SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Algebra, SPEC2 2012 VCAA 2 MC

A rectangle is drawn so that its sides lie on the lines with equations   , ,   and  .

An ellipse is drawn inside the rectangle so that it just touches each side of the rectangle.

The equation of the ellipse could be

A.  

B.  

C.  

D.  

E.  

Show Answers Only

Show Worked Solution

SPEC2 2012 VCAA 2 MC Answer 1_1

Vectors, SPEC1 2013 VCAA 7

The position vector    of a particle moving relative to an origin at time seconds is given by

where the components are measured in metres.

  1. Show that the cartesian equation of the path of the particle is     (2 marks)

  2. Sketch the path of the particle on the axes below, labelling any asymptotes with their equations.   (2 marks)

     

        
     VCAA 2013 spec 7b
     

  3. Find the speed of the particle, in , when    (2 marks)

Show Answers Only
  2.  
Show Worked Solution
a.  
 
 
 

  

 

b.   

♦ Mean mark part (b) 45%.

 

♦ Mean mark part (c) 39%.

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)

Show Answers Only

Show Worked Solution

a.  
   

 

b.   
   

 

 
 

 


 

c.   
   

 

 
 

Mechanics, SPEC1 2013 VCAA 1

A body of mass 10 kg is held in place on a smooth plane inclined at 30° to the horizontal by a tension force, newtons, acting parallel to the plane.

  1. On the diagram below, show all other forces acting on the body and label them.  (1 mark)
     

     

                   VCAA 2013 spec 1a
     

  2. Find the value of   (2 marks)
Show Answers Only
  1.  

Show Worked Solution
a.   

 

b.   
 
 
 
   

Statistics, SPEC2 2016 VCAA 19 MC

A random sample of 100 bananas from a given area has a mean mass of 210 grams and a standard deviation of 16 grams.

Assuming the standard deviation obtained from the sample is a sufficiently accurate estimate of the population standard deviation, an approximate 95% confidence interval for the mean mass of bananas produced in this locality is given by

A.   

B.   

C.   

D.   

E.   

Show Answers Only

Show Worked Solution

 

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)

Show Answers Only
Show Worked Solution
a.   
  
 

 

b.   

 
   
   
   
   

 

c.   
 
 
   
   
   

 

Mean mark 51%.

 
 
 

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)

Show Answers Only

  1.  

Show Worked Solution

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)

Show Answers Only

  1.  

     

  3.  

Show Worked Solution
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)

Show Answers Only


Show Worked Solution

a.i.   

 

a.ii. 

 
 
 

  


a.iii. 

 

 

b.   


 

 

 

c.   


 


 

 

d.   

   

   
 

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)

Show Answers Only
  5.  
Show Worked Solution
a.   

 

 
 

 

 

b.   

 

♦♦ Mean mark part (c) 31%.

c.   
 

 

d.
    
   
 
 

 

 

e.   

 

f.   

♦♦♦ Mean mark 1%!


 

  

 
 

 

♦♦ Mean mark 31%.

g.   
   

 

 

 

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)

Show Answers Only
Show Worked Solution

a.i.   

 

 

a.ii.   
 
 
 
 

 

b.   
 

 

 
   
 

 

   

 

c.   

 


 

 
 
 

 

d.   


 


 


 

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)

Show Answers Only

  1.  

Show Worked Solution

a.   

 
 
   

 

b.   

 

c.  


 


  

Mechanics, SPEC2-NHT 2018 VCAA 3

A 200 kg crate rests on a smooth plane inclined at to the horizontal. An external force of newtons acts up the plane, parallel to the plane, to keep the crate in equilibrium.

  1. On the diagram below, draw and label all forces acting on the crate.  (1 mark)

 

 

  1. Find in terms of .  (1 mark)

The magnitude of the external force is changed to 780 N and the plane is inclined at  .

    1. Taking the direction down the plane to be positive, find the acceleration of the crate.  (2 marks)
    2. On the axes below, sketch the velocity–time graph for the crate in the positive direction for the first four seconds of its motion.  (1 mark)

       

       
       
    3. Calculate the distance the crate travels, in metres, in its first four seconds of motion.  (1 mark)

Starting from rest, the crate slides down a smooth plane inclined at    degrees to the horizontal.

A force of    newtons, up the plane and parallel to the plane, acts on the crate.

  1. If the momentum of the crate is 800 kg ms¯¹ after having travelled 10 m, find the acceleration, in ms¯², of the crate.  (2 marks)
  2. Find the angle of inclination, , of the plane if the acceleration of the crate down the plane is 0.75 ms¯².  Give your answer in degrees, correct to one decimal place.  (2 marks)
Show Answers Only
Show Worked Solution
a.   

