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Mechanics, EXT2 M1 EQ-Bank 4

A torpedo with a mass of 80 kilograms has a propeller system that delivers a force of on the torpedo, at maximum power. The water exerts a resistance on the torpedo proportional to the square of the torpedo's velocity .

  1. Explain why 
     
    where is a positive constant.  (1 mark)

  2. If the torpedo increases its velocity from    to  , show that the distance it travels in this time, , is given by
     
           (3 marks)

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ii.   
 
 
 
   

 


 


 

 
 

Mechanics, EXT2 M1 EQ-Bank 1

A canon ball of mass 9 kilograms is dropped from the top of a castle at a height of metres above the ground.

The canon ball experiences a resistance force due to air resistance equivalent to  , where is the speed of the canon ball in metres per second. Let    and the displacement, metres at time seconds, be measured in a downward direction.

  1. Show the equation of motion is given by
     
           (1 mark)

  2. Show, by integrating using partial fractions, that
     
           (5 marks)

     

  3. If the canon hits the ground after 4 seconds, calculate the height of the castle, to the nearest metre.  (3 marks)

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Functions, 2ADV F2 EQ-Bank 16

  has been produced by three successive transformations: a translation, a dilation and then a reflection.

  1. Describe each transformation and state the equation of the graph after each transformation.  (2 marks)

  2. Sketch the graph.  (1 mark)

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ii.   

Statistics, 2ADV S3 EQ-Bank 2

Let  denote a normal random variable with mean 0 and standard deviation 1.

The random variable has the probability density function

   where 

The diagram shows the graph of  .
 

 
 

  1. Complete the table of values for the given function, correct to four decimal places.  (1 mark)
     

     
  2. Use the trapezoidal rule and 5 function values in the table in part i. to estimate

     

            (2 marks)

  3. The weights of Rhodesian ridgebacks are normally distributed with a mean of 48 kilograms and a standard deviation of 6 kilograms.

     

    Using the result from part ii., calculate the probability of a randomly selected Rhodesian ridgeback weighing less than 36 kilograms.  (2 marks)

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iii. 

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Statistics, 2ADV S2 EQ-Bank 3

The table below lists the average life span (in years) and average sleeping time (in hours/day) of 9 animal species.
 


 

  1. Using sleeping time as the independent variable, calculate the least squares regression line. (1 mark)

  2. A wallaby species sleeps for 4.5 hours, on average, each day.

     

    Use your equation from part i to predict its expected life span, to the nearest year.   (1 mark)

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i.   

COMMENT: Issues here? YouTube has short and excellent help videos – search your calculator model and topic – eg. “fx-82 regression line” .


 

ii.   

 

Statistics, 2ADV S2 EQ-Bank 2

The table below lists the average body weight (in kilograms) and average brain weight (in grams) of nine animal species.
 


 

A least squares regression line is fitted to the data using body weight as the independent variable.

  1. Calculate the equation of the least squares regression line. (1 mark)

  2. If dingos have an average body weight of 22.3 kilograms, calculate the predicted average brain weight of a dingo using your answer to part i.   (1 mark)

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i.   

COMMENT: Know this critical calculator skill!.

 

ii.   

 

Statistics, 2ADV S2 EQ-Bank 1

The arm spans (in cm) and heights (in cm) for a group of 13 boys have been measured. The results are displayed in the table below.
 

CORE, FUR2 2008 VCAA 4

The aim is to find a linear equation that allows arm span to be predicted from height.

  1. What will be the independent variable in the equation?  (1 mark)

  2. Assuming a linear association, determine the equation of the least squares regression line that enables arm span to be predicted from height. Write this equation in terms of the variables arm span and height. Give the coefficients correct to two decimal places.  (2 marks)

  3. Using the equation that you have determined in part b., interpret the slope of the least squares regression line in terms of the variables height and arm span.  (1 mark)

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a.  

COMMENT: Calculator skills for finding the least squares regression line were required in NESA sample exam – know this critical skill well!

 

b.   

 

c.   

NETWORKS, FUR2-NHT 2019 VCAA 3

The zoo’s management requests quotes for parts of the new building works.

