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Binomial, EXT1 2008 HSC 6c

Let    and    be positive integers with  .

  1. Use the binomial theorem to expand  , and hence write down the term of
     
  2.  which is independent of  .  (2 marks)
     

  3. Given that
     
  4. ,
     
    apply the binomial theorem and the result of part(i) to find a simpler expression for
     
  5. .  (3 marks)

 

 

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(i)  

 

(ii)  


 


 


 

Polynomials, EXT2 2015 HSC 12b

The polynomial    has roots    and    where  and    are real and  

  1. By evaluating    and  , find all the roots of     (3 marks)
  2. Hence, or otherwise, find one quadratic polynomial with real coefficients that is a factor of     (1 mark)

 

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(i)  

 

 

(ii)   
   

 

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%.
   
 
 

 

Mechanics, EXT2 2011 HSC 5a

A small bead of mass    is attached to one end of a light string of length  . The other end of the string is fixed at height    above the centre of a sphere of radius  , as shown in the diagram. The bead moves in a circle of radius    on the surface of the sphere and has constant angular velocity  . The string makes an angle of    with the vertical.

Three forces act on the bead: the tension force    of the string, the normal reaction force    to the surface of the sphere, and the gravitational force  .

  1. By resolving the forces horizontally and vertically on a diagram, show that
  2. and
    1.   (2 marks)
  3. Show that
    1.   (2 marks)
  4. Show that the bead remains in contact with the sphere if  
    1.   (2 marks)

 

 

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(i)  

♦ Mean mark part (i) 50%.

 

 

(ii)      

 

(iii) 

 
 
 

Harder Ext1 Topics, EXT2 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

Harder Ext1 Topics, EXT2 2014 HSC 16a

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

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

 

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

Harder Ext1 Topics, EXT2 2014 HSC 16a 

 

Copy or trace the diagram into your writing booklet.

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

Harder Ext1 Topics, EXT2 2014 HSC 16a Answer

 
  βαγ
 

 

(ii)  

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

α

β

γ

 

αβγ
αβγ
αβγ

 

 

♦ Mean mark 44%.

(iii)  

αβ

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) 

 
 

 

 

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) 

 

(iii)  

 

 

 
 
 
 
 
 

 

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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Show Worked Solution

(i)

 

(ii) 

 

(iii) 

 

 
 

 

(iv)

 2UA HSC 2007 10ai

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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(i)

 

(ii) 

 
 

 

(iii) 

 

 

(iv) 

 

 

 

 

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) 

 

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) 

     
 
 

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
Show Worked Solution

(i)  

 
 
 
 

 

(ii) 
 
 

 

(iii) 

 
 
 
 

 

(iv) 
 

 

(v) 

²

FS Comm, 2UG 2015 HSC 27e

A megabyte (MB) file is to be downloaded at a rate of kilobits per second (kbps), where kilobit = bits.

How long would it take to download this file? Give your answer in minutes and seconds, correct to the nearest second.  (3 marks)

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♦ Mean mark 50%.

 
 

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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Data, 2UG 2015 HSC 26a

A farmer used the ‘capture‑recapture’ technique to estimate the number of chickens he had on his farm. He captured, tagged and released 18 of the chickens. Later, he caught 26 chickens at random and found that 4 had been tagged.

What is the estimate for the total number of chickens on this farm?  (2 marks)

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Probability, 2UG 2015 HSC 18 MC

A Student Representative Council (SRC) consists of five members. Three of the members are being selected to attend a conference.

In how many ways can the three members be selected?

(A)   

(B)   

(C)   

(D)   

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♦ Mean mark 37%.
 

Measurement, 2UG 2015 HSC 14 MC

Stockholm is located at and Darwin is located at . 

What is the time difference between Stockholm and Darwin? (Ignore time zones and daylight saving.)

(A)    minutes

(B)    minutes

(C)    minutes

(D)    minutes

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Plane Geometry, 2UA 2015 HSC 15b

The diagram shows which has a right angle at . The point is the midpoint of the side . The point is chosen on such that . The line segment is produced to meet the line at .

Copy or trace the diagram into your writing booklet.

  1. Prove that is similar to   (2 marks)
  2. Explain why is isosceles.  (1 mark)
  3. Show that   (2 marks)
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(i) 

 

♦ Mean mark 45%.

