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Statistics, STD2 S1 2007 HSC 24d

Barry constructed a back-to-back stem-and-leaf plot to compare the ages of his students.
 

 

  1. Write a brief statement that compares the distribution of the ages of males and females from this set of data.   (1 mark)

  2. What is the mode of this set of data?   (1 mark)

  3. Liam decided to use a grouped frequency distribution table to calculate the mean age of the students at Barry’s Ballroom Dancing Studio. 

     

    For the age group 30 - 39 years, what is the value of the product of the class centre and the frequency?   (2 marks)

  4. Liam correctly calculated the mean from the grouped frequency distribution table to be 39.5.

     

    Caitlyn correctly used the original data in the back-to-back stem-and-leaf plot and calculated the mean to be 38.2. 

     

    What is the reason for the difference in the two answers?   (1 mark)

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i.
 
 
 

 

ii.

 

iii.
   
 

 


 

iv.
 
 

Algebra, STD2 A1 2007 HSC 24b

The distance in kilometres () of an observer from the centre of a thunderstorm can be estimated by counting the number of seconds () between seeing the lightning and first hearing the thunder.

Use the formula    to estimate the number of seconds between seeing the lightning and hearing the thunder if the storm is 1.2 km away.   (1 mark)

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Statistics, STD2 S1 2007 HSC 24a

Consider the following set of scores:

 

  1. Calculate the mean of the set of scores.   (1 mark)

  2. What is the effect on the mean and on the median of removing the outlier?   (2 marks)

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  2.  

  3.  

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i. 


 

 

ii. 


 


 

Measurement, STD2 M1 2007 HSC 23b

A cylindrical water tank, of height 2 m, is placed in the ground at a school.

The radius of the tank is 3.78 metres. The hole is 2 metres deep. When the tank is placed in the hole there is a gap of 1 metre all the way around the side of the tank.

 

  1. When digging the hole for the water tank, what volume of soil was removed? Give your answer to the nearest cubic metre.  (3 marks)

  2. Sprinklers are used to water the school oval at a rate of 7500 litres per hour.   

     

    The water tank holds 90 000 litres when full. 

     

    For how many hours can the sprinklers be used before a full tank is emptied?   (1 mark)

  3. Water is to be collected in the tank from the roof of the school hall, which has an area of 400 m².

     

    During a storm, 20 mm of rain falls on the roof and is collected in the tank. 

     

    How many litres of water were collected?   (2 marks)

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  1. ³³
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i. 

 
  ³³

 

ii. 

 

iii.

²

Financial Maths, STD2 F1 2007 HSC 23a

Lilly and Rose each have money to invest and choose different investment accounts.

The graph shows the values of their investments over time.
 

 

  1. How much was Rose’s original investment?  (1 mark)

  2. At the end of  6 years, which investment will be worth the most and by how much?  (2 marks)

  3. Lilly’s investment will reach a value of  $20 000  first.
  4. How much longer will it take Rose’s investment to reach a value of  $20 000?   (1 mark)

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  3.  

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i. 
 

ii. 

  

iii. 

PATTERNS, FUR1 2009 VCAA 4 MC

The sum of the infinite geometric sequence  96, – 48, 24, –12, 6 . . .  is equal to

A.      

B.     

C.     

D.   

E.  

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PATTERNS, FUR1 2009 VCAA 3 MC

The first four terms of a geometric sequence are  6400, , 8100, – 9112.5

The value of  is

A.    

B.   

C.   

D.     

E.     

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Probability, STD2 S2 2007 HSC 2 MC

Each student in a class is given a packet of lollies. The teacher records the number of red lollies in each packet using a frequency table.
 

What is the relative frequency of a packet of lollies containing more than three red lollies?

  1.    
  2.    
  3.    
  4.    
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GEOMETRY, FUR1 2014 VCAA 3 MC

The diagram below shows the location of three boats, , and .

Boat is on a bearing of 110° from boat .

Boat is also on a bearing of 035° from boat .

Boat is due north of boat .

The angle is

A.   

B.   

C.   

D.   

E.   

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CORE*, FUR1 2014 VCAA 4 MC

On day 1, Vikki spends 90 minutes on a training program.

On each following day, she spends 10 minutes less on the training program than she did the day before.

Let    be the number of minutes that Vikki spends on the training program on day  .

