SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Networks, STD2 N2 2020 FUR1 5 MC

The network below shows the distances, in metres, between camp sites at a camping ground that has electricity.

The vertices to represent the camp sites.
 


 

The minimum length of cable required to connect all the camp sites is 53 m.

The value of , in metres, is at least

  1.  6
  2.  8
  3.  9
  4. 11
Show Answers Only

Show Worked Solution


 

Networks, STD2 N3 FUR1 2020 6

The activity network below shows the sequence of activities required to complete a project.

The number next to each activity in the network is the time it takes to complete that activity, in days.

What is the critical path and minimum completion time for this project.   (2 marks)

Show Answers Only

Show Worked Solution


 

 
 

NETWORKS, FUR1 2020 VCAA 6 MC

The activity network below shows the sequence of activities required to complete a project.

The number next to each activity in the network is the time it takes to complete that activity, in days.

 

The minimum completion time for this project, in days, is

  1. 18
  2. 19
  3. 20
  4. 21
  5. 22
Show Answers Only

Show Worked Solution


 

 
 

 

MATRICES, FUR1 2020 VCAA 9 MC

Five competitors, Andy (A), Brie (B), Cleo (C), Della (D) and Eddie (E), participate in a darts tournament.

Each competitor plays each of the other competitors once only, and each match results in a winner and a loser.

The matrix below shows the results of this darts tournament.

There are still two matches that need to be played.
 


 

A ‘1’ in the matrix shows that the competitor named in that row defeated the competitor named in that column.

For example, the ‘1’ in row 2, column 3 shows that Brie defeated Cleo.

A ‘…’ in the matrix shows that the competitor named in that row has not yet played the competitor named in that column.

The winner of this darts tournament is the competitor with the highest sum of their one-step and two-step dominances.

Which player, by winning their remaining match, will ensure that they are ranked first by the sum of their one-step and two-step dominances?

  1. Andie
  2. Brie
  3. Cleo
  4. Della
  5. Eddie
Show Answers Only

Show Worked Solution


 


 


 


 

CORE, FUR1 2020 VCAA 29 MC

The value of a van purchased for $45 000 is depreciated by % per annum using the reducing balance method.

After three years of this depreciation, it is then depreciated in the fourth year under the unit cost method at the rate of 15 cents per kilometre.

The value of the van after it travels 30 000 km in this fourth year is $26 166.24

The value of is

  1.   9
  2. 12
  3. 14
  4. 16
  5. 18
Show Answers Only

Show Worked Solution

 

CORE, FUR1 2020 VCAA 28 MC

The nominal interest rate for a loan is 8% per annum.

When rounded to two decimal places, the effective interest rate for this loan is not

  1. 8.33% per annum when interest is charged daily.
  2. 8.32% per annum when interest is charged weekly.
  3. 8.31% per annum when interest is charged fortnightly.
  4. 8.30% per annum when interest is charged monthly.
  5. 8.24% per annum when interest is charged quarterly.
Show Answers Only

Show Worked Solution

 
 

 
 

CORE, FUR1 2020 VCAA 19-20 MC

The time series plot below displays the number of airline passengers, in thousands, each month during the period January to December 1960.
 


 

Part 1

During 1960, the median number of monthly airline passengers was closest to

  1. 461 000
  2. 465 000
  3. 471 000
  4. 573 000
  5. 621 000

 
Part 2

During the period January to May 1960, the total number of airline passengers was 2 160 000.

The five-mean smoothed number of passengers for March 1960 is

  1. 419 000
  2. 424 000
  3. 430 000
  4. 432 000
  5. 434 000
Show Answers Only

Show Worked Solution

Part 1

♦ Mean mark part (1) 45%.

 
 


 

Part 2


 

CORE, FUR1 2020 VCAA 17-18 MC

Table 4 below shows the monthly rainfall for 2019, in millimetres, recorded at a weather station, and the associated long-term seasonal indices for each month of the year.
 