(ii) 

 

(iii) 

♦♦ Mean mark 23%.
 
 
 
 

 

Calculus, EXT1* C1 2015 HSC 15a

The amount of caffeine, , in the human body decreases according to the equation

 

where is measured in mg and is the time in hours.

  1. Show that    is a solution to  where is a constant.

     

    When , there are 130 mg of caffeine in Lee’s body.  (1 mark)

  2. Find the value of   (1 mark)

  3. What is the amount of caffeine in Lee’s body after 7 hours?   (1 mark)

  4. What is the time taken for the amount of caffeine in Lee’s body to halve?  (2 marks)

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i.
 
   
   

 

 

ii. 

 

iii. 

 
 
 

 

 

iv. 

 
 

 

Quadratic, 2UA 2015 HSC 12e

The diagram shows the parabola with focus A tangent to the parabola is drawn at

  1. Find the equation of the tangent at the point .   (2 marks)
  2. What is the equation of the directrix of the parabola?   (1 mark)
  3. The tangent and directrix intersect at .
    Show that lies on the -axis.   (1 mark)

  4. Show that is isosceles.   (1 mark)
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  3.  
Show Worked Solution
(i)   


 

 

(ii) 
 

(iii) 

 

 

(iv) 

Functions, 2ADV F1 2015 HSC 12b

The diagram shows the rhombus  .

The diagonal from the point    to the point lies on the line .

The other diagonal, from the origin to the point , lies on the line which has equation  .

2015 2ua 12b

  1. Show that the equation of the line    is  .  (2 marks)
  2. The lines    and    intersect at the point .
  3. Find the coordinates of  .  (2 marks)

 

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(i) 

 
   

 

 

(ii) 

 

 

Calculus, 2ADV C4 2006 HSC 9b

During a storm, water flows into a 7000-litre tank at a rate of litres per minute, where and is the time in minutes since the storm began.

  1. At what times is the tank filling at twice the initial rate?  (2 marks)

  2. Find the volume of water that has flowed into the tank since the start of the storm as a function of .  (1 mark)

  3. Initially, the tank contains 1500 litres of water. When the storm finishes, 30 minutes after it began, the tank is overflowing.

     

    How many litres of water have been lost?  (2 marks)

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i. 

 

ii. 
 
 

 

 

iii. 


 

 

 

 

Quadratic, 2UA 2006 HSC 7c

  1. Write down the discriminant of    where    is a constant.  (1 mark)
  2. Hence, or otherwise, find the values of    for which the parabola   does not intersect the line  .  (2 marks)

 

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(i) 

 
 
 

 

(ii) 

 

HSC quadratic

Measurement, 2UG 2015 HSC 26c

Two cities lie on the same meridian of longitude. One is 40° north of the other.

What is the distance between the two cities, correct to the nearest kilometre? (2 marks)

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♦ Mean mark 46%.

2UG 2015 23c Answer

Linear Functions, 2UA 2006 HSC 3a

In the diagram, are the points    respectively. The line    meets the y-axis at .

  1. Show that the equation of the line    is  .  (2 marks)
  2. Find the coordinates of the point .  (1 mark)
  3. Find the perpendicular distance of the point from the line  .  (1 mark)
  4. Hence, or otherwise, find the area of the triangle  .  (2 marks)
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  4.  
  5. ²
Show Worked Solution

(i) 

 
 
 

 

 

(ii) 

 

(iii) 

 
 
 

 

(iv)

 
 
 
 
 
  ²

Data, 2UG 2006 HSC 23b

This radar chart was used to display the average daily temperatures each month for two different towns.

2UG-2006-23b

  1. What is the average daily temperature of Town for April?  (1 mark)
  2. In which month do the average daily temperatures of the two towns have the greatest difference?  (1 mark)
  4. In which months is the average daily temperature in Town higher than in Town ?  (1 mark)
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(i)   

 

(ii)   

 

(iii)  

Algebra, STD2 A4 2004 HSC 26a

  1. The number of bacteria in a culture grows from 100 to 114 in one hour.

     

    What is the percentage increase in the number of bacteria?   (1 mark)

  2. The bacteria continue to grow according to the formula  , where    is the number of bacteria after    hours.

     

    What is the number of bacteria after 15 hours?   (1 mark)

  1. Use the values of from    to    to draw a graph of  .

     

    Use about half a page for your graph and mark a scale on each axis.   (4 marks)

  2. Using your graph or otherwise, estimate the time in hours for the number of bacteria to reach 300.   (1 mark)

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i.   

