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Mechanics, EXT2 2013 HSC 7 MC

The angular speed of a disc of radius    cm is    revolutions per minute.

What is the speed of a mark on the circumference of the disc?

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Proof, EXT2 P2 2009 HSC 8a

  1. Using the substitution  ­, or otherwise, show that
     
    ­­  (2 marks)

  2. Use mathematical induction to prove that, for integers  ,
     
    ­­  (3 marks) 

  3. Show that  ­  (2 marks)

  4. Hence find the exact value of

  5.  

      (2 marks)

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i.   

­­

 
 
 
 
 
 

 

ii.  

­
­
 
­
 

 

­­

­­

­­­
  ­­
  ­­
  ­
 
 
  ­
 

 

 

iii.   ­

­

­

­

 

 

  ­

­

 

iv.   

­

­

Complex Numbers, EXT2 N2 2009 HSC 7b

Let  

  1. Show that  , where    is a positive integer.  (2 marks)

  2. Let    be a positive integer. Show that
     

     
          (3 marks)

  3. Hence, or otherwise, prove that
     
       
     
    where    is a positive integer.  (2 marks)

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i.   
 
 
   
 
   

 

 

ii. 

 

iii. 

Mechanics, EXT2 M1 2009 HSC 7a

A bungee jumper of height 2 m falls from a bridge which is 125 m above the surface of the water, as shown in the diagram. The jumper’s feet are tied to an elastic cord of length   m. The displacement of the jumper’s feet, measured downwards from the bridge, is   m.
 


 

The jumper’s fall can be examined in two stages. In the first stage of the fall, where  , the jumper falls a distance of   m subject to air resistance, and the cord does not provide resistance to the motion. In the second stage of the fall, where  , the cord stretches and provides additional resistance to the downward motion.

  1. The equation of motion for the jumper in the first stage of the fall is

     

          

     

    where    is the acceleration due to gravity,    is a positive constant, and    is the velocity of the jumper.
     
      (1)  Given that    and    initially, show that

             (3 marks)

      (2)  Given that    and  , find the length,  , of the cord such that the jumper’s velocity is    when  . Give your answer to two significant figures.  (1 mark)

  2. In the second stage of the fall, where  , the displacement    is given by
     
         
     
    where    is the time in seconds after the jumper’s feet pass  .

     

    Determine whether or not the jumper’s head stays out of the water.  (4 marks)

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i. (1)  
 
 

 

 
 

 

 
 

 

i. (2)   

 

 

ii.   
 
   
   

 

 
    

 

 

 
 

 

Harder Ext1 Topics, EXT2 2009 HSC 6c

The diagram shows a circle of radius  , centred at the origin, . The line    is tangent to the circle at  , the line    is horizontal, and    lies on the line  .

  1. Find the length of    in terms of    (1 mark)
  2. The point    moves such that  .
  3. Show that the equation of the locus of    is
    1.   (2 marks)

  4. Find the focus, , of the parabola in part (ii).   (2 marks)
  5. Show that the difference between the length    and the length    is independent of    (2 marks)
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(i)   
 
 
 

 

(ii) 

 

 

(iii) 

♦♦♦ “Few” students answered part (iii) correctly (exact data unavailable).
 

 

 

  

(iv) 

♦♦♦ “Few” students answered part (iv) correctly (exact data unavailable).
MARKER’S COMMENT: Very few used the efficient focus-directrix definition here.
 

 

 

 

 
 

 

Polynomials, EXT2 2009 HSC 6b

Let  , where    is real. One zero of    is  .

  1. Show that if is a zero of    then    is a zero of     (1 mark)
  3. Suppose that    is a zero of    and    is not real.
  4. (1)   Show that    (2 marks)
  5. (2)   Show that    (2 marks)

 

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(i)  

 

(ii)(1) 

 

(ii)(2) 

Volumes, EXT2 2009 HSC 6a

The base of a solid is the region enclosed by the parabola    and the -axis. The top of the solid is formed by a plane inclined at   to the  -plane. Each vertical cross-section of the solid parallel to the -axis is a rectangle. A typical cross-section is shown shaded in the diagram.

Find the volume of the solid.   (3 marks)

 

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³

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  ³

Harder Ext1 Topics, EXT2 2009 HSC 5c

Let  

  1. Show that    for all     (2 marks)
  2. Hence, or otherwise, show that
    1.  for all     (2 marks)
  3. Hence, or otherwise, show that
    1.  for all   (1 mark)

 

 

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(i)  
 
 

 

 

(ii)   

 

 
 
 

 

 

(iii) 

 

Harder Ext1 Topics, EXT2 2009 HSC 5a

In the diagram    is the diameter of the circle. The chords    and    intersect at  . The point    lies on    such that    is perpendicular to  . The point    is the intersection of    produced and    produced.

