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Calculus, MET2 2014 VCAA 19 MC

Jake and Anita are calculating the area between the graph of    and the -axis between    and  

Jake uses a partitioning, shown in the diagram below, while Anita uses a definite integral to find the exact area.
 

VCAA 2014 19mc
 

The difference between the results obtained by Jake and Anita is

A.  

B.  

C.  

D.  

E.  

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Mechanics, EXT2* M1 2008 HSC 7

EXT1 2008 HSC 7
 

A projectile is fired from    with velocity    at an angle of inclination    across level ground. The projectile passes through the points    and  , which are both    metres above the ground, at times    and    respectively. The projectile returns to the ground at  .

The equations of motion of the projectile are

  and   . (Do NOT prove this.)

  1. Show that    AND   .  (2 marks)

Let  α  and  β. It can be shown that
 
        and   .  (Do NOT prove this.)

  1. Show that  .  (2 marks)

  2. Show that  .  (1 mark)

Let   and  .

  1. Show that    and  .  (2 marks)

Let the gradient of the parabola at    be .

  1. Show that  .  (3 marks)

  2. Show that  .  (2 marks)

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i.   

 

 
 

 

ii.   

 
 
 
 

 

iii. 

 
 
 

 

iv.   

 

 

 

 
 
 

 

v.   
 
 
 
 
 
 

 

vi.   

 
 
 

 

MATRICES, FUR1 2014 VCAA 3 MC

Regular customers at a hairdressing salon can choose to have their hair cut by Shirley, Jen or Narj.

The salon has 600 regular customers who get their hair cut each month.

In June, 200 customers chose Shirley (S) to cut their hair, 200 chose Jen (J) to cut their hair and 200 chose Narj (N) to cut their hair.

The regular customers’ choice of hairdresser is expected to change from month to month as shown in the transition matrix, , below.

 

In the long term, the number of regular customers who are expected to choose Shirley is closest to

A. 

B. 

C. 

D. 

E. 

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MATRICES, FUR1 2014 VCAA 4 MC

Two hundred and fifty people buy bread each day from a corner store. They have a choice of two brands of bread: Megaslice (M) and Superloaf (S).

The customers’ choice of brand changes daily according to the transition diagram below.

On a given day, 100 of these people bought Megaslice bread while the remaining 150 people bought Superloaf bread.

The number of people who are expected to buy each brand of bread the next day is found by evaluating the matrix product

VCAA MATRICES FUR1 2014 4iii

VCAA MATRICES FUR1 2014 4iv

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Calculus, EXT1 C1 2008 HSC 4a

A turkey is taken from the refrigerator. Its temperature is  5°C when it is placed in an oven preheated to  190°C.

Its temperature,  ° C, after    hours in the oven satisfies the equation

.

  1. Show that    satisfies both this equation and the initial condition.  (2 marks)

  2. The turkey is placed into the oven at 9 am. At 10 am the turkey reaches a temperature of  29°C. The turkey will be cooked when it reaches a temperature of 80°C.

     

    At what time (to the nearest minute) will it be cooked?  (3 marks)

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i.   
   
   
   

 

 

 

 

 

ii.   

 

 

 
 

 

Calculus, 2ADV C3 2004 HSC 10b

2004 10b

 
The diagram shows a triangular piece of land    with dimensions   metres,  metres and    metres, where  .

The owner of the land wants to build a straight fence to divide the land into two pieces of equal area. Let    and    be points on    and    respectively so that    divides the land into two pieces of equal area.

Let   metres, metres and   metres.

  1. Show that  .  (1 mark)

  2. Use the cosine rule in triangle    to show that
     
           (2 marks)

  3. Show that the value of    in the equation in part (ii) is a minimum when
     
         .  (4 marks)

  4. Show that the minimum length of the fence is   metres, where  . 

     

    (You may assume that the value of    given in part (iii) is feasible.)  (2 marks)

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i.    
 

 

 

ii. 

   

 

iii.  
 
   

 

 

 

iv. 

 
 
 
 
 
 
 
 
 

 

Calculus, 2ADV C3 2004 HSC 9c

Consider the function  , for  .

  1. Show that the graph of    has a stationary point at  .  (2 marks)

  2. By considering the gradient on either side of  , or otherwise, show that the stationary point is a maximum.  (1 mark)

  3. Use the fact that the maximum value of    occurs at    to deduce that    for all  .  (2 marks)

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i.     

 
 

 

 

 

ii.   

Geometry and Calculus, 2UA 2004 HSC 9c Answer

 

 

 

iii.  
   

MARKER’S COMMENT: Very challenging for most students. Successful students recognised the link to part (ii).

 

Calculus, 2ADV C4 2004 HSC 8b

The diagram shows the graph of the parabola  . The points    and    are on the parabola, and is the point where the tangent to the parabola at intersects the directrix.
 

