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Calculus, 2ADV C4 2022 HSC 28

The graph of the circle    is shown.

The interval connecting the origin, , and the point makes an angle with the positive -axis.
 

  1. By considering the value of , find the exact area of the shaded region, as shown on the diagram.  (2 marks)

Part of the hyperbola    which passes through the points and is drawn with the circle    as shown.
 

  1. Show that  .  (2 marks)

  2. Using parts (a) and (b), find the exact area of the region bounded by the hyperbola, the positive -axis and the circle as shown on the diagram.   (3 marks)

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a.   

 
   
   
   
   

 


Mean mark (a) 54%.

b.   

 
 
 

 

 
 
 
 
 
 

 

 


♦ Mean mark (b) 47%.
c.   
   
   
   

 

 
   
   

♦♦ Mean mark (c) 36%.

Calculus, 2ADV C3 SM-Bank 7

The graph of    for    is shown below.
 


 

  1. Calculate the area between the graph of   and the -axis.  (2 marks)

  2. For in the interval , show that the gradient of the tangent to the graph of    is  .  (1 mark)

The edges of the right-angled triangle are the line segments and , which are tangent to the graph of  , and the line segment , which is part of the horizontal axis, as shown below.

Let be the angle that makes with the positive direction of the horizontal axis.
 


 

  1. Find the equation of the line through and in the form  , for  .  (3 marks)

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i.  
   
   
   
   
   

 

ii.  
 
   
   

 

iii. 

♦♦♦ Mean mark (Vic) part (iii) 20%.
MARKER’S COMMENT: Most successful answers introduced a pronumeral such as    to solve.

 

 

   
 
   

 

 

Calculus, 2ADV C3 2009 HSC 10

  1. Show that the graph of    has no turning points.   (2 marks)

  2. Find the point of inflection of  .     (1 mark)

  3. i. Show that  for  .   (1 mark)

     

    ii. Let  .

     

        Use the result in part c.i. to show that    for all  .   (2 marks)

  1. Sketch the graphs of    and    for  .    (2 marks)

  2. Show that  .   (2 marks)

  3. Find the area enclosed by the graphs of    and  , and the straight line  .   (2 marks)

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  3. i. 

     

    ii.  

  4.  
        Geometry and Calculus, 2UA 2009 HSC 10 Answer
  6. ²
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a.   
♦♦ Mean mark 28% for all of Q10 (note that data for each question part is not available).
 

 
 

 

 

b.   

 

 
 

 

c.i.     
 
 

 

c.ii.  
 
 

MARKER’S COMMENT: When 2 graphs are drawn on the same set of axes, you must label them. 
 

d. 

Geometry and Calculus, 2UA 2009 HSC 10 Answer
e.   
 
 
 
 

 

f.   
   
   
   
   
   
    ²