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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.

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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Plane Geometry, 2UA 2011 HSC 6a

The diagram shows a regular pentagon . Sides and are produced to meet at .
  

  1. Find the size of .    (1 mark)

  2. Hence, show that is isosceles.    (2 marks)

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i.  

 
MARKER’S COMMENT: Very few students solved part (i) efficiently. Remember the general formula for the sum of internal angles equals (# sides – 2) x 90°.

 
ii. 

Algebra, STD2 A4 2011 HSC 28a

The air pressure, , in a bubble varies inversely with the volume, , of the bubble. 

  1. Write an equation relating , and , where is a constant.    (1 mark)

  2. It is known that when .

     

    By finding the value of the constant, , find the value of when .    (2 marks)

  3. Sketch a graph to show how varies for different values of .

     

    Use the horizontal axis to represent volume and the vertical axis to represent air pressure.   (2 marks)


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  3.  
     
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♦ Mean mark (i) 39%
COMMENT: Expressing the proportional relationship as the equation is a core skill here.
i.
   

 

ii.

 

  

♦ Mean mark (ii) 47%
 

  

♦♦ Mean mark (iii) 26%
COMMENT: An inverse relationship is reflected by a hyperbola on the graph.
iii.

Statistics, STD2 S1 2011 HSC 25b

The graph below displays data collected at a school on the number of students
in each Year group, who own a mobile phone.
 

2UG 2011 25b
 

  1. Which Year group has the highest percentage of students with mobile phones? (1 mark)

  2. Two students are chosen at random, one from Year 9 and one from Year 10.

     

    Which student is more likely to own a mobile phone?

     

    Justify your answer with suitable calculations. (2 marks)

  3. Identify a trend in the data shown in the graph. (1 mark)

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i.   

MARKER’S COMMENT: Many students ignore the instruction to use calculations and lose marks as a result.
 

ii.    
   
     
   

  

 

iii.   

 

Statistics, STD2 S1 2010 HSC 27b

The graphs show the distribution of the ages of children in Numbertown in 2000 and 2010.
  

  1. In 2000 there were 1750 children aged 0–18 years.

     

    How many children were aged 12–18 years in 2000?   (1 mark)

  2. The number of children aged 12–18 years is the same in both 2000 and 2010.

     

    How many children aged 0–18 years are there in 2010?    (1 mark)

  3. Identify TWO changes in the distribution of ages between 2000 and 2010. In your answer, refer to measures of location or spread or the shape of the distributions.   (2 marks)

  4. What would be ONE possible implication for government planning, as a consequence of this change in the distribution of ages?   (1 mark)

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i.   

ii.   

iii. 

iv. 

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i.    

♦ Mean mark (i) 45%

 

♦♦ Mean mark (ii) 25%

ii.   


 

iii. 

♦ Mean mark (iii) 35%
MARKER’S COMMENT: A number of students incorrectly identified “positive” skew as “negative” skew here.

iv. 

♦ Mean mark (iv) 46%
MARKER’S COMMENT: Answers should reflect the 1 mark allocation.

Probability, STD2 S2 2013 HSC 30b

In a class there are 15 girls (G) and 7 boys (B). Two students are chosen at random to be class representatives.

  1. Complete the tree diagram below.    (2 marks)
     
     

  2. What is the probability that the two students chosen are of the same gender?    (2 marks)

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  1.  
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i.  

ii.   
   
   
   
   
♦ Mean mark (ii) 40%.

Measurement, STD2 M6 2010 HSC 26d

Find the area of triangle , correct to the nearest square metre.   (3 marks)

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²    ²

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♦♦ Mean mark 32%.
TIP: The allocation of 3 marks to this question should flag the need for more than 1 step.
 
 

 
 
  ²²

Probability, STD2 S2 2011 HSC 24b

A die was rolled 72 times. The results for this experiment are shown in the table.
  

