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Mechanics, SPEC2 2020 VCAA 5

Two objects, each of mass kilograms, are connected by a light inextensible strings that passes over a smooth pulley, as shown below. The object on the platform is initially at point A and, when it is released, it moves towards point C. The distance from point A to point C is 10 m. The platform has a rough surface and, when it moves along the platform, the object experiences a horizontal force opposing the motion of magnitude newtons in the section AB and a horizontal force opposing the motion of magnitude newtons when it moves in the section BC.
 

  1. On the diagram above, mark all forces that act on each object once the object on the platform has been released and the system is in motion.  (2 marks)

The force is given by  .

  1.  i. Show that an expression for the acceleration, in , of the object on the platform, in terms of , as it moves from point A to point B is given by  .  (2 marks)
  2. ii. The system will only be in motion for certain values of .
  3.     Find these values of .  (1 mark)

Point B is midway between points A and C.

  1. Find, in terms of , the time taken, is seconds, for the object on the platform to reach point B.  (2 marks)
  2. Express, in terms of , the speed , in , of the object on the platform when it reaches point B.  (2 marks)
  3. When the object on the platform is at point B, the string breaks. The velocity of the object at point B is  . The force that opposes motion from point B to point C is  , where is the velocity of the object when it is a distance of metres from point B. The object on the platform comes to rest before point C.
  4. Find the object's distance from point C when it comes to rest. Give your answer in metres, correct to two decimal places.  (4 marks)
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  1.  
  2.  i.
  3. ii.
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a.   

 

b. i.   


 

 
♦ Mean mark (b)(ii) 45%.

 

b. ii.   

 

c.   

 

 

d.   

 
 

 

 

e.   

♦♦ Mean mark (e) 35%.
 
 

 

 

 

 

Vectors, SPEC2 2020 VCAA 4

A pilot is performing at an air show. The position of her aeroplane at time relative to a fixed origin is given by 

   ,

where    is a unit vector in a horizontal direction,   is a unit vector vertically up, displacement components are measured in metres and time is measured in seconds where  .

  1. Find the maximum speed of the aeroplane. Give your answer in .  (3 marks)

  2.  i. Use    to show that the cartesian equation of the path of the aeroplane is given by
  3.     .  (2 marks)

  4. ii. Sketch the path of the aeroplane on the axes provided below. Label the position of the aeroplane when  , using coordinates, and use an arrow to show the direction of motion of the aeroplane.  (3 marks)

  5.   

         
     

A friend of the pilot launches an experimental jet-powered drone to take photographs of the air show. The position of the drone at time relative to the fixed origin is given by  , where is in seconds and    is a unit vector in the same horizontal direction,   is a unit vector vertically up, and displacement components are measured in metres.

  1. Sketch the path of the drone on the axes provided in part b.ii. Using coordinates, label the points where the path of the drone crosses the path of the aeroplane, correct to the nearest metre.  (3 marks)

  2. Determine whether the drone will make contact with the aeroplane. Give reasons for your answer.  (3 marks)

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  2.  i.
  3. ii.
  4.  

     

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a.   
   
 
 

 

b. i.   


 


 

 

b. ii.   


 

 

c.   

 

 

 

d.   


 


  

Calculus, SPEC2 2020 VCAA 3

Let  .

  1. Find an expression for    and state the coordinates of the stationary points of  .  (2 marks)
  2. State the equation(s) of any asymptotes of  .  (1 mark)
  3. Sketch the graph of    on the axes provided below, labelling the local maximum stationary point and all points of inflection with their coordinates, correct to two decimal places.  (3 marks)
     
         

Let  , where  .

  1. Write down an expression for  .  (1 mark)
  2.  i. Find the non-zero values of for which  .  (1 mark)
  3. ii. Complete the following table by stating the value(s) of for which the graph of    has the given number of points of inflection.  (2 marks)
     
       
         
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  3.   
  5.  i.
  6. ii.
       
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a.   

 

 

b.   

 

c.   

 

d.   

 

e.i.   

♦♦♦ Mean mark (e)(ii) 9%.

 

e.ii.   

