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Complex Numbers, EXT2 N1 2015 HSC 12a

The complex number is such that  and  

Plot each of the following complex numbers on the same half-page Argand diagram.

  1.     (1 mark)

  2.     (1 mark)

  3.     (1 mark)

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i.   ­
­

 

ii.   ­
­
­

COMMENT: 12a(iii) had the lowest mean mark (55%) of any part within Q12 and deserves attention.
iii.  ­
­
­

Complex Numbers, EXT2 N1 2015 HSC 11b

Consider the complex numbers    and  

  1. Evaluate     (1 mark)

  2. Evaluate     (1 mark)

  3. Find the argument of     (1 mark)

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i.   
 

 

ii.   ­
­
­

 

iii.   ­
­
­

Proof, EXT2 P1 2006 HSC 8a

Suppose  

  1. Show that    (2 marks)

  2. Hence show that    (1 mark)

  3. By integrating the expressions in the inequality in part (ii) with respect to    from    to  , show that
     
          (2 marks)

  4. Hence show that for  
     
          (1 mark)

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i.   
 
 
 
 
 
 

 

ii.   

 

 

iii.  
 
 
 

 

iv.    

 

Harder Ext1 Topics, EXT2 2006 HSC 7a

The curves    and    intersect at a point    whose -coordinate is  

  1. Show that the curves intersect at right angles at    (3 marks)
  2. Show that
  3.   (2 marks)

 

 

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(i)   

 

 
 
 

 

 

(ii)    α α
  α α
    α
  αα
  αα

 

α
 

Mechanics, EXT2 M1 2006 HSC 6b

In an alien universe, the gravitational attraction between two bodies is proportional to  , where    is the distance between their centres.

A particle is projected upward from the surface of a planet with velocity    at time  .  Its distance    from the centre of the planet satisfies the equation

  1. Show that  , where    is the magnitude of the acceleration due to gravity at the surface of the planet and    is the radius of the planet.  (1 mark)

  2. Show that  , the velocity of the particle, is given by
     
           (3 marks)

  3. It can be shown that   (Do NOT prove this.)

     

    Show that if    the particle will not return to the planet.  (2 marks)

  4. If    the particle reaches a point whose distance from the centre of the planet is  , and then falls back.

  5.  

      (1)   Use the formula in part (ii) to find    in terms of    and    (1 mark)

  6.  

      (2)   Use the formula in part (iii) to find the time taken for the particle to return to the surface of the planet in terms of    and    (1 mark)

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  4.  

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i.   

­
­

 

ii.   
 
   
 

 

 

 

iii. 

 
 
 
 

 

 

 

 

 

iv. (1)  

­
­
­

 

iv. (2)  

 

Harder Ext1 Topics, EXT2 2006 HSC 6a

In    and  . The point    is chosen inside    so that  , as shown in the diagram.

  1. Show that
    1.   (1 mark)

  2. Hence show that
    1.   (2 marks)

  3. Prove the identity
    1.   (1 mark)

  4. Hence show that
  5.   (1 mark)

  6. Hence find the value of    when    is an isosceles right triangle.  (2 marks)
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(i)  

 

(ii)
   

 

 

 

(iii)  ­
­
­
­
­

 

(iv)  

♦♦ Mean mark sub 30% (exact % not available).
MARKER’S COMMENT: Very few students correctly calculated the values of

 

(v)  

 

Harder Ext1 Topics, EXT2 2006 HSC 5d

In a chess match between the Home team and the Away team, a game is played on each of board 1, board 2, board 3 and board 4.

On each board, the probability that the Home team wins is  , the probability of a draw is    and the probability that the Home team loses is  .

The results are recorded by listing the outcomes of the games for the Home team in board order. For example, if the Home team wins on board 1, draws on board 2, loses on board 3  and draws on board 4, the result is recorded as  .

