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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)

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  1.  i.
  2. ii.
  3.  i.
  4. ii.
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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)

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a.   

 

 
 
 
 

 

b.   
   
   

 

  

c.i.  
 
 
   
     
 
 
   
   

 

c.ii.  
   

 

d.   

 

 

e.   

♦♦ Mean mark part (e) 30%.

 
 

 

f.   

♦♦ Mean mark part (f) 32%.

`= 0.05 sqrt h“

  

 
 

Complex Numbers, SPEC2 2018 VCAA 2

  1. State the centre in the form  , where  , and the state the radius of the circle given by  , where  .   (1 mark)

  2. Graph the circle given by    on the Argand diagram below, labelling the intercepts with the vertical axis.   (2 marks)

     

The line given by    intersects the circle given by    in two places.

  1. Draw the line given by    on the Argand diagram in part c. Label the points of intersection with their coordinates.   (2 marks)

  2. Find the area of the minor segment enclosed by an arc of the circle given by    and part of the line given by  .   (3 marks)
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a.   


 

b.   
 
 

 

 

c.   

 

d.   

 

e.   

♦ Mean mark 39%.
MARKER’S COMMENT: Students who used standard formulae rather than definite integrals were more successful.


 

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)


     

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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)

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a.   
   
 
 

 

b.   

 

c.   
 
 
 

 

♦ Mean mark part (c) 35%.


 

 
 

Calculus, SPEC1 2015 VCAA 9

Consider the curve represented by 

  1. Find the gradient of the curve at any point    (2 marks)
  2. Find the equation of the tangent to the curve at the point    and find the equation of the tangent to the curve at the point

     

    Write each equation in the form    (2 marks)

  3. Find the acute angle between the tangent to the curve at the point    and the tangent to the curve at the point 

     

    Give your answer in the form  , where is a real constant  (2 marks)

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a.   
 
 
 

 

b.   

 

 

 
 

 

c.   

 
 
 
 

 

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)

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a.   

b. (i)  
  (ii)  

c.   

d.   

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a.   

 
 
 
 

 

b.i. 

 

b.ii.

 

c.  
   
   

 

d.   
 
 

 

 

♦ Mean mark part (d) 49%.

 
 
 
 
 

Trigonometry, SPEC1 2015 VCAA 7

  1. Solve .  (3 marks)

  2. Find    (2 marks)

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  1. or
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a. 

MARKER’S COMMENT: Many students expanded sin(2x) and then cancelled sin(x) on both sides which lost a set of solutions!


 

b. 

♦♦ Mean mark 26%.

when

at


 

 

Complex Numbers, SPEC1 2015 VCAA 4

  1. Find all solutions of  in cartesian form.   (3 marks)

  2. Find all solutions of    in cartesian form.   (1 mark)

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a.   

 
 
   
 
 
   
 
 

 

 

♦ Mean mark part (b) 44%.

b.   
 
     
   

 

Mechanics, SPEC1 2015 VCAA 2

A 20 kg parcel sits on the floor of a lift.

  1. The lift is accelerating upwards at 1.2 ms¯².

     

    Find the reaction force of the lift floor on the parcel in newtons.  (2 marks)

  2. Find the acceleration of the lift downwards in ms¯² so that the reaction of the lift floor on the parcel is 166 N.  (2 marks)
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a.    

 

 

 
 

 

b.   

 

 

 
 
 
 

Mechanics, SPEC1 2016 VCAA 1

A taut rope of length m suspends a mass of 20 kg from a fixed point . A horizontal force of newtons displaces the mass by 1 m horizontally so that the taut rope is then at an angle of    to the vertical.

  1. Show all the forces acting on the mass on the diagram below.  (1 mark)

 

 

  1. Show that  .  (1 mark)
  2. Find the magnitude of the tension force in the rope in newtons.  (2 marks)
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a.   

 

b.   
   
   

 

c.   

