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Calculus, SPEC2 2013 VCAA 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.  

E.  

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Calculus, SPEC1 2013 VCAA 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.   
 
 
 
     

 

 

 

(b)  

 
 

Graphs, SPEC1 2013 VCAA 4

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

  2. Sketch the graph of    over its maximal domain. Label the endpoints with their coordinates.   (2 marks)

     

     
              VCAA 2013 spec 4b
     

  3. Find the gradient of the tangent to the graph of    at    (2 marks)

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a. 

 

 

b.   

 

c.   

 

 

 
 
 

Vectors, SPEC1 2013 VCAA 3

The coordinates of three points are 

  1. Find    (1 mark)

  2. The points and are the vertices of a triangle.
  3. Prove that the triangle has a right angle at   (2 marks)

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

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a.  
   

 

b.   
   

 

 
 

 


 

c.   
   

 

 
 

Calculus, SPEC2 2014 VCAA 21 MC

The acceleration, in , of a particle moving in a straight line is given by  , where metres is its displacement from a fixed origin .

If the particle is at rest where  , the speed of the particle, in , where    is

A.           

B.     

C.         

D.     

E.   

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Mechanics, SPEC2 2014 VCAA 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.   10.0 m/s

E.   19.6 m/s

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Calculus, SPEC2 2015 VCAA 20 MC

An object is moving in a straight line, initially at . Sixteen seconds later, it is moving at in the opposite direction to its initial velocity.

Assuming that the acceleration of the object is constant, after 16 seconds the distance, in metres, of the object from its starting point is

A.   24

B.   48

C.   73

D.   96

E.   128

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Calculus, SPEC2 2015 VCAA 11 MC

The velocity–time graph for a body moving along a straight line is shown below.
 

SPEC2 2015 VCAA 11 MC
 

The body first returns to its initial position within the time interval

A.   (0, 0.5)

B.   (0.5, 1.5)

C.   (1.5, 2.5)

D.   (2.5, 3.5)

E.   (3.5, 5)

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Statistics, SPEC2 2016 VCAA 20 MC

The lifetime of a certain brand of batteries is normally distributed with a mean lifetime of 20 hours and a standard deviation of two hours. A random sample of 25 batteries is selected.

The probability that the mean lifetime of this sample of 25 batteries exceeds 19.3 hours is

A.   0.0401

B.   0.1368

C.   0.6103

D.   0.8632

E.   0.9599

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Statistics, SPEC2 2016 VCAA 18 MC

Oranges grown on a citrus farm have a mean mass of 204 grams with a standard deviation of 9 grams

Lemons grown on the same farm have a mean mass of 76 grams with a standard deviation of 3 grams.

The masses of the lemons are independent of the masses of the oranges.

The mean mass and standard deviation, in grams, respectively of a set of three of these oranges and two of these lemons are

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Mechanics, SPEC2 2016 VCAA 17 MC

A body of mass 3 kg is moving to the left in a straight line at 2 . It experiences a force for a period of time, after which it is then moving to the right at 2 .

The change in momentum of the particle, in kg , in the direction of the final motion is

A.   − 6

B.     0

C.     4

D.     6

E.   12

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Calculus, SPEC2 2014 VCAA 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.  

E.  

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Complex Numbers, SPEC2 2016 VCAA 6 MC

The points corresponding to the four complex numbers given by
 


 

are the vertices of a parallelogram in the complex plane.

Which one of the following statements is not true?

A.  The acute angle between the diagonals of the parallelogram is 

B.  The diagonals of the parallelogram have lengths 2 and 4

C. 

D. 

E.    for all four of   

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Vectors, SPEC1 2016 VCAA 8

The position of a body with mass 3 kg from a fixed origin at time    seconds, , is given by  , where components are in metres.

  1. Find an expression for the speed, in metres per second, of the body at time  .   (2 marks)

  2. Find the speed of the body, in metres per second, when  .   (1 mark)

  3. Find the maximum magnitude of the net force acting on the body in newtons.   (3 marks)

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a.   
  
 

 

b.   

 
   
   
   
   

 

c.   
 
 
   
   
   

 

Mean mark 51%.

 
 
 

Vectors SPEC1 2016 VCAA 5

Consider the vectors    and  , where    is a real constant.