  1. How many different recordings are possible?  (1 mark)
  2. Calculate the probability of the result which is recorded as    (1 mark)
  3. Teams score 1 point for each game won, ½ a point for each game drawn and 0 points for each game lost.
  4. What is the probability that the Home team scores more points than the Away team?  (3 marks)

 

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(i)   

 

(ii)   
   

 

(iii) 

 

STRATEGY: Recognising symmetry in probability outcomes can often provide a very efficient solution involving less calculation as shown in part (iii) here.

 

 
 
   

 

 

Mechanics, EXT2 2006 HSC 5c

A particle,  , of mass    is attached by two strings, each of length  , to two fixed points,    and  , which lie on a vertical line as shown in the diagram.

The system revolves with constant angular velocity    about  . The string    makes an angle    with the vertical. The tension in the string    is    and the tension in the string    is    where    and  . The particle is also subject to a downward force,  , due to gravity.

  1. Resolve the forces on    in the horizontal and vertical directions.  (2 marks)

  2. If  , find the value of    in terms of    and    (1 mark)
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(i)   

 

(ii)  

 

 

Harder Ext1 Topics, EXT2 2006 HSC 5b

  1. Show that     (1 mark)
  2. Hence, or otherwise, solve the equation
  3. for    (3 marks)

 

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(i)  
 
 

 

STRATEGY: Students who applied the rule in part (i) 3 times solved this problem in the most efficient way. Many however, only applied it twice and lost valuable time.

(ii)   

 

 

Volumes, EXT2 2006 HSC 5a

A solid is formed by rotating the region bounded by the curve    and the line    about the -axis. Use the method of cylindrical shells to find the volume of this solid.   (3 marks)

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³

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  ³

Harder Ext1 Topics, EXT2 2006 HSC 4d

In the acute-angled triangle    is the midpoint of    is the midpoint of    and    is the midpoint of  . The circle through    and    also cuts    at    as shown in the diagram.

Copy or trace the diagram into your writing booklet.

  1. Prove that    is a parallelogram.  (1 mark)
  2. Prove that    (1 mark)
  3. Prove that    (2 marks)
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(i)  


   

 

(ii)  

 

(iii)  
 

 

 

 

Conics, EXT2 2006 HSC 4c

Let    and    be three distinct points on the hyperbola  

  1. Show that the equation of the line,  , through  , perpendicular to  , is    (2 marks)
  2. Write down the equation of the line,  , through  , perpendicular to    (1 mark)
  3. The lines    and    intersect at  
  4. Show that    lies on the hyperbola.  (2 marks)

 

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(i)  
­
­

 

  

(ii) 

 

(iii) 

MARKER’S COMMENT: Many students had difficulty simplifying   Be clear on how to do this. 

 

 

 
 

Volumes, EXT2 2006 HSC 4b

The base of a solid is the parabolic region   shaded in the diagram.

Vertical cross-sections of the solid perpendicular to the -axis are squares.

Find the volume of the solid.  (3 marks)

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³

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MARKER’S COMMENT: A substantial number of students incorrectly treated this part as a solid of revolution question. Make sure you understand why it is not!

 

δ
 
 
  ³

Polynomials, EXT2 2006 HSC 3c

Two of the zeros of    are    and  , where    and    are real and  

  1. Find the values of    and  .  (3 marks)
  2. Hence, or otherwise, express    as the product of quadratic factors with real coefficients.  (1 mark)
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(i)   

STRATEGY: The observation that complex zeros occur in conjugate pairs for any with real coefficients was critical to making any progress here and is a regularly examined concept.

 

 

 

(ii) 

 
 

Conics, EXT2 2006 HSC 3b

The diagram shows the graph of the hyperbola

  1. Find the coordinates of the points where the hyperbola intersects the -axis.  (1 mark)
  2. Find the coordinates of the foci of the hyperbola.  (2 marks)
  3. Find the equations of the directrices and the asymptotes of the hyperbola.  (2 marks)
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(i)  

 

 

(ii) 

 

 

(iii)  
   
 
   

Functions, EXT1′ F1 2006 HSC 3a

The diagram shows the graph of  . The graph has a horizontal asymptote at  .
 


 

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.

ii. 

iii. 

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i.   

 

ii.   

 

iii.   

