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Statistics, EXT1 S1 2016 MET1 4

A paddock contains 10 tagged sheep and 20 untagged sheep. Four times each day, one sheep is selected at random from the paddock, placed in an observation area and studied, and then returned to the paddock.

  1. What is the probability that the number of tagged sheep selected on a given day is zero?  (1 mark)

  2. What is the probability that at least one tagged sheep is selected on a given day?  (1 mark)

  3. What is the probability that no tagged sheep are selected on each of six consecutive days?

     

    Express your answer in the form , where , and are positive integers.  (1 mark)

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a.  

 

 

b.   
   
   

 

c.  

 
 

Functions, 2ADV F2 SM-Bank 13

 

  1. Show that the function    is an odd function?  (1 mark) 

  2. Sketch  , labelling all intercepts and asymptotes.  (2 marks)

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i.   

 
 
 

 

 

ii.   


 

Functions, 2ADV F2 SM-Bank 10

Consider the function  .

  1. Identify the domain of  .   (1 mark)

  2. Sketch the graph  , showing all intercepts and asymptotes.   (3 marks)

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  2.  
     
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i.   

 

ii.

COMMENT: Note that    is a short way of writing as approaches 2 from the negative (or left-hand) side. This notation can save time when required.
 
 
 

 

  EXT1 2013 11d

 

Calculus, EXT1 C3 2017 SPEC1 8

A slope field representing the differential equation    is shown below.

  1. Sketch the solution curve of the differential equation corresponding to the condition    on the slope field above and, hence, estimate the positive value of when  . Give your answer correct to one decimal place.  (2 marks)
  2. Solve the differential equation    with the condition  . Express your answer in the form  , where , , and are integers.  (2 marks)

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  1.  

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a.   

♦♦ Mean mark part (a) 32%.
MARKER’S COMMENT: Solution curve should follow slope ticks and not cross them.

 

b.   
 
 

 

 

 

Calculus, EXT1 C3 SM-Bank 5

Bacteria are spreading over a Petri dish at a rate modelled by the differential equation

where    is the proportion of the dish covered after    hours.

Given 

  1. Show by integration that  , where    is a constant of integration.  (2 marks)

  2. If half of the Petri dish is covered by the bacteria at  , express    in terms of  .  (2 marks)

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a. 

 
 
 

  


b. 

 

 

Calculus, EXT1 C3 2018 VCE 8

A tank initially holds 16 L of water in which 0.5 kg of salt has been dissolved. Pure water then flows into the tank at a rate of 5 L per minute. The mixture is stirred continuously and flows out of the tank at a rate of 3 L per minute.

  1.  Show that the differential equation for , the number of kilograms of salt in the tank after minutes, is given by
      (1 mark)
      
  2.  Solve the differential equation given in part a. to find as a function of .
      
    Express your answer in the form  , where and are positive integers.  (3 marks)
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  1.  
  2.  
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a.

♦ Net mean mark of both parts 44%.

 

 

 

MARKER’S COMMENT: Many students took the common factor of 2 from  . This wasn’t necessary and complicated the arithmetic in part b.

 

b.   

 

COMMENT: A very challenging test of using exponential and log laws!

 
 

Calculus, EXT1 C1 2013 VCE 5

A container of water is heated to boiling point (100°C) and then placed in a room that has a constant temperature of 20°C. After five minutes the temperature of the water is 80°C.

  1. Use Newton’s law of cooling  , where is the temperature of the water at the time minutes after the water is placed in the room, to show that    (2 marks)

  2. Find the temperature of the water 10 minutes after it is placed in the room.  (3 marks)

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a.   
 
 
 
     

COMMENT: Recognise that exponential G+D models are simply differential equations with known solutions (syllabus p.58)!

 

 

 

(b)  

 
 

Calculus, EXT1 C3 2014 VCE 10 MC

A large tank initially holds 1500 L of water in which 100 kg of salt is dissolved. A solution containing 2 kg of salt per litre flows into the tank at a rate of 8 L per minute. The mixture is stirred continuously and flows out of the tank through a hole at a rate of 10 L per minute.

The differential equation for , the number of kilograms of salt in the tank after minutes, is given by

A.   

B.  

C.  

D.  

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Calculus, EXT1 C3 2013 VCE 13 MC

Water containing 2 grams of salt per litre flows at the rate of 10 litres per minute into a tank that initially contained 50 litres of pure water. The concentration of salt in the tank is kept uniform by stirring and the mixture flows out of the tank at the rate of 6 litres per minute.

If grams is the amount of salt in the tank minutes after the water begins to flow, the differential equation relating to is

A.  

B.  

C.  

D.  

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Vectors, EXT2 V1 2015 VCE 1

Consider the rhombus    shown below, where    and  , and is a positive real constant.
 

VCAA 2015 spec 1a
 

  1. Find    (1 mark)

  2. Show that the diagonals of the rhombus    are perpendicular.  (2 marks)

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i.   

 

 

ii.   

MARKER’S COMMENT: Vector notation was poor in many answers.


 


 

 
 

 

Vectors, EXT2 V1 2014 SPEC1 1

Consider the vector  , where    and    are unit vectors in the positive directions of the and axes respectively.