Copy or trace the diagram into your writing booklet.

  1. Show that    (2 marks)
  2. Show that    is a cyclic quadrilateral.  (2 marks)
  3. Show that    are collinear.  (2 marks)
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(i)   HSC 2009 5bii

 

(ii) 

 

 

(iii)  

 

Mechanics, EXT2 2009 HSC 4b

A light string is attached to the vertex of a smooth vertical cone. A particle    of mass    is attached to the string as shown in the diagram. The particle remains in contact with the cone and rotates with constant angular velocity    on a circle of radius  . The string and the surface of the cone make an angle of    with the vertical, as shown.

The forces acting on the particle are the tension, , in the string, the normal reaction, , to the cone and the gravitational force  .

  1. Resolve the forces on    in the horizontal and vertical directions.  (2 marks)
  2. Show that    and find a similar expression for   (2 marks)
  3. Show that if    then
    1.   (2 marks)
  4. For which values of    can the particle rotate so that  ?  (1 mark)
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(i)    HSC 2009 4bii

ω

 

(ii)  

 

αα

 

(iii) 

 
♦ Part (iv) proved challenging (exact data not available).

MARKER’S COMMENT: Most students did not realise  ω².

 

(iv) 

Conics, EXT2 2009 HSC 4a

The ellipse    has foci    and    where    is the eccentricity, with corresponding directrices    and  . The point    is on the ellipse. The points where the horizontal line through    meets the directrices are    and  , as shown in the diagram.

  1. Show that the equation of the normal to the ellipse at the point    is
    1.   (2 marks)

  2. The normal at    meets the -axis at  . Show that    has coordinates     (2 marks)

  3. Using the focus-directrix definition of an ellipse, or otherwise, show that
    1.   (2 marks)

  4. Let    and  
  5. By applying the sine rule to    and to  , show that     (2 marks)

 

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(i)   
 
 

 

 

(ii) 

 
 

 

 

(iii)                  

 

 

 

(iv)  

 

 

 

Volumes, EXT2 2009 HSC 3d

The diagram shows the region enclosed by the curves    and  

The region is rotated about the -axis.

Use the method of cylindrical shells to find the volume of the solid formed.   (3 marks)

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³

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δ
  δ
  δ
 
 
 
 
  ³

Functions, EXT1′ F1 2009 HSC 3a

The diagram shows the graph  
 


 

Draw separate one-third page sketches of the graphs of the following:

  1.    (2 marks)

  2.    (2 marks)

  3.    (2 marks)

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i.

HSC 2009 3aii

 

ii.  

iii.  

 

 

 

 

 

 

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i.   


 

HSC 2009 3aii

 

ii. 

   HSC 2009 3aiii

 

iii.   


 

 

 

 

 

 

 

Polynomials, EXT2 2010 HSC 7c

Let  , where    is an odd integer,  .

  1. Show that    has exactly two stationary points.  (1 mark)
  2. Show that    has a double zero at  .  (1 mark)
  3. Use the graph    to explain why    has exactly one real zero other than  .  (2 marks)
  4. Let  α  be the real zero of    other than  .
  5. Given that    for , or otherwise, show that  α.  (2 marks)

  6. Deduce that each of the zeros of    has modulus less than or equal to  .  (2 marks)
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(i)  
   

 

 

(ii)   
 

 

 

(iii)  

♦♦ Mean mark part (iii) 26%.

 

 Polynomials, EXT2 2010 HSC 7c Answer1

♦♦♦ Mean mark part (iv) 3%.
(iv)   
   
   
   

 

 
 
 
 

 

α

 

(v)  

♦♦♦ Mean mark part (v) 2%.

 

 

α

   
   
   

 

α

Graphs, EXT2 2010 HSC 7b

The graphs of    and    intersect at    and at  .

Using these graphs, or otherwise, show that    for  .  (1 mark)

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Graphs, EXT2 2010 HSC 7b1

Harder Ext1 Topics, EXT2 2010 HSC 7a

In the diagram    is a cyclic quadrilateral. The point    is on    such that  , and hence    is similar to  .

Harder Ext1 Topics, EXT2 2010 HSC 7ai

Copy or trace the diagram into your writing booklet.

