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Probability, STD2 S2 2012 HSC 12 MC

Two unbiased dice, each with faces numbered 1, 2, 3, 4, 5, 6, are rolled. 

What is the probability of a 6 appearing on at least one of the dice? 

  1.  
  2.  
  3.  
  4.  
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Show Worked Solution


 

♦♦♦ Mean mark 25%
COMMENT: The term “at least” should flag that calculating the probability of is likely to be the most efficient way to solve.

Statistics, STD2 S4 2012 HSC 11 MC

Which of the following relationships would most likely show a negative correlation?

  1. The population of a town and the number of hospitals in that town. 
  2. The hours spent training for a race and the time taken to complete the race. 
  3. The price per litre of petrol and the number of people riding bicycles to work. 
  4. The number of pets per household and the number of computers per household. 
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Show Worked Solution

♦ Mean mark 43%.

Financial Maths, STD2 F4 2012 HSC 9 MC

Tracy invests some money for 2 years at 4% per annum, compounded quarterly. 

2012 9 mc

Which figure from the table should Tracy use to calculate the value of her investment at the end of 2 years?

  1.   1.020 
  2.   1.082
  3.   1.083
  4.   1.369
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Show Worked Solution
♦♦♦ Mean mark 24%. The lowest MC mean mark in the 2012 exam.
COMMENT: A process to follow: 1-convert the annual rate to the rate per compounding period. 2-calculate the number of compounding periods.

 

Algebra, STD2 A2 2012 HSC 5 MC

The line    has intercepts    and  ,  where    and    are positive integers. 
  

What is the gradient of line   ? 

  1.  
  2.  
  3.  
  4.  
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Show Worked Solution
 
♦ Mean mark 45%
 

Probability, 2UG 2012 HSC 3 MC

A pair of players is to be selected from 6 people.

How many different pairs of players can be selected?

(A)   

(B)   

(C)   

(D)   

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Show Worked Solution
♦ Mean mark 40%
COMMENT: Calculating #combinations involving unordered pairs has proven very challenging to a majority of students in past exams. Ensure you understand this concept.

 

Measurement, STD2 M1 2013 HSC 25 MC

A net is made using four rectangles and two trapeziums. It is folded to form a solid.
  

 What is the volume of the solid, in cm3 ?

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Show Worked Solution

♦ Mean mark 35%
COMMENT: Note that in the “area” formula is different to the used in the “volume” formula.
 
  ²

 

³

Probability, STD2 S2 2013 HSC 18 MC

Two unbiased dice, each with faces numbered  1, 2, 3, 4, 5, 6,  are rolled.

What is the probability of obtaining a sum of 6?

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Show Worked Solution

♦♦ Mean mark 35%.

Financial Maths, STD2 F1 2013 HSC 11 MC

An enterprise agreement has the following annual salary arrangements:
 

George's employer pays 6% more than the enterprise agreement. He is on Step 3 and receives an allowance for Leader 2.

What is George's gross monthly pay?

  1. $4375.00
  2. $4412.50
  3. $4600.00
  4. $4637.50
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Show Worked Solution
♦ Mean mark 46%
 

 

 

 

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 2010 HSC 5a

A rainwater tank is to be designed in the shape of a cylinder with radius    metres and height    metres.

2010 5a

The volume of the tank is to be 10 cubic metres. Let    be the surface area of the tank, including its top and base, in square metres.

  1. Given that  ,   show that  .   (2 marks)

  2. Show that    has a minimum value and find the value of    for which the minimum occurs.  (3 marks)

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Show Worked Solution

i.   

MARKER’S COMMENT: Students MUST know the volume formula for a cylinder. Those that did and stated most often completed this question efficiently.

 

 

♦ Mean mark 44%
MARKER’S COMMENT: The “table method” or 1st derivative test for proving a minimum (i.e. showing how changes sign) was also quite successful.
ii.  
 
