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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

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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.  
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(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. 

Calculus, EXT1 C1 2006 HSC 5c

2006 5c
 

A hemispherical bowl of radius    is initially empty. Water is poured into it at a constant rate of  ³  per minute. When the depth of water in the bowl is  , the volume, ³, of water in the bowl is given by

    (Do NOT prove this)

  1. Show that    (2 marks)

  2. Hence, or otherwise, show that it takes 3.5 times as long to fill the bowl to the point where    as it does to fill the bowl to the point where    (2 marks)

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i. 

 
 
   

 

ii. 
 

 

 

 

 
 
 

 

 
 
 
 
 

 

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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  3.  

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ii. 


 

 
 

 

 

iii.

 EXT1 2006 2a

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 C1 2005 HSC 2d

A salad, which is initially at a temperature of 25°C, is placed in a refrigerator that has a constant temperature of 3°C. The cooling rate of the salad is proportional to the difference between the temperature of the refrigerator and the temperature, , of the salad. That is, satisfies the equation

 

where    is the number of minutes after the salad is placed in the refrigerator.

  1. Show that    satisfies this equation.  (1 mark)

  2. The temperature of the salad is 11°C after 10 minutes. Find the temperature of the salad after 15 minutes.  (3 marks)

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i. 
 
   
   

 

 

ii.   

 
 

 

 
 
 

 

Quadratic, 2UA 2007 HSC 7a

  1. Find the coordinates of the focus, , of the parabola  .  (2 marks)
  2. The graphs of    and the line    have only one point of intersection, . Show that the -coordinate of satisfies.
    1. .  (1 mark)
  3. Using the discriminant, or otherwise, find the value of .  (1 mark)
  4. Find the coordinates of .  (2 marks)
  5. Show that is parallel to the directrix of the parabola.  (1 mark)
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(i)   

 

(ii) 

 

 

(iii) 

 

(iv) 

 

 

(v) 

 

Calculus, 2ADV C3 2007 HSC 6b

Let  .

  1. Find the coordinates of the points where the curve crosses the axes.  (2 marks)

  2. Find the coordinates of the stationary points and determine their nature.  (4 marks)

  3. Find the coordinates of the points of inflection.  (1 mark)

  4. Sketch the graph of  , indicating clearly the intercepts, stationary points and points of inflection.  (3 marks)

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  4.  
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i. 

 

 

ii.   

 

 

 
 

 

 

iii. 

 

 

 

 

iv.

 2UA HSC 2007 6b

Calculus, EXT1* C1 2007 HSC 5b

A particle is moving on the -axis and is initially at the origin. Its velocity, metres per second, at time seconds is given by

  1. What is the initial velocity of the particle?  (1 mark)

  2. Find an expression for the acceleration of the particle.  (2 marks)

  3. Find the time when the acceleration of the particle is zero.  (1 mark)

  4. Find the position of the particle when .  (3 marks)

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i. 

 

ii.  
 
 
 
 

 

iii. 

 

 

iv. 

 
 

 


 

 
 
 

 

Plane Geometry, 2UA 2007 HSC 5a

In the diagram, is a regular pentagon. The diagonals and intersect at .

Copy or trace this diagram into your writing booklet.

  1. Show that the size of is .  (1 mark)
  2. Find the size of . Give reasons for your answer.  (2 marks)
  3. By considering the sizes of angles, show that is isosceles.  (2 marks)
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(i)

 

(ii) 

 

 

(iii) 

     
 
 

Trigonometry, 2ADV T1 2007 HSC 4c

 
An advertising logo is formed from two circles, which intersect as shown in the diagram.

The circles intersect at and and have centres at and .

The radius of the circle centred at is 1 metre and the radius of the circle centred at is metres. The length of is 2 metres.

  1. Use Pythagoras’ theorem to show that  .  (1 mark)

  2. Find    and  .  (2 marks)

  3. Find the area of the quadrilateral  .  (1 mark)

  4. Find the area of the major sector  .  (1 mark)

  5. Find the total area of the logo (the sum of all the shaded areas).  (2 marks)

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  3. ²
  4. ²
  5. ²
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i.

 
 
 

 

 

ii. 

 

iii. 

²

 

iv. 

 

 

²

 

v. 

 

²

 

²

Probability, 2ADV S1 2007 HSC 4b

Two ordinary dice are rolled. The score is the sum of the numbers on the top faces.

