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Calculus, 2ADV C3 2019 MET1 4

Given the function  ,

  1. State the domain and range of .  (1 mark)

  2. i.  Find the equation of the tangent to the graph of  at  .  (2 marks)

     

    ii. On the axes below, sketch the graph of the function  , labelling any asymptote with its equation.

     

        Also draw the tangent to the graph of    at  .  (4 marks)
     

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  1.  

  2. i. 

     

    ii.

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a.  
 

 

b.i.  
 
 

 

 

b.ii.  

Probability, MET1-NHT 2019 VCAA 8

A fair standard die is rolled 50 times. Let be a random variable with binomial distribution that represents the number of times the face with a six on it appears uppermost.

  1. Write down the expression for  , where  .  (1 mark)

  2. Show that  .  (2 marks)

  3. Hence, or otherwise, find the value of for which    is the greatest.  (2 marks)

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a. 

 

b.  
   
   
   

 

c. 

 

Calculus, MET1-NHT 2019 VCAA 7

The shaded region in the diagram below is bounded by the vertical axis, the graph of the function with rule    and the horizontal line segment that meets the graph at  , where  .
 


 

Let    be the area of the shaded region.

  1. Show that  .   (3 marks)

  2. Determine the range of values of .   (2 marks)

    1. Express in terms of  , for a specific value of , the area bounded by the vertical axis, the graph of    and the horizontal axis.   (2 marks)

    2. Hence, or otherwise, find the area described in part c.i.   (1 mark)

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a.   

 
 
 

 

b.   


 

 

c.i. 

   
   
   
   

 

 

c.ii. 

 
 
 
 

Statistics, EXT1 S1 EQ-Bank 23

A light manufacturer knows that 6% of the light bulbs it produces are defective.

Light bulbs are supplied in boxes of 20 bulbs. Boxes are supplied in pallets of 120 boxes.

Calculate the probability that

  1. A box of light bulbs contains exactly 3 defective bulbs.  (1 mark)

  2. A box of light bulbs contains at least 1 defective bulb.  (1 mark)

  3. A pallet contains between 90 and 95 (inclusive) boxes with at least 1 defective bulb (use the probability table attached).  (3 marks)

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Show Worked Solution

i.   

 

 

ii.   
   
   

 

iii.   


 


 


 

 
 

Statistics, EXT1 S1 EQ-Bank 21

A biased coin has a 0.6 chance of landing on heads. The coin is tossed 15 times.

  1. Calculate the probability of obtaining 7, 8 or 9 heads using binomial probability distribution.
  2. Give your answer correct to 3 decimal places.  (2 marks)

  3. Calculate the probability of obtaining 7, 8 or 9 heads using normal approximation to the binomial distribution and the probability table attached.
  4. Give your answer correct to 3 decimal places.  (2 marks)

  5. Could this binomial distribution be reasonably approximated with a normal distribution? Support your answer with a brief calculation.  (1 mark)

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i.   
   
   

 

ii.   

 

COMMENT: A quick sketch of the normal distribution curve is often helpful when using probability tables in this context.
 

 
 
 

 

iii.   

COMMENT: is also common in assessing if normal approximation is reasonable.

Statistics, SPEC2-NHT 2019 VCAA 6

A paint company claims that the mean time taken for its paint to dry when motor vehicles are repaired is 3.55 hours, with a standard deviation of 0.66 hours.

Assume that the drying time for the paint follows a normal distribution and that the claimed standard deviation value is accurate.

  1. Let the random variable    represent the mean time taken for the paint to dry for a random sample of 36 motor vehicles.

     

    Write down the mean and standard deviation of  .   (2 marks)

At a car crash repair centre, it was found that the mean time taken for the paint company's paint to dry on randomly selected vehicles was 3.85 hours. The management of this crash repair centre was not happy and believed that the claim regarding the mean time taken for the paint to dry was too low. To test the paint company's claim, a statistical test was carried out.

  1. Write down suitable null and alternative hypotheses and respectively to test whether the mean time taken for the paint to dry is longer than claimed.   (1 mark)

  2. Write down an expression for the    value of the statistical test and evaluate it correct to three decimal places.   (2 marks)

  3. Using a 1% level of significance, state with a reason whether the crash repair centre is justified in believing that the paint company's claim of mean time taken for its paint to dry of 3.55 hours is too low.  (1 mark)

  4. At the 1% level of significance, find the set of sample mean values that would support the conclusion that the mean time taken for the paint to dry exceeded 3.55 hours. Give your answer in hours, correct to three decimal places.  (2 marks)

  5. If the true time taken for the paint to dry is 3.83 hours, find the probability that the paint company's claim is not rejected at the 1% level of significance, assuming the standard deviation for the paint to dry is still 0.66 hours. Give your answer correct to two decimal places.  (1 mark)

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a.   
 
 

 

b.   