 

 

 
 

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 2011 HSC 6b

The diagram shows the trajectory of a ball thrown horizontally, at speed m/s, from the top of a tower metres above the ground level.
 

2011 6b
 

The ball strikes the ground at an angle of  45°, metres from the base of the tower, as shown in the diagram. The equations describing the trajectory of the ball are

  and   ,    (DO NOT prove this)

where is the acceleration due to gravity, and is time in seconds.

  1. Prove that the ball strikes the ground at time  
     
           seconds.    (1 mark)

  2. Hence, or otherwise, show that  .    (2 marks)

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  1.  
Show Worked Solution

i.  

 

ii.  

,

 

 

♦♦ Mean mark 33%
STRATEGY: The key to unlocking this proof is understanding that , where   is the angle of trajectory at impact of a projectile, equals  .

 

 
 

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 C4 2009 HSC 7a

The acceleration of a particle is given by

,

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

Initially its velocity is  and its displacement is 5 m.

  1. Show that the displacement of the particle is given by
  2. .   (2 marks) 

  3. Find the time when the particle comes to rest.    (3 marks)

  4. Find the displacement  when the particle comes to rest.    (1 mark)

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Show Worked Solution

i.   

 

 

 
 

 

 

 

ii.  

 
 

 


 


  

iii. 

ALGEBRA TIP: Helpful identity  . Easily provable as follows:



.

Calculus, EXT1* C1 2010 HSC 7a

The acceleration of a particle is given by

,

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

Initially the particle is at the origin with a velocity of  .

  1. Show that the velocity of the particle is given by

     

      .    (2 marks)

  2. Find the time when the particle first comes to rest.    (2 marks)

  3. Find the displacement,  ,  of the particle in terms of  .    (2 marks)

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Show Worked Solution
i.   
 
 

 

 

 

ii.  

 

 

 

iii.   
 
 

 

 

Calculus, 2ADV C4 2012 HSC 15b

The velocity of a particle is given by

,

where    is the displacement in metres and    is the time in seconds. Initially the particle is 3 m to the right of the origin.

  1. Find the initial velocity of the particle.    (1 mark)

  2. Find the maximum velocity of the particle.    (1 mark)

  3. Find the displacement, ,  of the particle in terms of  .    (2 marks)

  4. Find the position of the particle when it is at rest for the first time.    (2 marks)

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Show Worked Solution

i.    :

 
 

 

 

ii.  

♦♦ Mean mark 29%
MARKER’S COMMENT: Solution 2 is more efficient here. Using the -1 and +1 limits of trig functions can be very a effective way to calculate max/min values.


 

,  ,  , …

 

 

 

 

iii.   
 
 

 

 

 

iv.  

♦ Mean mark 50%
MARKER’S COMMENT: Many students found    but failed to gain full marks by omitting to find  . Remember that for calculus, angles are measured in radians, NOT degrees!

 

 

 
 

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 2008 HSC 5c

Light intensity is measured in lux. The light intensity at the surface of a lake is 6000 lux. The light intensity,    lux, a distance    metres below the surface of the lake is given by

where  ,  and    are constants.

  1. Write down the value of  .   (2 marks)

  2. The light intensity 6 metres below the surface of the lake is 1000 lux. Find the value of  .    (2 marks)

  3. At what rate, in lux per metre, is the light intensity decreasing 6 metres below the surface of the lake?    (2 marks)

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Show Worked Solution

i.   

 

ii.   

MARKER’S COMMENT: Many students used the “log” function on their calculator rather than the function. BE CAREFUL!
 
 

 

iii. 

ALGEBRA TIP: Tidy your working by calculating using and then only substituting for in part (ii) at the final stage.

 

 
 

 

Calculus, EXT1* C1 2009 HSC 6b

Radium decays at a rate proportional to the amount of radium present. That is, if    is the amount of radium present at time  ,  then  ,  where    is a positive constant and    is the amount present at  . It takes 1600 years for an amount of radium to reduce by half.