  1. Find the vector resolute of    in the direction of  .  (2 marks)

  2. Find the value of    if the vectors are linearly dependent.  (2 marks)

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  2.  

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a.   


 

 
     

 

b.   

 

 
   
   

Calculus, SPEC1 2016 VCAA 4

Chemicals are added to a container so that a particular crystal will grow in the shape of a cube. The side length of the crystal,   millimetres,   days after the chemicals were added to the container, is given by  .

Find the rate at which the surface area, square millimetres, of the crystal is growing one day after the chemicals were added. Give your answer in square millimetres per day.  (4 marks)

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  ²  

Statistics, SPEC2-NHT 2017 VCAA 6

A bank claims that the amount it lends for housing is normally distributed with a mean of $400 000 and a standard deviation of $30 000.

A consumer organisation believes that the average loan amount is higher than the bank claims.

To check this, the consumer organisation examines a random sample of 25 loans and finds the sample mean to be $412 000.

  1. Write down the two hypotheses that would be used to undertake a one-sided test.   (1 mark)

  2. Write down an expression for the value for this test and evaluate it to four decimal places.   (2 marks)

  3. State with a reason whether the bank’s claim should be rejected at the 5% level of significance.   (1 mark)

  4. What is the largest value of the sample mean that could be observed before the bank’s claim was rejected at the 5% level of significance? Give your answer correct to the nearest 10 dollars.   (1 mark)

  5. If the average loan made by the bank is actually $415 000 and not $400 000 as originally claimed, what is the probability that a random selection of 25 loans has a sample mean that is at most $410 000? Give your answer correct to three decimal places.   (2 marks)

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a.   

 

b.   


 

 

c.   

 

d.   

 

e.   

Mechanics, SPEC2-NHT 2017 VCAA 5

A 5 kg mass is initially held at rest on a smooth plane that is inclined at 30° to the horizontal. The mass is connected by a light inextensible string passing over a smooth pulley to a 3 kg mass, which in turn is connected to a 2 kg mass.

The 5 kg mass is released from rest and allowed to accelerate up the plane.

Take acceleration to be positive in the directions indicated.
 

  1. Write down an equation of motion, in the direction of motion, for each mass.   (3 marks)
  2. Show that the acceleration of the 5 kg mass is  .  (1 mark)
  3. Find the tensions    and    in the string in terms of  .  (2 marks)
  4. Find the momentum of the 5 kg mass, in kg ms­, after it has moved 2 m up the plane, giving your answer in terms of .  (2 marks)
  5. A resistance force    acting parallel to the inclined plane is added to hold the system in equilibrium, as shown in the diagram below.
     

     


     

     

    Find the magnitude of    in terms of  .  (2 marks)

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  3.  

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b.   

 

c.   
 

 

 

 

d.   

 
   
   
 

 

 

e.   

 
 

Vectors, SPEC2-NHT 2017 VCAA 4

A cricketer hits a ball at time    seconds from an origin at ground level across a level playing field.

The position vector  , from , of the ball after seconds is given by
 
  ,
 
where,    is a unit vector in the forward direction,   is a unit vector vertically up and displacement components are measured in metres.

  1. Find the initial velocity of the ball and the initial angle, in degrees, of its trajectory to the horizontal.   (2 marks)

  2. Find the maximum height reached by the ball, giving your answer in metres, correct to two decimal places.   (2 marks)

  3. Find the time of flight of the ball. Give your answer in seconds, correct to three decimal places.   (1 mark)

  4. Find the range of the ball in metres, correct to one decimal place.   (1 mark)

  5. A fielder, more than 40 m from , catches the ball at a height of 2 m above the ground.

     

    How far horizontally from is the fielder when the ball is caught? Give your answer in metres, correct to one decimal place.   (2 marks)

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a.   


 

 
 

 

b. 

 

 

 

 

c.   

 

d.   
 
 


e.   

 
 

  

 

 


 

Calculus, SPEC2-NHT 2017 VCAA 3

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.