 

Part 1

The deseasonalised rainfall for May 2019 is closest to

  1.   71.3 mm
  2.   75.8 mm
  3.   86.1 mm
  4.   88.1 mm
  5. 113.0 mm

 
Part 2

The six-mean smoothed monthly rainfall with centring for August 2019 is closest to

  1. 67.8 mm
  2. 75.9 mm
  3. 81.3 mm
  4. 83.4 mm
  5. 86.4 mm
Show Answers Only

Show Worked Solution

Part 1


 


 

Part 2


 

CORE, FUR1 2020 VCAA 15-16 MC

Table 3 below shows the long-term mean rainfall, in millimetres, recorded at a weather station, and the associated long-term seasonal indices for each month of the year.

The long-term mean rainfall for December is missing.
 


 

Part 1

To correct the rainfall in March for seasonality, the actual rainfall should be, to the nearest per cent

  1. decreased by 26%
  2. increased by 26%
  3. decreased by 35%
  4. increased by 35%
  5. increased by 74%

 
Part 2

The long-term mean rainfall for December is closest to

  1. 64.7 mm
  2. 65.1 mm
  3. 71.3 mm
  4. 76.4 mm
  5. 82.0 mm
Show Answers Only

Show Worked Solution

Part 1

 


 

Part 2

 

CORE, FUR1 2020 VCAA 14 MC

In a study, the association between the number of tasks completed on a test and the time allowed for the test, in hours, was found to be non-linear.

The data can be linearised using a log10 transformation applied to the variable number of tasks.

The equation of the least squares line for the transformed data is

log10 (number of tasks) = 1.160 + 0.03617 × time

This equation predicts that the number of tasks completed when the time allowed for the test is three hours is closest to

  1. 13
  2. 16
  3. 19
  4. 25
  5. 26
Show Answers Only

Show Worked Solution
 

 

 
 

CORE, FUR1 2020 VCAA 10-12 MC

The data in Table 2 was collected in a study of the association between the variables frequency of nightmares (low, high) and snores (no, yes).
 


 

Part 1

The variables in this study, frequency of nightmares (low, high) and snores (no, yes), are

  1. ordinal and nominal respectively.
  2. nominal and ordinal respectively.
  3. both numerical.
  4. both ordinal.
  5. both nominal.

 
Part 2

The percentage of participants in the study who did not snore is closest to

  1. 42.0%
  2. 43.5%
  3. 49.7%
  4. 52.2%
  5. 56.5%

 
Part 3

Of those people in the study who did snore, the percentage who have a high frequency of nightmares is closest to

  1.   7.5%
  2. 17.1%
  3. 47.8%
  4. 52.2%
  5. 58.0%
Show Answers Only

Show Worked Solution

Part 1


 

Part 2

 
 


 

Part 3

 
 

CORE, FUR1 2020 VCAA 9 MC

The lifetime of a certain brand of light globe, in hours, is approximately normally distributed.

It is known that 16% of the light globes have a lifetime of less than 655 hours and 50% of the light globes have a lifetime that is greater than 670 hours.

The mean and the standard deviation of this normal distribution are closest to

A.     mean = 655 hours standard deviation = 10 hours
B.     mean = 655 hours standard deviation = 15 hours
C.     mean = 670 hours standard deviation = 10 hours
D.     mean = 670 hours standard deviation = 15 hours
E.     mean = 670 hours standard deviation = 20 hours
Show Answers Only

Show Worked Solution

CORE, FUR1 2020 VCAA 8 MC

The wing length of a species of bird is approximately normally distributed with a mean of 61 mm and a standard deviation of 2 mm.

