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Plane Geometry, EXT1 2013 HSC 3 MC

The points  ,    and    lie on a circle with centre  , as shown in the diagram.

The size of    is    radians. 

What is the size of    in radians? 

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♦ Mean mark 49%

 

Binomial, EXT1 2010 HSC 7b

The binomial theorem states that 
 

 
  

  1. Show that  
    1. .   (1 mark)
  2.  
  3. Hence, or otherwise, find the value of
     

    1. .   (1 mark)
    2.  
  4. Show that
    1.  .   (2 marks)
    2.  
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Show Worked Solution

(i)   

 

 

(ii)   

♦ Mean mark 37%
 

(iii) 
 


 

Plane Geometry, EXT1 2010 HSC 5c

In the diagram,    is tangent to both the circles at  .

The points    and    are on the larger circle, and the line    is tangent to the smaller circle at  . The line    intersects the smaller circle at  .

 

Plane Geometry, EXT1 2010 HSC 5c

Copy or trace the diagram into your writing booklet 

  1. Explain why     (1 mark)
  2. Explain why     (1 mark)
  3. Hence show that    bisects  .    (2 marks)
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Show Worked Solution
(i)    Plane Geometry, EXT1 2010 HSC 5c Answer

 

♦ Mean mark 47%
(ii)   
   

 

(iii)  

 

♦♦ Mean mark 22%
IMPORTANT: Recognising that angles in the alternate segment are equal is an examiner favourite. LOOK FOR IT!

 

Calculus, EXT1 C2 2010 HSC 5b

Let    for  . 

  1. By differentiating  , or otherwise, show that    for  .   (3 marks)

  2. Given that    is an odd function, sketch the graph  .     (1 mark)

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  2. Inverse Functions, EXT1 2010 HSC 5b Answer

 

 

 

 

 

 

Show Worked Solution
♦ Mean mark 48%
MARKER’S COMMENT: The most common errors were not being able to differentiate and not recognising the significance of .
i.   
 
   
   
   

 

 

 

 
 

 

 

♦ Mean mark 35%
ii.   
 

Inverse Functions, EXT1 2010 HSC 5b Answer

Trigonometry, EXT1 T3 2010 HSC 4b

  1. Express    in the form  ,
     
    where    and  .    (3 marks)

  2. Hence, or otherwise, solve  ,   
  3. for  .    (2 marks)

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Show Worked Solution
i.   
 
 
 
 

 

 
 

 

 

♦ Mean mark part (ii) 49%
MARKER’S COMMENT: Many students did not check their answers against the stated domain for .
ii.   
 
 
 

 

Inverse Functions, EXT1 2010 HSC 3b

Let  .  The diagram shows the graph  .
 

 Inverse Functions, EXT1 2010 HSC 3b
 

  1. The graph has two points of inflection.  
  2. Find the    coordinates of these points.   (3 marks)
  3.  
  4. Explain why the domain of    must be restricted if    is to have an inverse function.    (1 mark)
  5. Find a formula for    if the domain of    is restricted to  .   (2 marks)
  6. State the domain of  .    (1 mark)
  7. Sketch the curve  .    (1 mark)
  8. (1)   Show that there is a solution to the equation    between    and  .   (1 mark)
  9. (2)   By halving the interval, find the solution correct to one decimal place.   (1 mark)

 

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  1.  
  2.  
  5.  

  6.  
  8.  
  9. Inverse Functions, EXT1 2010 HSC 3b Answer
  10. (1)  
  11. (2)  
Show Worked Solution
(i)   
 
 
   
   

 

 
   
 
   
COMMENT: It is also valid to show that is an even function and if a P.I. exists at , there must be another P.I. at .
 

 

 

(ii)  
 
 
 

 

(iii)  

 

 

 


 

♦ Parts (iv) and (v) were poorly answered with mean marks of 39% and 49% respectively.
(iv)  

 

(v) 

Inverse Functions, EXT1 2010 HSC 3b Answer

 

(vi)(1)  

 
 

 

♦ Mean mark 37%.
MARKER’S COMMENT: Better responses showed the change in sign between and as shown in the solution.

 

(vi)(2)  
   

 

Combinatorics, EXT1 A1 2010 HSC 3a

At the front of a building there are five garage doors. Two of the doors are to be painted red, one is to be painted green, one blue and one orange. 

  1. How many possible arrangements are there for the colours on the doors?   (1 mark)

  2. How many possible arrangements are there for the colours on the doors if the two red doors are next to each other?    (1 mark)

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Show Worked Solution
i.   
   