    1. Express    in partial fraction form.   (1 mark)

    2. Hence show by integration that  , where    is a constant of integration.   (2 marks)

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

After one hour, a toxin is added to the Petri dish, which harms the bacteria and reduces their rate of growth. The differential equation that models the rate of growth is now

  for 

  1. Find the limiting value of  , which is the maximum possible proportion of the Petri dish that can now be covered by the bacteria. Give your answer correct to three decimal places.   (2 marks)

  2. The total time,   hours, measured from time  , needed for the bacteria to cover 80% of the Petri dish is given by
     

     


     

     

    where  .

     

     

    Find the values of  and , giving the value of correct to two decimal places.   (2 marks)

  1. Given that    when  , use Euler’s method with a step size of 0.5 to estimate the value of when  . Give your answer correct to three decimal places.   (3 marks)

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a.i.   

 

a.ii. 

 
 
 

  


a.iii. 

 

 

b.   


 

 

 

c.   


 


 

 

d.   

   

   
 

Statistics, SPEC2 2017 VCAA 6

A dairy factory produces milk in bottles with a nominal volume of 2 L per bottle. To ensure most bottles contain at least the nominal volume, the machine that fills the bottles dispenses volumes that are normally distributed with a mean of 2005 mL and a standard deviation of 6 mL.

  1. Find the percentage of bottles that contain at least the nominal volume of milk, correct to one decimal place.   (1 marks)

Bottles of milk are packed in crates of 10 bottles, where the nominal total volume per crate is 20 L.

  1. Show that the total volume of milk contained in each crate varies with a mean of 20 050 mL and a standard deviation of    mL.   (2 marks)

  2. Find the percentage, correct to one decimal place, of crates that contain at least the nominal volume of 20 L.   (1 mark)

  3. Regulations require at least 99.9% of crates to contain at least the nominal volume of 20 L.
  4. Assuming the mean volume dispensed by the machine remains 2005 mL, find the maximum allowable standard deviation of the bottle-filling machine needed to achieve this outcome. Give your answer in millilitres, correct to one decimal place.   (3 marks)

  5. A nearby dairy factory claims the milk dispensed into its 2 L bottles varies normally with a mean of 2005 mL and a standard deviation of 2 mL.
  6. When authorities visit the nearby dairy factory and check a random sample of 10 bottles of milk, they find the mean volume to be 2004 mL.
  7. Assuming that the standard deviation of 2 mL is correct, carry out a one-sided statistical test and determine, stating a reason, whether the nearby dairy’s claim should be accepted at the 5% level of significance.   (2 marks)

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  4. σ
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a.   

 

 

b.   

 
     
  σ σ
   
  σ

 

c.  

 

 

d.  

σ

 
 
 

 

σ
σ

 
σ

 

e.  

 

  

Vectors, SPEC2 2017 VCAA 5

On a particular morning, the position vectors of a boat and a jet ski on a lake    minutes after they have started moving are given by    and    respectively for  , where distances are measured in kilometres. The boat and the jet ski start moving at the same time. The graphs of their paths are shown below.

  1. On the diagram above, mark the initial positions of the boat and the jet ski, clearly identifying each of them. Use arrows to show the directions in which they move.   (2 marks)

    1. Find the first time for    when the speeds of the boat and the jet ski are the same.   (2 marks)

    2. State the coordinates of the boat at this time.   (1 mark)

    1. Write down an expression for the distance between the jet ski and the boat at any time .   (1 mark)

    2. Find the minimum distance separating the boat and the jet ski. Give your answer in kilometres, correct to two decimal places.   (1 mark)

  4. On another morning, the boat’s position vector remained the same but the jet skier considered starting from a different location with a new position vector given by  , where is a real constant. Both vessels are to start at the same time.
    Assuming the vessels would collide shortly after starting, find the time of the collision and the value of .   (3 marks)

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  2.   
  3.   
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a.   


  

 

b.i.   
 
   

 

♦ Mean mark (b)(ii) 48%.

b.ii.   
   

 

c.i.   
 
   

 

c.ii. 

♦♦♦ Mean mark part (c)(ii) 15%.

 

d.   

 
 
 
 

 

♦ Mean mark part (d) 41%.
 

