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Algebra, STD2 A4 2012 HSC 30c

In 2010, the city of Thagoras modelled the predicted population of the city using the equation

.

That year, the city introduced a policy to slow its population growth. The new predicted population was modelled using the equation

.

In both equations, is the predicted population and is the number of years after 2010.  

The graph shows the two predicted populations.
 

  1. Use the graph to find the predicted population of Thagoras in 2030 if the population policy had NOT been introduced.   (1 mark)

  2. In each of the two equations given, the value of is 3 000 000.

     

    What does represent?   (1 mark)

  3. The guess-and-check method is to be used to find the value of , in  .

     

    (1) Explain, with or without calculations, why 1.05 is not a suitable first estimate for .  (1 mark)

     

    (2) With    and  , use the guess-and-check method and the equation    to estimate the value of to two decimal places. Show at least TWO estimate values for , including calculations and conclusions.  (2 marks)

  4. The city of Thagoras was aiming to have a population under 7 000 000 in 2050. Does the model indicate that the city will achieve this aim?

     

    Justify your answer with suitable calculations.  (2 marks)

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  1.  
  2.  
  3.  

     

Show Worked Solution

i.   

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii.    


 

♦ Mean mark part (ii) 48%

iii. (1) 

   
 

iii. (2) 

 

 

 

 

 

iv. 

 

 

Algebra, STD2 A4 2012 HSC 30b

A golf ball is hit from point to point , which is on the ground as shown. Point is 30 metres above the ground and the horizontal distance from point to point is  300 m.
 

The path of the golf ball is modelled using the equation 

 

where 

is the height of the golf ball above the ground in metres, and 

is the horizontal distance of the golf ball from point in metres.

The graph of this equation is drawn below.

  

  1. What is the maximum height the ball reaches above the ground?    (1 mark)

  2. There are two occasions when the golf ball is at a height of 35 metres.

     

    What horizontal distance does the ball travel in the period between these two occasions?   (1 mark)

  3. What is the height of the ball above the ground when it still has to travel a horizontal distance of 50 metres to hit the ground at point ?   (1 mark)

  4. Only part of the graph applies to this model.

     

    Find all values of that are not suitable to use with this model, and explain why these values are not suitable.   (2 marks)

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Show Worked Solution

i.   

COMMENT: With a mean mark of 92% in (i), a classic example of low hanging fruit in later questions.

 

ii.  

 

 

iii.   

MARKER’S COMMENT: Responses for (iii) in the range    were deemed acceptable estimates read off the graph.

 

iv.   

♦♦♦ Mean mark (iv) 12%
MARKER’S COMMENT: Many students did not refer to the domain as unsuitable to the model.

Statistics, STD2 S1 2009 HSC 21 MC

The mean of a set of ten scores is 14. Another two scores are included and the new mean is 16.

What is the mean of the two additional scores?

  1.    4
  2.    16
  3.    18
  4.    26
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Show Worked Solution
♦♦♦ Mean mark 28%.

 

 

 

 

Algebra, STD2 A4 2013 HSC 22 MC

Leanne wants to build a rectangular vegetable garden in her backyard. She has 20 metres of fencing and will use a wall as one side of the garden. The plan for her garden is shown, where    metres is the width of her garden.
 

 

Which equation gives the area,  ,  of the vegetable garden?

  1.    
  2.    
  3.    
  4.    
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Show Worked Solution
♦♦♦ Mean mark 24% (lowest mean of any MC question in 2013 exam)

 

Statistics, STD2 S5 2013 HSC 20 MC

There are  60 000  students sitting a state-wide examination. If the results form a normal distribution, how many students would be expected to score a result between 1 and 2 standard deviations above the mean?

You may assume for normally distributed data that:

    • 68% of scores have  -scores between  – 1 and 1
    • 95% of scores have  -scores between  – 2 and 2
    • 99.7% of scores have  -scores between  – 3 and 3.
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Show Worked Solution
♦♦ Mean mark 24% 


 

 

Quadratic, 2UA 2012 HSC 16c

The circle  ,  where    and  ,  lies inside the parabola  .  The circle touches the parabola at exactly two points located symmetrically on opposite sides of the  -axis, as shown in the diagram.