 

♦ Mean mark 50%
MARKER’S COMMENT: Drawing a diagram was a successful strategy for many students in this part.
ii.   
 

Calculus, EXT1 C1 2011 HSC 7a

The diagram shows two identical circular cones with a common vertical axis.  Each cone has height cm and semi-vertical angle 45°.
 

2011 7a

The lower cone is completely filled with water. The upper cone is lowered vertically into the water as shown in the diagram. The rate at which it is lowered  is given by  

,

where cm is the distance the upper cone has descended into the water after seconds.

As the upper cone is lowered, water spills from the lower cone. The volume of water remaining in the lower cone at time is  cm³.

  1. Show that  .   (1 mark)

  2. Find the rate at which is changing with respect to time when  .     (2 marks)

  3. Find the rate at which is changing with respect to time when the lower cone has lost    of its water. Give your answer in terms of .   (2 marks)

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  2. ³
  3. ³
Show Worked Solution

i.   

 ♦ Mean mark 42% 

 

 
 

 

 

ii. 

 

  

  ³

 

iii. 

♦♦♦ Mean mark 12%
MARKER’S COMMENT: Many unsuccessful answers attempted to find an alternate version of . Part (ii) directed students directly toward the correct strategy.

   

  ³

 
³

Statistics, EXT1 S1 2011 HSC 6c

A game is played by throwing darts at a target. A player can choose to throw two or three darts.

Darcy plays two games. In Game 1, he chooses to throw two darts, and wins if he hits the target at least once. In Game 2, he chooses to throw three darts, and wins if he hits the target at least twice.

The probability that Darcy hits the target on any throw is  , where  .

  1. Show that the probability that Darcy wins Game 1 is  .    (1 mark)

  2. Show that the probability that Darcy wins Game 2 is  .     (1 mark)

  3. Prove that Darcy is more likely to win Game 1 than Game 2.    (2 marks)

  4. Find the value of    for which Darcy is twice as likely to win Game 1 as he is to win Game 2.    (2 marks)

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Show Worked Solution

i. 

♦ Mean mark 47%.


 

 
 
 

 

ii. 

♦ Mean mark 40%.

 
 

 

♦♦♦ Mean mark 17%.
COMMENT: It is critical for students to utilise the limits of    to answer part (iii).

iii. 

 
   
   

 


 

iv. 

♦♦ Mean mark 21%.
MARKER’S COMMENT: BE CAREFUL in formulating your equation here. Many students multiplied the wrong side by 2.
 

  

 
 

 

Trig Ratios, EXT1 2011 HSC 5a

In the diagram,    is a point on the unit circle    at an angle    from the positive  -axis, where  . The line through    and    intersects the line    at  . The points    and   are on the  -axis.
 

 
 

  1. Use the fact that    and    are similar to show that
     
  2.      .  (2 marks)
  3.  
  4. Show that  .    (1 mark)
  5.  
  6. Show that  .   (1 mark)
  7.  
  8. Hence, or otherwise, show that  .   (1 mark)
  9.  
  10. Hence, or otherwise, solve  , where  .   (2 marks)

 

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Show Worked Solution
(i)

 

♦ Mean mark 41%
MARKER’S COMMENT: When questions direct you to use a certain fact for a proof, use that fact!

 
 

 

(ii)   

♦ Mean mark 35%
 
 
 
 
 

 

(iii)  

♦♦♦ Mean mark 11%

 
 
 

 

(iv)  

♦♦ Mean mark 28%

 

 
 
 

 

 

♦ Mean mark 45%
(v)   
 
 

Plane Geometry, EXT1 2011 HSC 4b

In the diagram, the vertices of    lie on the circle with centre  . The point    lies on    such that    is isosceles and  .

Copy or trace the diagram into your writing booklet. 

  1. Explain why  .    (1 mark)
  2. Prove that    is a cyclic quadrilateral.    (2 marks)
  3. Let    be the midpoint of    and    the centre of the circle through  and  . 
  4. Show that    and    are collinear.   (1 mark)
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  3.  
Show Worked Solution
(i)

 

♦ Mean mark 38%
STRATEGY: Proving part (ii) by showing opposite angles are supplementary also worked but was more time consuming.

(ii)  

 

 

(iii)

 

♦♦♦ Mean mark 7%. 2nd hardest question in the 2011 paper!

 

 

Quadratic, EXT1 2011 HSC 3b

The diagram shows two distinct points    and    on the parabola  .  The point    is the intersection of the tangents to the parabola at    and  . 

