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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%.
 

 

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) 

 

 

 

 

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
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(i) 

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

  

(ii)  αα

αα
α α
α
α

Conics, EXT2 2010 HSC 3d

The diagram shows the rectangular hyperbola  , with  .

Conics, EXT2 2010 HSC 3d

The points  ,  and    are points on the hyperbola,
with  .

  1. The line    is the line through    perpendicular to  .
  2. Show that the equation of    is
    1. .   (2 marks)

  4. The line    is the line through    perpendicular to  .
  5. Write down the equation of  .   (1 mark)
  6. Let    be the point of intersection of the lines    and  .
  7. Show that    is the point  .   (2 marks)
  8. Give a geometric description of the locus of  .   (1 mark)
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(i) 

Conics, EXT2 2010 HSC 3d Answer1
 
 
 

 

 

 

 

(ii)  
   
   
   

 

 

 

 

(iii) 

 

 

 

(iv) 

♦♦ Mean mark part (iv) 1%.
MARKER’S COMMENT: Most students found the correct locus, although very few indicated that it was only the branch in the 1st quadrant.

 

 

Calculus, EXT2 C1 2011 HSC 7b

Let   

  1. Use the substitution    to show that
     
            (2 marks)

     

  2. Hence, find the value of  .  (3 marks)

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i.   

 
 
 

 

ii.  

 

 

 

♦ Mean mark 36%.

 

 
 
 

Mechanics, EXT2 M1 2011 HSC 6a

Jac jumps out of an aeroplane and falls vertically. His velocity at time    after his parachute is opened is given by  , where    and    is positive in the downwards direction. The magnitude of the resistive force provided by the parachute is  , where    is a positive constant. Let    be Jac’s mass and    the acceleration due to gravity. Jac’s terminal velocity with the parachute open is  

Jac’s equation of motion with the parachute open is

   (Do NOT prove this.)

  1. Explain why Jac’s terminal velocity    is given by  
     
           (1 mark)

  2. By integrating the equation of motion, show that    and    are related by the equation
     
           (3 marks)

  3. Jac’s friend Gil also jumps out of the aeroplane and falls vertically. Jac and Gil have the same mass and identical parachutes.

     

    Jac opens his parachute when his speed is   Gil opens her parachute when her speed is   Jac’s speed increases and Gil’s speed decreases, both towards  

     

    Show that in the time taken for Jac's speed to double, Gil's speed has halved.  (3 marks)

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i.  

  

ii. 

  ­

  

♦ Mean mark part (ii) 39%.

­
 
 
 
 
 

 

 

iii. 

 
 
 

 

 

♦♦♦ Mean mark part (iii) 16%.

 
 
 

 

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 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)   

 

Conics, EXT2 2011 HSC 3d

The equation    represents a hyperbola.

  1. Find the eccentricity    (1 mark)
  2. Find the coordinates of the foci.  (1 mark)
  3. State the equations of the asymptotes.  (1 mark)
  4. Sketch the hyperbola.  (1 mark)
  5. For the general hyperbola  
  6. ,
  8. describe the effect on the hyperbola as    (1 mark)

 

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(i)  

 

 

 

(ii)  

 

(iii) 

 

(iv)   HSC 2011 3di

 

♦♦♦ Mean mark part (v) 24%.

 

(v)  

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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i.   

 

ii. 


 

 

 

iii.   

 

Complex Numbers, EXT2 N2 2011 HSC 2b

On the Argand diagram, the complex numbers    and    form a rhombus.
 


 

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

  2. An interior angle, , of the rhombus is marked on the diagram.

     

    Find the value of   (2 marks)

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i.   
 

 

ii.  


 

 

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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(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%.

 

 

Harder Ext1 Topics, EXT2 2012 HSC 13b

The diagram shows  . The point    is on   so that    bisects  . The point    is on    produced so that  .

Harder Ext1 Topics, EXT2 2012 HSC 13b 

  1. Show that  .   (1 mark)
  2. Show that  .   (2 marks)

 

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(i)   α
  α
  α  

 

 

(ii) 

Harder Ext1 Topics, EXT2 2012 HSC 13b Answer

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. 

 
 
 
 

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) 

Show Worked Solution

(i)   

 

 

 

(ii) 

 

 

(iii)

   

 

 

 
 

Mechanics, EXT2 M1 2013 HSC 15d

A ball of mass is projected vertically into the air from the ground with initial velocity  . After reaching the maximum height it falls back to the ground. While in the air, the ball experiences a resistive force , where is the velocity of the ball and is a constant.

The equation of motion when the ball falls can be written as

      (Do NOT prove this.)

  1. Show that the terminal velocity   of the ball when it falls is
  2.      (1 mark)

  3. Show that when the ball goes up, the maximum height    is
  4.      (3 marks)

  5. When the ball falls from height    it hits the ground with velocity  .
  6. Show that    (2 marks)

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i. 

 

ii. 

♦ Mean mark 43%.

MARKER’S COMMENT: More than half of students incorrectly wrote the equation to solve as
 
 
 

 

 

iii. 