 

c.   
 
 
 

 

d.   

 

e.   

 

f.   
 
 

Mechanics, SPEC2-NHT 2019 VCAA 5

A pallet of bricks weighing 500 kg sits on a rough plane inclined at an angle of  α to the horizontal, where  α. The pallet is connected by a light inextensible cable that passes over a smooth pulley to a hanging container of mass kilograms in which there is 10 L of water. The pallet of bricks is held in equilibrium by the tension newtons in the cable and a frictional resistance force of 50 newtons acting up and parallel to the plane. Take the weight force exerted by 1 L of water to be newtons.
  


 

  1. Label all forces acting on both the pallet of bricks and the hanging container on the diagram above, when the pallet of bricks is in equilibrium as described.   (1 mark)
  2. Show that the value of is 80.  (3 marks)

Suddenly the water is completely emptied from the container and the pallet of bricks begins to slide down the plane. The frictional resistance force of 50 newtons acting up the plane continues to act on the pallet.

  1. Find the distance, in metres, travelled by the pallet after 10 seconds.  (3 marks)
  2. When the pallet reaches a velocity of  , water is poured back into the container at a constant rate of 2 L per second, which in turn retards the motion of the pallet moving down the plane. Let    be the time, in seconds, after the container begins to fill. 
  3.   i. Write down, in terms of  , an expression for the total mass of the hanging container and the water it contains after seconds. Give your answer in kilograms.  (1 mark)
  4.  ii. Show that the acceleration of the pallet down the plane is given by    for  .  (2 marks)
  5. iii. Find  the velocity of the pallet when  . Give your answer in metres per second, correct to one decimal place.  (2 marks)
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  1.  
  4.   i.
     ii.
    iii.
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a.

 
b.   


 


 

αα
 

 

c.   

α
 
 

 

d.i.   
 

d.ii.   

 

d.iii.   

 

Calculus, SPEC2-NHT 2019 VCAA 3


 

The vertical cross-section of a barrel is shown above. The radius of the circular base (along the -axis) is 30 cm and the radius of the circular top is 70 cm. The curved sides of the cross-section shown are parts of the parabola with rule  . The height of the barrel is 50 cm.
 
a.  i.   Show that the volume of the barrel is given by  .  (1 marks)
     ii.  Find the volume of the barrel in cubic centimetres.  (1 marks)    

The barrel is initially full of water. Water begins to leak from the bottom of the barrel such that    cubic centimetres per second, where after    seconds the depth of the water is    centimetres, the volume of water remaining in the barrel is    cubic centimetres and the uppermost surface area of the water is    square centimetres.
 
b.     Show that  ?  (2 marks)
c.      Find    in terms of  . Express your answer in the form  , where    and    are positive integers.  (3 marks)
d.     Using a definite integral in terms of  , find the time, in hours, correct to one decimal place, taken for the barrel to empty.  (2 marks)

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a.  i.

      ii. 

b.   

c.   

d.   

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a.i.
 
 
     
 

 

a.ii.
   
   

 

b.   

 
 

 

c.   

 
 
 

  
 
d.   

 
 

Calculus, SPEC2-NHT 2019 VCAA 2

Consider the function with rule  .

  1. State the equations of the asymptotes of the graph of .  (2 marks)

  2. State the coordinates of the stationary points and the point of inflection. Give your answers correct to two decimal places.  (2 marks)

  3. Sketch the graph of from    to    (endpoint coordinates are not required) on the set of axes below, labeling the turning points and the point of inflection with their coordinates correct to two decimal places. Label the asymptotes with their equations.  (3 marks)

Consider the function with rule    where  .

  1. For what values of will have no stationary points?  (2 marks)

  2. For what value of will the graph of have a point of  inflection located on the -axis?  (1 marks)

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i.     

 ii.   
iii.

iv. 

v. 

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i.   


  

ii.   


 


 

iii.

 

iv.   

 

 

v.   

Vectors, EXT1 V1 SM-Bank 30

A canon ball is fired from a castle wall across a horizontal plane at ms−1.

Its position vector  seconds after it is fired from its origin is given by  .

  1. If the projectile hits the ground at a distance 8 times the height at which it was fired, show that it initial velocity is given by
     
            (2 marks)

  2. Show that the total distance the canon ball travels can be expressed as
     
                (2 marks)

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i.   

 

 

 

ii.   

 
 

 

 

Vectors, EXT1 V1 2019 SPEC2-N 4

A snowboarder at the Winter Olympics leaves a ski jump at an angle of degrees to the horizontal, rises up in the air, performs various tricks and then lands at a distance down a straight slope that makes an angle of 45° to the horizontal, as shown below.

Let the origin of a cartesian coordinate system be at the point where the snowboarder leaves the jump, with a unit vector in the positive direction being represented by    and a unit vector in the positive direction being represented by  . Distances are measured in metres and time is measured in seconds.