 

  1. Show that the equation of the tangent to the parabola at    is  .   (2 marks)
  2. Using part  , write down the equation of the tangent to the parabola at  .     (1 mark)
  3. Show that the tangents at    and    intersect at
    .   (2 marks)
  5. Describe the locus of    as    varies, stating any restriction on the  -coordinate.   (2 marks)

 

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Show Worked Solution
(i)

 

 

(ii) 

MARKER’S COMMENT: Many students derived this equation rather than substituting the new parameter, costing them valuable time. This is a benefit of using the parametric approach.

 

(iii) 

 

 

(iv) 

♦♦ Mean mark of 22%. 
MARKER’S COMMENT: Many students stated the locus as rather than realising it had to be a straight line since ½, and that simply restricted the values of .

Mechanics, EXT2* M1 2011 HSC 3a

The equation of motion for a particle undergoing simple harmonic motion is 

 ,

where is the displacement of the particle from the origin at time , and is a positive constant.

  1. Verify that  , where and are constants, is a solution of the equation of motion.    (1 mark)

  2. The particle is initially at the origin and moving with velocity . 

     

    Find the values of and in the solution  .    (2 marks)

  3. When is the particle first at its greatest distance from the origin?   (1 mark)

  4. What is the total distance the particle travels between    and  ?   (1 mark)

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Show Worked Solution
i.  
 
 
   
   

 

ii.   

  
♦♦ Mean mark part (iii) 47%
 

iii.   

 

 

iv. 

♦♦ Mean mark 22%
MARKER’S COMMENT: Many students found the displacement at rather than the distance travelled while another common error found distance as 2 x amplitude.

 

 

Calculus, EXT1 C1 2012 HSC 14c

A plane takes off from a point . It flies due north at a constant angle to the horizontal. An observer is located at , 1 km from , at a bearing 060° from .

Let km be the distance from to the plane and let km be the distance from the observer to the plane. The point is on the ground directly below the plane.
 

2012 14c
 

  1. Show that  .   (3 marks)

  2. The plane is travelling at a constant speed of 360 km/h.
  3. At what rate, in terms of  , is the distance of the plane from the observer changing 5 minutes after take-off?    (2 marks)

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Show Worked Solution

i. 

♦ Mean mark 42%
IMPORTANT: Students should always be looking for opportunities to use the identity to clean up calculations with trig functions.

 

 

 

 
 

 

ii.  

♦♦♦ Mean mark 14%

 

 

 

 

Mechanics, EXT2* M1 2012 HSC 14b

A firework is fired from , on level ground, with velocity metres per second at an angle of inclination  . The equations of motion of the firework are

 and. (Do NOT prove this.) 

The firework explodes when it reaches its maximum height.
 

2012 14b
 

  1. Show that the firework explodes at a height of    metres.   (2 marks)

  2. Show that the firework explodes at a horizontal distance of    metres from  .    (1 mark)

  3. For best viewing, the firework must explode at a horizontal distance between 125 m and 180 m from , and at least 150 m above the ground.

     

    For what values of    will this occur?   (3 mark)

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Show Worked Solution

i.   


 

 

 
 

 

ii. 

 
 

 

iii. 

♦♦ Mean mark 28%
MARKER’S COMMENT: Many students ignored the findings in earlier parts and tried unsuccessfully to solve inequalities that still contained .

 

 

 

 

 

 

 

.

Plane Geometry, EXT1 2012 HSC 14a

The diagram shows a large semicircle with diameter    and two smaller semicircles with diameters    and  , respectively, where    is a point on the diameter  . The point    is the centre of the semicircle with diameter  .

The line perpendicular to    through    meets the largest semicircle at the point  . The points    and    are the intersections of the lines    and    with the smaller semicircles. The point    is the intersection of the lines    and  .

Copy or trace the diagram into your writing booklet.  

  1. Explain why    is a rectangle.   (1 mark)
  2. Show that    and    are congruent.     (2 marks)
  3. Show that the line    is a tangent to the semicircle with diameter  .    (1 mark)
  4.  
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Show Worked Solution
(i)

♦ Mean mark 49%

 

 

 

(ii)  

♦ Mean mark part (iii) 41%
STRATEGY: The congruency proof in part (ii) provided the critical information to answer this efficiently. Keep previous parts of questions front and centre of your mind when working on a solution.

 

(iii)  

 

Geometry and Calculus, EXT1 2012 HSC 13d

The concentration of a drug in the blood of a patient    hours after it was administered is given by

where    is measured in .  

