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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. 

 


 

 

Harder Ext1 Topics, EXT2 2013 HSC 10 MC

A hostel has four vacant rooms. Each room can accommodate a maximum of four people.

In how many different ways can six people be accommodated in the four rooms?

  1. 4020
  2. 4068
  3. 4080
  4. 4096
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♦♦♦ Mean mark 15%.
STRATEGY: The group of 5 can be in any of the four rooms, and note that the remaining person can then be in any of the 3 remaining rooms.

 

Harder Ext1 Topics, EXT2 2009 HSC 8c

A game is being played by    people, , sitting around a table. Each person has a card with their own name on it and all the cards are placed in a box in the middle of the table. Each person in turn, starting with  , draws a card at random from the box. If the person draws their own card, they win the game and the game ends. Otherwise, the card is returned to the box and the next person draws a card at random. The game continues until someone wins.

Let    be the probability that    wins the game.

Let  .

  1. Show that    (1 mark)
  2. Let    be a fixed positive integer and let    be the probability that    wins in no more than    attempts.
  3. Use  
  4. to show that, if    is large,  is approximately equal to    (3 marks)

 

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(i)   

 

 

(ii)  
 
 
 

 

 

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. 

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.
 

 

 

 

 
 

 

Integration, EXT2 2010 HSC 8

Let

 and  ,

where    is an integer,  .  (Note that , .)

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

  3. Using integration by parts on  , or otherwise, show that
    1.  for  .   (1 mark)

  4. Use integration by parts on the integral in part (ii) to show that
    1.  for  .   (3 marks)

  5. Use parts (i) and (iii) to show that
    1.  for  .   (1 mark)

  6. Show that
    1. .   (2 marks)

  7. Use the fact that
  8.  for    to show that

    1. .   (1 mark)

  9. Show that
  10. .   (1 mark)

  12. From parts (vi) and (vii) it follows that
    1. .

  13. Use the substitution    in this inequality to show that
  14.  
    1. .   (2 marks)

  15. Use part (v) to deduce that
    1. .   (1 mark)

  16. What is
  17. ?   (1 mark)

 

 

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♦♦ Mean mark part (i) 21%.
(i)   
 
 
 
 
 
 
 
   

 

 
 
 

 

♦♦ Mean mark part (ii) 32%.

 

(ii)   
 
 
 
 
 
 
♦♦♦ Mean mark part (iii) 8%.

 

(iii)   
 
   
   
   
 
 

 

 
 
 
 
 
♦♦♦ Mean mark part (iv) 13%.

 

(iv)  
 
   
   
   
♦♦♦ Mean mark part (v) 9%.

 

(v)  
 
   

 
♦♦♦ Mean mark part (vi) 8%.

 

(vi) 
 

 

 

(vii)  
 
 

 

♦♦♦ Mean mark part (vii) 2%.

 

♦♦♦ Mean mark part (viii) 4%.
(viii)  
 
 

 

 
 
 

 

 
 

 

♦♦♦ Mean mark part (ix) 3%.

(ix) 

,

 

 

♦♦♦ Mean mark part (x) 19%.

 

(x) 

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

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%.

 

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

Harder Ext1 Topics, EXT2 2010 HSC 4c

Let    be a real number, .

Show that, for every positive real number  , there is a positive real number    such that  .   (3 marks) 

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♦♦♦ Mean mark 13%.

 

 

 

 

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.

 

 

Harder Ext1 Topics, EXT2 2010 HSC 3c

Two identical biased coins are each more likely to land showing heads than showing tails.

The two coins are tossed together, and the outcome is recorded. After a large number of trials it is observed that the probability that the two coins land showing a head and a tail is  .

What is the probability that both coins land showing heads?   (2 marks)

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Show Worked Solution

♦♦♦ Mean mark 19%.

 

Polynomials, EXT2 2011 HSC 8c

Let be a root of the complex monic polynomial

Let    be the maximum value of  

  1. Show that
  2.   (2 marks)

  4. Hence, show that for any root    of  
    1.   (3 marks)

  6. Let  , where the real numbers    satisfy  for all  , and  
  8. Using parts (i) and (ii), or otherwise, show that    has no real solutions.  (3 marks)
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(i)  

♦♦♦ Mean mark part (i) 7%.