 

b.   

 

c.i.  
 
   
 

 

c.ii.  

 

c.iii. 


 

d.  
 
 

 

 
 
 

 

e.  

Complex Numbers, SPEC2-NHT 2018 VCAA 2

In the complex plane, is the line given by  .

  1. Show that the cartesian equation of is given by  .   (2 marks)

  2.  Find the point(s) of intersection of and the graph of the relation    in cartesian form.   (2 marks)

  3. Sketch and the graph of the relation    on the Argand diagram below.   (2 marks)

 

 

The part of the line in the fourth quadrant can be expressed in the form  .

  1. State the value of .   (1 mark)

  2. Find the area enclosed by and the graphs of the relations    and  .   (2 marks)

  3. The straight line can be written in the form , where  .

     

    Find in the form  , where    is the principal argument of .   (2 marks)

Show Answers Only
Show Worked Solution

a.   

 

b.  
 
 

 

 

 

c.   

 

d.  


 

e.   

²

 

f.   
 
   
   

Calculus, SPEC2-NHT 2018 VCAA 1

Consider the function    with rule  .

  1.  Sketch the graph of    over its maximal domain on the set of axes below. Label the endpoints with their coordinates.   (3 marks)

     


 

  1. A vase is to be modelled by rotating the graph of    about the -axis to form a solid of revolution, where units of measurement are in centimetres.
    1.  Write down a definite integral in terms of that gives the volume of the vase.   (2 marks)

    2.  Find the volume of the vase in cubic centimetres.  (1 mark)

  3. Water is poured into the vase at a rate of 20 cm³ s¯¹.
  4. Find the rate, in centimetres per second, at which the depth of the water is changing when the depth is    cm.  (3 marks)

  5. The vase is placed on a table. A bee climbs from the bottom of the outside of the vase to the top of the vase.
  6. What is the minimum distance the bee will need to travel? Give your answer in centimetres, correct to one decimal place.  (1 mark)

Show Answers Only

  2.  i.
  3. ii.

Show Worked Solution

a.  

 

b.i.  
 
 
 
 

 

 
   

 

b.ii.  

 

c.  ³

 


 

 
 

 

d.   

 

Calculus, SPEC2 2017 VCAA 1

Let  , where is the maximal domain of  .

  1.   i. Find the equations of any asymptotes of the graph of  .   (1 mark)

  2.  ii. Find    and state the coordinates of any stationary points of the graph of  , correct to two decimal places.  (2 marks)

  3. iii. Find the coordinates of any points of inflection of the graph of  , correct to two decimal places.  (2 marks)

  4. Sketch the graph of    from    and    on the axes provided below, marking all stationary points, points of inflection and intercepts with axes, labelling them with their coordinates. Show any asymptotes and label them with their equations.  (3 marks)

     

     

  5. The region , bounded by the graph of  , the -axis and the line  , is rotated about the -axis to form a solid of revolution. The line  , where  , divides the region into two regions such that, when the two regions are rotated about the -axis, they generate solids of equal volume.
  6. i.  Write down an equation involving definite integrals that can be used to determine .  (2 marks)

  7. ii. Hence, find the value of , correct to two decimal places.  (1 mark)

Show Answers Only

  1. i. 
  2. ii.
  3. iii.
  4.  
  5. i. 
  6. ii.

Show Worked Solution

a.i.   

♦♦ Mean mark 36%.


 

a.ii.   

 

 

 

a.iii. 


 

 

b.   

 

c.i.   
 

 

 

c.ii.  

Statistics, SPEC2 2018 VCAA 6

The heights of mature water buffaloes in northern Australia are known to be normally distributed with a standard deviation of 15 cm. It is claimed that the mean height of the water buffaloes is 150 cm.

To decide whether the claim about the mean height is true, rangers selected a random sample of 50 mature water buffaloes. The mean height of this sample was found to be 145 cm.

A one-tailed statistical test is to be carried out to see if the sample mean height of 145 cm differs significantly from the claimed population mean of 150 cm.

Let denote the mean height of a random sample of 50 mature water buffaloes.

  1. State suitable hypotheses and for the statistical test.   (1 mark)

  2. Find the standard deviation of .   (1 mark)

  3. Write down an expression for the value of the statistical test and evaluate your answer to four decimal places.   (2 marks)

  4. State with a reason whether should be rejected at the 5% level of significance.   (1 mark)

  5. What is the smallest value of the sample mean height that could be observed for to be not rejected? Give your answer in centimetres, correct to two decimal places.   (1 mark)

  6. If the true mean height of all mature water buffaloes in northern Australia is in fact 145 cm, what is the probability that will be accepted at the 5% level of significance? Give your answer correct to two decimal places.   (1 mark)

  7. Using the observed sample mean of 145 cm, find a 99% confidence interval for the mean height of all mature water buffaloes in northern Australia. Express the values in your confidence interval in centimetres, correct to one decimal place.   (1 mark)

Show Answers Only
  1.  

