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Vectors, EXT2 V1 2024 HSC 14e

The diagram shows triangle .

The point lies on so that  .

The point lies on so that  .

The point is the intersection of and .

The point is chosen on so that and are collinear.
 

Let    and    where is a real number.

  1. Show that  .   (2 marks)

Writing  , where is a real number, it can be shown that  .  (Do NOT prove this.)

  1. Show that  .   (2 marks)

  2. Find in terms of and .   (2 marks)

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i.   

 
   
   
   

 

ii.   

.

   

   

     
 
 

 
iii.    

Show Worked Solution

i.   

 
   
   
   

  

ii.   

.

   

   

     
 
 

 

iii.   

♦ Mean mark (iii) 45%.
 

 
 
   

 

 
 
 

 

Vectors, EXT2 V1 2020 HSC 15b

The point divides the interval so that  . The position vectors of and are and respectively, as shown in the diagram.
 

  1. Show that  .  (2 marks)

     

  2. Prove that  .  (1 mark)

Let be a parallelogram with    and  . The point is the midpoint of and is the intersection of and , as shown in the diagram.
  
 
         
 

  1. Show that  .  (3 marks)

  2. Using parts (ii) and (iii), or otherwise, prove that is the point that divides the interval in the ratio 2 :1.   (1 mark)

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Show Worked Solution

i.  

 

 

ii.   
   
   
   
   
   

 

iii. 

♦ Mean mark part (iii) 50%.
 

 


 

 
 
Mean mark part (iv) 51%.

 

iv.