Using the 68–95–99.7% rule, for a random sample of 10 000 of these birds, the number of these birds with a wing length of less than 57 mm is closest to

  1.   50
  2. 160
  3. 230
  4. 250
  5. 500
Show Answers Only

Show Worked Solution


 


 

CORE, FUR1 2020 VCAA 7 MC

Data relating to the following five variables was collected from insects that were caught overnight in a trap:

    • colour
    • name of species
    • number of wings
    • body length (in millimetres)
    • body weight (in milligrams)

The number of these variables that are discrete numerical variables is

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
Show Answers Only

Show Worked Solution

CORE, FUR1 2020 VCAA 6 MC

A percentaged segmented bar chart would be an appropriate graphical tool to display the association between month of the year (January, February, March, etc.) and the

  1. monthly average rainfall (in millimetres).
  2. monthly mean temperature (in degrees Celsius).
  3. annual median wind speed (in kilometres per hour).
  4. monthly average rainfall (below average, average, above average).
  5. annual average temperature (in degrees Celsius).
Show Answers Only

Show Worked Solution

CORE, FUR1 2020 VCAA 1-3 MC

The times between successive nerve impulses (time), in milliseconds, were recorded.

Table 1 shows the mean and the five-number summary calculated using 800 recorded data values.
 


 

Part 1

The difference, in milliseconds, between the mean time and the median time is

  1.  10
  2.  70
  3.  150
  4.  220
  5.  230

 
Part 2

Of these 800 times, the number of times that are longer than 300 milliseconds is closest to

  1. 20
  2. 25
  3. 75
  4. 200
  5. 400

 
Part 3

The shape of the distribution of these 800 times is best described as

  1. approximately symmetric.
  2. positively skewed.
  3. positively skewed with one or more outliers.
  4. negatively skewed.
  5. negatively skewed with one or more outliers.
Show Answers Only

Show Worked Solution

Part 1

 


 

Part 2


 

Part 3

♦ Mean mark 50%.

 
 

 

Calculus, SPEC1 2020 VCAA 9

Consider the curve defined parametrically by

where .

  1. can be written in the form  where    and    are real numbers.
  2. Show that    and  .  (2 marks)

  3. Find the arc length, , of the curve from    to  . Give your answer in the form  , where  .  (3 marks)

Show Answers Only

Show Worked Solution

a.  
 
 
   

 

♦♦ Mean mark part (b) 31%.

 

b.  
   
   
   
   
   
   

Calculus, SPEC1 2020 VCAA 7

Consider the function defined by

where and are real numbers.

  1. Given that    and    are continuous over  , show that    and  .  (2 marks)

  2. Find the area enclosed by the graph of the function, the x-axis and the lines    and  .  (3 marks)

Show Answers Only

  2. ²

Show Worked Solution

a. 

 

 

 

b.   

 
 
  ²

Calculus, EXT1 C2 2020 SPEC1 6

Let  .

  1. Show that  .  (1 mark)

  2. Hence, show that the graph of    has a point of inflection at  .  (2 marks)

  3. Sketch the graph of    on the axes provided below. Label any asymptotes with their equations and the point of inflection with its coordinates.   (2 marks)

Show Answers Only
Show Worked Solution
a.   
   

 

b.   

 

c.   

Calculus, SPEC1 2020 VCAA 6

Let  .

  1. Show that  .  (1 mark)

  2. Hence, show that the graph of    has a point of inflection at  .  (2 marks)

  3. Sketch the graph of    on the axes provided below. Label any asymptotes with their equations and the point of inflection with its coordinates.   (2 marks)

Show Answers Only

Show Worked Solution

a.   
   

 

b.   

♦ Mean mark part (b) 42%.

 

c.   

Vectors, SPEC1 2020 VCAA 5

Let    and  , where is an integer.

The vector resolute of    in the direction of    is  .

  1. Find the value of .   (3 marks)

  2. Find the component of    that is perpendicular to  .   (1 mark)

Show Answers Only

Show Worked Solution

a.   
 
 
 

 

♦♦ Mean mark part (b) 28%.

 

b.   
   
   

Mechanics, SPEC1 2020 VCAA 1

A 2 kg mass is initially at rest on a smooth horizontal surface. The mass is then acted on by two constant forces that cause the mass to move horizontally. One force has magnitude 10 N and acts in a direction 60° upwards from the horizontal, and the other force has magnitude 5 N and acts in a direction 30° upwards from the horizontal, as shown in the diagram below.
 