Complex Numbers, EXT2 N2 2020 SPEC2 2

Two complex numbers, and , are defined as    and  .

  1. Express the relation    in the cartesian form  , where  .  (3 marks)

  2. Plot the points that represent and and the relation on the Argand diagram below.  (2 marks)
     
         
     
  3. State a geometrical interpretation of the graph of    in relation to the points that represent and .  (1 mark)

  4.  i. Sketch the ray given by    on the Argand diagram in part b.  (1 mark)

  5. ii. In Cartesian form, write down the function that describes the ray  .  (1 mark)

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  2.  
  4.  i.
  5. ii.
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a.   

 

b.   

 

c.   

 

d.i.   

 

d.ii.   

Complex Numbers, SPEC2 2020 VCAA 2

Two complex numbers, and , are defined as    and  .

  1. Express the relation    in the cartesian form  , where  .   (3 marks)

  2. Plot the points that represent and and the relation on the Argand diagram below.   (2 marks)

     
         
     

  3. State a geometrical interpretation of the graph of    in relation to the points that represent and .   (1 mark)

  4.  i. Sketch the ray given by    on the Argand diagram in part b.   (1 mark)

  5. ii. Write down the function that describes the ray  , giving the rule in cartesian form.   (1 mark)

  6. The points representing and and    lie on the circle given by  , where is the centre of the circle and is the radius.
  7. Find in the form  , where  , and find the radius .   (3 marks)

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  2.  
  4.  i.
  5. ii.
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a.   

 

b.   

 

c.   

 

d.i.   

♦♦ Mean mark (d.ii.) 25%.

 

d.ii.   

 

e.   

♦ Mean mark (e) 40%.

 


 

 
 

Vectors, SPEC2 2020 VCAA 1

A particle moves in the    plane such that its position in terms of and metres at seconds is given by the parametric equations

where

  1. Find the distance, in metres, of the particle from the origin when  .   (2 marks)

  2.   i. Express    in terms of and, hence, find the equation of the tangent to the path of the particle at    seconds.   (3 marks)

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

  4. iii. Find the magnitude of the acceleration, in , when  .   (2 marks)

  5. Find the time, in seconds, when the particle first passes through the origin.   (1 mark)

  6. Express the distance, metres, travelled by the particle from    to    as a definite integral and find this distance correct to three decimal places.   (2 marks)

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  2.   i.
  3.  ii.
  4. iii.
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a.   

 

 

b.i.   

 

 

 


  

b.ii.  
 
 
   

 

b.iii. 

♦ Mean mark (b)(iii) 48%.
 
 

 

c.   

 

d.   
   
   

Number, NAPX-p110532v02

Ana recycles 3 kilograms of plastic bottles in her first week of recycling.

She then recycles twice as much as she recycled the previous week for the next 5 weeks.

The total weight, in kilograms, of the plastic bottles she recycles in any given week will be

 
  always even
 
  always odd
 
  sometimes even or sometimes odd
 
  none of the above
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Number, NAPX-p110532v01

Leo puts $15 into his piggy bank to start saving money.

He then puts the same amount into the piggy bank at the end of each month.

The total amount of money in his piggy bank will be which of the following:

 
  Always an even number
 
  Always an odd number
 
  Sometimes odd sometimes even
 
  None of the above
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Algebra, NAPX-p116707v03

Louise donated money to two charities

She donated $125 to the first charity and donated $45 to the second charity.

Which number sentence could be used to find total amount of money Louise donated?

 
  120 + 40 = 160
 
  130 + 50 = 180
 
  120 + 40 + 5 = 165
 
  130 + 50 – 10 = 170

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Algebra, NAPX-p116707v02

Shane played two basketball games.

She scored 35 points in the first game.

In the second game she scored 25 points.

Which number sentence could be used to find the total number of points Shane scored in the first two games?

30 + 20 + 5 = 55 40 + 30 = 70 30 + 20 + 10 = 60 30 + 20 = 50
 
 
 
 
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Mechanics, EXT2 M1 2015 SPEC1 6

The acceleration ms¯² of a body moving in a straight line in terms of the velocity ms¯¹ is given by 

Given that    when  , where is the displacement of the body in metres, find the velocity of the body when    (4 marks)

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COMMENT: Strong log and exponential calculation ability required here.