2012 16c

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

(i)    

 

♦♦♦ Mean mark 18%.
MARKER’S COMMENT: Few students were successful in this part, with those that solved the simultaneous equations correctly often not realising that the    was a critical element in solving the problem.

 

(ii)  

♦♦♦ If you found this part difficult, don’t despair, it is a beast. With a mean mark of 0%, it ranks as one of the hardest questions in the history of the course.

 

Calculus, 2ADV C3 2008 HSC 10b

 

The diagram shows two parallel brick walls    and    joined by a fence from    to  .  The wall    is    metres long and  .  The fence    is    metres long.

A new fence is to be built from    to a point    somewhere on  .  The new fence    will cross the original fence    at  .

Let    metres, where  .

  1. Show that the total area,    square metres, enclosed by    and    is given by
     
         .   (3 marks)

  2. Find the value of    that makes    as small as possible. Justify the fact that this value of    gives the minimum value for  .  (3 marks)

  3. Hence, find the length of    when    is as small as possible.    (1 mark)

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Show Worked Solution
i.

♦♦♦ Low mean marks highlighted (although exact data not available before 2009).

 

 

 
 
 
 
 
 
 

 

ii.   

MARKER’S COMMENT: Students who could not complete part (i) are reminded that they can still proceed to part (ii) and attempt to differentiate the result given.
Note that  and are constants when differentiating. 

 

 

iii.   

 
 

 

Calculus, 2ADV C3 2009 HSC 9b

An oil rig,  ,  is 3 km offshore. A power station,  ,  is on the shore. A cable is to be laid from  to  .  It costs $1000 per kilometre to lay the cable along the shore and $2600 per kilometre to lay the cable underwater from the shore to  .

The point    is the point on the shore closest to  ,  and the distance    is 5 km.

The point    is on the shore, at a distance of    km from  ,  as shown in the diagram.


  

  1. Find the total cost of laying the cable in a straight line from    to    and then in a straight line from    to  .   (1 mark)

  2. Find the cost of laying the cable in a straight line from    to  .  (1 mark)

  3. Let    be the total cost of laying the cable in a straight line from    to  ,  and then in a straight line from  to  .
     
    Show that  .    (2 marks)

  4. Find the minimum cost of laying the cable.    (4 marks)

  5. New technology means that the cost of laying the cable underwater can be reduced to $1100 per kilometre.

     

    Determine the path for laying the cable in order to minimise the cost in this case.    (2 marks)

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Show Worked Solution
♦♦ Although specific data is unavailable for question parts, mean marks were 35% for Q9 in total.
i.   
 
 

 

 

ii.   

 
 

 

 
 

 

iii. 

 
 

 

iv.   

 

IMPORTANT: Tougher derivative questions often require students to deal with multiple algebraic constants. See Worked Solutions in part (iv).

 
MARKER’S COMMENT: Check the nature of the critical points in these type of questions. If using the first derivative test, make sure some actual values are substituted in.

 
   

 

 

v.   

 
 

 

 

 

MARKER’S COMMENT: Many students failed to interpret a correct calculation of    as providing no solution.

.

Calculus, 2ADV C3 2011 HSC 10b

A farmer is fencing a paddock using    metres of fencing. The paddock is to be in the shape of a sector of a circle with radius    and sector angle  in radians, as shown in the diagram.

2011 10b

  1. Show that the length of fencing required to fence the perimeter of the paddock is
      
       .    (1 mark)

  2. Show that the area of the sector is  .    (1 mark) 

  3. Find the radius of the sector, in terms of  , that will maximise the area of the paddock.    (2 marks)

  4. Find the angle    that gives the maximum area of the paddock.    (1 mark)

  5. Explain why it is only possible to construct a paddock in the shape of a sector if
     
            (2 marks)

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Show Worked Solution

i.   

 
 

 

ii.   .