 

 
 

 

Complex Numbers, SPEC2 2017 VCAA 4

  1. Express    in polar form.  (1 mark)

  2. Show that the roots of    are    and  .  (1 mark)

  3. Express the roots of    in terms of  .  (1 mark)

  4. Show that the cartesian form of the relation    is    (2 marks)

  5. Sketch the line represented by    and plot the roots of    on the Argand diagram below.  (2 marks)

     
       

  6. The equation of the line passing through the two roots of    can be expressed as  , where  .

     

    Find in terms of .  (1 mark)

  7. Find the area of the major segment bounded by the line passing through the roots of    and the major arc of the circle given by  .  (2 marks)

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a.   

 

 
 

 

 

b.   

 

♦♦ Mean mark part (c) 31%.

c.   
 

 

d.
    
   
 
 

 

 

e.   

 

f.   

♦♦♦ Mean mark 1%!


 

  

 
 

 

♦♦ Mean mark 31%.

g.   
   

 

 

 

Calculus, SPEC2 2017 VCAA 3

A brooch is designed using inverse circular functions to make the shape shown in the diagram below.
 


 

The edges of the brooch in the first quadrant are described by the piecewise function

  1. Write down the coordinates of the corner point of the brooch in the first quadrant.  (1 mark)

  2. Specify the piecewise function that describes the edges in the third quadrant.  (1 mark)

  3. Given that each unit in the diagram represents one centimetre, find the area of the brooch.
  4. Give your answer in square centimetres, correct to one decimal place.  (3 marks)

  5. Find the acute angle between the edges of the brooch at the origin. Give your answer in degrees, correct to one decimal place.  (3 marks)

  6. The perimeter of the brooch has a border of gold.
    Show that the length of the gold border needed is given by a definite integral of the form  , where  . Find the values of and .  (2 marks)

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a.   


 

b.   

♦ Mean mark 49%.

 

c.   
    ²

 

d.   


 


  

e. 
 

♦♦ Mean mark 27%.

  

Complex Numbers, SPEC2-NHT 2017 VCAA 2

One root of a quadratic equation with real coefficients is  .

    1. Write down the other root of the quadratic equation.   (1 mark)

    2. Hence determine the quadratic equation, writing it in the form  .   (2 marks)

  1. Plot and label the roots of     on the Argand diagram below.   (3 marks)

  

 

  1. Find the equation of the line that is the perpendicular bisector of the line segment joining the origin and the point  . Express your answer in the form  .   (2 marks)

  1. The three roots plotted in part b. lie on a circle.

     

    Find the equation of this circle, expressing it in the form  ,  where  .   (3 marks)

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a.i.   

 

 

a.ii.   
 
 
 
 

 

b.   
 

 

 
   
 

 

   

 

c.   

 


 

 
 
 

 

d.   


 


 


 

Calculus, SPEC2-NHT 2017 VCAA 1

  1. i.  Use an appropriate double angle formula with    to deduce a quadratic equation of the form  , where and are real values.   (2 marks)

  2. ii. Hence show that  .   (1 mark)

Consider  .

  1. Sketch the graph of on the axes below, labelling the end points with their coordinates.   (3 marks)


 

  1. The region between the graph of    and the -axis is rotated about the -axis to form a solid of revolution.
  2. i.  Write down a definite integral in terms of    that gives the volume of the solid formed.   (2 marks)

  3. ii. Find the volume of the solid, correct to the nearest integer.   (1 mark)

  1. A fish pond that has a shape approximately like that of the solid of revolution in part c. is being filled with water. When the depth is metres, the volume, , of water in the pond is given by

     

     

    If water is flowing into the pond at a rate of 0.03 m³ per minute, find the rate at which the depth is increasing when the depth is 0.6 m. Give your answer in metres per minute, correct to three decimal places.   (3 marks)

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  1. i. 
  2. ii.
  4. i. 
  5. ii.
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a.i.   

 

 

a.ii.   
 
 
 

 

 

b.   


 

 

c.i.   
 
 
 
 

 

 
   

 

c.ii. 

³

 

d.   
 
 
 
   

  

Statistics, SPEC2-NHT 2018 VCAA 7

According to medical records, the blood pressure of the general population of males aged 35 to 45 years is normally distributed with a mean of 128 and a standard deviation of 14. Researchers suggested that male teachers had higher blood pressures than the general population of males.