♦♦ Mean mark 14%.
 

 

 
 
 
 

 

Harder Ext1 Topics, EXT2 2013 HSC 15c

Eight cars participate in a competition that lasts for four days. The probability that a car completes a day is . Cars that do not complete a day are eliminated.

  1. Find the probability that a car completes all four days of the competition.  (1 mark)
  2. Find an expression for the probability that at least three cars complete all four days of the competition.  (2 marks)
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(i)   

 

 

(ii) 
 
 
♦♦ Mean mark 26%.

STRATEGY: The use of complementary events reduced the required calculations and subsequent errors in this part.

 

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. 
 
 

Functions, EXT1′ F1 2013 HSC 13b

The diagram shows the graph of a function
 

Sketch the following curves on separate half-page diagrams.

  1.     (2 marks)

  2.     (3 marks)

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i.   


 

ii. 

MARKER’S COMMENT: Correct working sketches such as and meant that students could obtain some marks, even if their final sketch was wrong. Note this important advice.


 

Conics, EXT2 2013 HSC 12d

The points and , where , lie on the rectangular hyperbola with equation  

The tangent to the hyperbola at intersects the -axis at and the -axis at . Similarly, the tangent to the hyperbola at intersects the -axis at and the - axis at .

  1. Show that the equation of the tangent at is   (2 marks)
  2. Show that are on a circle with centre   (2 marks)
  3. Prove that is parallel to   (1 mark)
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(i) 

 

 

 

(ii)   

 

 

 

(iii) 

 
 
 
 

 

Harder Ext1 Topics, EXT2 2014 HSC 16a

The diagram shows two circles and . The point is one of their points of intersection. The tangent to at meets at , and the tangent to at meets at .

The points and are chosen on so that is a diameter of and parallel to . Likewise, points and are chosen on so that is a diameter of and parallel to .

 

The points and lie on the tangents and , respectively, as shown in the diagram.

Harder Ext1 Topics, EXT2 2014 HSC 16a 

 

Copy or trace the diagram into your writing booklet.

  1. Show that .  (2 marks)
  2. Show that , and are collinear.  (3 marks)
  3. Show that is a cyclic quadrilateral. (1 mark)
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(i)

Harder Ext1 Topics, EXT2 2014 HSC 16a Answer

 
  βαγ
 

 

(ii)  

♦♦ Mean mark 31%.
MARKER’S COMMENT: Be vigilant that your reasoning does not implicitly assume or are straight lines.

α

β

γ

 

αβγ
αβγ
αβγ

 

 

♦ Mean mark 44%.

(iii)  

αβ

Conics, EXT2 2014 HSC 13c

The point   is the focus of the hyperbola   on the positive -axis.

The points    and    lie on the asymptotes of the hyperbola, where .

The point   is the midpoint of .

Conics, EXT2 2014 HSC 13c

  1. Show that lies on the hyperbola.  (1 mark)
  2. Prove that the line through and is a tangent to the hyperbola at .  (3 marks)
  3. Show that  .  (2 marks)
  4. If    and    have the same -coordinate, show that    is parallel to one of the asymptotes of the hyperbola.  (2 marks)
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(i)  

 
 
 
 

 

 

(ii)

 
 

 

   

 

 

 

 

(iii)  
   
 
 
 
 

 

 

(iv) 

 
 
 
 

Functions, EXT1′ F1 2014 HSC 12a

The diagram shows the graph of a function  .
 

Graphs, EXT2 2014 HSC 12a
 

Draw a separate half-page graph for each of the following, showing all asymptotes and intercepts.

  1.    (2 marks)

  2.    (2 marks)

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i.  

Graphs, EXT2 2014 HSC 12a Answer1

 

ii.  

Graphs, EXT2 2014 HSC 12a Answer2

Volumes, EXT2 2014 HSC 11e

The region enclosed by the curve    and the -axis is rotated about the -axis to form a solid.

Using the method of cylindrical shells, or otherwise, find the volume of the solid.  (3 marks)

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³

Show Worked Solution

MARKER’S COMMENT: The best responses included a sketch that clearly showed the axis of rotation and an understanding of the use of cylindrical shells, as per the worked solution.

Volumes, EXT2 2014 HSC 11e Answer

δ
 
 

 

 
 
 
  ³

Complex Numbers, EXT2 N1 2014 HSC 11a

Consider the complex numbers    and  .

  1. Express    in modulus–argument form.  (2 marks)

  2. Express in the form  , where    and    are real numbers.  (2 marks)

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i.  
    
 
 
   

 

ii.  
   
   
   
   

Statistics, EXT1 S1 2007 HSC 4a

In a large city, 10% of the population has green eyes.

  1. What is the probability that two randomly chosen people both have green eyes?  (1 mark)

  2. What is the probability that exactly two of a group of 20 randomly chosen people have green eyes? Give your answer correct to three decimal places.  (1 mark)

  3. What is the probability that more than two of a group of 20 randomly chosen people have green eyes? Give your answer correct to two decimal places.  (2 marks)

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i.       
 
   

 

ii.  