  1. What is the probability that the score is 10?  (2 marks)

  2. What is the probability that the score is not 10?  (1 mark)

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i. 2UA HSC 2007 4b

 

ii. 

Mechanics, EXT2* M1 2015 HSC 14a

A projectile is fired from the origin with initial velocity m s at an angle to the horizontal. The equations of motion are given by

.    (Do NOT prove this)
 

Calculus in the Physical World, EXT1 2015 HSC 14a
 

  1. Show that the horizontal range of the projectile is
     
         .  (2 marks)

A particular projectile is fired so that  .

  1. Find the angle that this projectile makes with the horizontal when
     
         .  (2 marks)


  2. State whether this projectile is travelling upwards or downwards when
     
          . Justify your answer.  (1 mark)

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i.   

 

 

 

 

♦♦ Mean mark part (ii) 28%.
ii.   
 
 
 

 

 
 
 

Calculus in the Physical World, EXT1 2015 HSC 14a Answer

α
 
 
 
α

 

 

iii. 

♦♦ Mean mark part (iii) 31%.

Financial Maths, 2ADV M1 2007 HSC 3b

Heather decides to swim every day to improve her fitness level.

On the first day she swims 750 metres, and on each day after that she swims metres more than the previous day. That is, she swims 850 metres on the second day, 950 metres on the third day and so on.

  1. Write down a formula for the distance she swims on the th day.  (1 mark)

  2. How far does she swim on the 10th day?  (1 mark)

  3. What is the total distance she swims in the first 10 days?  (1 mark)

  4. After how many days does the total distance she has swum equal the width of the English Channel, a distance of 34 kilometres?  (2 marks)

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i. 

 
 
 

 

ii. 
 

 

 

iii. 

 
 

 

 

iv. 

 

 
 
 
 
 

 

Linear Functions, 2UA 2007 HSC 3a

2007 3a

In the diagram, , and are the points , and respectively.

Copy or trace this diagram into your writing booklet.

  1. Find the distance .  (1 mark)
  2. Find the midpoint of .  (1 mark)
  3. Show that .  (2 marks)
  4. Find the midpoint of and hence explain why is a rhombus.  (2 marks)
  5. Hence, or otherwise, find the area of .  (1 mark)
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  5. ²
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(i)  

 
 
 
 

 

(ii) 
 
 

 

(iii) 

 
 
 
 

 

(iv) 
 

 

(v) 

²

Mechanics, EXT2* M1 2015 HSC 13a

A particle is moving along the -axis in simple harmonic motion. The displacement of the particle is metres and its velocity is ms. The parabola below shows as a function of .

2015 13a

  1. For what value(s) of is the particle at rest?  (1 mark)

  2. What is the maximum speed of the particle?  (1 mark)

  3. The velocity of the particle is given by the equation

           where , and are positive constants.

     

    What are the values of , and ?  (3 marks)

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i.   

 

ii.   

 

iii. 

♦ Mean mark 41%.

 
 

Trig Ratios, EXT1 2015 HSC 12d

A kitchen bench is in the shape of a segment of a circle. The segment is bounded by an arc of length 200 cm and a chord of length 160 cm. The radius of the circle is cm and the chord subtends an angle at the centre of the circle.
 

2015 12d
 

  1. Show that  .  (1 mark)
  2. Hence, or otherwise, show that  .  (2 marks)
  3. Taking    as a first approximation to the value of  , use one application of Newton’s method to find a second approximation to the value of  . Give your answer correct to two decimal places.  (2 marks)

 

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(i)   

 
 

 

(ii)   

 

 

(iii)  
 
 
   
 
   
 
 
 

Plane Geometry, EXT1 2015 HSC 12a

In the diagram, the points , , and are on the circumference of a circle, whose centre lies on . The chord intersects the diameter at . The tangent at passes through the point .

It is given that    and  .

 

 

Copy or trace the diagram into your writing booklet.

  1. What is the size of  ?  (1 mark)
  2. What is the size of  ?  (1 mark)
  3. Find, giving reasons, the size of  .  (2 marks)
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(i)   

Plane Geometry, EXT1 2015 HSC 12a Answer
 

 

(ii)    
   

 

(iii) 
   
   
   
   
   

Functions, EXT1 F2 2015 HSC 11f

Consider the polynomials    and  .

  1. Given that    is divisible by  , show that  .  (1 mark)

  2. Find all the zeros of    when  .  (2 marks)

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i.   
 

 

 

ii.