The position vector of the snowboarder    seconds after leaving the jump is given by


 


 

  1. Find the angle  .    (2 marks)

  2. Find the speed, in metres per second, of the snowboarder when she leaves the jump at .    (1 mark)

  3. Find the maximum height above reached by the snow boarder. Give your answer in metres, correct to one decimal place.    (2 marks)

  4. Show that the time spent in the air by the snowboarder is    seconds.    (3 marks)

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Show Worked Solution

a.   

 

 

b.    
   
   

 
c.   

 
   

 
d.   

 

Vectors, SPEC2-NHT 2019 VCAA 4

A snowboarder at the Winter Olympics leaves a ski jump at an angle of degrees to the horizontal, rises up in the air, performs various tricks and then lands at a distance down a straight slope that makes an angle of 45° to the horizontal, as shown below.

Let the origin of a cartesian coordinate system be at the point where the snowboarder leaves the jump, with a unit vector in the positive direction being represented by    and a unit vector in the positive direction being represented by  . Distances are measured in metres and time is measured in seconds.

The position vector of the snowboarder    seconds after leaving the jump is given by


 


 

  1. Find the angle  .    (2 marks)

  2. Find the speed, in metres per second, of the snowboarder when she leaves the jump at .   (1 mark)

  3. Show that the time spent in the air by the snowboarder is    seconds.   (3 marks)

  4. Find the total distance the snowboarder travels while airborne. Give your answer in metres, correct to two decimal places.   (2 marks)

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a.   

 

 

b.    
   
   

 
c.   

 
   

 
d.   

 


 
e.   

 

Complex Numbers, SPEC2-NHT 2019 VCAA 1

In the complex plane, is the with equation .

  1.  Verify that the point (0, 0) lies on .   (1 marks)

  2.  The line    can also be expressed in the form  , where  .

     

     Find  in cartesian form.   (2 marks)

  3.  Find, in cartesian form, the points(s) of intersection of    and the graph of  .   (2 marks)

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

     
     
         
     

  5.  Find the area of the sector defined by the part of    where  , the graph of    where  , and imaginary axis where  .   (1 marks)

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  5.  

     
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a.   

 

b. 
 
 
 
 
 
 


c.


 

 

d.   

 

e.


 

f.    

 

Calculus, 2ADV C4 SM-Bank 1 MC

A lift accelerates from rest at a constant rate until it reaches a speed of 3 ms−1. It continues at this speed for 10 seconds and then decelerates at a constant rate before coming to rest. The total travel time for the lift is 30 seconds.

The total distance, in metres, travelled by the lift is

  1.  45
  2.  60
  3.  75
  4.  90
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Show Worked Solution


 


 


 

Mechanics, SPEC2-NHT 2019 VCAA 15 MC

A lift accelerates from rest at a constant rate until it reaches a speed of 3 ms−1. It continues at this speed for 10 seconds and then decelerates at a constant rate before coming to rest. The total travel time for the lift is 30 seconds.

The total distance, in metres, travelled by the lift is

  1.  30
  2.  45
  3.  60
  4.  75
  5.  90
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Show Worked Solution


 


 

Statistics, SPEC2-NHT 2019 VCAA 19 MC

Bags of peanuts are packed by a machine. The masses of the bags are normally distributed with a standard deviation of three grams.

The minimum size of a sample required to ensure that the manufacturer can be 98% confident that the sample mean is within one gram of the population mean is

  1.  37
  2.  38
  3.  48
  4.  49
  5.  60
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Statistics, SPEC2-NHT 2019 VCAA 20 MC

Nitrogen oxide emissions for a certain type of car are known to be normally distributed with a mean of 0.875 g/km and a standard deviation of 0.188 g/km.

For two randomly selected cars, the probability that their nitrogen oxide emissions differ by more than 0.5 g/km is closest to

  1. 0.030
  2. 0.060
  3. 0.960
  4. 0.970
  5. 0.977
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Mechanics, SPEC2-NHT 2019 VCAA 17 MC

A ball is thrown vertically upwards with an initial velocity of  ms−1, and is subject to gravity and air resistance. The acceleration of the ball is given by  , where ms−1 is its velocity when it is at a height of metres above ground level.

The maximum height, in metres, reached by the ball is

  1.  
  2.  
  3.  
  4.  
  5.  
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Mechanics, SPEC2-NHT 2019 VCAA 16 MC

An object of mass 2 kg is travelling horizontally in a straight line at a constant velocity of magnitude 2 ms−1.  The object is hit in such a way that it deflects 30° from its original path, continuing at the same speed in a straight line.

The magnitude, correct to two decimal places, of the change of momentum, in kg ms−1, of the object is

  1. 0.00
  2. 0.24
  3. 1.04
  4. 1.46
  5. 2.07
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Vectors, EXT2 V1 2019 SPEC2 4

The base of a pyramid is the parallelogram    with vertices at points    and  . The apex (top) of the pyramid is located at  .