A difference equation that can be used to model this situation for    is

A.  
B.  
C.  
D.  
E.  

 

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PATTERNS, FUR1 2014 VCAA 3 MC

A city has a population of 100 000 people in 2014.
Each year, the population of the city is expected to increase by 4%.
In 2018, the population is expected to be closest to

A.    

B.    

C.   

D.   

E.   

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CORE, FUR1 2014 VCAA 10-11 MC

The seasonal indices for the first 11 months of the year, for sales in a sporting equipment store, are shown in the table below.

Part 1

The seasonal index for December is

A. 

B. 

C. 

D. 

E. 

 

Part 2

In May, the store sold $213 956 worth of sporting equipment.

The deseasonalised value of these sales was closest to

A. 

B. 

C. 

D. 

E. 

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CORE, FUR1 2011 VCAA 9-10 MC

The length of a type of ant is approximately normally distributed with a mean of 4.8 mm and a standard deviation of 1.2 mm.

Part 1

From this information it can be concluded that around 95% of the lengths of these ants should lie between

A.   

B.   

C.   

D.   

E.   

 

Part 2

A standardised ant length of    corresponds to an actual ant length of

A.   

B.   

C.   

D.   

E.   

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CORE, FUR1 2009 VCAA 4-6 MC

The percentage histogram below shows the distribution of the fertility rates (in average births per woman) for 173 countries in 1975.

Part 1

In 1975, the percentage of these 173 countries with fertility rates of 4.5 or greater was closest to

A.    

B.     

C.     

D.    

E.    

 

Part 2

In 1975, for these 173 countries, fertility rates were most frequently

A.   less than 2.5

B.   between 1.5 and 2.5

C.   between 2.5 and 4.5 

D.   between 6.5 and 7.5 

E.   greater than 7.5 

 

Part 3

Which one of the boxplots below could best be used to represent the same fertility rate data as displayed in the percentage histogram?

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♦ Mean mark 43%.
MARKERS’ COMMENT: A systemic approach where students calculated the median, and was most successful.

CORE, FUR1 2008 VCAA 1-4 MC

The box plot below shows the distribution of the time, in seconds, that 79 customers spent moving along a particular aisle in a large supermarket.
 

     2008 1-4

Part 1

The longest time, in seconds, spent moving along this aisle is closest to

A.   

B.   

C.   

D.   

E.   

 

Part 2

The shape of the distribution is best described as

A.   symmetric.

B.   negatively skewed.

C.   negatively skewed with outliers.

D.   positively skewed.

E.   positively skewed with outliers.

 

Part 3

The number of customers who spent more than 90 seconds moving along this aisle is closest to

A.   

B.   

C.   

D.   

E.   

 

Part 4

From the box plot, it can be concluded that the median time spent moving along the supermarket aisle is

A.   less than the mean time.

B.   equal to the mean time.

C.   greater than the mean time

D.   half of the interquartile range.

E.   one quarter of the range.

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♦ Mean mark 43%.
MARKERS’ COMMENT: Note that the outliers are already accounted for in the boxplot.

 

PATTERNS, FUR1 2010 VCAA 5 MC

A team of swimmers was training.
Claire was the first swimmer for the team and she swam 100 metres.
Every other swimmer in the team swam 50 metres further than the previous swimmer.
Jane was the last swimmer for the team and she swam 800 metres.

The total number of swimmers in this team was

A.     

B.   

C.   

D.   

E.   

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PATTERNS, FUR1 2012 VCAA 3-4 MC

 Use the following information to answer Parts 1 and 2.

As part of a savings plan, Stacey saved $500 the first month and successively increased the amount that she saved each month by $50. That is, in the second month she saved $550, in the third month she saved $600, and so on.

Part 1

The amount Stacey will save in the 20th month is

A. 

B. 

C. 

D.  

E. 

 

Part 2

The total amount Stacey will save in four years is

A. 

B.  

C. 

D. 

E. 

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PATTERNS, FUR1 2010 VCAA 3 MC

The prizes in a lottery form the terms of a geometric sequence with a common ratio of 0.95.
If the first prize is $20 000, the value of the eighth prize will be closest to

A.     

B.     

C.     

D.     

E.     

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PATTERNS, FUR1 2010 VCAA 2 MC

The first three terms of a geometric sequence are 

The fourth term in this sequence would be

A.   