♦ Mean mark 41%.
TIP: Area of a sector the percentage of the circle that the sector angle accounts for, i.e. radians. .
 
 
 

 

iii. 

♦♦ Mean mark 29%
MARKER’S COMMENT: The second derivative test proved much more successful and easily proven in this part. Make sure you are comfortable choosing between it and the 1st derivative test depending on the required calcs.

 

 

iv.   

♦♦♦ Mean mark 20%
 

 

v. 

♦♦♦ Mean mark 4%. A BEAST!
MARKER’S COMMENT: When asked to ‘explain’, students should support their answer with a mathematical argument.

 

Mechanics, EXT2* M1 2009 HSC 6a

Two points, and ,  are on cliff tops on either side of a deep valley. Let and be the vertical and horizontal distances between and as shown in the diagram. The angle of elevation of from is  ,  so that  .
 

2009 6a
 

At time ,  projectiles are fired simultaneously from and .  The projectile from is aimed at , and has initial speed at an angle of    above the horizontal. The projectile from is aimed at and has initial speed at an angle    below the horizontal.

The equations of motion for the projectile from are

  and   ,

and the equations for the motion of the projectile from are

  and   ,     (DO NOT prove these equations.)

  1. Let be the time at which  .
     
    Show that  .   (1 mark)

  2. Show that the projectiles collide.    (2 marks)

  3. If the projectiles collide on the line  ,  where  ,  show that
     
         .    (1 mark)

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  1.  
Show Worked Solution

i.  

 
 

 

ii.  

♦♦♦ Mean mark data not available although “few” students were able to complete this part.
MARKER’S COMMENT: Better responses showed  ,  or    which eliminated  ½  from the algebra. Substituting  was a key element in completing this proof.
 
 
 
 
 
 

 

.

 

iii. 

MARKER’S COMMENT: Students must show sufficient working in proofs to convince markers they have derived the answer themselves.
 
 
 

Mechanics, EXT2* M1 2010 HSC 6b

A basketball player throws a ball with an initial velocity    m/s at an angle of    to the horizontal. At the time the ball is released its centre is at  , and the player is aiming for the point  as shown on the diagram. The line joining    and    makes an angle    with the horizontal, where  .
 

6b
 

Assume that at time    seconds after the ball is thrown its centre is at the point  ,  where

.      (DO NOT prove this.)

  1. If the centre of the ball passes through    show that
     
             (3 marks)

    (2) What happens to    as   ?    (1 mark)

  2. For a fixed value of  ,  let  .
     
    Show that     when    (2 marks)

  3. Using part (a)(ii)* or otherwise show that  ,   when  .    (1 mark)

     

    *Please note for the purposes of this question, part (a)(ii) showed that  when  ,  then   

  4. Explain why    is a minimum when       (2 marks)

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  1.  
  2. (1)  

     

    (2)  

  4.  
Show Worked Solution

i.   

♦ Mean mark 43%
IMPORTANT: It is critical to use the relationship  to achieve the required proof.

 

 
 
 

 

 

ii. (1) 

♦ Mean marks of 37% and 32% for parts (ii)(1) and (ii)(2) respectively.

ii. (2) 

 

iii.  .

 
 

 

♦♦ Mean mark part (iii) 21%
MARKER’S COMMENT: Many students failed to treat  as a constant when differentiating.

 

iv. 

 

♦♦♦ Mean mark 14%
MARKER’S COMMENT: Linking part (a)(ii) to this solution was a feature of the more efficient and successful approaches.

 

 

 

v.   

♦♦♦ Mean mark 9%
MARKER’S COMMENT: Only a small minority of students  explained correctly that    is a MIN when    is a MAX.

 

 

Mechanics, EXT2* M1 2013 HSC 13c

Points and are located metres apart on a horizontal plane. A projectile is fired from towards with initial velocity m/s at angle to the horizontal.

At the same time, another projectile is fired from towards with initial velocity m/s at angle to the horizontal, as shown on the diagram.

The projectiles collide when they both reach their maximum height.
 