Four businesses each submit quotes for four different tasks.

Each business will be offered only one task.

The quoted cost, in $100 000, of providing the work is shown in Table 1 below.
  


 

The zoo’s management wants to complete the new building works at minimum cost.

The Hungarian algorithm is used to determine the allocation of tasks to businesses.

The first step of the Hungarian algorithm involves row reduction; that is, subtracting the smallest element in each row of Table 1 from each of the elements in that row.

The result of the first step is shown in Table 2 below.
 


 

The second step of the Hungarian algorithm involves column reduction; that is, subtracting the smallest element in each column of Table 2 from each of the elements in that column.

The results of the second step of the Hungarian algorithm are shown in Table 3 below. The values of Task 1 are given as and .
 


 

  1. Write down the values of and .   (1 mark)

  2. The next step of the Hungarian algorithm involves covering all the zero elements with horizontal or vertical lines. The minimum number of lines required to cover the zeros is three.

     

    Draw these three lines on Table 3 above.   (1 mark)

  3. An allocation for minimum cost is not yet possible.

     

    When all steps of the Hungarian algorithm are complete, a bipartite graph can show the allocation for minimum cost.

     

    Complete the bipartite graph below to show this allocation for minimum cost.   (1 mark)

     

  1. Business 4 has changed its quote for the construction of the pathways. The new cost is $1 000 000. The overall minimum cost of the building works is now reduced by reallocating the tasks.

     

    How much is this reduction?   (1 mark)

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c. 
 


 

 

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Networks, STD2 N3 2019 FUR2-N 2

The construction of the new reptile exhibit is a project involving nine activities, to .

The directed network below shows these activities and their completion times in weeks.
 


 

  1. Which activities have more than one immediate predecessor?  (1 mark)

  2. Write down the critical path for this project.  (1 mark)

  3. What is the latest start time, in weeks, for activity ?  (1 mark)

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a.   

 

b.   
 

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c.  
   

NETWORKS, FUR2-NHT 2019 VCAA 1

A zoo has an entrance, a cafe and nine animal exhibits: bears , elephants , giraffes , lions , monkeys , penguins , seals , tigers and zebras .

The edges on the graph below represent the paths between the entrance, the cafe and the animal exhibits. The numbers on each edge represent the length, in metres, along that path. Visitors to the zoo can use only these paths to travel around the zoo.
  

 
 

  1. What is the shortest distance, in metres, between the entrance and the seal exhibit ?   (1 mark)

  2. Freddy is a visitor to the zoo. He wishes to visit the cafe and each animal exhibit just once, starting and ending at the entrance.
  3. i. What is the mathematical term used to describe this route?   (1 mark)

  4. ii. Draw one possible route that Freddy may take on the graph below.   (1 mark)


 

A reptile exhibit will be added to the zoo.

A new path of length 20 m will be built between the reptile exhibit and the giraffe exhibit .

A second new path, of length 35 m, will be built between the reptile exhibit and the cafe.

  1. Complete the graph below with the new reptile exhibit and the two new paths added. Label the new vertex and write the distances on the new edges.   (1 mark)

     


 

  1. The new paths reduce the minimum distance that visitors have to walk between the giraffe exhibit and the cafe.

     

    By how many metres will these new paths reduce the minimum distance between the giraffe exhibit and the cafe?   (1 mark)

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MATRICES, FUR2-NHT 2019 VCAA 1

A total of six residents from two towns will be competing at the International Games.

Matrix , shown below, contains the number of male and the number of female athletes competing from the towns of Gillen and Haldaw .

 

  1. How many of these athletes are residents of Haldaw?   (1 mark)

Each of the six athletes will compete in one event: table tennis, running or basketball.

Matrices and , shown below, contain the number of male and female athletes from each town who will compete in table tennis and running respectively.
 

            Table tennis                        Running             
 

 

  1. Matrix contains the number of male and female athletes from each town who will compete in basketball.

     

    Complete matrix below.   (1 mark)

Matrix contains the cost of one uniform, in dollars, for each of the three events: table tennis , running and basketball .