 

iii. 

Mechanics, EXT2* M1 2007 HSC 2d

A skydiver jumps from a hot air balloon which is 2000 metres above the ground. The velocity, metres per second, at which she is falling    seconds after jumping is given by  .

  1. Find her acceleration ten seconds after she jumps. Give your answer correct to one decimal place.  (2 marks)

  2. Find the distance that she has fallen in the first ten seconds. Give your answer correct to the nearest metre.  (2 marks)

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i.     
   
 
   
   

 

 
 
 

 

ii. 

Functions, EXT1 F2 2007 HSC 2c

The polynomial    has a zero at  . When    is divided by  , the remainder is .

Find the values of    and  .  (3 marks)

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Linear Functions, EXT1 2007 HSC 1d

The graphs of the line    and the curve    intersect at  . Find the exact value, in radians, of the acute angle between the line and the tangent to the curve at the point of intersection.  (3 marks)

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Mechanics, EXT2* M1 2004 HSC 6b

A fire hose is at ground level on a horizontal plane. Water is projected from the hose. The angle of projection, , is allowed to vary. The speed of the water as it leaves the hose, metres per second, remains constant. You may assume that if the origin is taken to be the point of projection, the path of the water is given by the parametric equations

where    is the acceleration due to gravity.  (Do NOT prove this.)

  1. Show that the water returns to ground level at a distance  metres from the point of projection.   (2 marks)

This fire hose is now aimed at a 20 metre high thin wall from a point of projection at ground level 40 metres from the base of the wall. It is known that when the angle    is 15°, the water just reaches the base of the wall.  
 

Calculus in the Physical World, EXT1 2004 HSC 6b
 

  1. Show that  .  (1 mark)

  2. Show that the cartesian equation of the path of the water is given by
     
         .  (2 marks)

  3. Show that the water just clears the top of the wall if
     
         .  (2 marks)

  4. Find all values of    for which the water hits the front of the wall.  (2 marks)

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i.   
 

 

 

 
 

 

 

ii.   

 

iii. 

 

 
 
 
 

 

iv.   

 

v.   

Calculus in the Physical World, EXT1 2004 HSC 6b Answer

 

 
 

 
 
       
       

 

 

Inverse Functions, EXT1 2004 HSC 5b

The diagram below shows a sketch of the graph of  , where    for  .
 

Inverse Functions, EXT1 2004 HSC 5b
 

  1. Copy or trace this diagram into your writing booklet.
    On the same set of axes, sketch the graph of the inverse function,  .  (1 mark)
  2.  
  3. State the domain of  .  (1 mark)

  4. Find an expression for    in terms of  .  (2 marks)

  5. The graphs of    and    meet at exactly one point  .
  6. Let  α  be the -coordinate of  . Explain why  α  is a root of the equation
     

    1. .  (1 mark)
    2.  
  7. Take 0.5 as a first approximation for  α. Use one application of Newton’s method to find a second approximation for  α.  (2 marks) 

 

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  4.  
Show Worked Solution
(i)   

Inverse Functions, EXT1 2004 HSC 5b Answer

(ii)   

 

(iii) 

 

 

 

 

(iv)   

 

α

 

(v)   
 
 
   
 
   

 

α α
 
 
 
 

Quadratic, EXT1 2004 HSC 4b

The two points    and    are on the parabola  .

  1. The equation of the tangent to    at an arbitrary point    on the parabola is   .  (Do not prove this.)
  3. Show that the tangents at the points    and    meet at  , where    is the point  .  (2 marks)
  4. As    varies, the point    is always chosen so that    is a right angle, where    is the origin.
  6. Find the locus of  .  (2 marks)
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(i)   

Quadratic, EXT1 2004 HSC 4b Answer

 

 

 

 
 

 

 

(ii)  

 
 
 

 

 

Calculus, EXT1 C1 2004 HSC 3c

A ferry wharf consists of a floating pontoon linked to a jetty by a 4 metre long walkway. Let    metres be the difference in height between the top of the pontoon and the top of the jetty and let    metres be the horizontal distance between the pontoon and the jetty.
 

2004 3c
 

  1. Find an expression for    in terms of  .  (1 mark)

When the top of the pontoon is 1 metre lower than the top of the jetty, the tide is rising at a rate of 0.3 metres per hour.

  1. At what rate is the pontoon moving away from the jetty?  (3 marks)

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i.   

 

ii.   

 
 

 

 
 
 
 

 

Functions, EXT1 F2 2004 HSC 3b

Let  , where    is a polynomial and    and    are real numbers.

When    is divided by    the remainder is  .

When    is divided by    the remainder is  .

  1. What is the value of  ?  (1 mark)

  2. What is the remainder when    is divided by  ?  (2 marks)

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i.   

 

 (ii)   

 

 
 
 

Combinatorics, EXT1 A1 2004 HSC 2e

A four-person team is to be chosen at random from nine women and seven men.

  1. In how many ways can this team be chosen?  (1 mark)

  2. What is the probability that the team will consist of four women?  (1 mark)

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i.   


 

 ii.