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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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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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iii.   EXT1 2007 7bi

 
 
 

 

 
 
 

 

 

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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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(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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Mechanics, EXT2* M1 2007 HSC 6a

A particle moves in a straight line. Its displacement, metres, after seconds is given by

.

  1. Prove that the particle is moving in simple harmonic motion about    by showing that   .  (2 marks)

  2. What is the period of the motion?  (1 mark)

  3. Express the velocity of the particle in the form  α, where  α  is in radians.  (2 marks)

  4. Hence, or otherwise, find all times within the first    seconds when the particle is moving at metres per second in either direction.  (2 marks)

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iii. 

α

αα
αα
α
α
 

 

α
α

 

(iv) 

 

 

Quadratic, EXT1 2007 HSC 5d

2007 5d

The diagram shows a point    on the parabola  . The normal to the parabola at    intersects the parabola again at  .

The equation of    is  . (Do NOT prove this.)

  1. Prove that  .  (1 mark)
  2. If the chords    and    are perpendicular, show that  .  (2 marks)
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(i)   

 
 
 

 

 

(ii) 

 

 

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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(i)  

 
 
 

 

 

(ii)  

 

 

 

 

 
 
 
 
 
 
 

 

(iii)    
 
   
   

 

 

 

(iv)  

 

 

 

(v)  
 

 

 

 
 
 
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Statistics, EXT1 S1 2006 HSC 6b

In an endurance event, the probability that a competitor will complete the course is    and the probability that a competitor will not complete the course is   Teams consist of either two or four competitors. A team scores points if at least half its members complete the course.

  1. Show that the probability that a four-member team will have at least three of its members not complete the course is    (1 mark)

  2. Hence, or otherwise, find an expression in terms of    only for the probability that a four-member team will score points.  (2 marks)

  3. Find an expression in terms of    only for the probability that a two-member team will score points.  (1 mark)

  4. Hence, or otherwise, find the range of values of    for which a two-member team is more likely than a four-member team to score points.  (2 marks)

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ii. 

 

iii.   

 

iv.   

 

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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Plane Geometry, EXT1 2007 HSC 4c

EXT1 2007 4c

The diagram shows points , , and on a circle. The lines    and    are perpendicular and intersect at  . The perpendicular to    through    meets    at    and    at  .

Copy or trace this diagram into your writing booklet.

  1. Prove that  .  (3 marks)
  2. Prove that    bisects  .  (2 marks)
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(i)   

 

MARKER’S COMMENT: Many students did not copy the diagram into their answer! Efficient solutions labelled angles with a symbol.

 

 

 

(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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Mechanics, EXT2* M1 2007 HSC 3c

A particle is moving in a straight line with its acceleration as a function of    given by  . It is initially at the origin and is travelling with a velocity of 1 metre per second.

  1. Show that  .  (2 marks)

  2. Hence show that  .  (2 marks)

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MARKER’S COMMENT: Most candidates “neglected” to consider the two cases that  .

 

 

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Functions, EXT1 F1 2007 HSC 3b

  1. Find the vertical and horizontal asymptotes of the hyperbola   

     

    and hence sketch the graph of  .  (3 marks)

  2. Hence, or otherwise, find the values of    for which
     
    .  (2 marks)

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Geometry and Calculus, EXT1 2007 HSC 3b Answer

 

ii. 

 

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

 

 
 

 
 
       
       

 

 

Plane Geometry, EXT1 2004 HSC 6a

Plane Geometry, EXT1 2004 HSC 6a

The points    and    are placed on a circle of radius    such that    and    meet at  . The lines    and    are produced to meet at  , and    is a cyclic quadrilateral.

Copy or trace this diagram into your writing booklet.

  1. Find the size of  , giving reasons for your answer.  (2 marks)

  2. Find an expression for the length of    in terms of  .  (1 mark)
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(i)   

Plane Geometry, EXT1 2004 HSC 6a Answer

 

 

 

(ii)   

 

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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(i)   

Inverse Functions, EXT1 2004 HSC 5b Answer

(ii)   

 

(iii) 

 

 

 

 

(iv)   

 

α

 

(v)   
 
 
   
 
   

 

α α
 
 
 
 

Mechanics, EXT2* M1 2004 HSC 5a

A particle is moving along the -axis, starting from a position    metres to the right of the origin (that is,    when  ) with an initial velocity of    and an acceleration given by

.

  1. Show that  .  (2 marks)

  2. Hence find an expression for    in terms of  .  (3 marks)

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ii.   
 
 
   

 

 

Combinatorics, EXT1 A1 2004 HSC 4c

Katie is one of ten members of a social club. Each week one member is selected at random to win a prize.

  1. What is the probability that in the first 7 weeks Katie will win at least 1 prize?  (1 mark)

  2. Show that in the first 20 weeks Katie has a greater chance of winning exactly 2 prizes than of winning exactly 1 prize.  (2 marks)

  3. For how many weeks must Katie participate in the prize drawing so that she has a greater chance of winning exactly 3 prizes than of winning exactly 2 prizes?  (2 marks)

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i. 


 

ii. 


 


 

 

iii. 


 

 

Trig Ratios, EXT1 2004 HSC 3d

Trig Ratios, EXT1 2004 HSC 3d
 

The length of each edge of the cube    is 2 metres. A circle is drawn on the face    so that it touches all four edges of the face. The centre of the circle is    and the diagonal    meets the circle at    and  .