 
 

 

(ii) 

♦♦♦ Mean mark part (ii) 3%.

 

 

 

 

(iii)   
 

 

♦♦♦ Mean mark part (iii) 2%.

 

 

 

 
 

 

 

HSC 2011 8d

 

 

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%.

 

 
 

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 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%.

 
 
 

 

Conics, EXT2 2011 HSC 5c

The diagram shows the ellipse  , where  . The line    is the tangent to the ellipse at the point  . The foci of the ellipse are    and  . The perpendicular to    through    meets    at the point  . The lines    and   meet at the point  .

Copy or trace the diagram into your writing booklet.

  1. Use the reflection property of the ellipse at    to prove that    (2 marks)
  2. Explain why    (1 mark)
  3. Hence, or otherwise, prove that    lies on the circle    (3 marks)
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(i)  

♦♦♦ Mean mark 21%.

 

 

 

♦ Mean mark part (ii) 46%.

(ii) 

 

 

(iii)   

♦♦♦ Mean mark 5%.
   
 
 

 

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)  
 

 

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)  

Proof, EXT2 P1 2012 HSC 16c

Let    be an integer where  . Integers from    to    inclusive are selected randomly one by one with repetition being possible. Let    be the probability that exactly    different integers are selected before one of them is selected for the second time, where  .

  1. Explain why  .   (2 marks)

  2. Suppose  . Show that  .   (2 marks)

  3. Show that if   then the integers    and    satisfy  .   (2 marks)

  4. Hence show that if    is not a perfect square, then    is greatest when    is the closest integer to  . 

     

    You may use part (iii) and also that    if  .   (2 marks)

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i.   

♦♦♦ Mean mark part (i) 7%.

 

 
 

♦ Mean mark part (ii) 38%.

 

ii.
 
 
 
 

 

♦♦♦ Mean mark part (iii) 10%.
iii.  
 
 
 
 

 

 

 

iv. 

 

♦♦♦ Mean mark part (iii) 1%.

 

Harder Ext1 Topics, EXT2 2012 HSC 16a

  1. In how many ways can    identical yellow discs and    identical black discs be arranged in a row?  (1 mark)

  2. In how many ways can 10 identical coins be allocated to 4 different boxes?  (1 mark)
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(i)  

♦ Mean mark part (i) 43%.

 

(ii)  

♦♦♦ Mean mark part (ii) 1%!

 

Harder Ext1 Topics, EXT2 2012 HSC 16a Answer1

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%.

 

 

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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i.
 
 
 

 

ii. 

♦♦ Mean mark part (ii) 28%.

 

 
 
 

 

 

iii.  

♦ Mean mark part (iii) 39%.

 

 

 

iv. 

♦♦♦ Mean mark part (iv) just 2%!

 

 

Volumes, EXT2 2012 HSC 14c

The solid    is cut from a quarter cylinder of radius    as shown. Its base is an isosceles triangle    with  . The length of    is    and the midpoint of    is  .

The cross-sections perpendicular to    are rectangles. A typical cross-section is shown shaded in the diagram.

Volumes, EXT2 2012 HSC 14c

Find the volume of the solid  .  (4 marks)

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³

Show Worked Solution

Volumes, EXT2 2012 HSC 14c Answer1

 

Volumes, EXT2 2012 HSC 14c Answer2

 

 

δδ
 
   

 
 
  ³

Mechanics, EXT2 2013 HSC 16b

A small bead    of mass    can freely move along a string. The ends of the string are attached to fixed points    and  , where    lies vertically above  . The bead undergoes uniform circular motion with radius    and constant angular velocity    in a horizontal plane.

The forces acting on the bead are the gravitational force and the tension forces along the string. The tension forces along    and    have the same magnitude  .

The length of the string is    and  , where  . The horizontal plane through    meets   at  . The midpoint of    is    and  . The parameter    is chosen so that  

 

  1. What information indicates that    lies on an ellipse with foci    and  , and with eccentricity  ?  (1 mark)

  2. Using the focus–directrix definition of an ellipse, or otherwise, show that
  3.   (1 mark)
  5. Show that
  6.   (2 marks)
  8. By considering the forces acting on    in the vertical direction, show that
    1.   (2 marks)
  9. Show that the force acting on    in the horizontal direction is
    1.   (3 marks)
  10. Show that
    1.   (1 mark)

 

 

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Show Worked Solution

(i)   

♦♦ Mean mark 22%.