  1. Find the value of  .   (2 marks)

  2. A factory site is contaminated with radium. The amount of radium on site is currently three times the safe level.

     

    How many years will it be before the amount of radium reaches the safe level.    (2 marks)

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Show Worked Solution

i.   

MARKER’S COMMENT: Students must be familiar with “half-life” and the algebra required. i.e. using within their calculations.

 

 

ii.   

IMPORTANT: Know how to use the memory function on your calculator to store the exact value of found in part (i) to save time.
 
 

 

Calculus, EXT1* C1 2011 HSC 10a

The intensity, ,  measured in  watt/m2,  of a sound is given by

,

where    is the loudness of the sound in decibels.

  1. If the loudness of a sound at a concert is 110 decibels, find the intensity of the sound. Give your answer in scientific notation.   (1 mark)

  2. Ear damage occurs if the intensity of a sound is greater than  watt/m2.

     

    What is the maximum loudness of a sound so that no ear damage occurs?    (2 marks)

  3. By how much will the loudness of a sound have increased if its intensity has doubled?     (2 marks)

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Show Worked Solution

i.    

MARKER’S COMMENT: Note that is not correct scientific notation.
 
 
 

 

ii.   

 
 

 

 

iii. 

♦♦ Mean mark 32%
MARKER’S COMMENT: Actual values can help here. Calculate the intensity at (which equals ) and then find when this intensity level is doubled (to ).

 
 

 

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

Calculus, 2ADV C4 2010 HSC 5c

The diagram shows the curve  , for  .

The area under the curve between    and    is  . The area under the curve between    and    is  .
 

2010 5c
 

The areas    and    are each equal to square unit.

Find the values of    and  .     (3 marks)

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Show Worked Solutions
IMPORTANT: Note when , the definition of a log means that . Many students failed to earn an easy 3rd mark by recognising this.

 

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.  

 

 

Financial Maths, 2ADV M1 2008 HSC 5b

Consider the geometric series

  1. For what values of does this series have a limiting sum?    (2 marks)

  2. The limiting sum of this series is .

     

    Find the value of .     (2 marks)

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Show Worked Solutions

i.   

   

 

 

ii. 

Calculus, EXT1* C1 2013 HSC 14a

The velocity of a particle moving along the -axis is given by  , where is the displacement from the origin in metres and is the time in seconds. Initially the particle is 5 metres to the right of the origin.

  1. Show that the acceleration of the particle is constant.  (1 mark)

  2. Find the time when the particle is at rest.  (1 mark)

  3. Show that the position of the particle after 7 seconds is 26 metres to the right of the origin.  (2 marks)

  4. Find the distance travelled by the particle during the first 7 seconds.   (2 marks)

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  1.  
Show Worked Solutions

i.   

 

 

 ii.  

 

 

 iii.

 
 

 

 

 

 
 

 

 

 iv.  

♦ Mean mark 37%.
IMPORTANT: Students can also draw a number line that shows where a particle starts from, changes direction and finishes in order to reduce errors.

,

 
 

 


 

Proof, EXT2* P2 2009 HSC 7a

  1. Use differentiation from first principles to show that  .   (1 mark)

  2. Use mathematical induction and the product rule for differentiation to prove that  
     
         for all positive integers .   (2 marks)

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Show Worked Solution

i.  

MARKER’S COMMENT: Students are reminded to use the number of marks allocated to a question as a guide to the work involved in answering it.
 
 

 

ii. 

 

 

IMPORTANT: Critical to read carefully and apply the product rule to prove the induction.

 
 
 
 
 
 

 

.

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 .
 
 
 

  

.

L&E, 2ADV E1 2008 HSC 7a

Solve     (3 marks)

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Show Worked Solution
 
IMPORTANT: Students should recognise this equation as a quadratic, and the best responses substituted with a variable such as .
   
 
MARKER’S COMMENT: Many responses incorrectly stated that there is no solution to  or could not find given .