  1. Show that    is similar to  .  (2 marks)
  2. Using the fact that  , 
  3. show that  .  (2 marks)
  5. A regular pentagon of side length    is inscribed in a circle, as shown in the diagram.

Harder Ext1 Topics, EXT2 2010 HSC 7aii

  1. Let    be the length of a chord in the pentagon.
  3. Use the result in part (ii) to show that  .  (2 marks)

 

 

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(i)    Harder Ext1 Topics, EXT2 2010 HSC 7a Answer
   

 

 

(ii)  

♦♦ Mean mark part (ii) 31%.

 

 

  

(iii) 

♦♦ Mean mark part (iii) 24%.

 

Polynomials, EXT2 2010 HSC 6c

  1. Expand    using the binomial theorem.   (1 mark)

  2. Expand    using de Moivre’s theorem, and hence show that

    1. .   (3 marks)

  3. Deduce that

  4.  is one of the solutions to

    1. .   (1 mark)

  5. Find the polynomial    such that  .   (1 mark)

  6. Find the value of    such that  .   (1 mark)

  7. Hence find an exact value for
    1. .   (1 mark)

 

 

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(i)   

 

(ii) 

 
 
 
 

 

(iii)  

 

 

(iv) 

 

(v)   

 

(vi)  

♦♦ Mean mark part (vi) 27%.
 

 

Harder Ext1 Topics, EXT2 2010 HSC 5c

A TV channel has estimated that if it spends    on advertising a particular program it will attract a proportion    of the potential audience for the program, where

and   is a given constant.

  1. Explain why
     has its maximum value when  .   (1 mark)

  2. Using
  3. , or otherwise, deduce that
  4.  
  5.   for some constant  .   (3 marks)
  6.  

  8. The TV channel knows that if it spends no money on advertising the program then the audience will be one-tenth of the potential audience.
  9. Find the value of the constant    referred to in part (c)(ii).   (1 mark)

  10. What feature of the graph
     is determined by the result in part (c)(i)?   (1 mark)

  11.  
  12. Sketch the graph
      (1 mark) 

 

 

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(i)  

♦♦ Mean mark part (i) 32%.

 

♦ Mean mark part (ii) 41%.

 

(ii)  

 

 

(iii)  
♦♦ Mean mark part (iv) 25%.

 

(iv)  

 

(v)  

♦♦ Mean mark part (v) 12%.

Harder Ext1 Topics, EXT2 2010 HSC 5c

Conics, EXT2 2010 HSC 5a

The diagram shows two circles,  and  , centred at the origin with radii    and  , where  .

The point    lies on    and has coordinates  .

The point    is the intersection of    and  .

The point    is the intersection of the horizontal line through    and the vertical line through  .

Conics, EXT2 2010 HSC 5a

  1. Write down the coordinates of  .   (1 mark)
  2. Show that    lies on the ellipse
    .   (1 mark)

  3. Find the equation of the tangent to the ellipse
     at  .   (2 marks)

  4. Assume that is not on the -axis.
  5. Show that the tangent to the circle    at  , and the tangent to the ellipse
     at  , intersect at a point on the -axis.   (2 marks)
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(i)  

 

(ii) 

 
 
 

 

 

(iii)  

 

 

 

 
 

 

 

(iv) 

 

 

 

 

Harder Ext1 Topics, EXT2 2010 HSC 4d

A group of    people is to be divided into discussion groups.

  1. In how many ways can the discussion groups be formed if there are    people in one group, and    people in another?   (1 mark)
  2. In how many ways can the discussion groups be formed if there are    groups containing    people each?   (2 marks) 
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(i)   
   

 

(ii)  

♦ Mean mark part (ii) 35%.

Mechanics, EXT2 2010 HSC 4b

A bend in a highway is part of a circle of radius  , centre  . Around the bend the highway is banked at an angle  α  to the horizontal.

A car is travelling around the bend at a constant speed  . Assume that the car is represented by a point    of mass  . The forces acting on the car are a lateral force  , the gravitational force    and a normal reaction    to the road, as shown in the diagram.

Mechanics, EXT2 2010 HSC 4b

  1. By resolving forces, show that
    αα.   (3 marks)
  2. Find an expression for    such that the lateral force    is zero.   (1 mark)
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  2. α
Show Worked Solution
(i) 

Mechanics, EXT2 2010 HSC 4b Answer
αα
αα
αα
ααα α
ααα α
αα αα
αα

  

(ii)  αα

αα
α α
α
α

Volumes, EXT2 2010 HSC 3b

The region shaded in the diagram is bounded by the -axis and the curve  .