  1. Initially the concentration of the drug in the blood of the patient increases until it reaches a maximum, and then it decreases. Find the time when this maximum occurs.   (3 marks)
  3. Taking    as a first approximation, use one application of Newton’s method to find approximately when the concentration of the drug in the blood of the patient reaches  .   (2 marks)
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(i)   
 
   

 

♦ Mean mark 45% 
MARKER’S COMMENT: Only a minority of students could successfully apply Newton’s method in this instance. A must know area asked every year!
(ii)   
 
 
 
 
 
 
 

 

Mechanics, EXT2* M1 2012 HSC 13c

A particle is moving in a straight line according to the equation

, 

where is the displacement in metres and is the time in seconds.

  1. Prove that the particle is moving in simple harmonic motion by showing that  satisfies an equation of the form  .  (2 marks)

  2. When is the displacement of the particle zero for the first time?    (3 marks)

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Show Worked Solution

i. 

 
 
 

 

ii. 

♦ Mean mark 42%.
IMPORTANT: The critical insight required to solve is to realise that the cosine of the difference between 2 angles, i.e. , applies.

 

 Calculus in the Physical World, EXT1 2012 HSC 13c Answer

 

 

 
 

Quadratic, EXT1 2012 HSC 12d

Let    be a fixed point on the  -axis with  . The point    is on the  -axis. The point    is on the  -axis so that    is right-angled with the right angle at  . The point    is chosen so that    is a rectangle as shown in the diagram.

2012 12d

  1. Show that    lies on the parabola given parametrically by          (2 marks)
    1. and. 
  2. Write down the coordinates of the focus of the parabola in terms of  .    (1 mark)
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Show Worked Solution
♦♦ Mean mark 30%.
IMPORTANT: The highlighted right-angle in this question should flag the potential use of .

 

 

(ii)   
 

 

Statistics, EXT1 S1 2012 HSC 12c

Kim and Mel play a simple game using a spinner marked with the numbers  1, 2, 3, 4 and 5.
 

2012 12c
 

The game consists of each player spinning the spinner once. Each of the five numbers is equally likely to occur.

The player who obtains the higher number wins the game.

If both players obtain the same number, the result is a draw.

  1. Kim and Mel play one game. What is the probability that Kim wins the game?   (1 mark)

  2. Kim and Mel play six games. What is the probability that Kim wins exactly three games?    (2 marks)

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Show Worked Solution

i. 

 

♦ Mean mark 37%

 

 

 

ii. 


 

Calculus, 2ADV C3 2009 HSC 10

  1. Show that the graph of    has no turning points.   (2 marks)

  2. Find the point of inflection of  .     (1 mark)

  3. i. Show that  for  .   (1 mark)

     

    ii. Let  .

     

        Use the result in part c.i. to show that    for all  .   (2 marks)

  1. Sketch the graphs of    and    for  .    (2 marks)

  2. Show that  .   (2 marks)

  3. Find the area enclosed by the graphs of    and  , and the straight line  .   (2 marks)

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  3. i. 

     

    ii.  

  4.  
        Geometry and Calculus, 2UA 2009 HSC 10 Answer
  6. ²
Show Worked Solution
a.   
♦♦ Mean mark 28% for all of Q10 (note that data for each question part is not available).
 

 
 

 

 

b.   

 

 
 

 

c.i.     
 
 

 

c.ii.  
 
 

MARKER’S COMMENT: When 2 graphs are drawn on the same set of axes, you must label them. 
 

d. 

Geometry and Calculus, 2UA 2009 HSC 10 Answer
e.   
 
 
 
 

 

f.   
   
   
   
   
   
    ²

Calculus, 2ADV C3 2009 HSC 8a

Geometry and Calculus, 2UA 2009 HSC 8a

The diagram shows the graph of a function  . 

  1. For which values of    is the derivative,  , negative?    (1 mark)

  2. What happens to    for large values of  ?    (1 mark)

  3. Sketch the graph  .     (2 marks)

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  1.  


  2.  

  3.   
    Geometry and Calculus, 2UA 2009 HSC 8a Answer

Show Worked Solution

i.   

♦♦ Exact data not available.

 

ii.   

 

♦♦ Exact data not available.
MARKER’S COMMENT: Poorly drawn graphs with axes not labelled and inaccurate scales were common.

iii.

  Geometry and Calculus, 2UA 2009 HSC 8a Answer

Trigonometry, 2ADV T3 2009 HSC 7b

Between 5 am and 5 pm on 3 March 2009, the height, , of the tide in a harbour was given by

where    is in metres and    is in hours, with    at 5 am. 