    1. For which event will the total cost of uniforms for the athletes be $1030?   (1 mark)

    2. Write a matrix calculation, that includes matrix , to show that the total cost of uniforms for the event named in part c.i. is contained in the matrix answer of [1030].   (1 mark)

  1. Matrix and matrix are two new matrices where    and:
  • matrix is a    matrix
  • element total cost of uniforms for all female athletes from Gillen
  • element total cost of uniforms for all female athletes from Haldaw
  • element total cost of uniforms for all male athletes from Gillen
  • element total cost of uniforms for all male athletes from Haldaw
     
  1. Complete matrix with the missing values.   (1 mark)

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a. 
 

b. 
 

c.i. 
 

c.ii. 
 

d. 

CORE, FUR2-NHT 2019 VCAA 6

Marlon plays guitar in a band.

He paid $3264 for a new guitar.

The value of Marlon's guitar will be depreciated by a fixed amount for each concert that he plays.

After 25 concerts, the value of the guitar will have decreased by $200.

  1. What will be the value of Marlon's guitar after 25 concerts?   (1 mark)

  2. Write a calculation that shows that the value of Marlon's guitar will depreciate by $8 per concert.   (1 mark)

  3. The value of Marlon's guitar after concerts, , can be determined using a rule.

     

    Complete the rule below by writing the appropriate numbers in the boxes provided.   (1 mark)

     
                – ×
     

  4. The value of the guitar continues to be depreciated by $8 per concert.

     

    After how many concerts will the value of Marlon's guitar first fall below $2500?   (2 marks)

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b.   
 

c.   
 

d.   

 

 

CORE, FUR2-NHT 2019 VCAA 4

The scatterplot below plots the variable life span, in years, against the variable sleep time, in hours, for a sample of 19 types of mammals.
 

On the assumption that the association between sleep time and life span is linear, a least squares line is fitted to this data with sleep time as the explanatory variable.

The equation of this least squares line is

life span = 42.1 – 1.90 × sleep time

The coefficient of determination is 0.416

  1. Draw the graph of the least squares line on the scatterplot above.   (1 mark)

  2. Describe the linear association between life span and sleep time in terms of strength and direction.   (2 marks)

  3. Interpret the slope of the least squares line in terms of life span and sleep time.   (2 marks)

  4. Interpret the coefficient of determination in terms of life span and sleep time.   (1 mark)

  5. The life of the mammal with a sleep time of 12 hours is 39.2 years.
  6. Show that, when the least squares line is used to predict the life span of this mammal, the residual is 19.9 years.   (2 marks)

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CORE, FUR2-NHT 2019 VCAA 3

The life span, in years, and gestation period, in days, for 19 types of mammals are displayed in the table below.
 

  1. A least squares line that enables life span to be predicted from gestation period is fitted to this data.
  2. Name the explanatory variable in the equation of this least squares line.   (1 mark)

  3. Determine the equation of the least squares line in terms of the variables life span and gestation period.
  4. Round the numbers representing the intercept and slope to three significant figures.   (2 marks)

  5. Write the value of the correlation rounded to three decimal places.   (1 mark)

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a.   
 

b.   


 

c.   

CORE, FUR2-NHT 2019 VCAA 1

The table below displays the average sleep time, in hours, for a sample of 19 types of mammals.
 

  1. Which of the two variables, type of mammal or average sleep time, is a nominal variable?   (1 mark)

  2. Determine the mean and standard deviation of the variable average sleep time for this sample of mammals.
  3. Round your answer to one decimal place.   (1 mark)

  4. The average sleep time for a human is eight hours.
  5. What percentage of this sample of mammals has an average sleep time that is less than the average sleep time for a human.
  6. Round your answer to one decimal place.   (1 mark)

  7. The sample is increase in size by adding in the average sleep time of the little brown bat.
  8. Its average sleep time is 19.9 hours.
  9. By how many many hours will the range for average sleep time increase when the average sleep time for the little brown bat is added to the sample?   (1 mark)

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Statistics, EXT1 S1 EQ-Bank 10

Four cards are placed face down on a table. The cards are made up of a Jack, Queen, King and Ace.