Volumes, EXT2 2010 HSC 3b

The shaded region is rotated about the line  .

Find the volume generated.   (4 marks)

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 ³

Show Worked Solution

 
 
 
 
 
  ³

Complex Numbers, EXT2 N2 2010 HSC 2d

Let    where  .

On the Argand diagram the point represents  , the point represents    and the point represents  .
 

Complex Numbers, EXT2 2010 HSC 2d
 

Copy or trace the diagram into your writing booklet.

  1. Explain why the parallelogram    is a rhombus.   (1 mark)

  2. Show that  .   (1 mark)

  3. Show that  .   (2 marks)

  4. By considering the real part of  , or otherwise deduce that

  5.  

        .   (1 mark) 

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i.   

Complex Numbers, EXT2 2010 HSC 2d Answer 

 

 

ii. 


 

 

iii. 

♦♦ Mean mark part (iii) 26%.

 
 

 

iv.   
   

♦♦ Mean mark part (iv) 44%.


 

Harder Ext1 Topics, EXT2 2011 HSC 8b

A bag contains seven balls numbered from to . A ball is chosen at random and its number is noted. The ball is then returned to the bag. This is done a total of seven times.

  1. What is the probability that each ball is selected exactly once?  (1 mark)
  2. What is the probability that at least one ball is not selected?  (1 mark)
  3. What is the probability that exactly one of the balls is not selected?  (2 marks)
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(i)  

♦♦ Mean mark part (i) 32%.

 

(ii)  

 

(iii) 

♦♦♦ Mean mark part (iii) 2%.

 

 
 

Volumes, EXT2 2011 HSC 7a

The diagram shows the graph of    for  

The area bounded by  , the line    and the  -axis is rotated about the line   to form a solid.

Use the method of cylindrical shells to find the volume of the solid.  (4 marks)

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³

Show Worked Solution

 

 
 

 

 

 

 
 
 
 
  ³

Graphs, EXT2 2011 HSC 6b

Let be a function with a continuous derivative.

  1. Prove that    has a stationary point at    if    or    (2 marks)
  2.  
  3. Without finding  , explain why    has a horizontal point of inflection at    if    and    (1 mark)
  4.  
  5. The diagram shows the graph   
     

    1. HSC 2011 6bi
       
  6. Copy or trace the diagram into your writing booklet. 
  7.  
    On the diagram in your writing booklet, sketch the graph  , clearly distinguishing it from the graph    (3 marks)

 

 

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(i)  

 

 

 

♦♦♦ Mean mark part (ii) 0%!
A BEAST!

(ii)   

 

 

(iii) 


 


 


 

Mechanics, EXT2 2011 HSC 5a

A small bead of mass    is attached to one end of a light string of length  . The other end of the string is fixed at height    above the centre of a sphere of radius  , as shown in the diagram. The bead moves in a circle of radius    on the surface of the sphere and has constant angular velocity  . The string makes an angle of    with the vertical.

Three forces act on the bead: the tension force    of the string, the normal reaction force    to the surface of the sphere, and the gravitational force  .

  1. By resolving the forces horizontally and vertically on a diagram, show that
  2. and
    1.   (2 marks)
  3. Show that
    1.   (2 marks)
  4. Show that the bead remains in contact with the sphere if  
    1.   (2 marks)

 

 

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(i)  

♦ Mean mark part (i) 50%.

 

 

(ii)      

 

(iii) 

 
 
 

Harder Ext1 Topics, EXT2 2011 HSC 4c

A mass is attached to a spring and moves in a resistive medium. The motion of the mass satisfies the differential equation

where    is the displacement of the mass at time  

  1. Show that, if    and    are both solutions to the differential equation and    and    are constants, then
  2. is also a solution.  (2 marks)
  5. A solution of the differential equation is given by    for some values of  , where    is a constant.
  6. Show that the only possible values of    are    and    (2 marks)
  9. A solution of the differential equation is
  10. When  , it is given that    and  .
  11. Find the values of    and    (3 marks)
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(i)  

♦♦♦ Mean mark 18%.

 

 
 
 
 

 

(ii)  

 

 

(iii)  
 

 

Harder Ext1 Topics, EXT2 2011 HSC 4b

In the diagram,    is a cyclic quadrilateral. The point    lies on the circle through the points  and    such that  . The line    meets the line    at the point  . The point    lies on the line    such that  

Copy or trace the diagram into your writing booklet.