Show Worked Solution
a.   
 

 

b.   
   
   

 

c.   

 

d.   

 

e. 

♦ Mean mark part (e) 48%.

 

f.   

♦♦♦ Mean mark part (f) 11%.

 
   

 

g.   

Statistics, SPEC1-NHT 2017 VCAA 9

The random variables and are independent with    and  .

  1. The random variable is such that  .
  2.  i. Find .   (1 mark)

  3. ii. Find the standard deviation of .   (1 mark)

  1. Researchers have reason to believe that the mean of has decreased. They collect a random sample of 64 observations of and find that the sample mean is 
  2.  i. State the null hypothesis and the alternative hypothesis that should be used to test that the mean has decreased.   (1 mark)

  3. ii. Calculate the mean and standard deviation for a distribution of sample means, , for samples of 64 observations.   (1 mark)

Show Answers Only
  1.  i.
  2. ii.
  3.  i.
  4. ii.
Show Worked Solution
a.i.   
   
   
   

 

a.ii.   
   
   
   

 

 

 

b.i.   
 

 

b.ii.   
 

 

 

Calculus, SPEC2 2018 VCAA 3

Part of the graph of    is shown below.
 


 

The curve shown is rotated about the -axis to form a volume of revolution that is to model a fountain, where length units are in metres.

  1. Show that the volume, cubic metres, of water in the fountain when it is filled to a depth of metres is given by  .   (2 marks)

  2. Find the depth when the fountain is filled to half's its volume. Give your answer in metres, correct to two decimal places.   (2 marks)

The fountain is initially empty. A vertical jet of water in the centre fills the fountain at a rate of 0.04 cubic metres per second and, at the same time, water flows out from the bottom of the fountain at a rate of    cubic metres per second when the depth is metres.

  1.  i. Show that  .   (2 marks)

  2. ii. Find the rate, in metres per second, correct to four decimal places, at which the depth is increasing when the depth is 0.25 m.   (1 mark)

  3. Express the time taken for the depth to reach 0.25 m as a definite integral and evaluate this integral correct to the nearest tenth of a second.   (2 marks)

  4. After 25 seconds the depth has risen to 0.4 m.
    Using Euler's method with a step size of five seconds, find an estimate of the depth 30 seconds after the fountain began to fill. Give your answer in metres, correct to two decimal places.   (2 marks)

  5. How far from the top of the fountain does the water level ultimately stabilise? Give your answer in metres, correct to two decimal places.   (2 marks)

Show Answers Only
Show Worked Solution

a.   

 

 
 
 
 

 

b.   
   
   

 

  

c.i.  
 
 
   
     
 
 
   
   

 

c.ii.  
   

 

d.   

 

 

e.   

♦♦ Mean mark part (e) 30%.

 
 

 

f.   

♦♦ Mean mark part (f) 32%.

`= 0.05 sqrt h“

  

 
 

Calculus, SPEC2 2018 VCAA 1

Consider the function  , where  .

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

  2. Sketch the graph of    on the axes below, labelling any endpoints and the -intercept with their coordinates.  (3 marks)


      

     
     

  3. Find    for  , expressing your answer in the form  .  (1 mark)

  4. Write down    for  , expressing your answer in the form  .  (1 mark)

  5. The derivative    can be expressed in the form  over its maximal domain.
  1. Find the maximal domain of  .  (1 mark)

  2. Find  , expressing your answer as a piecewise (hybrid) function.  (1 mark)

  3. Sketch the graph of  on the axes below.  (2 marks)


     

Show Answers Only

Show Worked Solution

a.   

 

 

b.   

 

c.   

 

 

d.   
   

 

e.i. 

♦♦ Mean mark part (e)(i) 21%.

 

e.ii. 

♦ Mean mark part (e)(ii) 49%, part (e)(iii) 48%.

 

e.iii.   

Calculus, SPEC1 2017 VCAA 10

  1.  Show that  , where  .   (1 mark)

  2.  State the maximum domain and the range of  .   (2 marks)

  3.  Find the volume of the solid of revolution generated when the region bounded by the graph of  , and the lines    and  , is rotated about the -axis.   (4 marks)

Show Answers Only
Show Worked Solution
a.   
   
 
 

 

b.   

 

c.   
 
 
 

 

♦ Mean mark part (c) 35%.