  1. Find the values of    and  .  (2 marks)

  2. Find the cosine of the angle between the vectors    and  .  (2 marks)

  3. Find the area of the base of the pyramid.  (2 marks)

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  3. ²
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i.   
   

 

 

ii.   

 
 

 

iii.   
 

  

  

  ²

Statistics, SPEC1-NHT 2019 VCAA 3

The number of cars per day making a U-turn at a particular location is known to be normally distributed with a standard deviation of 17.5. In a sample of 25 randomly selected days, a total of 1450 cars were observed making the U-turn.

  1. Based on this sample, calculate an approximate 95% confidence interval for the number of cars making the U-turn each day. Use an integer multiple of the standard deviation in your calculations.   (3 marks)

  2. The average number of U-turns made at the location is actually 60 per day.

     

    Find an approximation, correct to two decimal places, for the probability that on 25 randomly selected days the average number of U-turns is less than 53.   (1 mark)

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a.      

σ

 

 

b.     μσ
 
         
 
 
         
 

 
 
 

 

Trigonometry, 2ADV* T1 2011 HSC 24c

A ship sails 6 km from to on a bearing of 121°. It then sails 9 km to .  The
size of angle is 114°.
 

2UG 2011 24c

Copy the diagram into your writing booklet and show all the information on it.

  1. What is the bearing of from ?   (1 mark)

  2. Find the distance . Give your answer correct to the nearest kilometre.   (2 marks)

  3. What is the bearing of from ? Give your answer correct to the nearest degree.   (3 marks)

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  1.  
  2.  
  3.  
Show Worked Solution
STRATEGY: This deserves repeating again: Draw North-South parallel lines through major points to make the angle calculations easier!
i.     2011 HSC 24c

 

 
 

   

 

ii.   

 
 
 

 

iii.   

MARKER’S COMMENT: The best responses showed clear working on the diagram.
 
 

 

 
 
 

Trigonometry, 2ADV* T1 2017 HSC 30c

The diagram shows the location of three schools. School is 5 km due north of school , school is 13 km from school and is 135°.
 


 

  1. Calculate the shortest distance from school to school , to the nearest kilometre.  (2 marks)

  2. Determine the bearing of school from school , to the nearest degree.  (3 marks)

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(i)  

 
 
 

 

(ii)   

♦♦ Mean mark part (ii) 31%.
 

 

Trigonometry, 2ADV* T1 2007 26a

The diagram shows information about the locations of towns  ,    and  .
 

 
 

  1. It takes Elina 2 hours and 48 minutes to walk directly from Town    to Town  .

     

    Calculate her walking speed correct to the nearest km/h.    (1 mark)

  2. Elina decides, instead, to walk to Town    from Town    and then to Town  .

     

    Find the distance from Town    to Town  . Give your answer to the nearest km.   (2 marks)

  3. Calculate the bearing of Town    from Town  .   (1 mark)

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i. 

 

 

 
 

 

ii.  

 
 

 

 

iii.  

 

Statistics, STD2 S1 EQ-Bank 4

A high school conducted a survey asking students what their favourite Summer sport was.

The Pareto chart shows the data collected.
 


 

  1. What percentage of students chose Hockey as their favourite Summer sport?  (1 mark)

  2. What percentage of students chose Touch Football as their favourite Summer sport?  (1 mark)

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i.   

 


 

 
   

 

ii.  

 

GRAPHS, FUR2 2019 VCAA 3

Members of the association will travel to a conference in cars and minibuses:

  • Let be the number of cars used for travel.
  • Let be the number of minibuses used for travel.
  • A maximum of eight cars and minibuses in total can be used.
  • At least three cars must be used.
  • At least two minibuses must be used.

The constraints above can be represented by the following three inequalities.
 


 

  1. Each car can carry a total of five people and each minibus can carry a total of 10 people.

      

    A maximum of 60 people can attend the conference.

      

    Use this information to write Inequality 4.   (1 mark)

The graph below shows the four lines representing Inequalities 1 to 4.

Also shown on this graph are four of the integer points that satisfy Inequalities 1 to 4. Each of these integer points is marked with a cross (✖).
 


 

  1. On the graph above, mark clearly, with a circle (o), the remaining integer points that satisfy Inequalities 1 to 4.  (1 mark)

Each car will cost $70 to hire and each minibus will cost $100 to hire.

  1. What is the cost for 60 members to travel to the conference?  (1 mark)
  2. What is the minimum cost for 55 members to travel to the conference?  (1 mark)
  3. Just before the cars were booked, the cost of hiring each car increased.

      

    The cost of hiring each minibus remained $100.

      

    All original constraints apply.