  1. What is the period of the function  ?    (1 mark)

  2. What was the value of    at low tide, and at what time did low tide occur?     (2 marks)

  3. A ship is able to enter the harbour only if the height of the tide is at least 1.35 m.

     

    Find all times between 5 am and 5 pm on 3 March 2009 during which the ship was able to enter the harbour.    (3 marks)

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Show Worked Solution

i.   

 
 

 

 

ii. 

IMPORTANT: Using for a minimum here is very effective and time efficient. This property of trig functions is often very useful in harder questions.

 

 

 

 


iii. 

   

 

Trig Calculus, 2UA 2009 HSC 7b Answer

 

Calculus, 2ADV C1 2009 HSC 6c

The diagram illustrates the design for part of a roller-coaster track. The section    is a straight line with slope 1.2, and the section    is a straight line with slope  – 1.8. The section    is a parabola  . The horizontal distance from the  -axis to    is 30 m.
 

2009 6c
 

In order that the ride is smooth, the straight line sections must be tangent to the parabola at    and at  .  

  1. Find the values of    and    so that the ride is smooth.   (3 marks)

  2. Find the distance  , from the vertex of the parabola to the horizontal line through  , as shown on the diagram.     (2 marks)

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Show Worked Solution
i.   
 
 
 
♦♦♦ Exact data unavailable.
MARKER’S COMMENT: Successful students equated the curve gradient to the straight section as a requirement for a smooth ride.
 


 

 

 

 

 

 

ii.   
 

 

 

 

 

 
 

 

 
 

 

 

Trigonometry, 2ADV T1 2009 HSC 5c

The diagram shows a circle with centre and radius 2 centimetres. The points    and    lie on the circumference of the circle and  .
 

2009 5c  
 

  1. There are two possible values of    for which the area of    is    square centimetres. One value is  .

     

    Find the other value.    (2 marks)

  2. Suppose that  .

     

    (1)  Find the area of sector     (1 mark)

     

    (2)  Find the exact length of the perimeter of the minor segment bounded by the chord    and the arc  .   (2 marks)

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

 

 

 

ii. (1)   
   
   
    ² 

 

ii. (2)   
 
 
 

 

 
 

 

Probability, 2ADV S1 2009 HSC 5b

On each working day James parks his car in a parking station which has three levels. He parks his car on a randomly chosen level. He always forgets where he has parked, so when he leaves work he chooses a level at random and searches for his car. If his car is not on that level, he chooses a different level and continues in this way until he finds his car.    

  1. What is the probability that his car is on the first level he searches?     (1 mark)

  2. What is the probability that he must search all three levels before he finds his car?   (1 mark)

  3. What is the probability that on every one of the five working days in a week, his car is not on the first level he searches?    (1 mark)

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Show Worked Solution

i. 
 

ii. 

♦♦ Mean marks of 31% and 39% for part (ii) and (iii) respectively.


 

iii. 

Plane Geometry, 2UA 2009 HSC 4c

In the diagram,    is a right-angled triangle, with the right angle at  . The midpoint of    is  , and  .

Plane Geometry, 2UA 2009 HSC 4c_1

Copy or trace the diagram into your writing booklet. 

  1. Prove that    is similar to  .   (2 marks)
  2. What is the ratio of    to  ?    (1 mark)
  3. Prove that    is isosceles.   (2 marks)
  4. Show that    can be divided into two isosceles triangles.   (1 mark)
  5. Plane Geometry, 2UA 2009 HSC 4c_2
  6. Copy or trace this triangle into your writing booklet and show how to divide it into four isosceles triangles.   (1 mark)
  7.  
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  5. Plane Geometry, 2UA 2009 HSC 4c_2 Answer1
Show Worked Solution
(i)   
 
 
 

 

(ii)  
   

 

 

(iii)

Plane Geometry, 2UA 2009 HSC 4c_1 Answer
 

 

(iv)  
   
   

 

(v)

Plane Geometry, 2UA 2009 HSC 4c_2 Answer1
♦♦ Mean mark 25%.
MARKER’S COMMENT: Use a ruler, draw large diagrams and always pay careful attention to previous parts of the question!

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).

 

Calculus, 2ADV C3 2010 HSC 9b

Let    be a function defined for  , with  . 

The diagram shows the graph of the derivative of  ,  . 

The shaded region    has area    square units. The shaded region    has area    square units. 

  1. For which values of    is    increasing?  (1 mark)

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

  3. Find the value of  .   (1 mark)

  4. Draw a graph of    for  .   (2 marks)

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  4.  
  5.  
Show Worked Solution
i.   
 
 

 

ii.  
 
 
 
 
 

 

♦♦♦ Parts (ii) and (iii) proved particularly difficult for students with mean marks of 12% and 11% respectively.