B.   

C.   

D.   

E.   

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CORE, FUR1 2010 VCAA 7-9 MC

The height (in cm) and foot length (in cm) for each of eight Year 12 students were recorded and displayed in the scatterplot below.
A least squares regression line has been fitted to the data as shown.
 

Part 1

By inspection, the value of the product-moment correlation coefficient for this data is closest to

 

Part 2

The explanatory variable is foot length.

The equation of the least squares regression line is closest to

  1. height = –110 + 0.78 × foot length.
  2. height = 141 + 1.3 × foot length.
  3. height = 167 + 1.3 × foot length.
  4. height = 167 + 0.67 × foot length.
  5. foot length = 167 + 1.3 × height.

 

Part 3

The plot of the residuals against foot length is closest to

CORE, FUR1 2010 VCAA 7-9 MCab

CORE, FUR1 2010 VCAA 7-9 MCcd

CORE, FUR1 2010 VCAA 7-9 MCe

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♦♦ Mean mark 35%.
STRATEGY: An alternate but less efficient strategy could be to find 2 points and calculate the gradient and then use the point gradient formula.

 

CORE, FUR1 2011 VCAA 1-3 MC

The histogram below displays the distribution of the percentage of Internet users in 160 countries in 2007.
 

Part 1

The shape of the histogram is best described as

A.   approximately symmetric.

B.   bell shaped. 

C.   positively skewed.

D.   negatively skewed.

E.   bi-modal.

 

Part 2

The number of countries in which less than 10% of people are Internet users is closest to

A.   

B.   

C.   

D.   

E.   

 

Part 3

From the histogram, the median percentage of Internet users is closest to

A.   

B.   

C.   

D.   

E.   

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Show Worked Solution

 

 

♦ Mean mark 45%.

Trig Ratios, EXT1 2008 HSC 6a

From a point    due south of a tower, the angle of elevation of the top of the tower  , is 23°. From another point  , on a bearing of 120° from the tower, the angle of elevation of    is 32°. The distance    is 200 metres.
 

Trig Ratios, EXT1 2008 HSC 6a 
 

  1. Copy or trace the diagram into your writing booklet, adding the given information to your diagram.  (1 mark)
  2. Hence find the height of the tower.   (3 marks)
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  1. Trig Ratios, EXT1 2008 HSC 6a Answer

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(i) 

Trig Ratios, EXT1 2008 HSC 6a Answer 

 

(ii) 

 

 


 

 
 
 
 

CORE, FUR1 2014 VCAA 7 MC

The parallel boxplots below summarise the distribution of population density, in people per square kilometre, for 27 inner suburbs and 23 outer suburbs of a large city.
 

Which one of the following statements is not true?

  1. More than 50% of the outer suburbs have population densities below 2000 people per square kilometre. 
  2. More than 75% of the inner suburbs have population densities below 6000 people per square kilometre. 
  3. Population densities are more variable in the outer suburbs than in the inner suburbs.
  4. The median population density of the inner suburbs is approximately 4400 people per square kilometre.
  5. Population densities are, on average, higher in the inner suburbs than in the outer suburbs.
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CORE, FUR1 2014 VCAA 3-5 MC

The following table shows the data collected from a sample of seven drivers who entered a supermarket car park. The variables in the table are:

distance – the distance that each driver travelled to the supermarket from their home
 

    • sex – the sex of the driver (female, male)
    • number of children – the number of children in the car
    • type of car – the type of car (sedan, wagon, other)
    • postcode – the postcode of the driver’s home.
       

Part 1

The mean,  , and the standard deviation, , of the variable, distance, are closest to

A. 

B. 

C. 

D. 

E. 

 

Part 2

The number of categorical variables in this data set is

A. 

B.  

C.  

D. 

E.  

 

Part 3

The number of female drivers with three children in the car is

A. 

B. 

C. 

D. 

E.  

 

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♦♦ Mean mark 29%.
MARKER’S NOTE: Postcodes here are categorical variables. Ask yourself “Does it make sense to calculate the mean of this variable?” If the answer is “No”, the variable is categorical.

 

CORE, FUR1 2014 VCAA 2 MC

The time spent by shoppers at a hardware store on a Saturday is approximately normally distributed with a mean of 31 minutes and a standard deviation of 6 minutes.