 

Graphs, SPEC2 2014 VCAA 3 MC

The features of the graph of the function with rule   include

A.   asymptotes at    and 

B.   asymptotes at    and 

C.   an asymptote at    and a point of discontinuity at 

D.   an asymptote at    and a point of discontinuity at 

E.   an asymptote at    and a point of discontinuity at 

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Mechanics, SPEC1 2014 VCAA 8

A body of mass 5 kg is held in equilibrium by two light inextensible strings. One string is attached to a ceiling at and the other to a wall at . The string attached to the ceiling is at an angle to the vertical and has tension newtons, and the other string is horizontal and has tension newtons. Both strings are made of the same material.

  1. i. Resolve the forces on the body vertically and horizontally, and express in terms of  .  (2 marks)
  2. ii. Express in terms of  .  (1 mark)
  3.  Show that    for  .  (1 mark)
  4.  The type of string used will break if it is subjected to a tension of more than 98 N.

     

     Find the maximum allowable value of    so that neither string will break.  (3 marks)

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  1.  i.
  2. ii.
  3.  
  4.  
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a.i.   

 

a.ii.   
   

 

b.   
 

♦ Mean mark part (b) 40%.

 

 

c.   

♦ Mean mark part (c) 46%.

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)

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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)


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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)

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a.   


 

b.

   

 

c.

Complex Numbers, SPEC1 2014 VCAA 3

Let  be a function of a complex variable, defined by the rule  .

  1.  Given that    is a solution of  , write down a quadratic factor of  .   (2 marks)

  2.  Given that the other quadratic factor of    has the form  , find all solutions of    in a cartesian form.   (3 marks)

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a.   

 

b.   
 

 

 

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)

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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)

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a.   
   
 

 

Mean mark part (b) 51%.

b.   
 
   
 

 

c.   

 
 

Vectors, SPEC1 2015 VCAA 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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a.   

 

 

b.   

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


 


 

 
 

 

Mechanics, SPEC2 2017 VCAA 16 MC

An object of mass 20 kg, initially at rest, is pulled along a rough horizontal surface by a force of 80 N acting at an angle of 40° upwards from the horizontal. A friction force of 20 N opposes the motion.

After the pulling force has acted for 5 seconds, the magnitude of the momentum, in kg ms−1, of the object is closest to

  1.     10
  2.     40
  3.   160
  4.   210
  5. 4100
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Show Worked Solution


 

 

 

 

 

 

Vectors, SPEC1 2017 VCAA 9

A particle of mass 2 kg with initial velocity    ms−1 experiences a constant force for 10 seconds.

The particle's velocity at the end of the 10-second period    ms−1 .

  1. Find the magnitude of the constant force in newtons.   (2 marks)

  2. Find the displacement of the particle from its initial position after 10 seconds.   (3 marks)

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a.   
 

♦ Mean mark (a) 38%.

 

 
 

 

 

 

b.  

 

 


 

 
 

 

 

Calculus, SPEC1 2017 VCAA 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.   
 
 

 

 

 

 

Statistics, SPEC2-NHT 2018 VCAA 19 MC

A local supermarket sells apples in bags that have negligible mass. The stated mass of a bag of apples is 1 kg.

The mass of this particular type of apple is known to be normally distributed with a mean of 115 grams and a standard deviation of 7 grams. A particular bag contains nine randomly selected apples.

The probability that the nine apples in this bag have a total mass of less than 1 kg is

  1. 0.0478
  2. 0.1132
  3. 0.4265
  4. 0.5373
  5. 0.9522
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Statistics, SPEC2-NHT 2018 VCAA 18 MC

The heights of all six-year-old children in a given population are normally distributed. The mean height of a random sample of 144 six-year-old children from this population is found to be 115 cm.

If a 95% confidence interval for the mean height of all six-year-old children is calculated to be (113.8, 116.2) cm, the standard deviation used in this calculation is closest to

A.     1.20

B.     7.35

C.   15.09

D.   54.02

E.   88.13

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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)

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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)

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    1.  
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a.  

 

b.i.  
 
 
 
 
   

 

b.ii.  
 
 
 
     
 
   

 

c.