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Vectors, EXT1 V1 EQ-Bank 28

A projectile is fired from a canon at ground level with initial velocity ms−1 at an angle of 30° to the horizontal.

The equations of motion are    and  .

  1. Show that  .  (1 mark)

  2. Show that  .  (2 marks)
  3. Hence find the Cartesian equation for the trajectory of the projectile.  (1 mark)

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i.   

 

ii.   


 

 

iii.   

 
 

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.   
 

  

  

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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
 
         
 
 
         
 

 
 
 

 

Mechanics, SPEC1-NHT 2019 VCAA 1

A 10 kg mass is placed on a rough plane that inclined at 30° to the horizontal, as shown in the diagram below. A force of 40 N is applied to the mass up the slope and parallel to the slope. There is also a frictional resistance force of magnitude that opposes  the motion of the mass.
 


 

  1. Find the magnitude of the frictional resistance force, in newtons, acting up the slope if the force is just sufficient to stop the mass from sliding down the slope.  (2 marks)
     
  2. An additional force of magnitude newtons is applied to the mass  up the slope and parallel to the slope. The sum of the additional force and the frictional resistance force of magnitude that now acts down the slope is such that it is just sufficient to stop the mass from sliding up the slope.  (2 marks)
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a.       

 

 

b.   

 
 

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.  
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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 2005 HSC 27c

2UG-2005-27c
 

The bearing of from is 250° and the distance of from is 36 km.

  1. Explain why    is  .   (1 mark)

  2. If    is 15 km due north of  , calculate the distance of    from  , correct to the nearest kilometre.   (3 marks)

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i. 

 

ii.   
 
 
 

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.  

 

Vectors, EXT2 V1 SM-Bank 22

are the vertices of a rectangular prism.
  


 

  1. Show that the internal diagonals of the prism, and , intersect.  (2 marks)

  2. Calculate the acute angle, , between the diagonals, to the nearest minute.  (2 marks)

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i.     
   

 


 

 

ii.   
 

 

 

Vectors, EXT2 V1 SM-Bank 15

Consider the two vector line equations

  1. Show that    and    intersect and determine the point of intersection . (2 marks)

  2. What is the acute angle between the vector lines, to the nearest minute.  (2 marks)

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i.   

 


  

 

ii.   

 
 

 

 

Vectors, EXT2 V1 SM-Bank 2

  1. Find values of  , ,   and    such that    is a vector equation of a line that passes through    and  .  (2 marks)

  2. Determine whether    is perpendicular to  .  (1 mark)

  3. Express    in Cartessian form.  (1 mark)

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  1. , or

     

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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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a. 
 

b. 
 

c.  

 

d.   

GRAPHS, FUR2 2019 VCAA 1

The graph below shows the membership numbers of the Wombatong Rural Women’s Association each year for the years 2008–2018.
 


 

  1. How many members were there in 2009?  (1 mark)
    1. Show that the average rate of change of membership numbers from 2013 to 2018 was − 6 members per year.  (1 mark)
    2. If the change in membership numbers continues at this rate, how many members will there be in 2021?  (1 mark)
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a. 

 

b.i.  
 
 
   

 

b.ii.  
   

GEOMETRY, FUR2 2019 VCAA 3

The following diagram shows a crane that is used to transfer shipping containers between the port and the cargo ship.
 


 

The length of the boom, , is 25 m. The length of the hoist, , is 15 m.

    1. Write a calculation to show that the distance is 20 m.  (1 mark)
    2. Find the angle .

        

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

  1. The diagram below shows a cargo ship next to a port. The base of a crane is shown at point .
     

      

      


     

      

    The base of the crane () is 20 m from a shipping container at point . The shipping container will be moved to point , 38 m from . The crane rotates 120° as it moves the shipping container anticlockwise from to .

      

    What is the distance , in metres?

      

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

  2. A shipping container is a rectangular prism.

      

    Four chains connect the shipping container to a hoist at point , as shown in the diagram below.
     

      

     
     

      

    The shipping container has a height of 2.6 m, a width of 2.4 m and a length of 6 m.

     

    Each chain on the hoist is 4.4 m in length.

      

    What is the vertical distance, in metres, between point and the top of the shipping container?

      

    Round your answer to the nearest metre.  (2 marks)

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a.i.  
   
   
   

 

a.ii.  
 
   
   

 

b. 

 
 

 

c.  


 


 

 

 

 
 

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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a.  

 

b.  
   

 

c.   

 
 
 

GEOMETRY, FUR2 2019 VCAA 1

The following diagram shows a cargo ship viewed from above.
 

 
The shaded region illustrates the part of the deck on which shipping containers are stored.

  1. What is the area, in square metres, of the shaded region?  (1 mark)

Each shipping container is in the shape of a rectangular prism.

Each shipping container has a height of 2.6 m, a width of 2.4 m and a length of 6 m, as shown in the diagram below.
 

  1. What is the volume, in cubic metres, of one shipping container?  (1 mark)
  2. What is the total surface area, in square metres, of the outside of one shipping container?  (1 mark)
  3. One shipping container is used to carry barrels. Each barrel is in the shape of a cylinder.

      

    Each barrel is 1.25 m high and has a diameter of 0.73 m, as shown in the diagram below.

      

    Each barrel must remain upright in the shipping container
     
     

      


     
    What is the maximum number of barrels that can fit in one shipping container?  (1 mark)

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a.  
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b.  
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c.  
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d.  
 
 

 

 

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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a. 
 

 

 


 

b. 
 

c.   


 

d.