      

    If the increase in the cost of hiring each car is more than dollars, then the maximum cost of transporting members to this conference can only occur when using six cars and two minibuses.

      

    Determine the value of  .  (1 mark)

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a. 

 

b.    

 

c.   

 

d.   

 

e.   

   

GRAPHS, FUR2 2019 VCAA 2

Each branch within the association pays an annual fee based on the number of members it has.

To encourage each branch to find new members, two new annual fee systems have been proposed.

Proposal 1 is shown in the graph below, where the proposed annual fee per member, in dollars, is displayed for branches with up to 25 members.
 


 

  1. What is the smallest number of members that a branch may have?  (1 mark)
  2. The incomplete inequality below shows the number of members required for an annual fee per member of $10.

      

    Complete the inequality by writing the appropriate symbol and number in the box provided.   (1 mark)
     

3 ≤ number of members  
 

 

Proposal 2 is modelled by the following equation.

annual fee per member = – 0.25 × number of members + 12.25

  1. Sketch this equation on the graph for Proposal 1, shown below.  (1 mark)

 

 

  1. Proposal 1 and Proposal 2 have the same annual fee per member for some values of the number of members.

      

    Write down all values of the number of members for which this is the case.  (1 mark)

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Show Worked Solution

a. 
 

b. 
 

c.  

 

d.   

GEOMETRY, FUR2 2019 VCAA 2

A cargo ship travels from Magadan (60° N, 151° E) to Sydney (34° S, 151° E).

  1. Explain, with reference to the information provided, how we know that Sydney is closer to the equator than Magadan.   (1 mark)
  2. Assume that the radius of Earth is 6400 km.

      

    Find the shortest great circle distance between Magadan and Sydney.

      

    Round your answer to the nearest kilometre.  (1 mark)

  3. The cargo ship left Sydney (34° S, 151° E) at 6 am on 1 June and arrived in Perth (32° S, 116° E) at 10 am on 11 June.

      

    There is a two-hour time difference between Sydney and Perth at that time of year.

      

    How many hours did it take the cargo ship to travel from Sydney to Perth?  (1 mark)

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Show Worked Solution
a.  

 

b.  
   

 

c.   

 
 
 

Statistics, STD2 S1 EQ-Bank 5

An island resort surveyed 400 guests by asking them on which continent they lived.

The table below shows the data collected.
 


 

Complete the Pareto chart below to show the data collected.  (3 marks)
 

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Show Worked Solution


 

Networks, STD2 N3 2019 FUR2 3

Fencedale High School is planning to renovate its gymnasium.

This project involves 12 activities, to .

The directed network below shows these activities and their completion times, in weeks.
 


 

The minimum completion time for the project is 35 weeks.

  1. Identify the critical path and state how many activities are on it?  (2 marks)

  2. Determine the latest start time of activity .  (1 mark)

  3. Which activity has the longest float time?  (1 mark)

It is possible to reduce the completion time for activities and by employing more workers.

  1.  The completion time for each of these five activities can be reduced by a maximum of two weeks.

      

    What is the minimum time, in weeks, that the renovation project could take?  (1 mark)

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Show Worked Solution

a. 
 

 

 


 

b. 
 

c.   


 

d.   


 

 

NETWORKS, FUR2 2019 VCAA 3

Fencedale High School is planning to renovate its gymnasium.

This project involves 12 activities, to .

The directed network below shows these activities and their completion times, in weeks.
 


 

The minimum completion time for the project is 35 weeks.

  1. How many activities are on the critical path?   (1 mark)

  2. Determine the latest start time of activity .   (1 mark)

  3. Which activity has the longest float time?   (1 mark)

It is possible to reduce the completion time for activities and by employing more workers.

  1.  The completion time for each of these five activities can be reduced by a maximum of two weeks.

      

    What is the minimum time, in weeks, that the renovation project could take?   (1 mark)

  2. The reduction in completion time for each of these five activities will incur an additional cost to the school.

     

    The table below shows the five activities that can have their completion times reduced and the associated weekly cost, in dollars.
      

           Activity                    Weekly cost ($)             
    3000
    2000
    2500
    1000
    4000

      
    The completion time for each of these five activities can be reduced by a maximum of two weeks.

      

    Fencedale High School requires the overall completion time for the renovation project to be reduced by four weeks at minimum cost.

     

    Complete the table below, showing the reductions in individual activity completion times that would achieve this.   (2 marks)

           Activity       

    		              Reduction in completion time             
            (0, 1 or 2 weeks)
     
     
     
     
     
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  5.  
             Activity       

    		          Reduction in completion time         
            (0, 1 or 2 weeks)
      0
      1
      2
      1
      1
Show Worked Solution

a. 
 

 

  


 

b. 
 

c.   


 

d.   


 

 

  
e.   