 

♦ Mean mark 40%.
(ii)  
   
   
   
   

 

♦ Mean mark 39%.

(iii)  
 
 
 

 

♦♦ Mean mark 43%.

(iv)

 
 

 

 

(v)  

♦♦♦ Mean mark 8%.


 
 
 
 

 

 
 
 
 
 

 

 
 

 

♦ Mean mark 38%.
(vi)  
 

Proof, EXT2 P1 2013 HSC 16a

  1. Find the minimum value of  , for    (2 marks)

  2. Hence, or otherwise, show that for  ,
     
          (1 mark)

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

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Show Worked Solution
♦ Mean mark 43%.

STRATEGY: The best responses carefully considered the domain restrictions, notably
i.
 
   
   
 

 

 

 

♦ Mean mark 47%.

ii. 
 
 
 
 
 

 

 

♦♦ Mean mark 19%.

STRATEGY: The best responses carefully considered the domain restrictions, notably
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 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.

 
 
 
 

Proof, EXT2 P1 2014 HSC 16b

Suppose is a positive integer. 

  1. Show that 
     
        .  (3 marks)

  2. Use integration to deduce that
     
    .  (2 marks)

  3. Explain why   .  (1 mark)

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Show Worked Solution

i.   

♦♦ Mean mark 12%.
STRATEGY: Applying the GP formula and simplifying the middle term is worth 2 full marks in this question.
 

 

 

 

 

ii.   

♦ Mean mark 43%.
 
 
 
 

 

 

iii. 

♦♦ Mean mark 28%.

Mechanics, EXT2 2014 HSC 15c

A toy aeroplane of mass is attached to a fixed point by a string of length . The string makes an angle  ø  with the horizontal. The aeroplane moves in uniform circular motion with velocity in a circle of radius in a horizontal plane.

Mechanics, EXT2 2014 HSC 15c

The forces acting on the aeroplane are the gravitational force  , the tension force    in the string and a vertical lifting force  , where is a positive constant.

  1. By resolving the forces on the aeroplane in the horizontal and the vertical directions, show that
    øø.  (3 marks)

  3. Part (i) implies that
    øø.   (Do NOT prove this.)
  5. Use this to show that
    1. ø  (2 marks)

  6. Show that
    øø is an increasing function of  ø  for  ø.  (2 marks)

  8. Explain why ø increases as increases.  (1 mark)
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Show Worked Solution

(i)    

 ø

ø
  øø
ø

 

øø
øø
øø
øø

 

♦♦♦ Mean mark 14%.
STRATEGY: The form of the required proof strongly hints that the quadratic formula is relevant – a clue that was ignored by many students. 
(ii)   øø
   ø ø
  ø ø
    øø 

 

ø

øø

ø

 

øø 

ø

ø

 

(iii)   øø
  øøøøøø
    øøø

 

♦♦ Mean mark 25%.

ø

øøø

 

øøø

 

(iv)  øø

♦♦♦ Mean mark 8%.
COMMENT: This question had the lowest mean mark in the entire 2014 exam.
 
  øø

 

øøø

ø

Complex Numbers, EXT2 N2 2014 HSC 15b

  1. Using de Moivre’s theorem, or otherwise, show that for every positive integer ,
     
        .  (2 marks)

  2. Hence, or otherwise, show that for every positive integer divisible by 4,
     
         (3 marks)

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Show Worked Solution
i.
    
 
     
 
 
   
 

 

 

ii.   

 

♦♦ Mean mark 22%.
MARKER’S COMMENT: Few students could link part (i) to solving part (ii).


 

 


 

Mechanics, EXT2* M1 2004 HSC 7a

The rise and fall of the tide is assumed to be simple harmonic, with the time between successive high tides being 12.5 hours. A ship is to sail from a wharf to the harbour entrance and then out to sea. On the morning the ship is to sail, high tide at the wharf occurs at 2 am. The water depths at the wharf at high tide and low tide are 10 metres and 4 metres respectively.

