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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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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.  
   
   
   
   
   
   
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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.  
 
   

 

Statistics, SPEC1-NHT 2018 VCAA 4

Throughout this question, use an integer multiple of standard deviations in calculations.

The standard deviation of all scores on a particular test is 21.0

  1. From the results of a random sample of students, a 95% confidence interval for the mean score for all students was calculated to be  .

     

    Calculate the mean score and the size of this random sample.  (2 marks)

  2. Determine the size of another random sample for which the endpoints of the 95% confidence interval for the population mean of the particular test would be 1.0 either side of the sample mean.  (2 marks)

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  1.  
  2.  
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a. 
 

 

 

b.  
 
 
   
   
   

Statistics, SPEC2 2018 VCAA 20 MC

The scores on the Mathematics and Statistics tests, expressed as percentages, in a particular year were both normally distributed. The mean and the standard deviation of the Mathematics test scores were 71 and 10 respectively, while the mean and the standard deviation of the statistics test scores were 75 and 7 respectively.

Assuming the sets of tests scores were independent of each other, the probability, correct to four decimal places, that a randomly chosen Mathematics score is higher than a randomly chosen Statistics score is

  1. 0.2877
  2. 0.3716
  3. 0.4070
  4. 0.7123
  5. 0.9088
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Statistics, SPEC2 2018 VCAA 19 MC

The gestation period of cats is normally distributed with mean    days and variance  .

The probability that a sample of five cats chosen at random has an average gestation period greater than 65 days is closest to

  1. 0.5000
  2. 0.7131
  3. 0.7734
  4. 0.8958
  5. 0.9532
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Statistics, SPEC2 2018 VCAA 18 MC

A 95% confidence interval for the mean height , in centimetres, of a random sample of 36 Irish setter dogs is 

The standard deviation of the height of the population of Irish setter dogs, in centimetres, correct to two decimal places, is

A.     2.26

B.     2.27

C.   13.60

D.   13.61

E.   62.87

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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)

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a.     

 

b.   
 
 
 
 
 

 

♦♦ Mean mark 30%.

 

c.  
 
 
 

 

 
 
 
 
 
 
 

Calculus, SPEC1 2018 VCAA 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
  2.   (1 mark)

  3. Solve the differential equation given in part a. to find as a function of .
  4. 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.   

 

 
 

Vectors, SPEC1 2018 VCAA 6

A particle of mass 2 kg moves under a force    so that its position vector    at anytime is given by  .  Distances are measured in metres and time is measured in seconds.

Find the change in momentum, in kg ms¯², from    to  .   (3 marks)

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Complex Numbers, SPEC1 2018 VCAA 2

  1. Show that  .   (1 mark)

  2. Evaluate  , giving your answer in the form  , where  .   (3 marks)

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a.  
   
 
     
 

 

b.   
   
   

 

   
 

 

 
 
 
 
 

Mechanics, SPEC1 2018 VCAA 1

Two objects of masses 5 kg and 8 kg are attached by a light inextensible string that passes over a smooth pulley. The 8 kg mass is on a smooth plane inclined at 30° to the horizontal. The 5 kg mass is hanging vertically, as shown in the diagram below.
 


 

  1. On the diagram above, show all forces acting on both masses.  (1 mark)
  2. Find the magnitude, in , and state the direction of the acceleration of the 8 kg mass.  (3 marks)
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a.  

b.   

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a.  

 

b.   
 
 

Calculus, MET1 2018 VCAA 8

Let  , where is a positive real constant.

  1.  Show that  .   (1 mark)

  2. Find the value of for which the graphs of    and    have exactly one point of intersection.   (3 marks)

Let  . The diagram below shows sections of the graphs of  and for  .
 


 

Let be the area of the region bounded by the curves  and the line  .

  1. Write down a definite integral that gives the value of .   (1 mark)

  2. Using your result from part a., or otherwise, find the value of such that  .   (3 marks)

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a.   

 

 

b.  

♦♦♦ Mean mark 7%.
MARKER’S COMMENT: Most students found the correct quadratic equation but solved for .

 

 

 

c.   
   
   

♦♦ Mean mark 29%.

 

d.   
 
 
 
 
 
 
 

Probability, MET1 2018 VCAA 6

Two boxes each contain four stones that differ only in colour.

Box 1 contains four black stones.

Box 2 contains two black stones and two white stones.

A box is chosen randomly and one stone is drawn randomly from it.

Each box is equally likely to be chosen, as is each stone.

  1.  What is the probability that the randomly drawn stone is black?  (2 marks)

  2.  It is not known from which box the stone has been drawn.
  3. Given that the stone that is drawn is black, what is the probability that it was drawn from Box 1?  (2 marks)

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a.   
   
   

 

b.   
   
   

Algebra, STD2 A2 2007 HSC 27b*

A clubhouse uses four long-life light globes for five hours every night of the year. The purchase price of each light globe is $6.00 and they each cost    per hour to run.

  1. Write an equation for the total cost () of purchasing and running these four light globes for one year in terms of  .    (2 marks)

  2. Find the value of    (correct to three decimal places) if the total cost of running these four light globes for one year is $250.   (1 mark)

  3. If the use of the light globes increases to ten hours per night every night of the year, does the total cost double? Justify your answer with appropriate calculations.   (1 mark)

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i. 

 
 
 

 

ii. 

 
 

 

iii. 

 
   

 

Measurement, STD2 M6 SM-Bank 4

The diagram shows three checkpoints A, B and C. Checkpoint C is due east of Checkpoint A. The bearing of Checkpoint B from Checkpoint A is N22°E and the bearing of Checkpoint C from Checkpoint B is S68°E. The distance between Checkpoint A and Checkpoint B is 42 kilometres.
 


 

  1. Mark the given information on the diagram and explain why is 90°.  (2 marks)

  2. Find the distance, to the nearest kilometre, between Checkpoint A and Checkpoint C.  (2 marks)

  3. If a runner is travelling 12.6 km/h, how long does it take her to travel between Checkpoint A and Checkpoint B, in hours and minutes?  (2 marks)

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i.   
 

 

ii.  

 
 

 

iii.   
   
   
   

Statistics, STD2 S5 SM-Bank 5

The diastolic measurement for blood pressure in 30-year-old people is normally distributed, with a mean of 82 and standard deviation of 16.

  1.   A person is considered to have low blood pressure if the diastolic measurement is 66 or less.

     

      What percentage of 30-year-old people have low blood pressure?  (1 mark)

  2.   Calculate the -score for a diastolic measurement of 70.  (1 mark)

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a.   
   

 

 

 

b.   
   

Algebra, STD2 A4 SM-Bank 7 MC

A computer application was used to draw the graphs of the equations

  and 

Part of the screen is shown.
 

Which row of the table correctly matches the equations with the lines drawn and identifies the solution when the equations are solved simultaneously?

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