 

Financial Maths, 2ADV M1 2009 HSC 8b

One year ago Daniel borrowed $350 000 to buy a house. The interest rate was 9% per annum, compounded monthly. He agreed to repay the loan in 25 years with equal monthly repayments of $2937.

  1. Calculate how much Daniel owed after his first monthly repayment.    (1 mark)

Daniel has just made his 12th monthly repayment. He now owes $346 095. The interest rate now decreases to 6% per annum, compounded monthly.

 

The amount  , owing on the loan after the th monthly repayment is now calculated using the formula  
 

  
where is the monthly repayment, and .   (DO NOT prove this formula.)

  1. Calculate the monthly repayment if the loan is to be repaid over the remaining 24 years (288 months).    (3 marks)

  2. Daniel chooses to keep his monthly repayments at $2937. Use the formula in part (ii) to calculate how long it will take him to repay the $346 095.   (3 marks)

  3. How much will Daniel save over the term of the loan by keeping his monthly repayments at $2937, rather than reducing his repayments to the amount calculated in part (ii)?   (1 mark)

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Show Worked Solutions

i.   

 
 

 

 

ii.   

 

MARKER’S COMMENT: Careless setting out and poor handwriting, especially where indexes were involved, was a major contributor to errors in this question.

 

 

 

 

iii. 

♦♦ A poorly answered question.
MARKER’S COMMENT: Many students struggled to handle the exponential and logarithm calculations in this question.
ALGEBRA TIP: Dividing by in part (iii) is equivalent to multiplying by 200, and cleans up working calculations (see Worked Solutions).
 

 

 

 

iv.  

 

Financial Maths, 2ADV M1 2010 HSC 9a

  1. When Chris started a new job, $500 was deposited into his superannuation fund at the beginning of each month. The money was invested at 0.5% per month, compounded monthly. 

     

    Let    be the value of the investment after 240 months, when Chris retires.

     

    Show that      (2 marks)

  2. After retirement, Chris withdraws $2000 from the account at the end of each month, without making any further deposits. The account continues to earn interest at 0.5% per month.

     

    Let    be the amount left in the account    months after Chris's retirement.

     

      (1)  Show that  .     (3 marks)

     

      (2)  For how many months after retirement will there be money left in the account?     (2 marks)

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

 

MARKER’S COMMENT: Common errors included using , and taking the first term of the GP as 1 instead of 1.005 (note that the $500 goes in at the start of the month and earns interest before it is included in ).
 
 
 

 

 

ii. (1) 

IMPORTANT: At the end of the month, earns interest for the month BEFORE any withdrawal is made. Many students mistakenly had .
 
 
 
 

 
 
 

 

ii. (2)  

♦ Mean mark 38%

,

 

Financial Maths, 2ADV M1 2011 HSC 8c

When Jules started working she began paying $100 at the beginning of each month into a superannuation fund.

The contributions are compounded monthly at an interest rate of 6% per annum.

She intends to retire after having worked for 35 years.

  1. Let    be the final value of Jules's superannuation when she retires after 35 years (420 months). Show that    to the nearest dollar.     (2 marks)

  2. Fifteen years after she started working Jules read a magazine article about retirement, and realised that she would need in her fund when she retires. At the time of reading the magazine article she had in her fund. For the remaining 20 years she intends to work, she decides to pay    into her fund at the beginning of each month. The contributions continue to attract the same interest rate of 6% per annum, compounded monthly.

  3.  

    At the end of    months after starting the new contributions, the amount in the fund is  .

  4.  

      (1)  Show that  .     (1 mark)

  5.  

      (2)  Find the value of    so that Jules will have $800 000 in her fund after the remaining 20 years (240 months).     (3 marks)

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  1.  
  2. (1)
    (2)
Show Worked Solutions
i.   
 
   
   
 
   
   
 

MARKER’S COMMENT: Common errors in this part included having the first term of the GP as 1 instead of 1.005 (note that the $100 goes in at the start of the month and earns interest before it is included in .
 