  1. Show that the water depth, metres, at the wharf is given by
     
         , where    is the number of hours after high tide.  (2 marks)  

  2. An overhead power cable obstructs the ship’s exit from the wharf. The ship can only leave if the water depth at the wharf is 8.5 metres or less.

     

    Show that the earliest possible time that the ship can leave the wharf is 4:05 am.  (2 marks)

  3. At the harbour entrance, the difference between the water level at high tide and low tide is also 6 metres. However, tides at the harbour entrance occur 1 hour earlier than at the wharf. In order for the ship to be able to sail through the shallow harbour entrance, the water level must be at least 2 metres above the low tide level.

     

    The ship takes 20 minutes to sail from the wharf to the harbour entrance and it must be out to sea by 7 am. What is the latest time the ship can leave the wharf?  (2 marks)

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i.   

 

 

 

 

 

  

ii.  

 Calculus in the Physical, EXT1 2004 HSC 7a Answer

 
 

 

 

iii. 

 
 

 

 

Mechanics, EXT2* M1 2007 HSC 7b

A small paintball is fired from the origin with initial velocity metres per second towards an eight-metre high barrier. The origin is at ground level, metres from the base of the barrier.

The equations of motion are

where    is the angle to the horizontal at which the paintball is fired and    is the time in seconds. (Do NOT prove these equations of motion)
 

  1. Show that the equation of trajectory of the paintball is
     
         , where  .  (2 mark)

  2. Show that the paintball hits the barrier at height    metres when
     
         .

     

    Hence determine the maximum value of  .  (2 marks)

  3. There is a large hole in the barrier. The bottom of the hole is metres above the ground and the top of the hole is metres above the ground. The paintball passes through the hole if    is in one of two intervals. One interval is  .

     

    Find the other interval.  (2 marks)

  4. Show that, if the paintball passes through the hole, the range is 
     
         
     
    Hence find the widths of the two intervals in which the paintball can land at ground level on the other side of the barrier.  (3 marks)

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i.   
 

 

 
 
 
 

 

ii. 

 

 
 
 
 

 

 

 

iii.   EXT1 2007 7bi

 
 
 

 

 
 
 

 

 

iv. 

 

 

 

 

 

L&E, EXT1 2007 HSC 7a

2007 7a

The graphs of the functions    and    have a common tangent at  , as shown in the diagram.

  1. By considering gradients, show that 
    1. . (1 mark)
  2.  
  3. Express    as a function of    by eliminating .   (2 marks)

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Show Worked Solution
(i)   
 

 

 

 

 

 

 

(ii) 

 

♦♦♦ “Vast majority” could not eliminate in part (ii).

 

Functions, EXT1 F1 2007 HSC 6b

Consider the function  .

  1. Show that    is increasing for all values of .  (1 mark)

  2. Show that the inverse function is given by
     
     (3 marks)

  3. Hence, or otherwise, solve  . Give your answer correct to two decimal places.  (1 mark)

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i.   
 

 

 
 

 

 

ii. 

 

 

 

 

  

iii.   
 
 

 

 
 
 

Trig Calculus, EXT1 2006 HSC 7

A gutter is to be formed by bending a long rectangular metal strip of width so that the cross-section is an arc of a circle.

Let be the radius of the arc and the angle at the centre, , so that the cross-sectional area, , of the gutter is the area of the shaded region in the diagram on the right.

  1. Show that, when  , the cross-sectional area is
    2.  (2 marks)

  2. The formula in part (i) for    is true for        (Do NOT prove this.)
  3. By first expressing    in terms of    and  , and then differentiating, show that
  4. for    (3 marks)

  5. Let  
  6. By considering  , show that    for     (3 marks)

  8. Show that there is exactly one value of    in the interval   for which
    1.   (2 marks)

  9. Show that the value of    for which    gives the maximum cross-sectional area. Find this area in terms of   (2 marks)

 

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  5. ²
Show Worked Solution

(i)  

 
 
 

 

 

(ii)  

 

 

 

 

 
 
 
 
 
 
 

 

(iii)    
 
   
   

 

 

 

(iv)  

 

 

 

(v)  
 

 

 

 
 
 
  ²

Mechanics, EXT2* M1 2006 HSC 6a

Two particles are fired simultaneously from the ground at time  

Particle 1 is projected from the origin at an angle  , with an initial velocity  

Particle 2 is projected vertically upward from the point  , at a distance    to the right of the origin, also with an initial velocity of  
 


 

It can be shown that while both particles are in flight, Particle 1 has equations of motion:

and Particle has equations of motion:

   Do NOT prove these equations of motion.