  1. Explain why  .  (1 mark)
  2. Show that   metres.  (1 mark)
  3. Calculate the size of    to the nearest degree.  (1 mark)
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(i)   

Trig Ratios, EXT1 2004 HSC 3d Answer

 

(ii)   

Trig Ratios, EXT1 2004 HSC 3d Answer2

 
 
 

 

 

 

(iii)

Trig Ratios, EXT1 2004 HSC 3d Answer3

 

 
 

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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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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 ii. 

Trigonometry, EXT1 T3 2004 HSC 2d

  1. Write    in the form  α, where    and  απ.  (2 marks)

  2. Hence, or otherwise, solve the equation    for  π. 

     

    Give your answers correct to three decimal places.  (2 marks)

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i.   α

αα
αα

α

α

 
α
α

 

 

ii.   
 
 
 
 
   

Mechanics, EXT2* M1 2006 HSC 4c

A particle is moving so that  

Initially    and the velocity, , is  

  1. Show that    (2 marks)

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

  3. It can be shown that for some constant  ,
     
                (Do NOT prove this.)
     
    Using this equation and the initial conditions, find    as a function of    (2 marks)

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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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(ii) 

 

 

 

 

(iii)  

 

 

Combinatorics, EXT1 A1 2006 HSC 3c

Sophie has five coloured blocks: one red, one blue, one green, one yellow and one white. She stacks two, three, four or five blocks on top of one another to form a vertical tower.

  1. How many different towers are there that she could form that are three blocks high?  (1 mark)

  2. How many different towers can she form in total?  (2 marks)

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Polynomials, EXT1 2006 HSC 3b

  1. By considering  , show that the curve    and the line    meet at a point    whose -coordinate is between    and  .  (1 mark)

  2. Use one application of Newton’s method, starting at  , to find an approximation to the -coordinate of  . Give your answer correct to two decimal places.  (2 marks)

 

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(i)     
 
   
 
   

 

 

(ii)
 
 
 

 

 
 
 

Quadratic, EXT1 2006 HSC 2c

2006 2c

The points    and    lie on the parabola  . The chord    is perpendicular to the axis of the parabola. The chord    meets the axis of the parabola at  .

The equation of the chord    is       (Do NOT prove this.)

The equation of the tangent at    is              (Do NOT prove this.) 

  1. Find the coordinates of    (1 mark)
  2. The tangents at    and    meet at the point  . Show that the coordinates of    are    (2 marks)
  3. Show that    is perpendicular to the axis of the parabola.  (1 mark)
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(i)   

 

(ii)   

 

 

 

 
 

 

 

(iii) 

 

 

 

Calculus, EXT1 C2 2006 HSC 2a

Let  

  1. State the domain and range of the function    (2 marks)

  2. Find the gradient of the graph of    at the point where    (2 marks)

  3. Sketch the graph of    (2 marks)

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 EXT1 2006 2a

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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Statistics, EXT1 S1 2005 HSC 6a

There are five matches on each weekend of a football season. Megan takes part in a competition in which she earns one point if she picks more than half of the winning teams for a weekend, and zero points otherwise. The probability that Megan correctly picks the team that wins any given match is .

  1. Show that the probability that Megan earns one point for a given weekend is  0.7901, correct to four decimal places.  (2 marks)

  2. Hence find the probability that Megan earns one point every week of the eighteen-week season. Give your answer correct to two decimal places.  (1 mark)

  3. Find the probability that Megan earns at most 16 points during the eighteen-week season. Give your answer correct to two decimal places.  (2 marks)

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ii. 

 

iii. 


 


 

Mechanics, EXT2* M1 2005 HSC 5c

A particle moves in a straight line and its position at time    is given by

  1. Express    in the form    where    is in radians.  (2 marks)

  2. The particle is undergoing simple harmonic motion. Find the amplitude and the centre of the motion.  (2 marks)

  3. When does the particle first reach its maximum speed after time  ?  (1 mark)

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Plane Geometry, EXT1 2005 HSC 5b

Two chords of a circle,  and  , intersect at  . The perpendiculars to    at   and    at    intersect at  . The line    meets    at  , as shown in the diagram.

  1. Explain why    is a cyclic quadrilateral.  (1 mark)
  2. Prove that    (2 marks)
  3. Deduce that    is perpendicular to    (1 mark)
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(i) 

(ii)

(iii)

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(i)

 

  

 

(ii) 

 

(iii)  

Calculus, EXT1 C3 2005 HSC 5a

Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve  , the -axis and the line    is rotated about the -axis.  (3 marks)

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COMMENT: Michael Wells (1st in state Ext2) would derive this formula in his working from the identity in ~5 seconds every time he used it in an exam.
  

 
 
 
 
 
 
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Quadratic, EXT1 2005 HSC 4c

The points    and    lie on the parabola  .

The equation of the normal to the parabola at    is    and the equation of the normal at    is similarly given by  

  1. Show that the normals at    and    intersect at the point    whose coordinates are
    1.   (2 marks)

  2. The equation of the chord    is
    1. (Do NOT show this.)
  4. If the chord    passes through  , show that    (1 mark)

  6. Find the equation of the locus of    if the chord    passes through    (2 marks)
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(i)  

 

ext1 2005 4c

 

 

 
 
 

 

 
 

 

 

(ii) 

 

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Plane Geometry, EXT1 2005 HSC 3d

In the circle centred at    the chord    has length  . The point    lies on    and    has length  . The chord    passes through  .

Let the length of    be    and the length of    be  .

  1. Show that    (2 marks)
  2. Find the length of the shortest chord that passes through    (2 marks)
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(i)   

 

 

 

(ii)  

MARKER’S COMMENT: A majority of students tried to differentiate this equation, not realising that is a variable, NOT a constant.

 

 

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”.