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii. 

 
 

 

iii. 

 

iv. 

Measurement, 2UG MM6 SM-Bank 01 MC

 

Mapupu and Minoha are two towns on the equator.

The longitude of Mapupu is and the longitude of Minoha is .

How far apart are these two towns if the radius of Earth is approximately ?

(A)  

(B)  

(C)  

(D)  

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  º

 

Functions, EXT1 F2 2008 HSC 2c

The polynomial    is given by  , where    and    are constants.

The three zeros of    are  ,    and  .

Find the value of  .   (3 marks) 

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MARKER’S COMMENT: Many students displayed significant inefficiencies in solving simultaneous equations.

 

 

Measurement, 2UG 2008 HSC 28b

A tunnel is excavated with a cross-section as shown.
 

 
 

  1. Find an expression for the area of the cross-section using TWO applications of Simpson’s rule.  (2 marks)
  2. The area of the cross-section must be 600 m2. The tunnel is 80 m wide.   
  3. If the value of increases by 2 metres, by how much will change?   (2 marks)

 

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  2.  
Show Worked Solution
(i)   
   
   

 

(ii)    ²
 

 

 

Linear Functions, 2UA 2008 HSC 3a

2008 3a

In the diagram,    is a quadrilateral. The equation of the line    is  . 

  1. Show that    is a trapezium by showing that    is parallel to  .  (2 marks)
  2. The line    is parallel to the  -axis. Find the coordinates of  .   (1 mark)
  3. Find the length of  .   (1 mark)
  4. Show that the perpendicular distance from    to    is  .   (2 marks)
  5. Hence, or otherwise, find the area of the trapezium  .   (2 marks)
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  5. ²
Show Worked Solution
(i)   

 
 

 

 

(ii)   
 
 

 

(iii)  
 
 
 

 

(iv)  

 

 
 
 

 

(v)   
   

 

 

 
 
 
 
  ²

Binomial, EXT1 2009 HSC 6b

  1. Sum the geometric series  
  3.  
  4. and hence show that
     
    1. .   (3 marks)
    2.  
  6. Consider a square grid with    rows and    columns of equally spaced points.
  7.  
    1. 2009 6b
  8. The diagram illustrates such a grid. Several intervals of gradient  , whose endpoints are a pair of points in the grid, are shown. 
  9.  
  10. (1)   Explain why the number of such intervals on the line    is equal to  .   (1 mark)
  11. (2)   Explain why the total number,  , of such intervals in the grid is given by
  12.  
      1. .   (1 mark)
    2.  
  14. Using the result in part (i), show that 
    1. .   (3 marks)

 

 

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  2. (1)
  3. (2)
  4.  
Show Worked Solution

(i)   

♦♦ Exact data unavailable although mean marks for all Q6 was < 25%.
MARKER’S COMMENT: A common mistake was to assume the number of terms in the series was  .

 

 
 
 

 

 

 

 

 

 

(ii)(1)  
 
 
   
     (2)  
 
 
 
 
 

 

 
 

 

♦♦♦ Exact data unavailable although very few completed part (iii) correctly. 
MARKER’S COMMENT: Note that even if part (i) wasn’t solved, the proof as stated in the question can be used to solve part (iii).
(iii)
   
 
 
 
 
 

Trig Calculus, EXT1 2009 HSC 3b

  1. On the same set of axes, sketch the graphs of  
    1.  and  , for  .    (2 marks)
  3. Use your graph to determine how many solutions there are to the equation    for  .     (1 mark)
  5. One solution of the equation    is close to  . Use one application of Newton’s method to find another approximation to this solution. Give your answer correct to three decimal places.   (3 marks)

 

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Show Worked Solution
(i)   

 

(ii)  

 

(iii)  

MARKER’S COMMENT: Better responses defined and and evaluated each for before calculating Newton’s formula, as done in the Worked Solution.

 

 
 

 

 
 
 
 

Plane Geometry, 2UA 2010 HSC 10a

In the diagram    is an isosceles triangle with  . The point    on the interval    is chosen so that  . Let  ,    and  .
 