Let    be the distance between the particles at time  

  1. Show that, while both particles are in flight,
     
           (2 marks)

  2. An observer notices that the distance between the particles in flight first decreases, then increases.

     

    Show that the distance between the particles in flight is smallest when
     
           and that this smallest distance is    (3 marks)

  3. Show that the smallest distance between the two particles in flight occurs while Particle 1 is ascending if  
     
           (1 mark)

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i.   

 

 
 
 
 
 

 

ii.  

 
 

 

 

 
 
 
 
 
 
 
 
 

 

iii.   

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

 

 
 

 
 
       
       

 

 

Mechanics, EXT2* M1 2006 HSC 4b

A particle is undergoing simple harmonic motion on the -axis about the origin. It is initially at its extreme positive position. The amplitude of the motion is 18 and the particle returns to its initial position every 5 seconds.

  1. Write down an equation for the position of the particle at time    seconds.  (2 marks)

  2. How long does the particle take to move from a rest position to the point halfway between that rest position and the equilibrium position?  (3 marks)

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i.   

♦♦♦ “Very poorly” answered (exact data unavailable).

 

 

ii.   

 

 

Plane Geometry, EXT1 2006 HSC 3d

The points    and    lie on a circle. The line    is tangent to the circle at    with    chosen so that    is perpendicular to  . The point    on    is chosen so that    is perpendicular to    as shown in the diagram.

  1. Show that    is a cyclic quadrilateral.  (1 mark)
  2. Show that    (1 mark)
  3. Hence, or otherwise, show that    is parallel to    (2 marks)
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(i)   

 

 

(ii) 

 

 

 

 

(iii)  

 

 

Mechanics, EXT2* M1 2005 6b

An experimental rocket is at a height of  5000 m, ascending with a velocity of    at an angle of  45°  to the horizontal, when its engine stops.
 

 
After this time, the equations of motion of the rocket are:

where is measured in seconds after the engine stops. (Do NOT show this.)

  1. What is the maximum height the rocket will reach, and when will it reach this height?  (2 marks)

  2. The pilot can only operate the ejection seat when the rocket is descending at an angle between 45° and 60° to the horizontal. What are the earliest and latest times that the pilot can operate the ejection seat?  (3 marks)

  3. For the parachute to open safely, the pilot must eject when the speed of the rocket is no more than  . What is the latest time at which the pilot can eject safely?  (2 marks)

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  1.  

Show Worked Solution

i.

    

 

 

 
 

 

 

ii. 

 

 

 

 
 
 

 

 

iii.

 
 

 

Polynomials, EXT1 2005 HSC 3a

  1. Show that the function    has a zero between    and    (1 mark)
  2. Use the method of halving the interval to find an approximation to this zero of  , correct to one decimal place.  (2 marks)

 

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(i)   
 
 
 
 

 

 

(ii) 

 
 

 

♦♦♦ Part (ii) was “very poorly done”.

 

 

 

Calculus, 2ADV C3 2007 HSC 10b

The noise level, , at a distance metres from a single sound source of loudness is given by the formula

Two sound sources, of loudness and are placed metres apart.
 

The point lies on the line between the sound sources and is metres from the sound source with loudness

  1. Write down a formula for the sum of the noise levels at in terms of .  (1 mark)

  2. There is a point on the line between the sound sources where the sum of the noise levels is a minimum.

     

    Find an expression for in terms of , and if is chosen to be this point.  (4 marks)

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i.  

 

ii. 

 

 

 

 

 

Calculus in the Physical World, 2UA 2007 HSC 10a

An object is moving on the -axis. The graph shows the velocity, , of the object, as a function of time, . The coordinates of the points shown on the graph are  . The velocity is constant for  .
 