If 2850 shoppers are expected to visit the store on a Saturday, the number of shoppers who are expected to spend between 25 and 37 minutes in the store is closest to

A.   16

B.   68

C.   460

D.   1900

E.   2400

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Show Worked Solution
 
 

 

 

 

 

 

CORE, FUR1 2012 VCAA 6 MC

The table below shows the percentage of households with and without a computer at home for the years 2007, 2009 and 2011.
 


 

In the year 2009, a total of   households were surveyed.

The number of households without a computer at home in 2009 was closest to

A.      
B.    
C.    
D.    
E.    
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CORE, FUR1 2010 VCAA 5-6 MC

The lengths of the left feet of a large sample of  Year 12 students were measured and recorded. These foot lengths are approximately normally distributed with a mean of 24.2 cm and a standard deviation of 4.2 cm.

Part 1

A Year 12 student has a foot length of 23 cm.
The student’s standardised foot length (standard score) is closest to

A.   –1.2

B.   –0.9

C.   –0.3

D.    0.3

E.     1.2

 

Part 2

The percentage of students with foot lengths between 20.0 and 24.2 cm is closest to

A.   16%

B.   32%

C.   34%

D.   52%

E.   68%

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Show Worked Solution

   
 
 

 

  
 
 

 

CORE, FUR1 2010 VCAA 4 MC

The passengers on a train were asked why they travelled by train. Each reason, along with the percentage of passengers who gave that reason, is displayed in the segmented bar chart below.
 

CORE, FUR1 2010 VCAA 4 MC
 

The percentage of passengers who gave the reason ‘no car’ is closest to

A.   

B.   

C.   

D.  

E.   

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CORE, FUR1 2009 VCAA 8 MC

An animal study was conducted to investigate the relationship between exposure to danger during sleep (high, medium, low) and chance of attack (above average, average, below average). The results are summarised in the percentage segmented bar chart below.
 


 

The percentage of animals whose exposure to danger during sleep is high, and whose chance of attack is below average, is closest to

A.     

B.  

C.   

D.   

E.   

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CORE, FUR1 2009 VCAA 1-3 MC

The back-to-back ordered stem plot below shows the female and male smoking rates, expressed as a percentage, in 18 countries.
 

  

Part 1

For these 18 countries, the lowest female smoking rate is

A.       

B.       

C.       

D.     

E.     

 

Part 2

For these 18 countries, the interquartile range (IQR) of the female smoking rates is

A.      

B.     

C.   

D.   

E.   

 

Part 3

For these 18 countries, the smoking rates for females are generally

A.   lower and less variable than the smoking rates for males.

B.   lower and more variable than the smoking rates for males.

C.   higher and less variable than the smoking rates for males.

D.   higher and more variable than the smoking rates for males.

E.   about the same as the smoking rates for males.

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Show Worked Solution

 

   

 

 

PATTERNS, FUR1 2013 VCAA 3 MC

The first time a student played an online game, he played for 18 minutes.
Each time he played the game after that, he played for 12 minutes longer than the previous time.
After completing his 15th game, the total time he had spent playing these 15 games was

A.    minutes 

B.   minutes  

C. minutes 

D.  minutes  

E. minutes

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CORE, FUR1 2013 VCAA 12-13 MC

The time series plot below displays the number of guests staying at a holiday resort during summer, autumn, winter and spring for the years 2007 to 2012 inclusive.
 

CORE, FUR1 2013 VCAA 12-13 MC_1
 

 Part 1

Which one of the following best describes the pattern in the time series?

A.  random variation only

B.  decreasing trend with seasonality

C.  seasonality only

D.  increasing trend only

E.  increasing trend with seasonality 

 

Part 2

The table below shows the data from the times series plot for the years 2007 and 2008. 
 

CORE, FUR1 2013 VCAA 12-13 MC_2
 

Using four-mean smoothing with centring, the smoothed number of guests for winter 2007 is closest to

A.  

B.  

C.  

D.  

E.   

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CORE, FUR1 2013 VCAA 5-6 MC

The time, in hours, that each student spent sleeping on a school night was recorded for 1550 secondary-school students. The distribution of these times was found to be approximately normal with a mean of 7.4 hours and a standard deviation of 0.7 hours.
 