  1. Prove that    is a cyclic quadrilateral.  (2 marks)
  2. Explain why    (1 mark)
  3. Prove that    is a tangent to the circle through the points    and    (2 marks)
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(i)  

 

 

(ii) 

 

(iii)   

 

Complex Numbers, EXT2 N2 2011 HSC 4a

Let    and    be real numbers with  . Let    be a complex number such that

     

  1. Prove that    (2 marks)

  2. Hence, describe the locus of all complex numbers    such that    (1 mark)

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i.   
 
 
 
 
 

  

♦ Mean mark part (ii) 44%.

ii. 

Volumes, EXT2 2011 HSC 3b

The base of a solid is formed by the area bounded by    and    for  

Vertical cross-sections of the solid taken parallel to the -axis are in the shape of isosceles triangles with the equal sides of length unit as shown in the diagram.

Find the volume of the solid.   (3 marks)

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³

Show Worked Solution

 
 
 
 
 
 
 

³

Functions, EXT1′ F1 2011 HSC 3a

  1.  Draw a sketch of the graph
     
       for      (1 mark)

  2.  Find     (1 mark)

  3.  Draw a sketch of the graph
     
           for      (2 marks)

  4.  

    (Do NOT calculate the coordinates of any turning points.)  

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i. 

ii. 

iii. 

Show Worked Solution
i.  

 

ii.   
 
 

 

iii.   

Complex Numbers, EXT2 N2 2011 HSC 2d

  1. Use the binomial theorem to expand    (1 mark)

  2. Use de Moivre’s theorem and your result from part (i) to prove that
     
          (3 marks)

  3. Hence, or otherwise, find the smallest positive solution of
     
          (2 marks)

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Show Worked Solution

i.   

 

ii. 


 

 

 

iii.   

 

Proof, EXT2 P2 2012 HSC 16b

  1. Show that   for    and  .  (1 mark)

  2. Use mathematical induction to prove 
     
          for all positive integers .  (3 marks)

  3. Find  .  (1 mark)

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Show Worked Solution

i.   

 

 

 

ii. 

 
 

 

 

 
 
 
 
 
 
 
 
 

 

 

iii.  
   
   
   

Polynomials, EXT2 2012 HSC 15b

Let , where is real.

Let  α, where    and    are real.

Suppose that  α  and  α  are zeros of  , where  αα.

  1. Explain why  α  and  α  are zeros of  .   (1 mark)

  2. Show that  .   (1 mark)

  3. Hence show that if    has a real zero then
    1. or     (2 marks)

  4. Show that all zeros of    have modulus .   (2 marks)
  5. Show that  .   (1 mark)
  6. Hence show that  .   (2 marks) 
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Show Worked Solution

(i)  

αααα

αα

α
α
 
 
 
  α

 

αα

 

(ii)  
   
   
   
     

 

(iii)  

♦♦♦ Mean mark part (iii) 11%.

 

 

  

(iv) 

♦♦♦ Mean mark part (iv) 20%.
αααα 
αα
α
α

 

(v)  

♦♦♦ Mean mark part (v) 20%.
αααα
 
 
 

 

 

(vi)  α

♦♦♦ Mean mark part (vi) 2%.

 

 

Proof, EXT2 P1 2012 HSC 15a

  1. Prove that  , where    and  .   (1 mark)

  2. If  ,  show that  .   (2 marks)

  3. Let    and    be positive integers with  .
     
    Prove that   (2 marks)

  4. For integers  , prove that
     
        .   (1 mark)

     
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Show Worked Solution
i.
 
 
 

 

ii. 

♦♦ Mean mark part (ii) 28%.

 

 
 
 

 

 

iii.  

♦ Mean mark part (iii) 39%.

 

 

 

iv. 

♦♦♦ Mean mark part (iv) just 2%!

 

 

Harder Ext1 Topics, EXT2 2012 HSC 14d

The diagram shows points    and    on a circle. The tangents to the circle at    and    meet at the point  . The point    is on the circle inside  . The point    lies on    so that  . The points    and    lie on    and    respectively so that    and  .

Harder Ext1 Topics, EXT2 2012 HSC 14d

Copy or trace the diagram into your writing booklet.

  1. Show that    and    are similar.  (2 marks)
  2. Show that  .  (2 marks)
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(i)   

Harder Ext1 Topics, EXT2 2012 HSC 14d Answer

 

(ii)  

   
   
 
 

Functions, EXT1′ F1 2012 HSC 14b

The diagram shows the graph  . The line    is an asymptote.
 