 


 

         Activity       

          Reduction in completion time         
        (0, 1 or 2 weeks)
  0
  1
  2
  1
  1

NETWORKS, FUR2 2019 VCAA 2

Fencedale High School offers students a choice of four sports, football, tennis, athletics and basketball.

The bipartite graph below illustrates the sports that each student can play.
  
 


  

Each student will be allocated to only one sport.

  1. Complete the table below by allocating the appropriate sport to each student.   (1 mark)

 

         Student                                 Sport                         
    Blake  
    Charli  
    Huan  
    Marco  

 

  1. The school medley relay team consists of four students, Anita, Imani, Jordan and Lola.

      

    The medley relay race is a combination of four different sprinting distances: 100 m, 200 m, 300 m and 400 m, run in that order.

      

    The following table shows the best time, in seconds, for each student for each sprinting distance.
     

      Best time for each sprinting distance (seconds)
         Student             100 m               200 m               300 m               400 m       
      Anita 13.3 29.6 61.8 87.1
      Imani 14.5 29.6 63.5 88.9
      Jordan 13.3 29.3 63.6 89.1
      Lola 15.2 29.2 61.6 87.9

      
     
    The school will allocate each student to one sprinting distance in order to minimise the total time taken to complete the race.

      

    To which distance should each student be allocated?

      

    Write your answers in the table below.  (2 marks)

     

           Student                                Sprinting distance (m)                         
      Anita  
      Imani  
      Jordan  
      Lola  
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a.           Student                                 Sport                         
    Blake   Tennis
    Charli   Football
    Huan   Basketball
    Marco   Athletics

 

b.           Student                Sprinting distance (m)         
    Anita 400
    Imani 200
    Jordan 100
    Lola 300
Show Worked Solution

a.   


 

         Student                                 Sport                         
    Blake   Tennis
    Charli   Football
    Huan   Basketball
    Marco   Athletics

 

b.   


 

     Student        100 m         200 m         300 m         400 m    
    Anita 0 2.3 2.1 1.1
    Imani 0 1.1 2.6 1.7
    Jordan 0 2 3.9 3.1
    Lola 0 0 0 0

 


 

     Student        100 m         200 m         300 m         400 m    
    Anita 0 1.2 1 0
    Imani 0 0 1.5 0.6
    Jordan 0 0.9 2.8 2
    Lola 1.1 0 0 0

 

         Student                Sprinting distance (m)         
    Anita 400
    Imani 200
    Jordan 100
    Lola 300

MATRICES, FUR2 2019 VCAA 3

On Sunday, matrix is used when calculating the expected number of visitors at each location every hour after 10 am. It is assumed that the park will be at its capacity of 2000 visitors for all of Sunday.

Let be the state matrix that shows the number of visitors at each location at 10 am on Sunday.

The number of visitors expected at each location at 11 am on Sunday can be determined by the matrix product

 

 

  1. Safety restrictions require that all four locations have a maximum of 600 visitors.
  2. Which location is expected to have more than 600 visitors at 11 am on Sunday?   (1 mark) 
  3. Whenever more than 600 visitors are expected to be at a location on Sunday, the first 600 visitors can stay at that location and all others will be moved directly to Ground World .
  4. State matrix contains the number of visitors at each location hours after 10 am on Sunday, after the safety restrictions have been enforced.
  5. Matrix can be determined from the matrix recurrence relation
  7. where matrix shows the required movement of visitors at 11 am.
    1. Determine the matrix .   (1 mark)

    2. State matrix can be determined from the new matrix rule
    4. where matrix shows the required movement of visitors at 12 noon.
    5. Determine the state matrix .   (1 mark)

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  2.  i.
    ii.
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a.  
 
 
   

 

 

b.i.  
     
 
   
 

 

b.ii.  
   

MATRICES, FUR2 2019 VCAA 2

The theme park has four locations, Air World , Food World , Ground World and Water World .

The number of visitors at each of the four locations is counted every hour.

By 10 am on Saturday the park had reached its capacity of 2000 visitors and could take no more visitors.

The park stayed at capacity until the end of the day

The state matrix, , below, shows the number of visitors at each location at 10 am on Saturday.
 


 

  1. What percentage of the park’s visitors were at Water World at 10 am on Saturday?   (1 mark)

Let be the state matrix that shows the number of visitors expected at each location hours after 10 am on Saturday.

The number of visitors expected at each location hours after 10 am on Saturday can be determined by the matrix recurrence relation below.
 


 

  1. Complete the state matrix, , below to show the number of visitors expected at each location at 11 am on Saturday.   (1 mark)

 

 

  1. Of the 300 visitors expected at Ground World at 11 am, what percentage was at either Air World or Food World at 10 am?   (1 mark)

  2. The proportion of visitors moving from one location to another each hour on Sunday is different from Saturday.

     

    Matrix , below, shows the proportion of visitors moving from one location to another each hour after 10 am on Sunday.