2013 13c
 

The equations of motion of a projectile fired from the origin with initial velocity  m/s at angle    to the horizontal are

  and   .        (DO NOT prove this.)

  1. How long does the projectile fired from    take to reach its maximum height?  (2 marks)

  2. Show that  .    (1 mark)

  3. Show that  .    (2 marks)

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Show Worked Solution

i.   

 

 

 

 

ii.  

IMPORTANT: Part (i) in this example leads students to a very quick solution. Always look to previous parts for clues to direct your strategy.

 

 

iii 

 
 

 

♦ Mean mark 39%
IMPORTANT: Students should direct their calculations by reverse engineering the required result. in the proof means that   will appear in the working calculations.
 
 
 
 
 
 
 

 

Calculus, 2ADV C3 2012 HSC 16b

The diagram shows a point    on the unit circle    at an angle    from the positive  -axis, where  .

The tangent to the circle at    is perpendicular to  , and intersects the  -axis at  ,  and the line    intersects the  -axis at  .
 

 
 

  1. Show that the equation of the line    is  .    (2 marks)

  2. Find the length of    in terms of  .    (1 mark)

  3. Show that the area,  ,  of the trapezium    is given by 
     
             (2 marks)

  4. Find the angle    that gives the minimum area of the trapezium.    (3 marks)

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Show Worked Solution
i.    

♦♦ Mean mark 20%

 

ii.  

 

 

 

iii. 

 

               

♦♦ Mean mark 24%

 

 
  ²

 

iv.  

 
 
 
 
Mean mark 19%
IMPORTANT: Look for any opportunity to use the identity → often a key to simplifying difficult trig equations.
 

 

 

IMPORTANT: Is the 1st or 2nd derivative test easier here? Examiners have often made one significantly easier than the other.

.

Calculus, 2ADV C3 2013 HSC 14b

Two straight roads meet at    at an angle of 60°.  At time    car    leaves    on one road, and car    is 100km from    on the other road.  Car    travels away from    at a speed of 80 km/h, and car    travels towards    at a speed of 50 km/h.
 

2013 14b

The distance between the cars at time    hours is    km.

  1. Show that  .   (2 marks)

  2. Find the minimum distance between the cars.    (3 marks)

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Show Worked Solution

i.   

♦♦ Mean mark 26%

 
 
   

 

ii.   

♦♦ Mean mark 27%
ALGEBRA TIP: Finding the derivative of (rather than making the subject), makes calculations much easier. ENSURE you apply the test to confirm a minimum.

 
 
 

 

Calculus, EXT1* C1 2013 HSC 16b

Trout and carp are types of fish. A lake contains a number of trout. At a certain time, 10 carp are introduced into the lake and start eating the trout. As a consequence, the number of trout,  ,  decreases according to

,

where is the time in months after the carp are introduced.

The population of carp,  ,  increases according to  .

  1. How many trout were in the lake when the carp were introduced?    (1 mark)

  2. When will the population of trout be zero?    (1 mark)

  3. Sketch the number of trout as a function of time.     (1 marks)

  4. When is the rate of increase of carp equal to the rate of decrease of trout?    (3 marks)

  5. When is the number of carp equal to the number of trout?    (2 marks)

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  3.  
     
    2UA 2013 HSC 16b Answer
Show Worked Solution

i.   

MARKER’S COMMENT: A number of students did not equate in this part.

 

ii.   

NOTE: The last line of the solution isn’t necessary but is included as good practice as a check that the answer matches the exact question asked.
 
 

 

 

iii.   2UA 2013 HSC 16b Answer

♦♦ Mean mark 33% for part (iii)

 

iv.   


  

♦♦♦ Mean mark 20%
COMMENT: Students who progressed to received 2 full marks in this part. Show your working!

 
 

 

 
 

 

v.   

♦♦♦ Mean mark 4%!
MARKER’S COMMENT: Correctly applying substitution to exponentials to form a solvable quadratic proved very difficult for almost all students.

