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Probability, STD2 S2 2008 HSC 26b

The retirement ages of two million people are displayed in a table.
 

 
 

  1. What is the relative frequency of the 51–55 year retirement age?  (1 mark)

  2. Describe the distribution.   (1 mark)

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  2.  


     

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i. 

 

ii. 

Probability, STD2 S2 2008 HSC 26a

Cecil invited 175 movie critics to preview his new movie. After seeing the movie, he conducted a survey. Cecil has almost completed the two-way table.
 

  1. Determine the value of  .   (1 mark)

  2. A movie critic is selected at random.

     

    What is the probability that the critic was less than 40 years old and did not like the movie?  (2 marks)

  3. Cecil believes that his movie will be a box office success if 65% of the critics who were surveyed liked the movie.

     

    Will this movie be considered a box office success? Justify your answer.  (1 mark)

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i.  

 
ii. 


 


 

iii.   

 

 

Probability, STD2 S2 2008 HSC 25b

In a drawer there are 30 ribbons. Twelve are blue and eighteen are red.

Two ribbons are selected at random.

  1. Copy and complete the probability tree diagram.  (1 mark)
     

     
  2. What is the probability of selecting a pair of ribbons which are the same colour?  (2 marks)

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  1.  
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i. 

ii. 

 

Probability, STD2 S2 2008 HSC 24b

Three-digit numbers are formed from five cards labelled  1,  2,  3,  4  and  5.

  1. How many different three-digit numbers can be formed?  (1 mark)

  2. If one of these numbers is selected at random, what is the probability that it is odd?   (1 mark)

  3. How many of these three-digit numbers are even?   (1 mark)

  4. What is the probability of randomly selecting a three-digit number less than 500 with its digits arranged in descending order?   (2 marks)

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i. 

♦ Mean mark 45%.

 

ii. 

 

iii. 

♦ Mean mark 48%.

 

iv. 

♦♦♦ Mean mark 10%.

Financial Maths, STD2 F1 2008 HSC 24a

Bob is employed as a salesman. He is offered two methods of calculating his income.

Bob’s research determines that the average sales total per employee per month is $15 670. 

  1. Based on his research, how much could Bob expect to earn in a year if he were to choose Method 1?   (2 marks)

  2. If Bob were to choose a method of payment based on the average sales figures, state which method he should choose in order to earn the greater income. Justify your answer with appropriate calculations.   (3 marks)

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i.   

 
 

 

ii. 

 
 
 

 

Statistics, STD2 S1 2008 HSC 23e

In a survey, 450 people were asked about their favourite takeaway food. The results are displayed in the bar graph.

2008 23e
 
How many people chose pizza as their favourite takeaway food?   (2 marks)

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COMMENT: This question required measurement of the actual image on the exam. The same methodology works here.


 

Measurement, STD2 M7 2008 HSC 23c

An alcoholic drink has 5.5% alcohol by volume. The label on a 375 mL bottle says it contains 1.6 standard drinks.

  1. How many millilitres of alcohol are in a 375 mL bottle? (1 mark)

  2. It is recommended that a fully-licensed male driver should have a maximum of one standard drink every hour.

     

    Express this as a rate in millilitres per minute, correct to one decimal place.    (2 marks)

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i.   
   

 

ii.   


 

 
 

Measurement, STD2 M7 2008 HSC 20 MC

A point lies between a tree, 2 metres high, and a tower, 8 metres high. is 3 metres away from the base of the tree.

From , the angles of elevation to the top of the tree and to the top of the tower are equal.
 

What is the distance, , from to the top of the tower?

  1. 9 m
  2. 9.61 m
  3. 12.04 m
  4. 14.42 m
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Probability, STD2 S2 2008 HSC 16 MC

A bag contains some marbles. The probability of selecting a blue marble at random from this bag is  .

Which of the following could describe the marbles that are in the bag?

  1.      blue,    red
  2.      blue,    red
  3.      blue,    red,    green
  4.      blue,    red,    green 
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Statistics, STD2 S1 2008 HSC 13 MC

The height of each student in a class was measured and it was found that the mean height was 160 cm.

Two students were absent. When their heights were included in the data for the class, the mean height did not change.

Which of the following heights are possible for the two absent students?

  1.    155 cm and 162 cm
  2.    152 cm and 167 cm
  3.    149 cm and 171 cm
  4.    143 cm and 178 cm
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Statistics, STD2 S4 2008 HSC 12 MC

A scatterplot is shown.
 

Which of the following best describes the correlation between    and  ?

  1. Positive
  2. Negative 
  3. Positively skewed
  4. Negatively skewed
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Measurement, STD2 M1 2008 HSC 11 MC

The diagram shows the floor of a shower. The drain in the floor is a circle with a diameter of 10 cm.

What is the area of the shower floor, excluding the drain?
 

 
 

  1. 9686 cm²
  2. 9921 cm²
  3. 9969 cm²
  4. 10 000 cm²
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COMMENT: Students should see that answers are all in cm², and therefore use cm as the base unit for their calculations. 
 
 
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Statistics, STD2 S1 2008 HSC 10 MC

The marks for a Science test and a Mathematics test are presented in box-and-whisker plots.
 

 Which measure must be the same for both tests?

  1. Mean
  2. Range
  3. Median
  4. Interquartile range
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Statistics, STD2 S1 2008 HSC 8 MC

What is the median of the following set of scores?
 

 
 

  1.    12
  2.    13
  3.    14
  4.    15
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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  ?

 
 

  1.     
  2.    
  3.    
  4.    
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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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  ³

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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  1. ³
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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)   
 

 

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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  5.  
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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.