 
 

 

 

♦ Mean mark 34% for part (ii)(1)

ii. (1)  

 
 
 

 

ii. (2)  

♦ Mean mark 49%

  ..

Financial Maths, 2ADV M1 2011 HSC 5a

The number of members of a new social networking site doubles every day. On Day 1 there were 27 members and on Day 2 there were 54 members.

  1. How many members were there on Day 12?     (1 mark)

  2. On which day was the number of members first greater than 10 million?     (2 marks)

  3. The site earns 0.5 cents per member per day. How much money did the site earn in the first 12 days? Give your answer to the nearest dollar.     (2 marks)

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Show Worked Solutions
MARKER’S COMMENT: Better responses stated the general term, before any substitution was made, as shown in the worked solutions.

i.  

 

 

ii.   

MARKER’S COMMENT: Many elementary errors were made by students in dealing with logarithms. BE VIGILANT.

 

 

iii.  


 

♦ Mean mark 44%.
NOTE: This question can also be easily solved by making the total sum of members (each day) and then multiplying by 0.5 cents.
 
 
 

 

Financial Maths, 2ADV M1 2012 HSC 15c

Ari takes out a loan of $360 000. The loan is to be repaid in equal monthly repayments, , at the end of each month, over 25 years (300 months). Reducible interest is charged at 6% per annum, calculated monthly.

Let    be the amount owing after the th repayment.

  1. Write down an expression for the amount owing after two months, .   (1 mark)

  2. Show that the monthly repayment is approximately $2319.50.   (2 marks)

  3. After how many months will the amount owing, , become less than $180 000.   (3 marks)

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  1.  
Show Worked Solutions
i.   
 
 
 

 

ii. 

 

 

iii.   
♦ Mean mark 38%
MARKER’S COMMENT: Challenging calculations using logarithms are common in this topic. A high percentage of students consistently struggle in this area.

 

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.

 

 
 
 

 

.

Financial Maths, 2ADV M1 2013 HSC 13d

A family borrows $500 000 to buy a house. The loan is to be repaid in equal monthly instalments. The interest, which is charged at 6% per annum, is reducible and calculated monthly. The amount owing after    months, , is given by

(DO NOT prove this)

where    is the amount borrowed,   and    is the monthly repayment.

  1. The loan is to be repaid over 30 years. Show that the monthly repayment is $2998 to the nearest dollar.     (2 marks)

  2. Show that the balance owing after 20 years is $270 000 to the nearest thousand dollars.             (1 mark)

After 20 years the family borrows an extra amount, so that the family then owes a total of $370 000. The monthly repayment remains $2998, and the interest rate remains the same.

  1. How long will it take to repay the $370 000?     (2 marks)

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Show Worked Solutions

i.    

 

 

 ii.     

 
 
 
MARKER’S COMMENT: Within the GP formula, many students incorrectly wrote the last term as rather than . Note .

 

 

iii. 

♦♦ Mean mark 33%
COMMENT: Another good examination of working with logarithms. Students should understand why they must ’round up’ their answer in this question.
 

 

Financial Maths, 2ADV M1 2013 HSC 12c

Kim and Alex start jobs at the beginning of the same year. Kim's annual salary in the first year is $30,000 and increases by 5% at the beginning of each subsequent year. Alex's annual salary in the first year is $33,000, and increases by $1,500 at the beginning of each subsequent year.

  1. Show that in the 10th year, Kim's annual salary is higher than Alex's annual salary.     (2 marks)

  2. In the first 10 years how much, in total, does Kim earn?     (2 marks)

  3. Every year, Alex saves of her annual salary. How many years does it take her to save $87,500?     (3 marks)

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Show Worked Solutions

i.    

 


 

 

ii.   

 
 

 

 

iii.   


 

♦ Mean mark 45%.
IMPORTANT: Using the AP sum formula to create and then solve a quadratic in is challenging and often examined. Students need to solve and interpret the solutions.