     

     

     
    Matrix is similar to matrix but has the first two rows of matrix interchanged.

     

  3. The matrix product that will generate matrix from matrix is
  5. where matrix is a binary matrix.
  6. Write down matrix .   (1 mark)
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a.   

 

b.  
   

 

c. 


 

d.  
 

 

Calculus, EXT2 C1 2002 HSC 2b

For  ...

let  .

  1.  Show that  .   (1 mark)

  2.  Show that, for  ,
     
         .   (3 marks)

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i.         

 
 
 
 

 

ii.
   
   

 

 
   

 

 
 

 

Calculus, EXT2 C1 2008 HSC 3c

For  , let  .   

  1. Show that for ,
     
         .  (2 marks)

  2. Hence, or otherwise, calculate  .   (2 marks)

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i.
   
   
   

 

   
 

 

 
 
 

 

ii.       

 
 
 
 

CORE, FUR2 2019 VCAA 9

Phil would like to purchase a block of land.

He will borrow $350 000 to make this purchase.

Interest on this loan will be charged at the rate of 4.9% per annum, compounding fortnightly.

After three years of equal fortnightly repayments, the balance of Phil’s loan will be $262 332.33.

  1. What is the value of each fortnightly repayment Phil will make?

     

    Round your answer to the nearest cent.   (1 mark)

  2. What is the total interest Phil will have paid after three years?

     

    Round your answer to the nearest cent.   (1 mark)

  3. Over the next four years of his loan, Phil will make monthly repayments of $3517.28 and will be charged interest at the rate of 4.8% per annum, compounding monthly.

     

    Let be the balance of the loan months after these changes apply.

     

     

    Write down a recurrence relation, in terms of  and , that could be used to model the balance of the loan over these four years.   (2 marks)

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a. 

 


  

b.  
   
   

  
c. 

CORE, FUR2 2019 VCAA 8

Phil invests $200 000 in an annuity from which he receives a regular monthly payment.

The balance of the annuity, in dollars, after months, , can be modelled by the recurrence relation

  1. What monthly payment does Phil receive?   (1 mark)

  2. Show that the annual percentage compound interest rate for this annuity is 4.2%.   (1 mark)

At some point in the future, the annuity will have a balance that is lower than the monthly payment amount.

  1. What is the balance of the annuity when it first falls below the monthly payment amount?

     

    Round your answer to the nearest cent.   (1 mark)

  2. If the payment received each month by Phil had been a different amount, the investment would act as a simple perpetuity.

     

    What monthly payment could Phil have received from this perpetuity?   (1 mark)

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Show Worked Solution

a. 

b.  
 

  
c. 

 

 

  

d. 

 

CORE, FUR2 2019 VCAA 7

Phil is a builder who has purchased a large set of tools.

The value of Phil’s tools is depreciated using the reducing balance method.

The value of the tools, in dollars, after years, , can be modelled by the recurrence relation shown below.

 

  1. Use recursion to show that the value of the tools after two years, , is $48 600.   (1 mark)

  2. What is the annual percentage rate of depreciation used by Phil?   (1 mark)

  3. Phil plans to replace these tools when their value first falls below $20 000.

     

    After how many years will Phil replace these tools?   (1 mark)

  4. Phil has another option for depreciation. He depreciates the value of the tools by a flat rate of 8% of the purchase price per annum.

     

    Let be the value of the tools after years, in dollars.

     

    Write down a recurrence relation, in terms of and , that could be used to model the value of the tools using this flat rate depreciation.   (1 mark)

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a.  
 
 

  
b. 

 
c. 

 
d. 

CORE, FUR2 2019 VCAA 6

The total rainfall, in millimetres, for each of the four seasons in 2015 and 2016 is shown in Table 5 below.

  1. The seasonal index for winter is shown in Table 6 below.

     

    Use the values in Table 5 to find the seasonal indices for summer, autumn and spring.

  2. Write your answers in Table 6, rounded to two decimal places.   (2 marks)

     


  3. The total rainfall for each of the four seasons in 2017 is shown in Table 7 below.

     


    Use the appropriate seasonal index from Table 6 to deseasonalise the total rainfall for winter in 2017.

     

    Round your answer to the nearest whole number.   (1 mark)

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a.  

 

b.  
   

CORE, FUR2 2019 VCAA 5

The scatterplot below shows the atmospheric pressure, in hectopascals (hPa), at 3 pm (pressure 3 pm) plotted against the atmospheric pressure, in hectopascals, at 9 am (pressure 9 am) for 23 days in November 2017 at a particular weather station.
 

A least squares line has been fitted to the scatterplot as shown.

