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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.   
   
   

Vectors, SPEC1 2012 VCAA 9

The position of a particle at time    is given by

  1. Find the velocity of the particle at time     (1 mark)

  2. Find the speed of the particle at time    in the form  , where and are positive integers.   (2 marks)

  3. Show that at time     (2 marks)

  4. Find the angle in terms of , between the vector    and the vector    at time     (2 marks)

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a. 

 
   
   

 

b.   
   
 
   
   
   
   
   

♦ Mean mark part (c) 41%.

c.   
 
 
   
   
   

 

d.    

♦♦ Mean mark part (d) 32%.

 

 
 
 

Vectors, SPEC2 2018 VCAA 4

Two yachts, and , are competing in a race and their position vectors on a certain section of the race after time    hours are given by

where displacement components are measured in kilometres from a given reference buoy at origin .

  1. Find the cartesian equation of the path for each yacht.   (2 marks)

  2. Show that the two yachts will not collide if they follow these paths.   (2 marks)

  3. Find the coordinates of the point where the paths of the two yachts cross. Give your coordinates correct to three decimal places.   (2 marks)

One of the rules for the race is that the yachts are not allowed to be within 0.2 km of each other. If this occurs there is a time penalty for the yacht that is travelling faster.

  1. For what values of is yacht travelling faster than yacht ?   (2 marks)

  2. If yacht does not alter its course, for what period of time will yacht be within 0.2 km of yacht ? Give your answer in minutes, correct to one decimal place.   (2 marks)

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Show Worked Solution

a. 

 

 

 

b. 

 


 

 

c.  
 
 
 
 
   

 

d.   
 
   
     
 
 
   

  
 

♦ Mean mark part (d) 41%.

 

 

e.   
   

 


 

♦ Mean mark part (e) 37%.