 

  1. Using Simpson’s rule, estimate the distance travelled between    and  .  (2 marks)
  2. The object is initially at the origin. During which time(s) is the displacement of the object decreasing?  (1 mark)
  3. Estimate the time at which the object returns to the origin. Justify your answer.  (2 marks)
  4. Sketch the displacement, , as a function of time.  (2 marks)
Show Answers Only

 

 

 

 

 

 

 

 

Show Worked Solution

(i)

 

(ii) 

 

(iii) 

 

 
 

 

(iv)

 2UA HSC 2007 10ai

Financial Maths, 2ADV M1 2007 HSC 9c

Mr and Mrs Caine each decide to invest some money each year to help pay for their son’s university education. The parents choose different investment strategies.

Mr Caine makes 18 yearly contributions of $1000 into an investment fund. He makes his first contribution on the day his son is born, and his final contribution on his son’s seventeenth birthday. His investment earns 6% compound interest per annum.

  1. Find the total value of Mr Caine’s investment on his son’s eighteenth birthday.  (3 marks)

Mrs Caine makes her contributions into another fund. She contributes $1000 on the day of her son’s birth, and increases her annual contribution by 6% each year. Her investment also earns 6% compound interest per annum.

  1. Find the total value of Mrs Caine’s investment on her son’s third birthday (just before she makes her fourth contribution).  (2 marks)

  2. Mrs Caine also makes her final contribution on her son’s seventeenth birthday. Find the total value of Mrs Caine’s investment on her son’s eighteenth birthday.  (1 mark)

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Show Worked Solution

i. 

 
 
 
 

 
 

 

 
 
 

 

 

ii. 

 
 
 
 
 
 

 

iii. 

 
 

 

Geometry and Calculus, EXT1 2005 HSC 7b

Let  ,  where  .

  1. Show that    has stationary points at
    1.   (1 mark)

  2. Show that    has exactly one zero when
    1.   (2 marks)

  3. By observing that  , deduce that    does not have a zero in the interval    when
    1.   (1 mark)

  4. Let  ,  where  
  6. By calculating    and applying the result in part (iii), or otherwise, show that    does not have any stationary points.  (3 marks)
  7. Hence, or otherwise, deduce that    has an inverse function.  (1 mark)
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(i)  

 

 

(ii)  

 

 

 

(iii) 

 

 

(iv)  

 

 
 

 

 

 

(v)  

Calculus, EXT1 C1 2005 HSC 7a

An oil tanker at    is leaking oil which forms a circular oil slick. An observer is measuring the oil slick from a position  , 450 metres above sea level and 2 kilometres horizontally from the centre of the oil slick.
 


 

  1. At a certain time the observer measures the angle, , subtended by the diameter of the oil slick, to be 0.1 radians. What is the radius, , at this time?  (2 marks)

  2. At this time,  radians per hour. Find the rate at which the radius of the oil slick is growing.  (2 marks)

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i.

 

 
 

 

 
 

 

ii. 

   
 

  

 
 

Linear Functions, 2UA 2005 HSC 10b

Xuan and Yvette would like to meet at a cafe on Monday. They each agree to come to the cafe sometime between 12 noon and 1 pm, wait for 15 minutes, and then leave if they have not seen the other person.

Their arrival times can be represented by the point    in the Cartesian plane, where    represents the fraction of an hour after 12 noon that Xuan arrives, and    represents the fraction of an hour after 12 noon that Yvette arrives.

Thus    represents Xuan arriving at 12:20 pm and Yvette arriving at 12:24 pm. Note that the point    lies somewhere in the unit square    and    as shown in the diagram.
 

Probability, 2UA 2005 HSC 10b
 

  1. Explain why Xuan and Yvette will meet if
  2.  or  .  (1 mark)
  3.  
  4. The probability that they will meet is equal to the area of the part of the region given by the inequalities in part (i) that lies within the unit square    and  .
     