Part 1

The time that 95% of these students spent sleeping on a school night could be 

A.  less than 6.0 hours.   

B.  between 6.0 and 8.8 hours.

C.  between 6.7 and 8.8 hours.

D.  less than 6.0 hours or greater than 8.8 hours.

E.  less than 6.7 hours or greater than 9.5 hours.

 

Part 2

The number of these students who spent more than 8.1 hours sleeping on a school night was closest to

A.       16

B.     248

C.   1302

D.   1510

E.   1545

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Show Worked Solution

            
   
 

 

 
 

 
 
 

 

 

 

CORE, FUR1 2013 VCAA 3-4 MC

The heights of 82 mothers and their eldest daughters are classified as 'short', 'medium' or 'tall'. The results are displayed in the frequency table below.
 

CORE, FUR1 2013 VCAA 3-4 MC

 
 Part 1

The number of mothers whose height is classified as 'medium' is

 A.    

B.   

C.  

D.  

E.  

 

Part 2

Of the mothers whose height is classified as 'tall', the percentage who have eldest daughters whose height is classified as 'short' is closest to

A.    

B.    

C.  

D.  

E.  

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Show Worked Solution

       

 

 

♦ Mean mark 45%.
MARKER’S COMMENT: Many students obtained the wrong base of 82 for this percentage calculation.
 
 

 

CORE, FUR1 2013 VCAA 1-2 MC

The following ordered stem plot shows the percentage of homes connected to broadband internet for 24 countries in 2007.
 

   CORE, FUR1 2013 VCAA 1-2 MC 
 

 Part 1

The number of these countries with more than 22% of homes connected to broadband internet in 2007 is

A.    

B.    

C.  

D.  

E.  

 

Part 2

Which one of the following statements relating to the data in the ordered stem plot is not true?

A.  The minimum is 16%.

B.  The median is 30%.

C.  The first quartile is 23.5%

D.  The third quartile is 32%.

E.  The maximum is 38%.

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Show Worked Solution

     

 

 

 
 

 

Functions, EXT1 F2 2008 HSC 2c

The polynomial    is given by  , where    and    are constants.

The three zeros of    are  ,    and  .

Find the value of  .   (3 marks) 

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MARKER’S COMMENT: Many students displayed significant inefficiencies in solving simultaneous equations.

 

 

Statistics, STD2 S1 2008 HSC 26d

The graph shows the predicted population age distribution in Australia in 2008.
 

 

  1. How many females are in the 0–4 age group?  (1 mark)

  2. What is the modal age group?   (1 mark)

  3. How many people are in the 15–19 age group?   (2 marks)

  4. Give ONE reason why there are more people in the 80+ age group than in the 75–79 age group.   (1 mark)

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  5.  

Show Worked Solution
i.   
   

 

ii.   

 

iii.  
   

 

 

 

 

 

iv.  
 

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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Show Worked Solution

i.  

 
ii. 


 


 

iii.   

 

 

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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Show Worked Solution

i.   

 
 

 

ii. 

 
 
 

 

Statistics, STD2 S1 2008 HSC 23a

You are organising an outside sporting event at Mathsville and have to decide which month has the best weather for your event. The average temperature must be between 20°C and 30°C, and average rainfall must be less than 80 mm.

The radar chart for Mathsville shows the average temperature for each month, and the table gives the average rainfall for each month.
 

VCAA 2008 23a
 

  1. If you consider only the temperature data, there are a number of possible months for holding the event. Name ONE of these months.  (1 mark)

  2. If both rainfall and temperature data are considered, which month is the best month for the sporting event?   (1 mark)

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Show Worked Solution

i. 
 

ii. 

Measurement, STD2 M6 2008 HSC 14 MC

Danni is flying a kite that is attached to a string of length 80 metres. The string makes an angle of 55° with the horizontal.

How high, to the nearest metre, is the kite above Danni’s hand?
 

VCAA 2008 14 mc
 

  1.    46 m
  2. 66 m
  3. 98 m
  4. 114 m
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2UG 2008 14MC ans

 

Financial Maths, STD2 F1 2008 HSC 7 MC

Luke’s normal rate of pay is $15 per hour. Last week he was paid for 12 hours, at time-and-a-half.

How many hours would Luke need to work this week, at double time, to earn the same amount?

  1. 4
  2. 6
  3. 8
  4. 9
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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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