 
 

  1. Show that    is similar to  .    (2 marks)
  2. Show that  .     (1 mark)
  3. Show that  .   (2 marks)
  4. Deduce that  .   (1 mark) 
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Show Worked Solution
MARKER’S COMMENT: When dealing with multiple triangles, the best performing students label all angles clearly and unambiguously.
(i)


 

 

♦♦ Mean mark 25%.
MARKER’S COMMENT: The similarity proof in part (i) should have immediately alerted students to using similar ratios of sides to solve part (ii). Many did not follow this obvious hint.

(ii) 


 

 

 

(iii) 

 
 
 

 

(iv) 

♦♦♦ Parts (iii) and (iv) mean marks – 18% and just 4% respectively.
IMPORTANT: Limits of trig functions are often extremely useful in proving inequalities (see Worked Solutions).

 

Plane Geometry, 2UA 2011 HSC 6a

The diagram shows a regular pentagon . Sides and are produced to meet at .
  

  1. Find the size of .    (1 mark)

  2. Hence, show that is isosceles.    (2 marks)

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Show Worked Solution
i.  

 
MARKER’S COMMENT: Very few students solved part (i) efficiently. Remember the general formula for the sum of internal angles equals (# sides – 2) x 90°.

 
ii. 

Statistics, 2ADV 2011 HSC 5b

Kim has three red shirts and two yellow shirts. On each of the three days, Monday, Tuesday and Wednesday, she selects one shirt at random to wear. Kim wears each shirt that she selects only once.

  1. What is the probability that Kim wears a red shirt on Monday?    (1 mark)
  2. What is the probability that Kim wears a shirt of the same colour on all three days?     (1 mark)
  3. What is the probability that Kim does not wear a shirt of the same colour on consecutive days?   (2 marks)
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Show Worked Solution
i.   
   

 

MARKER’S COMMENT: Students could also have drawn a probability tree setting out this problem over 3 different days (stages).
ii.   
 

 

iii.  

♦ Mean mark 47%.
MARKER’S COMMENT: Students should clearly write what probability they are finding in words or symbols before showing calculations.

Probability, 2UG 2010 HSC 26c

Tai plays a game of chance with the following outcomes. 

•  chance of winning   

•  chance of winning   

•  chance of losing   

The game has a    entry fee. 

What is his financial expectation from this game?    (2 marks)

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Show Worked Solution
♦♦ Mean mark 31%
MARKER’S COMMENT: A common error was not to deduct the chance of losing or the entry fee. BE CAREFUL!


 

Measurement, 2UG 2011 HSC 24c

A ship sails 6 km from to on a bearing of 121°. It then sails 9 km to .  The
size of angle is 114°.

2UG 2011 24c

Copy the diagram into your writing booklet and show all the information on it.

  1. What is the bearing of from ?   (1 mark)
  2. Find the distance . Give your answer correct to the nearest kilometre.   (2 marks)
  3. What is the bearing of from ? Give your answer correct to the nearest degree.   (3 marks)

 

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  1.  
  2.  
  3.  
Show Worked Solution
♦♦ Mean mark 24% 
STRATEGY: This deserves repeating again: Draw North-South parallel lines through major points to make the angle calculations easier.
(i)     2011 HSC 24c

 

 
 

  

 

(ii)   

♦ Mean mark 39%
 
 
 

 

(iii)    

♦♦♦ Mean mark 15%
MARKER’S COMMENT: The best responses clearly showed what steps were taken with working on the diagram. Note that all North/South lines are parallel.
 
 

 

 
 
 

Calculus in the Physical World, 2UA 2008 HSC 6b

The graph shows the velocity of a particle,    metres per second, as a function of time,    seconds.
 


 

  1. What is the initial velocity of the particle?   (1 mark)
  2. When is the velocity of the particle equal to zero?    (1 mark)
  3. When is the acceleration of the particle equal to zero?    (1 mark)
  4. By using Simpson's Rule with five function values, estimate the distance travelled by the particle between    and  .   (3 marks)
  5.  
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Show Worked Solution

(i)    

 

(ii)    

 

(iii) 

 

(iv)   

MARKER’S COMMENT: Less errors were made by students using a table and the given formula. Note however, that the formula produced the most errors.