 

  1. Find the normal reaction force, in newtons, that the surface exerts on the mass.  (2 marks)
  2. Find the acceleration of the mass, in ms−2, after it begins to move.  (2 marks)
  3. Find how far the mass travels, in metres, during the first four seconds of motion.   (1 mark)
Show Answers Only
Show Worked Solution
a.  

 

b.   
 
 
Mean mark part (c) 51%.

 

c.   
   
   

Calculus, MET1 2020 VCAA 8

Part of the graph of  , where  , is shown below.
 


 

The graph of has a minimum at the point , as shown above.

  1. Find the coordinates of the point .   (2 marks)

  2. Using  , show that    has an antiderivative  .   (1 mark)

  3. Find the area of the region that is bounded by , the lines    and the horizontal axis for  , where is the -intercept of .   (2 marks)

  4. Let    for  .

     

    i. Find the value of for which    is a tangent to the graph of .   (1 mark)

    ii. Find all values of for which the graphs of and do not intersect.   (2 marks)

Show Answers Only
  4. i. 
  5. ii. 
Show Worked Solution

a.   

 

 
 

 

 

b.   
 
 

 

c.   

 
 
 
 

 

d.i.   

 

 

d.ii.

 


 

Calculus, MET1 2020 VCAA 7

Consider the function    and the point  . Part of the graph    is shown below.
 

   
 

  1. Show the point is not on the graph of  .   (1 mark)

  2. Consider a point    to be a point on the graph of .

     

      i. Find the slope of the line connecting points and in terms of .   (1 mark)

     ii. Find the slope of the tangent to the graph of at point in terms of .   (1 mark)

    iii. Let the tangent to the graph of at    pass through point .

     

    Find the values of .   (2 marks)

     iv. Give the equation of one of the lines passing through point that is tangent to the graph of .   (1 mark)

  3. Find the value of , that gives the shortest possible distance between the graph of the function of    and point .   (2 marks)

Show Answers Only
  2.   i. 
     ii. 
    iii. 
    iv. 
Show Worked Solution

a.   

 

b.i.   

 

 

b.ii.   


  

b.iii.   

 

  

b.iv.   

 

 

c.   

Functions, EXT1 F1 2020 MET1 6

, where 

  1. Find  , and state its domain.  (2 marks)

The graph of  , where  , is shown on the axes below.
 

     
 

  1. On the axes above, sketch the graph of    over its domain. Label the endpoints and point(s) of intersection with  , giving their coordinates.  (2 marks)

Show Answers Only

  1.  

  2.  
       
Show Worked Solution

a.   

 
 
 

 

 

b.     
  

Calculus, MET1 2020 VCAA 6

Let  , where  .

  1. Find the domain and the rule for  , the inverse function of  .   (2 marks)

The graph of  , where  , is shown on the axes below.
 

     
 

  1. On the axes above, sketch the graph of    over its domain. Label the endpoints and point(s) of intersection with the function  , giving their coordinates.   (2 marks)

  2. Find the total area of the two regions: one region bounded by the functions  and , and the other region bounded by    and the line  . Give your answer in the form  , where  .   (4 marks)

Show Answers Only

  1.  

  2.  
       
  3.  ²
Show Worked Solution
a.   

 

 

 

b.     

 

c.     
 
 
 
 
 
  ²

Statistics, EXT1 S1 2020 MET1 5

For a certain population the probability of a person being born with the specific gene SPGE1 is .

The probability of a person having this gene is independent of any other person in the population having this gene.

In a randomly selected group of four people, what is the probability that three or more people have the SPGE1 gene?  (2 marks)

Show Answers Only

Show Worked Solution

 
 
 

Probability, MET1 2020 VCAA 5

For a certain population the probability of a person being born with the specific gene SPGE1 is .

The probability of a person having this gene is independent of any other person in the population having this gene.

  1. In a randomly selected group of four people, what is the probability that three or more people have the SPGE1 gene?  (2 marks)

  2. In a randomly selected group of four people, what is the probability that exactly two people have the SPGE1 gene, given that at least one of those people has the SPGE1 gene? Express your answer in the form  , where, .  (2 marks)

Show Answers Only

Show Worked Solution

a.   