Mechanics, SPEC2 2020 VCAA 18 MC

A particle of mass kilogram hangs from a string that is attached to a fixed point. The particle is acted on by a horizontal force of magnitude newtons. The system is in equilibrium when the string makes an angle to the horizontal, as shown in the diagram below. The tension in the string has magnitude newtons.
 


 

The value of    is

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Vectors, SPEC2 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.   
 

 

 

Calculus, EXT1 C3 2020 SPEC2 10

A tank initially contains 300 grams of salt that is dissolved in 50 L of water. A solution containing 15 grams of salt per litre of water is poured into the tank at a rate of 2 L per minute and the mixture in the tank is kept well stirred. At the same time, 5 L of the mixture flows out of the tank per minute.

Find the differential equation,  , where  represents the mass, in grams, of salt in the tank at time minutes, for a non-zero volume of mixture.   (2 marks)

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Calculus, SPEC2 2020 VCAA 10 MC

A tank initially contains 300 grams of salt that is dissolved in 50 L of water. A solution containing 15 grams of salt per litre of water is poured into the tank at a rate of 2 L per minute and the mixture in the tank is kept well stirred. At the same time, 5 L of the mixture flows out of the tank per minute.

A differential equation representing the mass, grams, of salt in the tank at time minutes, for a non-zero volume of mixture is

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Calculus, SPEC2 2020 VCAA 3 MC

A train is travelling from Station A to Station B. The train starts from rest at Station A and travels with constant acceleration for 30 seconds until it reaches a velocity of 10 ms−1. It then travels at this velocity for 200 seconds. The train then slows down, with constant acceleration, and stops at Station B having travelled for 260 seconds in total. Let ms−1 be the velocity of the train at time seconds. The velocity as a function of is given by

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Measurement, STD2 M6 FUR1 2011 2


 

The point  on building  is visible from the point  on building , as shown in the diagram above.

Building  is 16 metres taller than building .

The horizontal distance between point  and point  is 23 metres.

Calculate the angle of depression of point from point . Give your answer correct to 1 decimal place.   (2 marks)

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GEOMETRY, FUR1 2020 VCAA 8 MC

A cake is in the shape of a rectangular prism, as shown in the diagram below.
 

The cake is cut in half to create two equal portions.

The cut is made along the diagonal, as represented by the dotted line.

The total surface area, in square centimetres, of one portion of the cake is

  1. 132
  2. 180
  3. 192
  4. 212
  5. 264
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GEOMETRY, FUR1 2020 VCAA 7 MC

Cape Town has latitude 34° S and longitude 18° E.

Stockholm has latitude 59° N and longitude 18° E.

Assume that the radius of Earth is 6400 km.

The shortest distance along the meridian between Cape Town and Stockholm, in kilometres, is closest to

  1.   2011
  2.   3798
  3.   4021
  4.   6590
  5. 10 388
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GEOMETRY, FUR1 2020 VCAA 6 MC

The cities of Lhasa (30° N, 91° E), Chengdu (31° N, 104° E) and Shanghai (31° N, 121° E) are all in the same time zone.

Assume that 15° of longitude equates to a one-hour time difference.

Which one of the following statements is true?

  1. The sun rises at the same time in all three cities.
  2. The sun rises in Lhasa approximately one hour before it rises in Chengdu.
  3. The sun rises in Lhasa approximately two hours after it rises in Shanghai.
  4. Sunrise in each of the three cities is approximately 15 minutes apart.
  5. The sun rises in Shanghai and Chengdu at the same time but it rises earlier in Lhasa.
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GEOMETRY, FUR1 2020 VCAA 5 MC

The pie chart below displays the results of a survey.
 


 

Eighty per cent of the people surveyed selected ‘agree’.

Twenty per cent of the people surveyed selected ‘disagree’.

The radius of the pie chart is 16 mm.

The area of the sector representing ‘agree’, in square millimetres, is closest to

  1.   80
  2. 161
  3. 483
  4. 643
  5. 804
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