 

 
 

Calculus, SPEC1 2015 VCAA 8

  1.  Show that    (2 marks)


  2.  The graph of    is shown below
     
         
     
  3. i.  Write down the equations of the asymptotes.  (1 mark)

  4. ii. On the axes above, sketch the graph of  , labelling any asymptotes with their equations.  (1 mark)

  5.  Find    (1 mark)

  6.  Find the area enclosed by the graph of  , the -axis and the line    (2 marks)

Show Answers Only

a.   

b. (i)  
  (ii)  

c.   

d.   

Show Worked Solution

a.   

 
 
 
 

 

b.i. 

 

b.ii.

 

c.  
   
   

 

d.   
 
 

 

 

♦ Mean mark part (d) 49%.

 
 
 
 
 

Calculus, SPEC1 2014 VCAA 7

Consider  .

  1.  Write down the range of  .  (1 mark)

  2.  Show that  .  (1 mark)

  3.  Hence evaluate the area enclosed by the graph of  , the -axis and the lines    and  .  (3 marks)

Show Answers Only

Show Worked Solution

a. 

♦♦♦ Mean mark 4%.


 


 
 

 

b.  

 

 

c.   
   
   
   
   
   
   
   
   

Calculus, SPEC1 2014 VCAA 6

  1.  Verify that  .   (1 mark)

Part of the graph of    is shown below.
 


  

  1.  The region enclosed by the graph of    and the lines    and    is rotated about the -axis

     

     Find the volume of the resulting solid of revolution.   (4 marks)


Show Answers Only

Show Worked Solution

a.   
   
   

 

b.  

 
 
 

 


 

 
 
 
 
 

Calculus, SPEC1 2014 VCAA 5

  1. For the function with rule  , Find the value of such that  .   (1 mark)

  2. Use an appropriate substitution in the form    to find an equivalent definite integral for

     

           in terms of only.   (3 marks)

  3. Hence evaluate  , giving your answer in the form  .   (1 mark)

Show Answers Only
Show Worked Solution

a.   


 

b.

   

 

c.

Vectors, SPEC1 2014 VCAA 2

The position vector of a particle at time    is given by

  1.  Show that the cartesian equation of the path followed by the particle is  .  (1 mark)

  2.  Sketch the path followed by the particle on the axes below, labelling all important features.  (2 marks)


 

 

  1.  Find the speed of the particle when  .  (2 marks)

Show Answers Only

Show Worked Solution

a.   
 
 

 

b.   

 

 

 

c.   
 
   
 
   

Vectors, SPEC1 2014 VCAA 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  .
  4. Given that    is perpendicular to  find the value of .  (2 marks)

Show Answers Only

Show Worked Solution

a.   
   
 

 

Mean mark part (b) 51%.

b.   
 
   
 

 

c.   

 
 

Calculus, SPEC1-NHT 2018 VCAA 9

  1.    i. Given that  , show that  .   (2 marks)

  2.  ii. Show that  .  (1 mark)

  3. iii. Hence, show that  , given that  , where  .   (1 mark)

  4.  Find the gradient of the tangent to the curve    at  .   (2 marks)

  5.  A solid of revolution is formed by rotating the region between the graph of  , the horizontal axis, and the lines    and    about the horizontal axis.
  6. Find the volume of the solid of revolution.   (3 marks)

Show Answers Only

Show Worked Solution

a.i.  
 
 
 
 
 

 

a.ii.  
 
 
 

 

a.iii.   

 
 
 
   
   

 

 

b.  
 
   

 

 
 
 

 

c.  
   
   
   
   
   
   
    ³

Complex Numbers, SPEC1-NHT 2018 VCAA 8

A circle in the complex plane is given by the relation  .

  1. Sketch the circle on the Argand diagram below.   (1 mark)

 

  1. i.  Write the equation of the circle in the form    and show that the gradient of a tangent to the circle can be expressed as  .   (2 marks)

  2. ii. Find the gradient of the tangent to the circle where    in the first quadrant of the complex plane.   (1 mark)

  3. Find the equations of all rays that are perpendicular to the circle in the form  .   (2 marks)

Show Answers Only
    1.  
Show Worked Solution
a.  

 

b.i.  
 
 
 
 
   

 

b.ii.  
 
 
 
     
 
   

 

c.  
 
   

 

Calculus, SPEC1 2018 VCAA 9

A curve is specified parametrically by  .

  1.  Show that the cartesian equation of the curve is  .   (2 marks)

  2.  Find the -coordinates of the points of intersection of the curve    and the line  .   (1 mark)

  3.  Find the volume of the solid of revolution formed when the region bounded by the curve and the line is rotated about the -axis.   (2 marks)

Show Answers Only

Show Worked Solution

a.     

 

b.   
 
 
 
 
 

 

♦♦ Mean mark 30%.

 

c.