 

 

iii.  
   

 
 
 

 

 

  

♦♦ Mean mark 28%
EXAM TIP: Clearly identify THE EXTREMES when given a defined domain. In this case, the origin is obvious graphically, and the other extreme at , is CLEARLY LABELLED! 
iv.  2UA HSC 2010 9bi

Calculus, 2ADV C3 2010 HSC 8d

Let  , where is a constant.

Find the values of for which is an increasing function.   (2 marks)

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♦♦ Mean mark 28%.
MARKER’S COMMENT: The arithmetic required to solve   proved the undoing of many students.

 


 

 

Probability, 2ADV S1 2010 HSC 8b

Two identical biased coins are tossed together, and the outcome is recorded.

After a large number of trials it is observed that the probability that both coins land showing heads is  0.36.

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

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♦♦ Mean mark 28%.
NOTE: The most common error was . Ensure you understand why this does not apply.
 

 

 
 

 

 

Calculus, 2ADV C1 2010 HSC 7b

The parabola shown in the diagram is the graph  . The points    and    are on the parabola.
 

 
 

  1.  Find the equation of the tangent to the parabola at .   (2 marks)

  2.  Let be the midpoint of  .

     

    There is a point on the parabola such that the tangent at is parallel to  .

     

    Show that the line    is vertical.   (2 marks)  

  3. The tangent at meets the line at .

     

    Show that the line is a tangent to the parabola.  (2 marks)

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i.
 

 

 

 

 

 

♦ Mean mark 37%.
IMPORTANT: The critical understanding required for this question is that the gradient of needs to be equated to the gradient function (i.e. ).

ii.   

 

 

 

 

 

 

 

 

iii.    

♦♦ Mean mark 29%.

 

 

 

Trigonometry, 2ADV T1 2010 HSC 6b

The diagram shows a circle with centre and radius 5 cm.

The length of the arc is 9 cm. Lines drawn perpendicular to and at and respectively meet at  .
 

  1. Find    in radians.   (1 mark)

  2. Prove that    is congruent to  .   (2 marks)

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

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  4. ²
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i.   
 
 

 

ii.

 MARKER’S COMMENT: Know the difference between the congruency proof of and . Incorrect identification will lose a mark.


 

iii.   
 
 
♦♦ Mean mark below 30%.
MARKER’S COMMENT: Many students struggled to work in radians. Make sure you understand this concept.
 
 

 

iv.   
♦ Mean mark 35%.
 
 
 
  ²

 

 
  ²

 

 
  ²

Calculus, 2ADV C3 2010 HSC 6a

Let  .

  1. Show that the graph    has no stationary points.   (2 marks)

  2. Find the values of    for which the graph    is concave down, and the values for which it is concave up.    (2 marks)

  3. Sketch the graph  ,  indicating the values of the    and   intercepts.   (2 marks)

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  2.  

     

  3.  
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i. 

 

 

 
 
 

 

 

ii. 

MARKER’S COMMENT: The significance of the sign of the second derivative was not well understood by most students.

 

♦♦ Mean mark 33%.
MARKER’S COMMENT: Students are reminded to bring a ruler to the exam and use it to draw the axes for graphing and to help with an appropriate scale.

iii. 

 

 

Calculus, 2ADV C4 2010 HSC 5b

  1. Prove that  .   (1 mark)

  2. Hence prove that  .     (1 mark)

  3. Hence, use the identity    to find the exact value of

     

          .   (2 marks)

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i.   


 

 
 
 
 

 

ii.    

♦♦ Mean mark 31%.
 
 
 

 

iii. 

♦ Mean mark 37%.

Calculus, 2ADV C3 2011 HSC 9c

The graph    in the diagram has a stationary point when  , a point of inflection when  , and a horizontal asymptote  .
 

 Geometry and Calculus, 2UA 2011 HSC 9c
 

Sketch the graph   , clearly indicating its features at    and at  , and the shape of the graph as  .   (3 marks)

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Geometry and Calculus, 2UA 2011 HSC 9c Answer

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♦ Mean mark 43%
IMPORTANT: Examiners regularly ask questions that require the graphing of an given the  graph and vice-versa. KNOW IT!

Geometry and Calculus, 2UA 2011 HSC 9c Answer

Plane Geometry, 2UA 2012 HSC 16a

The diagram shows a triangle    with sides    and  . The points  ,    and    lie on the sides  ,    and  , respectively, so that    is a rhombus with sides of length  .

 

  1. Prove that    is similar to  .    (2 marks)
  2. Find an expression for    in terms of    and  .    (2 marks)
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(i)

♦ Mean mark 47%.