  1. Find the value of  .   (1 mark)

  2. What was the relative frequency of obtaining a 4.   (1 mark)

  3. If the die was unbiased, which number was obtained the expected number of times?   (1 mark)

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i.     
 
 
♦ Mean mark 38%
IMPORTANT: Many students confused ‘relative frequency’ with ‘frequency’ and incorrectly answered 8.
ii.     
 

 

iii. 
 

Statistics, STD2 S1 2011 HSC 17 MC

The heights of the players in a basketball team were recorded as 1.8 m, 1.83 m, 1.84 m, 1.86 m and 1.92 m. When a sixth player joined the team, the average height of the players increased by 1 centimetre.

What was the height of the sixth player?

  1.   1.85 m
  2.   1.86 m
  3.   1.91 m
  4.   1.93 m
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Statistics, STD2 S1 2011 HSC 14 MC

A data set of nine scores has a median of 7.

The scores  6, 6, 12 and 17  are added to this data set.

What is the median of the data set now?

  1. 6
  2. 7
  3. 8
  4. 9
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Algebra, STD2 A4 2009 HSC 28c

The height above the ground, in metres, of a person’s eyes varies directly with the square of the distance, in kilometres, that the person can see to the horizon.

A person whose eyes are 1.6 m above the ground can see 4.5 km out to sea.

How high above the ground, in metres, would a person’s eyes need to be to see an island that is 15 km out to sea? Give your answer correct to one decimal place.   (3 marks)

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♦♦ Mean mark 22%
CRITICAL STEP: Reading the first line of the question carefully and establishing the relationship is the key part of solving this question.

 

 

 
 

Statistics, STD2 S4 2009 HSC 28b

The height and mass of a child are measured and recorded over its first two years. 

This information is displayed in a scatter graph. 
 

  1. Describe the correlation between the height and mass of this child, as shown in the graph.   (1 mark)

  2. A line of best fit has been drawn on the graph.

     

    Find the equation of this line.   (2 marks)

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  1.  

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i. 

♦ Mean mark 48%. 

 

ii. 

♦♦♦ Mean mark 18%. 
MARKER’S COMMENT: Many students had difficulty due to the fact the horizontal axis started at  and not the origin.
 
 
 

 

 

Probability, STD2 S2 2009 HSC 27c

In each of three raffles, 100 tickets are sold and one prize is awarded.

Mary buys two tickets in one raffle. Jane buys one ticket in each of the other two raffles.

Determine who has the better chance of winning at least one prize. Justify your response using probability calculations.   (4 marks)  

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♦♦ Mean mark 31%.
MARKER’S COMMENT: Very few students calculated Jane’s chance of winning correctly. Note the use of “at least” in the question. Finding (complement) is the best strategy here.

Algebra, STD2 A1 2013 HSC 29a

Sarah tried to solve this equation and made a mistake in Line 2. 

 
Copy the equation in Line 1 and continue your solution to solve this equation for .

Show all lines of working.   (2 marks)

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♦♦ Mean mark 27%
STRATEGY: The RHS of the equation increases from 1 to 15 (from Line 1 to Line 2), indicating both sides must have been multiplied by 15.

Measurement, STD2 M1 2013 HSC 27d

A rectangular wooden chopping board is advertised as being 17 cm by 25 cm, with each side measured to the nearest centimetre.

  1. Calculate the percentage error in the measurement of the longer side.   (1 mark)

  2. Between what lower and upper limits does the actual area of the top of the chopping board lie?     (2 marks)

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i.   

♦♦ Mean mark 23%
MARKER’S COMMENT: Be aware that measurements accurate to the nearest cm have an absolute error for calculation purposes of 0.5 cm.

 
   
   

 

ii.   

♦ Mean mark 35%
 

 

 

 

Statistics, STD2 S1 2013 HSC 26f

Jason travels to work by car on all five days of his working week, leaving home at 7 am each day. He compares his travel times using roads without tolls and roads with tolls over a period of 12 working weeks.

He records his travel times (in minutes) in a back-to-back stem-and-leaf plot.
 

