?   (2 marks)

  3. What is the probability that three games are played in the series?   (2 marks)

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  1.    
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i. 2UA HSC 2008 7ci

 

MARKER’S COMMENT: A tree diagram with 8 outcomes is incorrect (i.e. no third game is played if 1 player wins the first 2 games). If outcomes cannot occur, do not draw them on a tree diagram.

 

ii. 

 

iii. 

 

Trigonometry, 2ADV T1 2008 HSC 7b

2008 7b

The diagram shows a sector with radius    and angle    where  .

The arc length is  . 

  1.  Show that  .   (2 marks)

  2.  Calculate the area of the sector when  .    (2 marks)

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i.   

 

 

 

ii.  
   
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Calculus, EXT1* C3 2008 HSC 6c

The graph of    is shown below.
 

2008 6c
 

The shaded region in the diagram is bounded by the curve  , the  -axis and the lines    and  .

Find the volume of the solid of revolution formed when the shaded region is rotated about the  -axis.  (3 marks)

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³

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Quadratic, 2UA 2008 HSC 4c

Consider the parabola  .

  1. Write down the coordinates of the vertex.  (1 mark)
  2. Find the coordinates of the focus.   (1 mark)
  3. Sketch the parabola.   (1 mark)
  4. Calculate the area bounded by the parabola and the line  .   (3 marks)
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(i)  

(ii)  

(iii)  

(iv)  ²

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(i)   

 

(ii)  
 
 

 

(iii) 2UA HSC 2008 4c 

 

(iv)  

 

 
 
 
 
 
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Calculus, 2ADV C4 2008 HSC 3b

  1. Differentiate    with respect to  .  (2 marks)

  2. Hence, or otherwise, evaluate  .    (2 marks)

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i.   
 
   

 

ii.   
 
 
 
 
 
 
 

Linear Functions, 2UA 2008 HSC 3a

2008 3a

In the diagram,    is a quadrilateral. The equation of the line    is  . 

  1. Show that    is a trapezium by showing that    is parallel to  .  (2 marks)
  2. The line    is parallel to the  -axis. Find the coordinates of  .   (1 mark)
  3. Find the length of  .   (1 mark)
  4. Show that the perpendicular distance from    to    is  .   (2 marks)
  5. Hence, or otherwise, find the area of the trapezium  .   (2 marks)
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(i)   

 
 

 

 

(ii)   
 
 

 

(iii)  
 
 
 

 

(iv)  

 

 
 
 

 

(v)   
   

 

 

 
 
 
 
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Financial Maths, STD2 F5 SM-Bank 2

The table below shows the present value of an annuity with a contribution of  $1.
 

  1. Fiona pays $3000 into an annuity at the end of each year for 4 years at 2% p.a., compounded annually.   What is the present value of her annuity?  (1 mark)

  2. If John pays $6000 into an annuity at the end of each year for 2 years at 4% p.a., compounded annually, is he better off than Fiona?  Use calculations to justify your answer.    (2 marks)

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i. 

 

 

ii. 

 

 

Mechanics, EXT2* M1 2014 HSC 14a

The take-off point on a ski jump is located at the top of a downslope. The angle between the downslope and the horizontal is  .  A skier takes off from with velocity m s−1 at an angle to the horizontal, where  .  The skier lands on the downslope at some point , a distance metres from .
 

2014 14a
 

The flight path of the skier is given by

,      (Do NOT prove this.)

where    is the time in seconds after take-off.

  1. Show that the cartesian equation of the flight path of the skier is given by
     
         .   (2 marks)

  2. Show that  
     
         .   (3 marks)

  3. Show that  
     
         .   (2 marks)

  4. Show that has a maximum value and find the value of for which this occurs.   (3 marks)

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i. 

 
 

 

ii. 

♦♦ Mean mark 28%
MARKER’S COMMENT: The key to finding and solving for is to realise you need the intersection of the Cartesian equation in (i) and the line .

 

 

 

 

iii. 

 
 

 

iv. 

♦ Mean mark 47%
IMPORTANT: Note that students can attempt every part of this question, even if they couldn’t successfully prove earlier parts.
 

 

 
 

 

Plane Geometry, EXT1 2014 HSC 13d

In the diagram,    is a diameter of a circle with centre  . The point    is chosen such that    is acute-angled. The circle intersects    and    at    and    respectively.

Copy or trace the diagram into your writing booklet.

  1. Why is   ?   (1 mark)
  2. Show that the line    is a tangent to the circle through  ,    and  .    (2 marks)
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(i)   

 

(ii)  

♦ Mean mark 37%
COMMENT: Labelling angles and etc.. can often make a proof clearer and less messy.

 

Calculus, EXT1 C1 2014 HSC 13b

One end of a rope is attached to a truck and the other end to a weight. The rope passes over a small wheel located at a vertical distance of  40 m above the point where the rope is attached to the truck.

The distance from the truck to the small wheel is , and the horizontal distance between them is  . The rope makes an angle with the horizontal at the point where it is attached to the truck.

The truck moves to the right at a constant speed of   , as shown in the diagram.
 


 

  1. Using Pythagoras’ Theorem, or otherwise, show that  .   (2 marks)

  2.  Show that  .    (1 mark)

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i.    

 
 
 

 

ii.   

 
 

Mechanics, EXT2* M1 2014 HSC 12a

 A particle is moving in simple harmonic motion about the origin, with displacement metres. The displacement is given by  , where is time in seconds. The motion starts when  .

  1. What is the total distance travelled by the particle when it first returns to the origin?   (1 mark)

  2. What is the acceleration of the particle when it is first at rest?    (2 marks)

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i.   

 

 

ii.   
   
 

 

 

 
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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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MARKER’S COMMENT: Students who drew a diagram and included their working calculations on it were the most successful.
(i) 
 
 
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(ii)   
 
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(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)   
 

 

Mechanics, EXT2* M1 2009 HSC 5a

The equation of motion for a particle moving in simple harmonic motion is given by

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

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

     

    where    and    is the amplitude of the motion.   (3 marks)

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

  3. Find the maximum acceleration of the particle.    (1 mark)

  4. The particle is initially at the origin. Write down a formula for    as a function of  , and hence find the first time that the particle’s speed is half its maximum speed.   (2 marks)

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i.   

 

 
 

 

ii.   
 

 

iii.  
 

 

iv.  
 

 

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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(i)   
 
   
   

 

 

(ii)   
   
 

 

 

(iii)  
 
 
 
 

 

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

 

 

 

(iv)   
 

 

EXT1 2009 4b

Statistics, EXT1 S1 2009 HSC 4a

A test consists of five multiple-choice questions. Each question has four alternative answers. For each question only one of the alternative answers is correct.

Huong randomly selects an answer to each of the five questions. 

  1. What is the probability that Huong selects three correct and two incorrect answers?   (2 marks)

  2. What is the probability that Huong selects three or more correct answers?    (2 marks)

  3. What is the probability that Huong selects at least one incorrect answer?  (1 mark)

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i.   


 

ii. 


 

iii. 

TIP: The use of “at least” should flag a good chance of applying to solve.

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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(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.

 

 
 

 

 
 
 
 

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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