 

 

 

(ii) 

♦ Mean mark 31%.

 

Calculus, 2ADV C3 2012 HSC 14a

A function is given by  . 

  1. Find the coordinates of the stationary points of    and determine their nature.   (3 marks)

  2. Hence, sketch the graph     showing the stationary points.   (2 marks)

  3. For what values of    is the function increasing?   (1 mark)

  4. For what values of    will    have no solution?   (1 mark)

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  2.  
        
        2UA HSC 2012 14ai
     
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i.
 
 

 

 

 

 

 

 

 
 
 

 

ii.  2UA HSC 2012 14ai
♦ Mean mark 42%
MARKER’S COMMENT: Be careful to use the correct inequality signs, and not carelessly include ≥ or ≤ by mistake.

 

iii.
 

 

iv.   

♦♦♦ Mean mark 12%.

Calculus, 2ADV C4 2011 HSC 6c

The diagram shows the graph   . 
  

2011 6c 
  

  1. State the coordinates of  .   (1 mark)

  2. Evaluate the integral  .    (2 marks)

  3. Indicate which area in the diagram,  ,  ,   or  , is represented by the integral
     
           .   (1 mark)

  4. Using parts (ii) and (iii), or otherwise, find the area of the region bounded by the curve    and the  -axis, between    and   .   (1 mark)

  5. Using the parts above, write down the value of
     
         .   (1 mark)

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  4. ²
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i. 

 

ii. 


 

iii. 
 

iv.   
 

 

  ²

 

MARKER’S COMMENT: “Using the parts above” in part (v) was ignored by many students. Important to know when finding an area and evaluating an integral may differ.

v. 

Real Functions, 2UA 2011 HSC 6b

A point    moves so that the sum of the squares of its distance from each of the points    and    is equal to 40.

Show that the locus of    is a circle, and state its radius and centre.   (3 marks)

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♦ Mean mark 39%.
MARKER’S COMMENT: Challenging question with many students unable to handle the algebra in expanding and completing the squares.

 
 

 

 

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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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.

Calculus, 2ADV C4 2011 HSC 4d

  1. Differentiate    with respect to  .   (2 marks)

  2. Hence, or otherwise, find  .    (2 marks)

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IMPORTANT: Some students might find calculations easier by rewriting the equation as .
i.   
   

 

 
 

 

ii.   
   
   

Functions, 2ADV F2 2013 HSC 15c

  1. Sketch the graph  .   (1 mark)

  2. Using the graph from part (i), or otherwise, find all values of    for which the equation    has exactly one solution.   (2 marks)

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  1.  
    2UA 2013 HSC 15c Answer
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i. 

♦ Mean mark 49%
MARKER’S COMMENT: Many students drew diagrams that were “too small”, didn’t use rulers or didn’t use a consistent scale on the axes!

2UA 2013 HSC 15c Answer

 

ii.

   2UA 2013 HSC 15c1 Answer

 

♦♦ Mean mark 25%.
COMMENT: Students need a clear graphical understanding of what they are finding to solve this very challenging, Band 6 question.

 
 

  

 


 

Trigonometry, 2ADV T1 2013 HSC 14c

2013 14c

 
The right-angled triangle    has hypotenuse  . The point    is on    such that  ,   and  .

Using the sine rule, or otherwise, find the exact value of  .   (3 marks)

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♦ Mean mark 36%.
STRATEGY TIP: The hint to use the sine rule should flag to students that they will be dealing in non-right angled trig (i.e. ) and to direct their energies at initially finding and .

 

 
 
 

 

 
 

 

Trigonometry, 2ADV T3 2013 HSC 13a

The population of a herd of wild horses is given by

where    is time in months. 

  1. Find all times during the first 12 months when the population equals 375 horses.    (2 marks)

  2. Sketch the graph of    for  .    (2 marks)

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i. 

   
 

 

 

♦ Mean mark 39%
ii. 2UA 2013 13a ans

Financial Maths, STD2 F4 2011 HSC 28b

Norman and Pat each bought the same type of tractor for $60 000 at the same time. The value of their tractors depreciated over time.

The salvage value , in dollars, of each tractor, is its depreciated value after years.

Norman drew a graph to represent the salvage value of his tractor.
 

 2011 28b

  1. Find the gradient of the line shown in the graph.   (1 mark)

  2. What does the value of the gradient represent in this situation?   (1 mark)

  3. Write down the equation of the line shown in the graph.   (1 mark)

  4. Find all the values of that are not suitable for Norman to use when calculating the salvage value of his tractor. Explain why these values are not suitable.   (2 marks)

Pat used the declining balance formula for calculating the salvage value of her tractor. The depreciation rate that she used was 20% per annum.

