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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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  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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Statistics, 2ADV S3 SM-Bank 8

The continuous random variable has a distribution with probability density function given by
 


 

where is a positive constant.

  1. Find the value of  .  (3 marks)

  2. Express    as a  definite integral. (Do not evaluate the definite integral.)  (1 mark)

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a.  
 
 
 
 
 

 

b.  

Probability, 2ADV S1 EQ-Bank 40

One bag contains red and green balls.

Kalyn randomly chooses one ball from the bag. Without replacement, he then chooses a second ball from the bag.

Complete the tree diagram below and then draw a probability distribution table for the number of red balls that could be drawn out of the bag.  (3 marks)
 

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Trigonometry, 2ADV T3 SM-Bank 16

Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point . The height of above the ground, , is modelled by  , where is the time in minutes after Sammy enters the capsule and is measured in metres.

Sammy exits the capsule after one complete rotation of the Ferris wheel.
 


 

  1. State the minimum and maximum heights of above the ground.  (1 mark)

  2. For how much time is Sammy in the capsule?  (1 mark)

  3. Find the rate of change of with respect to and, hence, state the value of at which the rate of change of is at its maximum.  (2 marks)

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i.   
     

 

ii.   

 

iii.   
 
   
     

 

 
 
   

Trigonometry, 2ADV T3 SM-Bank 13

On any given day, the depth of water in a river is modelled by the function

where is the depth of water, in metres, and    is the time, in hours, after 6 am. 

  1. Find the minimum depth of the water in the river.  (1 mark)

  2. Find the values of    for which  .  (2 marks)

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i.   

MARKER’S COMMENT: Students who used calculus to find the minimum were less successful.
 

 

ii.