 

 
 

Calculus, 2ADV C3 2011 HSC 7b

The velocity of a particle moving along the -axis is given by

,

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

  1. Show that the particle is initially at rest.     (1 mark)

  2. Show that the acceleration of the particle is always positive.     (1 mark)

  3. Explain why the particle is moving in the positive direction for all  .     (2 marks)

  4. As  , the velocity of the particle approaches a constant.

     

    Find the value of this constant.     (1 mark) 

  5. Sketch the graph of the particle's velocity as a function of time.     (2 marks)

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Show Worked Solution

i.   

 
 

 

 

MARKER’S COMMENT: Students whose working showed as , tended to score highly in this question.

ii.   

.

.
 

.
 

iii. 

♦♦♦ Mean mark 22%
COMMENT: Students found part (iii) the most challenging part of this question by far.


 

.
 

iv.   ,  


 

 

IMPORTANT: Use previous parts to inform this diagram. i.e. clearly show velocity was zero at    and the asymptote at  . 
v.   

Calculus in the Physical World, 2UA 2011 HSC 7b Answer

Financial Maths, 2ADV M1 2008 HSC 9b

Peter retires with a lump sum of $100 000. The money is invested in a fund which pays interest each month at a rate of 6% per annum, and Peter receives a fixed monthly payment from the fund. Thus the amount left in the fund after the first monthly payment is   .

  1. Find a formula for the amount, , left in the fund after monthly payments.   (2 marks)

  2. Peter chooses the value of so that there will be nothing left in the fund at the end of the 12th year (after 144 payments). Find the value of .   (3 marks)

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  1.  
Show Worked Solutions

i.   

MARKER’S COMMENT: Students who developed this answer from writing the calculations of , , to then generalising for were the most successful.
 
 
 
 

 
 
 

 

ii.  

 

 

Proof, EXT2* P2 2013 HSC 14a

  1. Show that for  .    (1 mark)

  2. Use mathematical induction to prove that for all integers  ,
     
        .   (3 marks)

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Show Worked Solution

i. 

♦♦ Mean mark 25%.
MARKER’S COMMENT: Student algebra was poor in creating a common denominator and justifying why the final expression is always negative proved challenging.
 
 

 

 

ii. 

 

 

♦♦ Mean mark of 26%.
IMPORTANT: Students have to carefully examine part (i) of this question to provide the information required to prove true for .
 
 
 

  

.

Calculus, 2ADV C4 2010 HSC 3b

  1. Sketch the curve  .   (1 mark)

  2. Use the trapezoidal rule with 3 function values to find an approximation to   (2 marks) 

  3. State whether the approximation found in part (ii) is greater than or less than the exact value of  . Justify your answer.   (1 mark)

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  2. ²
Show Worked Solutions
i. 2010 3b image - Simpsons
MARKER’S COMMENT: Many students failed to illustrate important features in their graph such as the concavity, -axis intercept and -axis asymptote (this can be explicitly stated or made graphically clear).

 

ii.   
 
 
 
  ²   

 

iii. 2010 13b image 2 - Simpsons

 

♦♦♦ Mean mark 12%.
MARKER’S COMMENT: Best responses commented on concavity and that the trapezia lay beneath the curve. Diagrams featured in the best responses.

 

Probability, 2ADV S1 2013 HSC 15d

Pat and Chandra are playing a game. They take turns throwing two dice. The game is won by the first player to throw a double six. Pat starts the game.

  1. Find the probability that Pat wins the game on the first throw.     (1 mark)

  2. What is the probability that Pat wins the game on the first or on the second throw?     (2 marks)

  3. Find the probability that Pat eventually wins the game.     (2 marks)

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Show Worked Solutions

i.   

 
 

 

ii.  

 

♦♦ Mean mark 33%
MARKER’S COMMENT: Many students did not account for Chandra having to lose when Pat wins on the 2nd attempt.

 

iii. 

 

♦♦♦ Mean mark 8%!
 COMMENT: Be aware that diminishing probabilities and within the Series and Applications are a natural cross-topic combination.

 

 
 
 

 

.