To investigate this, a random sample of 49 male teachers from this age group was obtained and found to have a mean blood pressure of 133.

  1. State two hypotheses and perform a statistical test at the 5% level to determine if male teachers belonging to the 35 to 45 years age group have higher blood pressures than the general population of males. Clearly state your conclusion with a reason.   (3 marks)

  2. Find a 90% confidence interval for the mean blood pressure of all male teachers aged 35 to 45 years using a standard deviation of 14. Give your answers correct to the nearest integer.   (1 mark)

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b.  

Statistics, SPEC2-NHT 2018 VCAA 6

A coffee machine dispenses coffee concentrate and hot water into a 200 mL cup to produce a long black coffee. The volume of coffee concentrate dispensed varies normally with a mean of 40 mL and a standard deviation of 1.6 mL.

Independent of the volume of coffee concentrate, the volume of water dispensed varies normally with a mean of 150 mL and a standard deviation of 6.3 mL.

  1. State the mean and the standard deviation, in millilitres, of the total volume of liquid dispensed to make a long black coffee.   (2 marks)

  2. Find the probability that a long black coffee dispensed by the machine overflows a 200 mL cup. Give your answer correct to three decimal places.   (1 mark)

  3. Suppose that the standard deviation of the volume of water dispensed by the machine can be adjusted, but that the mean volume of water dispensed and the standard deviation of the volume of coffee concentrate dispensed cannot be adjusted.
  4. Find the standard deviation of the volume of water dispensed that is needed for there to be only a 1% chance of a long black coffee overflowing a 200 mL cup. Give your answer in millilitres, correct to two decimal places.   (2 marks)

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a.   

 
 
   

 

b.   

 

c.  


 


  

Calculus, SPEC2-NHT 2018 VCAA 5

A horizontal beam is supported at its endpoints, which are 2 m apart. The deflection metres of the beam measured downwards at a distance metres from the support at the origin is given by the differential equation  .
 


 

  1. Given that both the inclination, , and the deflection, , of the beam from the horizontal at    are zero, use the differential equation above to show that  .   (2 marks)

  2. Find the angle of inclination of the beam to the horizontal at the origin . Give your answer as a positive acute angle in degrees, correct to one decimal place.   (2 marks)

  3. Find the value of , in metres, where the maximum deflection occurs, and find the maximum deflection, in metres.   (3 marks)

  4. Find the maximum angle of inclination of the beam to the horizontal in the part of the beam where  . Give your answer as a positive acute angle in degrees, correct to one decimal place.   (2 marks)

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a.  
 

 

 

 

 

 

b.   

 
 
 

 

c.   

 

 

 

d.   

 

 

 

Vectors, SPEC2-NHT 2018 VCAA 4

A basketball player aims to throw a basketball through a ring, the centre of which is at a horizontal distance of 4.5 m from the point of release of the ball and 3 m above floor level. The ball is released at a height of 1.75 m above floor level, at an angle of projection to the horizontal and at a speed of  . Air resistance is assumed to be negligible.
 


 

The position vector of the centre of the ball at any time, seconds, for  , relative to the point of release is given by 
 
,
 
where    is a unit vector in the horizontal direction of motion of the ball and    is a unit vector vertically up. Displacement components are measured in metres.

  1. For the player’s first shot at goal, and 
  2.   i. Find the time, in seconds, taken for the ball to reach its maximum height. Give your answer in the form  , where  and are positive integers.   (2 marks) 
  3.  ii. Find the maximum height, in metres, above floor level, reached by the centre of the ball.   (2 marks)

  4. iii. Find the distance of the centre of the ball from the centre of the ring one second after release. Give your answer in metres, correct to two decimal places.   (2 marks)

  5. For the player’s second shot at goal, .
    Find the possible angles of projection,  , for the centre of the ball to pass through the centre of the ring. Give your answers in degrees, correct to one decimal place.   (3 marks)

  6. For the player’s third shot at goal, the angle of projection is 
  7. Find the speed required for the centre of the ball to pass through the centre of the ring. Give your answer in metres per second, correct to one decimal place.   (2 marks)

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a.i.   
   
     

 

 

a.ii.  
   

 

 

a.iii. 

 
 
   
   

 

b.  


 

 

c.