2013 26f
 

  1. What is the modal travel time when he uses roads without tolls?  (1 mark)

  2. What is the median travel time when he uses roads without tolls?   (1 mark)

  3. Describe how the two data sets differ in terms of the spread and skewness of their distributions.   (2 marks)

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  6.  

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i. 

♦ Mean mark 36%
MARKER’S COMMENT: Finding a median proved challenging for many students. Take note!

 

ii. 

 
 

 

♦ Mean mark 39%

 

 iii. 

Probability, STD2 S2 2013 HSC 26c

The probability that Michael will score more than 100 points in a game of bowling is . 

  1. A commentator states that the probability that Michael will score less than 100 points in a game of bowling is  .

     

    Is the commentator correct? Give a reason for your answer.   (1 mark)

  2. Michael plays two games of bowling. What is the probability that he scores more than 100 points in the first game and then again in the second game?   (1 mark)

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♦♦♦ Mean mark 11%

i.  

 

♦ Mean mark 34%
ii.  
   

Statistics, STD2 S1 2013 HSC 26b

Write down a set of six data values that has a range of 12, a mode of 12 and a minimum value of 12.   (2 marks)

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Measurement, STD2 M6 2010 HSC 24d

The base of a lighthouse, , is at the top of a cliff 168 metres above sea level. The angle of depression from to a boat at is 28°. The boat heads towards the base of the cliff, , and stops at . The distance is 126 metres.
 

  1. What is the angle of depression from to , correct to the nearest degree?   (3 marks)

  2. How far did the boat travel from to , correct to the nearest metre?   (2 marks)

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♦♦ Mean mark 31%
i.   
 
     

 

 
 

 

ii.     

♦♦ Mean mark 31%
MARKER’S COMMENT: Solve efficiently by using right-angled trigonometry. Many students used non-right angled trig, adding to the calculations and the difficulty.
 
 

 

 
 
 

Algebra, STD2 A1 2010 HSC 24a

Fred tried to solve this equation and made a mistake in Line 2. 

Copy the equation in Line 1.

  1. Rewrite Line 2 correcting his mistake.   (1 mark)

  2. Continue your solution showing the correct working for Lines 3 and 4 to solve this equation for .   (1 mark)

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  1.  

  2.  

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i.   
 

 

ii.   
 

Probability, STD2 S2 2010 HSC 23c

On Saturday, Jonty recorded the colour of T-shirts worn by the people at his gym. The results are shown in the graph.

 

  1. How many people were at the gym on Saturday? (Assume everyone was wearing a T-shirt).   (1 mark)

  2. What is the probability that a person selected at random at the gym on Saturday, would be wearing either a blue or green T-shirt?   (1 mark)

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i.   
 

 

ii.   
 
 

Probability, STD2 S2 2010 HSC 20 MC

Lou and Ali are on a fitness program for one month. The probability that Lou will finish the program successfully is 0.7 while the probability that Ali will finish successfully is 0.6. The probability tree shows this information

 

What is the probability that only one of them will be successful ?

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♦ Mean mark 48%.

Statistics, STD2 S1 2010 HSC 16 MC

This back-to-back stem-and-leaf plot displays the test results for a class of 26 students.
 

2010 Q16 MC  
 

What is the median test result for the class?

  1.    
  2.    
  3.    
  4.    
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♦♦ Mean mark 35%

 

 

Algebra, STD2 A2 2009 HSC 24d

A factory makes boots and sandals. In any week

• the total number of pairs of boots and sandals that are made is 200
• the maximum number of pairs of boots made is 120
• the maximum number of pairs of sandals made is 150.

The factory manager has drawn a graph to show the numbers of pairs of boots () and sandals () that can be made.
 

 

  1. Find the equation of the line .   (1 mark)

  2. Explain why this line is only relevant between and for this factory.     (1 mark)

  3. The profit per week, , can be found by using the equation  .

     

    Compare the profits at and .     (2 marks)

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  2.  