Graphs, EXT2 2012 HSC 14b

  1. Use the above graph to draw a sketch of the graph   
     
       
     
    indicating all asymptotes and all  - and  -intercepts.  (2 marks)

  2. Write   in the form  .  (1 mark)

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i.    Graphs, EXT2 2012 HSC 14b Answer

 

ii.   
   
   
   

Conics, EXT2 2012 HSC 13c

Let    be a point on the hyperbola given parametrically by    and  , where    and    are positive. The foci of the hyperbola are    and    where    is the eccentricity. The point    is on the -axis so that    bisects .

Conics, EXT2 2012 HSC 13c

  1. Show that  .  (1 mark)
  2. It is given that  
  3. .
  5. Using this, or otherwise, show that the -coordinate of    is
    1. .  (2 marks)

  6. The slope of the tangent to the hyperbola at    is
    1. .   (Do NOT prove this.)
  8. Show that the tangent at    is the line  .  (1 mark)
     
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(i)  

♦ Mean mark 50%.

 

  

 

 

 
 
 
 
 
 
 
 
 

 

(ii)  

 

(iii)  

 
 
 

 

 

 

 

Mechanics, EXT2 M1 2012 HSC 13a

An object on the surface of a liquid is released at time    and immediately sinks. Let    be its displacement in metres in a downward direction from the surface at time    seconds.

The equation of motion is given by

,

where    is the velocity of the object.  

  1. Show that  .   (4 marks)

  2. Use   to show that  
     
            (2 marks)

     

  3. How far does the object sink in the first 4 seconds?   (2 marks)

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i.  
 
 
 
 
   

 

 

ii.  
 
 

 

 

 

iii. 

 
 
 
 

Complex Numbers, EXT2 N2 2012 HSC 12d

On the Argand diagram the points    and    correspond to the distinct complex numbers    and    respectively. Let    be a point corresponding to a third complex number  .

Points    and    are positioned so that    and  , labelled in an anti-clockwise direction, are right-angled and isosceles with right angles at    and  , respectively. The complex numbers    and    correspond to    and  , respectively.
 

Complex Numbers, EXT2 2012 HSC 12d1 
 

  1. Explain why  .  (1 mark)

  2. Find the locus of the midpoint of    as    varies.  (2 marks)

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i.
     

 

ii.  
 

 

Conics, EXT2 2012 HSC 12b

The diagram shows the ellipse    with  . The ellipse has focus    and eccentricity  . The tangent to the ellipse at    meets the -axis at  . The normal at    meets the -axis at  .  

Conics, EXT2 2012 HSC 12b

  1. Show that the tangent to the ellipse at    is given by the equation
    1. .   (2 marks) 
  2. Show that the -coordinate of    is  .   (2 marks)
  3. Show that    (2 marks) 
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(i)   

(ii) 

(iii) 

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(i)   

 

 

 

(ii) 

 

 

(iii)

   

 

 

 
 

Functions, EXT1′ F2 2013 HSC 15b

The polynomial    has remainder when divided by  . The polynomial has a double root at  

  1. Show that    (2 marks)

  2. Hence, or otherwise, find the slope of the tangent to the graph    when    (1 mark)

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i.  
 

 

   

 

 

 

ii. 
 
 

Harder Ext1 Topics, EXT2 2013 HSC 14d

A triangle has vertices and . The point lies on the interval such that and . The point lies on the interval such that , and .

  1. Prove that and are similar.  (1 mark)
  2. Prove that is a cyclic quadrilateral.  (1 mark)
  3. Show that .  (2 marks)
  4. Find the exact value of the radius of the circle passing through the points .  (2 marks)

 

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(i) 

 

 

 

 

(ii)  

 

(iii)  

♦♦ Mean mark 33%.

 
 

 
 

  

(iv)

 
   
   

 

♦♦♦ Mean mark 7%.

COMMENT: This part had the lowest mean mark in the entire 2013 exam.

 
 
 
 

Harder Ext1 Topics, EXT2 2013 HSC 13c

The points and lie on a circle of radius , forming a cyclic quadrilateral. The side is a diameter of the circle. The point is chosen on the diagonal so that . Let

  1. Show that   (2 marks)
  2. By considering , or otherwise, show that   (2 marks)
  3. Hence, show that   (1 mark)
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(i)  

 

 

 

(ii)

♦ Mean mark 49%.

 

 
 
 

 

 

(iii) 

♦ Mean mark 35%.