 
 
 

 

b.   
   
   
   

Probability, 2ADV S1 2012 MET1 2

A car manufacturer is reviewing the performance of its car model X. It is known that at any given six-month service, the probability of model X requiring an oil change is , the probability of model X requiring an air filter change is and the probability of model X requiring both is .

  1. State the probability that at any given six-month service model X will require an air filter change without an oil change.  (1 mark)

  2. The car manufacturer is developing a new model. The production goals are that the probability of model Y requiring an oil change at any given six-month service will be , the probability of model Y requiring an air filter change will be and the probability of model Y requiring both will be , where .
     
    Determine in terms of if the probability of model Y requiring an air filter change without an oil change at any given six-month service is 0.05.  (2 marks)

Show Answers Only
Show Worked Solution
a.   
 
   
   

 

 

b.   

Probability, MET1 2020 VCAA 2

A car manufacturer is reviewing the performance of its car model X. It is known that at any given six-month service, the probability of model X requiring an oil change is , the probability of model X requiring an air filter change is and the probability of model X requiring both is .

  1. State the probability that at any given six-month service model X will require an air filter change without an oil change.  (1 mark)

  2. The car manufacturer is developing a new model. The production goals are that the probability of model Y requiring an oil change at any given six-month service will be , the probability of model Y requiring an air filter change will be and the probability of model Y requiring both will be , where .
  3. Determine in terms of if the probability of model Y requiring an air filter change without an oil change at any given six-month service is 0.05.  (2 marks)

Show Answers Only

Show Worked Solution

a.   
 
   
   

 

 

b.   

Graphs, MET1 2020 VCAA 3

Shown below is part of the graph of a period of the function of the form  .
 


 

The graph is continuous for  .

Find the value of and the value of , where    and  .   (3 marks)

Show Answers Only

Show Worked Solution

 

 

Statistics, STD2 S1 2011 HSC 25a*

A study on the mobile phone usage of NSW high school students is to be conducted.

Data is to be gathered using a questionnaire.

The questionnaire begins with the three questions shown.

2UG 2011 25a

  1. Classify the type of data that will be collected in Q2 of the questionnaire.  (1 mark)

  2. Write a suitable question for this questionnaire that would provide discrete ordinal data.  (1 mark)

Show Answers Only
Show Worked Solution

i. 

 

ii. 

Mechanics, EXT2 M1 2018 SPEC1-N 1

A light inextensible string hangs over a frictionless pulley connecting masses of 3 kg and 7 kg, as shown below.
 


 

  1.  Draw all of the forces acting on the two masses on the diagram above. (1 mark)
  2.  Calculate the tension in the string.  (2 marks)
Show Answers Only
  1.  
Show Worked Solution
a.   

 

b.   
   
 
 
 
   

Mechanics, EXT2 M1 2012 SPEC2 20 MC

Particles of mass 3 kg and kg are attached to the ends of a light inextensible string that passes over a smooth pulley, as shown.
 


 

If the acceleration of the 3 kg mass is    upwards, then

A.    = 4.5

B.    = 6.0

C.    = 9.0

D.    = 13.5

E.    = 18.0

Show Answers Only

Show Worked Solution

 

Mechanics, EXT2 M1 2015 SPEC2 19 MC

A light inextensible string passes over a smooth pulley, as shown below, with particles of mass 1 kg and kg attached to the ends of the string.
 

SPEC2 2015 VCAA 19 MC

If the acceleration of the 1 kg particle is 4.9 upwards, then is equal to

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
Show Answers Only

Show Worked Solution


 

 

 

Mechanics, EXT2 M1 2014 SPEC2 20 MC

Particles of mass 3 kg and 5 kg are attached to the ends of a light inextensible string that passes over a fixed smooth pulley, as shown above. The system is released from rest.

Assuming the system remains connected, the speed of the 5 kg mass after two seconds is

A.     4.0 m/s

B.     4.9 m/s

C.     9.8 m/s

D.   19.6 m/s

Show Answers Only

Show Worked Solution