The equation of this line is

pressure 3 pm = 111.4 + 0.8894 × pressure 9 am

  1. Interpret the slope of this least squares line in terms of the atmospheric pressure at this weather station at 9 am and at 3 pm.   (1 mark)

  2. Use the equation of the least squares line to predict the atmospheric pressure at 3 pm when the atmospheric pressure at 9 am is 1025 hPa.
  3. Round your answer to the nearest whole number.   (1 mark)

  4. Is the prediction made in part b. an example of extrapolation or interpolation?   (1 mark)

  5. Determine the residual when the atmospheric pressure at 9 am is 1013 hPa.
  6. Round your answer to the nearest whole number.   (1 mark)

  7. The mean and the standard deviation of pressure 9 am and pressure 3 pm for these 23 days are shown in Table 4 below.

    1. Use the equation of the least squares line and the information in Table 4 to show that the correlation coefficient for this data, rounded to three decimal places, is  = 0.966   (1 mark)

    2. What percentage of the variation in pressure 3 pm is explained by the variation in pressure 9 am?
    3. Round your answer to one decimal place.   (1 mark)

  1. The residual plot associated with the least squares line is shown below.
     

    1. The residual plot above can be used to test one of the assumptions about the nature of the association between the atmospheric pressure at 3 pm and the atmospheric pressure at 9 am.
    2. What is this assumption?   (1 mark)

    3. The residual plot above does not support this assumption.
    4. Explain why.   (1 mark)

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Show Worked Solution

a.   

 

b.  
   

 

c. 

 

d.  
   
   
   
   

 

e.i.   

   
   
   

 

e.ii.  
 
   

 

f.i.  
 

 

f.ii. 

GRAPHS, FUR1 2019 VCAA 8 MC

Jenny and Alan’s house is 900 m from a supermarket.

Jenny is at the house and Alan is at the supermarket.

At 12 noon Jenny leaves the house and walks towards the supermarket.

At the same time, Alan leaves the supermarket and walks towards the house.

Jenny’s planned walk is modelled by the equation
 


 

where    is Jenny’s distance, in metres, from the house after    minutes.

Alan’s planned walk is modelled by the equation
 


 

where    is Alan’s distance, in metres, from the house after    minutes.

When they meet

  1. Jenny will have walked 359 m, to the nearest metre.
  2. Alan will have walked 360 m, to the nearest metre.
  3. Alan will have walked 382 m, to the nearest metre.
  4. Alan will have walked 518 m, to the nearest metre.
  5. Jenny will have walked 541 m, to the nearest metre.
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GRAPHS, FUR1 2019 VCAA 6 MC

A recipe for a fruit drink lists both pineapple juice and mango juice.

Let    be the number of millilitres of pineapple juice required to make the fruit drink.

Let    be the number of millilitres of mango juice required to make the fruit drink.

For every 200 mL of mango juice that is used, at least 300 mL of pineapple juice must be used.

The inequality representing this situation is

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Networks, STD2 N3 SM-Bank 44

A project involves nine activities, to .

The immediate predecessor(s) of each activity is shown in the table below.
 

             Activity            Immediate
     predecessor(s)     
 
 
 
 
 
 
 
 
 

 
A directed network for this project will require a dummy activity.

Sketch the network diagram, clearly identifying the dummy activity.   (3 marks)

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GEOMETRY, FUR1 2019 VCAA 8 MC

Four identical circles of radius are drawn inside a square, as shown in the diagram below

The region enclosed by the circles has been shaded in the diagram.
 

 
The shaded area can be found using

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GEOMETRY, FUR1 2019 VCAA 7 MC

A can of dog food is in the shape of a cylinder. The can has a circumference of 18.85 cm and a volume of 311 cm³.

The height of the can, in centimetres, is closest to

  1.   2.8
  2.   3.0
  3.   6.0
  4. 11.0
  5. 16.5
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Show Worked Solution
 

 

 

 

Networks, STD2 N3 SM-Bank 41

The directed network below shows the sequence of activities, to , that is required to complete a manufacturing process.

The time taken to complete each activity, in hours, is also shown.
  

  1. Determine the critical path of this network.   (2 marks)
  2. Identify the activities that have a float time of 10 hours.   (2 marks)

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i.   
 

 

 

ii.    

NETWORKS, FUR1 2019 VCAA 8 MC

The directed network below shows the sequence of activities, to , that is required to complete a manufacturing process.

The time taken to complete each activity, in hours, is also shown.
  

The number of activities that have a float time of 10 hours is

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
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NETWORKS, FUR1 2019 VCAA 7 MC

A project involves nine activities, to .

The immediate predecessor(s) of each activity is shown in the table below.
 

             Activity            Immediate
     predecessor(s)     
 
 
 
 
 
 
 
 
 

 
A directed network for this project will require a dummy activity.

The dummy activity will be drawn from the end of

  1. activity to the start of activity .
  2. activity to the start of activity .
  3. activity to the start of activity .
  4. activity to the start of activity .
  5. activity to the start of activity .
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