Conics, EXT2 2006 HSC 2d

The equation    corresponds to an ellipse in the Argand diagram.

  1. Write down the complex number corresponding to the centre of the ellipse.  (1 mark)

  2. Sketch the ellipse, and state the lengths of the major and minor axes.  (3 marks)

  3. Write down the range of values of  arg  for complex numbers    corresponding to points on the ellipse.  (1 mark)

 

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(i)  

♦♦ A “significant portion” of students could not do this question at all (exact data not available).

 
 

  

(ii)  

 

 

(iii) 

Conics, EXT2 2007 HSC 7c

The diagram shows an ellipse with eccentricity    and foci    and  .

The tangent at    on the ellipse meets the directrices at    and  . The perpendicular to the directrices through    meets the directrices at    and    as shown. Both    and    are right angles.

Let  

  1. Show that
  2. where    is the eccentricity of the ellipse.   (2 marks)

  3. By also considering    show that    (2 marks)

 

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(i)   
 

 

 

(ii) 

 

Conics, EXT2 2007 HSC 7b

In the diagram the secant    of the ellipse    meets the directrix at  . Perpendiculars from    and    to the directrix meet the directrix at    and    respectively. The focus of the ellipse which is nearer to    is at  .

Copy or trace this diagram into your writing booklet.

  1. Prove that
    1.   (1 mark)

  2. Prove that
    1.   (1 mark)

  3. Let  
  4. By considering the sine rule in  , and applying the results of part (i) and part (ii),
  5. show that    (2 marks)

  6. Let    approach    along the circumference of the ellipse, so that  
  7. What is the limiting value of    (1 mark)

 

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(i)  

 

 

(ii) 

 

 

(iii) 

 

 

 

 

(iv)    
 
 

Proof, EXT2 P1 2007 HSC 7a

  1. Show that    for     (2 marks)

  2. Let  .

     

    Show that the graph of    is concave up for     (2 marks)

  3. By considering the first two derivatives of  ,show that    for     (2 marks)

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i.   
 
MARKER’S COMMENT: A geometric proof using arc length and a right angled triangle caused problems as few students could deal with the case  .

 

 

 

ii.
 
 
 

 

 

iii. 

 


 

 

Mechanics, EXT2 M1 2007 HSC 6b

A raindrop falls vertically from a high cloud. The distance it has fallen is given by

where is in metres and is the time elapsed in seconds.

  1. Show that the velocity of the raindrop, metres per second, is given by
     
           (2 marks)

  2. Hence show that    (2 marks)

  3. Hence, or otherwise, show that    (2 marks)

  4. The physical significance of the 9.8 in part (iii) is that it represents the acceleration due to gravity.

     

    What is the physical significance of the term    (1 mark)

  5. Estimate the velocity at which the raindrop hits the ground.   (1 mark)

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i.   
 
   

 

ii.   
   
   
   
   

 

 

 

 

iii.  
   
   
   
   

 
iv. 
 

v.   

 

Proof, EXT2 P2 2007 HSC 6a

  1. Use the binomial theorem 

     

    to show that, for  ,
     
          (1 mark)

  2. Hence show that, for  ,
         
          (2 marks)

  3. Prove by induction that, for integers  ,

     

          (3 marks)

  4. Hence determine the limiting sum of the series

     

          (1 mark)

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i.  

 

ii.   
 
 
 
 
 

 

iii. 


 


 

 
 
 
 

 

 

iv. 

 

 

Polynomials, EXT2 2007 HSC 5d

In the diagram,    is a regular pentagon with sides of length . The perpendicular to    through    meets    at  

Copy or trace this diagram into your writing booklet.

  1. Let  .
  2. Use the cosine rule in    to show that     (2 marks)

  3. One root of    is  .
  4. Find the other roots of    and hence find the exact value of    (2 marks)

 

 

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(i)   
 
 

 

 

(ii) 

 
 

 

Calculus, EXT2 C1 2007 HSC 5c

  1. Write    in the form  , where    and  are real numbers.   (1 mark)

  2. Using the values of    and    found in part (i) and making the substitution  , or otherwise, evaluate  
     
           (2 marks)

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i.   
 