A gambler bets that she will choose the Queen in a random pick of one of the cards.

If this process is repeated 7 times, express the gambler's success as a Bernoulli random variable and calculate

  1. the mean.  (1 mark)

  2. the variance.  (1 mark)

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i.     

 
 

 

ii.  
   
   

GRAPHS, FUR1-NHT 2019 VCAA 2 MC

Two straight lines have the equations    and  .

These lines have one point of intersection.

Another line that also passes through this point of intersection has the equation

  1.  
  2.  
  3.  
  4.  
  5.  
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MATRICES, FUR1-NHT 2019 VCAA 1 MC

The number of individual points scored by Rhianna (), Suzy (), Tina (), Ursula () and Vicki () in five basketball matches is shown in matrix below.
 


 

Who scored the highest number of points and in which match?

  1. Suzy in match 
  2. Tina in match 
  3. Vicki in match 
  4. Ursula in match 
  5. Rhianna in match 
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Graphs, MET2-NHT 2019 VCAA 1

Parts of the graphs of    and    are shown on the axes below.
 


 

The two graphs intersect at three points,  (–2, 0),  (1, 0)  and  (, ). The point  (, )  is not shown in the diagram above.

  1. Find the values of and .   (2 marks)

  2. Find the values of such that  .   (1 mark)

  3. State the values of for which
    1.    (1 mark)

    2.    (1 mark)

  4. Show that    for all  .   (1 mark)

  5. Find the values of such that    has exactly one negative solution.   (2 marks)

  6. Find the values of such that    has no solutions.   (1 mark)

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CORE, FUR1-NHT 2019 VCAA 5-7 MC

The birth weights of a large population of babies are approximately normally distributed with a mean of 3300 g and a standard deviation of 550 g.

Part 1

A baby selected at random from this population has a standardised weight of 

Which one of the following calculations will result in the actual birth weight of this baby?
 

 

Part 2

Using the 68–95–99.7% rule, the percentage of babies with a birth weight of less than 1650 g is closest to

  1. 0.14%
  2. 0.15%
  3. 0.17%
  4. 0.3%
  5. 2.5%

 

Part 3

A sample of 600 babies was drawn at random from this population.

Using the 68–95–99.7% rule, the number of these babies with a birth weight between 2200 g and 3850 g is closest to

  1. 111
  2. 113
  3. 185
  4. 408
  5. 489
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CORE, FUR1-NHT 2019 VCAA 1-2 MC

The histogram and boxplot shown below both display the distribution of the birth weight, in grams, of 200 babies.
 

Part 1

The shape of the distribution of the babies’ birth weight is best described as

  1. positively skewed with no outliers.
  2. negatively skewed with no outliers.
  3. approximately symmetric with no outliers.
  4. positively skewed with outliers.
  5. approximately symmetric with outliers.

 

Part 2

The number of babies with a birth weight between 3000 g and 3500 g is closest to

  1. 30
  2. 32
  3. 37
  4. 74
  5. 80
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Calculus, 2ADV C3 2019 MET1 4

Given the function  ,

  1. State the domain and range of .  (1 mark)

  2. i.  Find the equation of the tangent to the graph of  at  .  (2 marks)

     

    ii. On the axes below, sketch the graph of the function  , labelling any asymptote with its equation.

     

        Also draw the tangent to the graph of    at  .  (4 marks)
     

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Calculus, MET1-NHT 2019 VCAA 4

A function has rule  .

  1. State the maximal domain of and the range of over its maximal domain.   (2 marks)

  2. i.  Find the equation of the tangent to the graph of at .   (2 marks)

     

    ii. On the axes below, sketch the graph of the function , labelling any asymptote with its equation.

  3.     Also draw the tangent to the graph of at  .   (4 marks)
      

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  2. i. 
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Calculus, MET1-NHT 2019 VCAA 3

  1. Evaluate    and  .   (2 marks)

  2. Show that  .   (1 mark)

  3. Use your answers to part a. and part b. to evaluate    in the form  , where  and are positive integers.   (1 mark)

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b.  
   
   

 

c.