  1.  Find the unit vector in the direction of  .  (1 mark)

  2.  Find the acute angle that    makes with the positive direction of the -axis.  (2 marks)

  3.  The vector  .

     

     Given that    is perpendicular to  find the value of .  (2 marks)

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i.   
   
 

 

Mean mark part (b) 51%.

ii.   
 
   
 

 

iii.   

 
 

Vectors, EXT2 V1 2013 SPEC1 3

The coordinates of three points are 

  1. Find    (1 mark)

  2. The points and are the vertices of a triangle. 

     

    Prove that the triangle has a right angle at   (2 marks)

  3. Find the length of the hypotenuse of the triangle.  (1 mark)

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i.  
   

 

ii.   
   

 

 
 

 


 

iii.   
   

 

 
 

Functions, EXT1 F1 SM-Bank 12

Given  , without calculus sketch a separate half page graph of the following functions, showing all asymptotes and intercepts.

  1.     (1 mark)

  2.     (2 marks)

  3.      (2 marks)

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  1.  
  2.   
  3.   
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i.   
   
   

 

 

ii.   

 

 

iii.

Functions, 2ADV F2 SM-Bank 6 MC

The graph of a function    is obtained from the graph of the function   by a reflection in the -axis followed by a dilation from the -axis by a factor of  .

Which one of the following is the function  ?

A.  

B.  

C.  

D.  

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COMMENT: Using “swap” terminology for dilations from the y-axis is simpler and more intelligible for students in our view.

 

 

 

Functions, 2ADV F2 SM-Bank 3

 

The diagram below shows part of the graph of the function with rule

, where , and are real constants.
 

    • The graph has a vertical asymptote with equation  .
    • The graph has a y-axis intercept at 1.
    • The point on the graph has coordinates  , where is another real constant.
       

      VCAA 2010 1b

  1. State the value of .  (1 mark)

  2. Find the value of .  (1 mark)

  3. Show that  .  (2 marks)

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i.  


 

  ii.  


 

iii.  

Algebra, STD2 A4 SM-Bank 2

Moses finds that for a Froghead eel, its mass is directly proportional to the square of its length.

An eel of this species has a length of 72 cm and a mass of 8250 grams.

What is the expected length of a Froghead eel with a mass of 10.2 kg? Give your answer to one decimal place.  (3 marks)

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Probability, 2ADV S1 SM-Bank 3

In a workplace of 25 employees, each employee speaks either French or German, or both.

If 36% of the employees speak German, and 20% speak both French and German.

  1. Calculate the probability one person chosen could speak German if they could speak French. Give your answer to the nearest percent.  (1 mark)

  2. Calculate the probability one person chosen could not speak French if they could speak German. Give your answer to the nearest percent.  (1 mark)

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i.   
 


 

 
 
 

 

ii.   
   
   
   

Financial Maths, 2ADV M1 SM-Bank 8

When placed in a pond, the length of a fish was 14.2 centimetres.

During its first month in the pond, the fish increased in length by 3.6 centimetres.

During its th month in the pond, the fish increased in length by centimetres, where  

Calculate the maximum length this fish can grow to.  (3 marks)

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Financial Maths, 2ADV M1 SM-Bank 6

Julie deposits some money into a savings account that will pay compound interest every month.

The balance of Julie’s account, in dollars, after months,  , can be modelled by the recurrence relation shown below.
 


 

  1.  Recursion can be used to calculate the balance of the account after one month.

     

    1. Write down a calculation to show that the balance in the account after one month, , is  $12 074.40. (1 mark)

    2. After how many months will the balance of Julie’s account first exceed $12 300  (1 mark)

  2.  A rule of the form    can be used to determine the balance of Julie's account after months.

    1. Complete this rule for Julie’s investment after months by writing the appropriate numbers in the boxes provided below. (1 mark)
       
balance = 
 
  × 
 
  n

 

    1. What would be the value of if Julie wanted to determine the value of her investment after three years?  (1 mark)

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a.i.   

a.ii. 

b.i 

b.ii. 

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a.i.  
   
   

 

a.ii.  
 
 

 

 
b.i.  

 
b.ii. 

Financial Maths, 2ADV M1 SM-Bank 5 MC

Shirley would like to purchase a new home. She will establish a loan for $225 000 with interest charged at the rate of 3.6% per annum, compounding monthly.

Each month, Shirley will pay only the interest charged for that month.

Let be the value of Shirley’s loan, in dollars, after months.

A recurrence relation that models the value of is

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Financial Maths, 2ADV M1 SM-Bank 4 MC

Each trading day, a share trader buys and sells shares according to the rule
 

 
 

where is the number of shares the trader owns at the start of the th trading day.

From this rule, it can be concluded that each day

A.    the trader sells 60% of the shares that she owned at the start of the day and then buys another 50 000 shares.

B.    the trader sells 40% of the shares that she owned at the start of the day and then buys another 50 000 shares.

C.    the trader sells 50 000 of the shares that she owned at the start of the day.

D.    the trader sells 60% of the 50 000 shares that she owned at the start of the day.

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