  1. What did Pat calculate the salvage value of her tractor to be after 14 years?   (2 marks)

  2. Using Pat’s method for depreciation, describe what happens to the salvage value of her tractor for all values of greater than 15.   (1 mark)

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  7.  
Show Worked Solution
♦♦♦ Mean mark 14%
COMMENT: The intercepts of both axes provide points where the gradient can be quickly found.
i.   
   
   

 

♦ Mean mark 37%

ii.   

 

♦♦ Mean mark 28%
COMMENT: Using the general form is quick here because you have the gradient (from part (i)) and the -intercept is obviously .
iii.  
 
 

 

iv.    

♦♦♦ Mean mark 20%

 

v.   
 
 

 

 

♦ Mean mark 37%
vi.  
 

Algebra, STD2 A4 2011 HSC 28a

The air pressure, , in a bubble varies inversely with the volume, , of the bubble. 

  1. Write an equation relating , and , where is a constant.    (1 mark)

  2. It is known that when .

     

    By finding the value of the constant, , find the value of when .    (2 marks)

  3. Sketch a graph to show how varies for different values of .

     

    Use the horizontal axis to represent volume and the vertical axis to represent air pressure.   (2 marks)


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  3.  
     
Show Worked Solution
♦ Mean mark (i) 39%
COMMENT: Expressing the proportional relationship as the equation is a core skill here.
i.
   

 

ii.

 

  

♦ Mean mark (ii) 47%
 

  

♦♦ Mean mark (iii) 26%
COMMENT: An inverse relationship is reflected by a hyperbola on the graph.
iii.

Financial Maths, STD2 F5 2011 HSC 27d

Josephine invests $50 000 for 15 years, at an interest rate of 6% per annum, compounded annually.

Emma invests $500 at the end of each month for 15 years, at an interest rate of 6% per annum, compounded monthly. 

Financial gain is defined as the difference between the final value of an investment and the total contributions.

Who will have the better financial gain after 15 years? Using the Table below* and appropriate formulas, justify your answer with suitable calculations.   (4 marks)
  2UG-2011-27d1

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♦ Mean mark 42%
COMMENT: Note that compound interest vs annuity comparisons are commonly tested.

 
 

 

 

 

 

 

 

 
 
 

 

Statistics, STD2 S5 2011 HSC 27c

Two brands of light bulbs are being compared. For each brand, the life of the light bulbs is normally distributed.

2011 27c

  1. One of the Brand B light bulbs has a life of 400 hours. 

     

    What is the  -score of the life of this light bulb?  (1 mark)

  2. A light bulb is considered defective if it lasts less than 400 hours. The following claim is made:

     

    ‘Brand A light bulbs are more likely to be defective than Brand B light bulbs.’

     

    Is this claim correct? Justify your answer, with reference to  -scores or standard deviations or the normal distribution.   (2 marks)

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i.   

 

ii.   

♦ Mean mark 42%
MARKER’S COMMENT: A number of students found drawing normal curves in their solution advantageous.

 

Measurement, 2UG 2011 HSC 27b

Pontianak has a longitude of 109°E, and Jarvis Island has a longitude of 160°W`.

Both places lie on the Equator. 

  1. Find the shortest distance between these two places, to the nearest kilometre. You may assume that the radius of the Earth is 6400 km.   (2 marks)
  2.  
  3. The position of Rabaul is 4° to the south and 48° to the west of Jarvis Island. What is the latitude and longitude of Rabaul?    (2 marks)
  4.  
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♦♦ Mean mark 29%
(i)
   
 
 
 
 

 

♦♦ Mean mark 33%
(ii)
 
 
 
 
 
 
 
 

Statistics, STD2 S1 2011 HSC 27a

A company sells handbags in Paris, New York and Florence. 

Use the data in the table to complete the area chart below.   (2 marks)

 2011 27a 

2UG 2011 27a

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 2011 27a

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♦♦ Mean mark 27%
COMMENT: Most common error was to not realise Area charts show cumulative data.

2011 27a

Financial Maths, 2UG 2011 HSC 26c

Furniture priced at  $20 000  is purchased. A deposit of 15% is paid. 

The balance is borrowed using a flat-rate loan at 19% per annum interest, to be repaid in equal monthly instalments over five years.

What will be the amount of each monthly instalment? Justify your answer with suitable calculations.   (4 marks) 

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Show Worked Solution

♦ Mean mark 44%