  5. Find the probability that they will meet.  (2 marks)
  6.  
  7. Xuan and Yvette agree to try to meet again on Tuesday. They agree to arrive between 12 noon and 1 pm, but on this occasion they agree to wait for minutes before leaving.
  8.  
  9. For what value of do they have a 50% chance of meeting?  (2 marks)

 

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  3.  
Show Worked Solution

(i)  


 

(ii)  


 


 

Probability, 2UA 2005 HSC 10b Answer
 

²
 


 

(iii)  ²

Probability-2UA-2005-HSC-10b-Answer2 

 
 
 
 

 

Quadratic, 2UA 2005 HSC 10a

2005 10a

The parabola    and the line    intersect at the points  αα  and  ββ  as shown in the diagram.

  1. Explain why  αβ  and  αβ.  (1 mark)
  2. Given that
  3. αβαβαβαβ, show that the distance    (2 marks)
  4. The point    lies on the parabola between    and  . Show that the area of the triangle    is given by    (2 marks)
  5.  
  6. The point    in part (iii) is chosen so that the area of the triangle    is a maximum.
  7. Find the coordinates of    in terms of  .  (2 marks)
  8.  
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(i)   

αβ

αβ

αβ  
αβ

 

 

(ii)   ααββ

αβαβ
  αβαβ
  αβαβαβ
 
 

 

(iii) 

 

 

(iv)
   
 
     

 

 

Combinatorics, EXT1 A1 2015 HSC 14c

Two players and play a series of games against each other to get a prize. In any game, either of the players is equally likely to win.

To begin with, the first player who wins a total of 5 games gets the prize.

  1. Explain why the probability of player getting the prize in exactly 7 games is  .  (1 mark)

  2. Write an expression for the probability of player getting the prize in at most 7 games.  (1 mark)

  3. Suppose now that the prize is given to the first player to win a total of games, where is a positive integer.

     

    By considering the probability that gets the prize, prove that
     

  4. .   (2 marks)

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i.   

♦♦♦ Mean mark 19%.


 

 

ii. 

♦♦♦ Mean mark 23%.

♦♦♦ Mean mark part (iii) 9%.

 

iii. 

 


 

 


 


 

Measurement, STD2 M6 2015 HSC 30e

From point , which is 1.8 m above the ground, a pulley at is used to lift a flat object . The lengths and are 5.4 m and 2.1 m respectively. The angle is 108°.
 

 

  1. Show that the length    is 6.197 m, correct to 3 decimal places.  (1 mark)

  2. Calculate , the height of the object above the ground.  (4 marks)

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i.   

2UG 2015 30e Answer

♦ Mean mark 41%.

 

 
 
 

 

ii.   

♦♦ Mean mark below 19%.
STRATEGY: Finding in part (i) and needing to find should flag the strategy of finding and using Pythagoras.

 

 
 

 

 

 

 
 
 

Algebra, STD2 A4 2015 HSC 29e

A diver springs upwards from a diving board, then plunges into the water. The diver’s height above the water as it varies with time is modelled by a quadratic function. Graphing software is used to produce the graph of this function.
 

Explain how the graph could be used to determine how high above the height of the diving board the diver was when he reached the maximum height.  (2 marks)

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Show Worked Solution

♦♦ Mean mark 19%.


 

Statistics, STD2 S1 2015 HSC 29d

Data from 200 recent house sales are grouped into class intervals and a cumulative frequency histogram is drawn.
 

2UG 2015 29d1 
 

  1. Use the graph to estimate the median house price.   (1 mark)

  2. By completing the table, calculate the mean house price.   (3 marks)

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i.   

2UG 2015 29d Answer

♦♦♦ Mean marks of 9% for part (i) and 34% for part (ii)!

 
ii.

 

Measurement, STD2 M7 2015 HSC 29c

The image shows a rectangular farm shed with a flat roof.
 

Measurement, 2UG 2015 HSC 29c

 
The real width of the shed indicated by the dotted line was measured using an online ruler tool, and found to be approximately 12 metres.

  1. By measurement and calculation, show that the area of the roof of the shed is approximately 216 m².  (2 marks)

  2. All the rain that falls onto this roof is diverted into a cylindrical water tank which has a diameter of 3.6 m. During a storm, 5 mm of rain falls onto the roof.

     

    Calculate the increase in the depth of water in the tank due to the rain that falls onto the roof during the storm.  (3 marks)

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Show Worked Solution

i.    

♦ Mean mark 35%.

 

 

  ²

 

ii.  

♦♦♦ Mean mark 14%.

³
 


 

³