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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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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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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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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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³

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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

 

 

 
 
 

 

 
 

 

 

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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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(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”.

 

 

 

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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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)
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 2UA HSC 2007 10ai

Probability, 2ADV S1 2007 HSC 9b

A pack of 52 cards consists of four suits with 13 cards in each suit.

  1. One card is drawn from the pack and kept on the table. A second card is drawn and placed beside it on the table. What is the probability that the second card is from a different suit to the first?  (1 mark)

  2. The two cards are replaced and the pack shuffled. Four cards are chosen from the pack and placed side by side on the table. What is the probability that these four cards are all from different suits?  (2 marks)

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Plane Geometry, 2UA 2007 HSC 8b

In the diagram, is parallel to , , , , and , and are equal.

Copy or trace this diagram into your writing booklet.

  1. Prove  that .  (2 marks)
  2. Prove  that .  (2 marks)
  3. Show that , , , are the first four terms of a geometric series.  (1 mark)
  4. Hence find the values of and .  (2 marks)
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Calculus, EXT1* C1 2007 HSC 8a

One model for the number of mobile phones in use worldwide is the exponential growth model,

,

where is the estimate for the number of mobile phones in use (in millions), and is the time in years after 1 January 2008.

  1. It is estimated that at the start of 2009, when  , there will be 1600 million mobile phones in use, while at the start of 2010, when  , there will be 2600 million. Find and .  (3 marks)

  2. According to the model, during which month and year will the number of mobile phones in use first exceed 4000 million?  (2 marks)

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Calculus, 2ADV C4 2007 HSC 7b

2ua 2007 7b
 

The diagram shows the graphs of    and  . The first two points of intersection to the right of the -axis are labelled    and  .

  1. Solve the equation    to find the -coordinates of    and  .  (2 marks)

  2. Find the area of the shaded region in the diagram.  (3 marks)

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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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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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 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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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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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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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. 

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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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)
   
 
     

 

 

Mechanics, EXT2* M1 2015 HSC 14b

A particle is moving horizontally. Initially the particle is at the origin moving with velocity .

The acceleration of the particle is given by  , where is its displacement at time  .

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

  2. Find an expression for as a function of .  (2 marks)

  3. Find the limiting position of the particle.  (1 mark)

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♦ Mean mark 49%.

 

♦ Mean mark 42%.
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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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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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♦ 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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Trig Ratios, EXT1 2015 HSC 12c

A person walks 2000 metres due north along a road from point to point . The point is due east of a mountain , where is the top of the mountain. The point is directly below point and is on the same horizontal plane as the road. The height of the mountain above point is metres.

From point , the angle of elevation to the top of the mountain is 15°.

From point , the angle of elevation to the top of the mountain is 13°.
 

Trig Ratios, EXT1 2015 HSC 12c
 

  1. Show that .  (1 mark)
  2. Hence, find the value of  .  (2 marks)
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(i)    

Trig Ratios, EXT1 2015 HSC 12c Answer1

 

 

(ii)   

 

 

Trig Ratios, EXT1 2015 HSC 12c Answer2 
 
 
 

Quadratic, EXT1 2015 HSC 12b

The points    and    lie on the parabola  .

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

  1. Show that if    is a focal chord then  .  (1 mark)
  2. If    is a focal chord and    has coordinates  , what are the coordinates of    in terms of  ?  (2 marks)

 

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(i)  
 

Quadtratic, EXT1 2015 HSC 12b Answer1

 

 

 

(ii) 

 

Statistics, STD2 S4 2015 HSC 28e

The shoe size and height of ten students were recorded.

  1. Complete the scatter plot AND draw a line of fit by eye.  (2 marks)
     
     
  2. Use the line of fit to estimate the height difference between a student who wears a size 7.5 shoe and one who wears a size 9 shoe.  (1 mark)

  3. A student calculated the correlation coefficient to be 1 for this set of data. Explain why this cannot be correct.  (1 mark)

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  2. `13\ text{cm  (or close given LOBF drawn)}


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i.    
      2UG 2015 28e Answer

ii.   ½

 

 

iii.   

♦ Mean mark (c) 39%.

Algebra, STD2 A1 2015 HSC 28d

The formula    is used to convert temperatures between degrees Fahrenheit and degrees Celsius .

Convert 3°C to the equivalent temperature in Fahrenheit.  (2 marks)

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Statistics, STD2 S1 2015 HSC 27d

In a small business, the seven employees earn the following wages per week:

  1.  Is the wage of $970 an outlier for this set of data? Justify your answer with calculations.  (3 marks)

  2.  Each employee receives a $20 pay increase.

     

     What effect will this have on the standard deviation?  (1 mark)

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ii.   

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♦ Mean mark (i) 39%.


ii.   

Measurement, STD2 M7 2015 HSC 27a

At a particular time during the day, a tower of height 19.2 metres casts a shadow. At the same time, a person who is 1.65 metres tall casts a shadow 5 metres long.

  

What is the length of the shadow cast by the tower at that time?  (2 marks)

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FS Comm, 2UG 2015 HSC 26g

Pat’s mobile phone plan is shown.

2015 26g

Last month Pat:

• made calls to the value of
• sent SMS messages
• sent MMS messages
• used GB of data.

What was the total of Pat’s phone bill for last month?  (3 marks)

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Probability, STD2 S2 2015 HSC 26e

The table shows the relative frequency of selecting each of the different coloured jelly beans from packets containing green, yellow, black, red and white jelly beans.

  1. What is the relative frequency of selecting a red jelly bean?  (1 mark)

  2. Based on this table of relative frequencies, what is the probability of NOT selecting a black jelly bean?  (1 mark)

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ii.