     

     

     

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i.    

♦♦♦ Mean mark part (i) 14%. 
Using is a less efficient but equally valid method, using    and  (-intercept).

 

ii. 

♦ Mean mark 49%

 

iii. 

♦ Mean mark 40%.
 
 

 
 

 

Statistics, STD2 S1 2009 HSC 24a

The diagram below shows a stem-and-leaf plot for 22 scores. 
 

2UG-2009-24a
 

  1.  What is the mode for this data?   (1 mark)

  2.  What is the median for this data?     (1 mark)

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i.   

 

ii.   


 

 

Measurement, STD2 M1 2009 HSC 23c

The diagram shows the shape and dimensions of a terrace which is to be tiled.
 

  1. Find the area of the terrace.   (2 marks)

  2. Tiles are sold in boxes. Each box holds one square metre of tiles and costs $55. When buying the tiles, 10% more tiles are needed, due to cutting and wastage.

     

    Find the total cost of the boxes of tiles required for the terrace.   (2 marks)

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i.    ²

ii.   

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i.
 
 
  ²

 

ii.
    ²

 

 

 

Measurement, STD2 M6 2009 HSC 23a

The point is 25 m from the base of a building. The angle of elevation from  to the top of the building is 38°.
 

  1. Show that the height of the building is approximately 19.5 m.   (1 mark)

  2. A car is parked 62 m from the base of the building.

     

    What is the angle of depression from the top of the building to the car?

     

    Give your answer to the nearest minute.   (2 marks)

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i.   

ii.   

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i. 

 
 

 

ii.  

♦♦ Mean mark 33%
MARKER’S COMMENT: If >30 “seconds”, round to the higher “minute”.
 
 
 
 

 

Algebra, STD2 A1 2009 HSC 16 MC

The time for a car to travel a certain distance varies inversely with its speed.

Which of the following graphs shows this relationship?
 

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♦ Mean mark 38%

Algebra, STD2 A2 2009 HSC 14 MC

If   , and    is increased by  2, what will be the corresponding increase in ?

  1.  
  2.  
  3.  
  4.  
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♦ Mean mark 50%.
STRATEGY: Substituting real numbers into the equation can work well in these type of questions. eg. If and when .

Measurement, STD2 M1 2009 HSC 11 MC

 What is the area of the shaded part of this quadrant, to the nearest square centimetre?  

  1. 34 cm²
  2. 42 cm²
  3. 50 cm²
  4. 193 cm²
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  ²

Financial Maths, STD2 F1 2009 HSC 10 MC

Billy worked for 35 hours at the normal hourly rate of pay and for five hours at double time. He earned $561.60 in total for this work.

What was the normal hourly rate of pay?

  1. $7.02
  2. $12.48
  3. $14.04
  4. $16.05
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Probability, STD2 S2 2009 HSC 9 MC

A wheel has the numbers 1 to 20 on it, as shown in the diagram. Each time the wheel is spun, it stops with the marker on one of the numbers.
 


 

 The wheel is spun 120 times.

 How many times would you expect a number less than 6 to be obtained?

  1.    
  2.    
  3.    
  4.    
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Financial Maths, STD2 F4 2009 HSC 6 MC

A house was purchased in 1984 for $35 000. Assume that the value of the house has increased by 3% per annum since then. 

Which expression gives the value of the house in 2009?  

  2.  
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Statistics, STD2 S1 2009 HSC 3 MC

The eye colours of a sample of children were recorded.

When analysing this data, which of the following could be found?

  1. Mean
  2. Median
  3. Mode
  4. Range
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Algebra, STD2 A4 2010 HSC 13 MC

The number of hours that it takes for a block of ice to melt varies inversely with the temperature. At 30°C it takes 8 hours for a block of ice to melt.

How long will it take the same size block of ice to melt at 12°C?  

  1. 3.2 hours
  2. 20 hours
  3. 26  hours
  4. 45 hours
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♦ Mean mark 50%