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Vectors, EXT2 V1 2024 HSC 16a

Consider the function  , where  . The value of has been chosen so that a circle can be drawn, centred at the origin, which has exactly two points of intersection with the graph of the function and so that the circle is never above the graph of the function. The point    is the point of intersection in the first quadrant, so    and  ,  as shown in the diagram.

The vector joining the origin to the point is perpendicular to the tangent to the graph of the function at that point. (Do NOT prove this.)

Show that  .   (4 marks)

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♦♦♦ Mean mark 8%.

 
   

 

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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.    

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i.   

 
   
   
   

  

ii.   

.

   

   

     
 
 

 

iii.   

♦ Mean mark (iii) 45%.
 

 
 
   

 

 
 
 

 

Vectors, EXT2 V1 EQ-Bank 11

Point is the circumcentre of triangle which is the centre of the circle that passes through each of its vertices.

Point is the centroid of triangle where the bisectors of each angle within the triangle intersect.

Point is such that .
 

Prove that     (5 marks)

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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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i.  

 

 

ii.   
   
   
   
   
   

 

iii. 

♦ Mean mark part (iii) 50%.
 

 


 

 
 
Mean mark part (iv) 51%.

 

iv.   

 

 

Vectors, EXT2 V1 2015 VCE 1

Consider the rhombus    shown below, where    and  , and is a positive real constant.
 

VCAA 2015 spec 1a
 

  1. Find    (1 mark)

  2. Show that the diagonals of the rhombus    are perpendicular.  (2 marks)

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i.   

 

 

ii.   

MARKER’S COMMENT: Vector notation was poor in many answers.