 
 

 

ii.   
 

 

Conics, EXT2 2007 HSC 5b

The points at    and    lie on the same branch of the hyperbola

The tangents at    and    meet at  

 

  1. Show that the equation of the tangent at    is
    1.   (2 marks)

  2. Hence show that the chord of contact,  , has equation
    1.   (2 marks)

  3. The chord    passes through the focus  , where    is the eccentricity of the hyperbola. Prove that    lies on the directrix of the hyperbola.   (1 mark)

 

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(i)   
 
 
 

 

 

(ii)  

 

(iii) 

Combinatorics, EXT1′ A1 2007 HSC 5a

A bag contains 12 red marbles and 12 yellow marbles. Six marbles are selected at random without replacement.

  1. Calculate the probability that exactly three of the selected marbles are red. Give your answer correct to two decimal places.   (1 mark)

  2. Hence, or otherwise, calculate the probability that more than three of the selected marbles are red. Give your answer correct to two decimal places.   (2 marks)

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i.   

 
 

 

ii.   

 
 
 

 

 
 
 

Polynomials, EXT2 2007 HSC 4d

The polynomial    has real coefficients. It has three distinct zeros,  

  1. Prove that     (3 marks)
  2. The polynomial does not have three real zeros. Show that two of the zeros are purely imaginary. (A number is purely imaginary if it is of the form  , with    real and  )   (2 marks)

 

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(i)   

 

 

(ii)  

α

α

Volumes, EXT2 2007 HSC 4c

The base of a solid is the region bounded by the curve  , the -axis and the lines    and  , as shown in the diagram.

Vertical cross-sections taken through this solid in a direction parallel to the -axis are squares. A typical cross-section,  , is shown.

Find the volume of the solid.  (3 marks)

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³

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δ δ
 
 
 
 
 
  ³

Volumes, EXT2 2012 HSC 9 MC

The diagram shows the graph    for  . The region bounded by the graph and the -axis is rotated about the line    to form a solid.

Volumes, EXT2 2012 HSC 9 MC

Which integral represents the volume of the solid?

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δ δ
  δ

Harder Ext1 Topics, EXT2 2007 HSC 4a

Two circles intersect at    and  .

The lines    and    pass through  , with    and    on one circle and    and    on the other circle, as shown in the diagram.

Copy or trace this diagram into your writing booklet.

Show that    (2 marks)

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(i)  

 

Mechanics, EXT2 2007 HSC 3d

A particle    of mass    undergoes uniform circular motion with angular velocity    in a horizontal circle of radius    about  . It is acted on by the force due to gravity,  , a force    directed at an angle    above the horizontal and a force    which is perpendicular to  , as shown in the diagram.

  1. By resolving forces horizontally and vertically, show that
    1.   (3 marks)

  2. For what values of    is     (1 mark)
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(i)      

 

 

 

 

(ii) 

 

Volumes, EXT2 2007 HSC 3c

Use the method of cylindrical shells to find the volume of the solid formed when the shaded region bounded by

is rotated about the -axis.   (4 marks)

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³

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(i)  

MARKER’S COMMENT: A “significant number” of students were confused about the radius of the cylindrical shell.
δ δ
 

 
 
 
  ³

Functions, EXT1′ F1 2007 HSC 3a

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

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

  1.     (1 mark)

  2.     (2 marks)

  3.       (2 marks)

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  1.  

     

  2.  
  3.  
Show Worked Solution
i.  
MARKER’S COMMENT: In part (ii), a significant number of students graphed  .
ii.

 

iii. 

Complex Numbers, EXT2 N2 2007 HSC 2d

The points    and    on the Argand diagram represent the complex numbers    and    respectively.

The triangles    and    are equilateral with unit sides, so  

Let  ­­

  1. Explain why     (1 mark)

  2. Show that     (1 mark)

  3. Show that   and  are the roots of     (2 marks)

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i. 

 

ii. 

 

 

  ­­­­
  ­­
 

 

iii. 

 

  ­­­­
  ­
 
 

 

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)