2004 8s
 

  1. Write down the equation of the directrix of the parabola  .  (1 mark)
  2. Find the equation of the tangent to the parabola at the point .  (2 marks)
  3. Show that is the point  .  (1 mark)
  4. Given that the equation of the line is  , find the area bounded by the line and the parabola.  (2 marks)
  5. Hence, or otherwise, find the shaded area bounded by the parabola, the tangent at and the line .  (3 marks)
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  4. ²
  5. ²
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(i)   
   
 
 

 

 

(ii)  
 
 
 
   

 

 

(iii)  

 

 

(iv)  
 

 

²

 

(v)  

   
 

 

 
 
 
 
 
 
 

 

 

²

  ²

Financial Maths, 2ADV M1 2004 HSC 7c

Betty decides to set up a trust fund for her grandson, Luis. She invests $80 at the beginning of each month. The money is invested at 6% per annum, compounded monthly.

The trust fund matures at the end of the month of her final investment, 25 years after her first investment. This means that Betty makes 300 monthly investments.

  1. After 25 years, what will be the value of the first $80 invested?  (2 marks)

  2. By writing a geometric series for the value of all Betty’s investments, calculate the final value of Luis’ trust fund.  (3 marks)

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i.   

 

 
 
 

 

ii.  
 
 
   
 

 

Calculus, EXT1* C1 2004 HSC 7b

At the beginning of 1991 Australia’s population was 17 million. At the beginning of 2004 the population was 20 million.

Assume that the population is increasing exponentially and satisfies an equation of the form  , where    and    are constants, and    is measured in years from the beginning of 1991.

  1. Show that    satisfies  .  (1 mark)

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

  3. Find the value of  .  (2 marks)

  4. Predict the year during which Australia’s population will reach 30 million.  (2 marks)

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i.  
 
   

 

ii. 

   
 

 

iii. 

 
 
 

 

iv. 

MARKER’S COMMENT: Many students had the correct calculations but didn’t answer the question by identifying the exact year and lost a valuable mark.
 
 

 
 

 

Probability, 2ADV S1 2004 HSC 6c

In a game, a turn involves rolling two dice, each with faces marked  0, 1, 2, 3, 4 and 5. The score for each turn is calculated by multiplying the two numbers uppermost on the dice.

  1. What is the probability of scoring zero on the first turn?  (2 marks)

  2. What is the probability of scoring or more on the first turn?  (1 mark)

  3. What is the probability that the sum of the scores in the first two turns is less than 45?  (2 marks)

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MARKER’S COMMENT: Students who drew up the table for the sample space were “overwhelmingly” more successful in all parts of this question.
i.    Probability, 2UA 2004 HSC 6c

 

ii. 

 

iii. 

 
 
 

 

 

Plane Geometry, 2UA 2004 HSC 6b

 

The diagram shows a right-angled triangle    with  . The point    is the midpoint of  , and    is the point where the perpendicular to    at    meets  .

  1. Show that  .  (2 marks)
  2. Suppose that it is also given that    bisects  . Find the size of    and hence find the exact ratio  .  (3 marks) 

 

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(i)    Plane Geometry, 2UA 2004 HSC 6b Answer

MARKER’S COMMENT: Markers strongly recommend that students copy the diagram in geometry questions and label it along with the workings of their solution.

 

 

(ii)  

   
 

 

♦♦ Very few students answered this part correctly.
MARKER’S COMMENT: Students who began by making were the most successful.

Calculus, 2ADV C3 2004 HSC 5b

A particle moves along a straight line so that its displacement, metres, from a fixed point is given by  , where    is measured in seconds.

  1. What is the initial displacement of the particle?  (1 mark)

  2. Sketch the graph of    as a function of    for  .  (2 marks)

  3. Hence, or otherwise, find when AND where the particle first comes to rest after  .  (2 marks)

  4. Find a time when the particle reaches its greatest magnitude of velocity. What is this velocity?  (2 marks)

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i.   

 
 

 

 

 ii. 

 

 Calculus in the Physical World, 2UA 2004 HSC 5b

 

iii.  
 

 

 

 
 
 

 

 

iv. 

   

 

 

 
 

 

Financial Maths, 2ADV M1 2004 HSC 5a

Clare is learning to drive. Her first lesson is 30 minutes long. Her second lesson is 35 minutes long. Each subsequent lesson is 5 minutes longer than the lesson before.

  1. How long will Clare’s twenty-first lesson be?  (1 mark)

  2. How many hours of lessons will Clare have completed after her twenty-first lesson?  (2 marks)

  3. During which lesson will Clare have completed a total of 50 hours of driving lessons?  (2 marks)

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i.   

 
 

 

 

ii. 

 
 
 
 

 

(iii) 
 

 
 
 
 

 

Calculus, EXT1* C3 2004 HSC 4c

2004 4c

In the diagram, the shaded region is bounded by the curve  , the coordinate axes and the line  . The shaded region is rotated about the -axis.

Calculate the exact volume of the solid of revolution formed.  (3 marks)

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³

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  ³

Calculus, MET2 2012 VCAA 1

A solid block in the shape of a rectangular prism has a base of width cm. The length of the base is two-and-a-half times the width of the base.

 VCAA 2012 1a

The block has a total surface area of 6480 sq cm.

  1. Show that if the height of the block is cm,     (2 marks)

  2. The volume, cm³, of the block is given by  
  3. Given that    and  , find the possible values of .   (2 marks)

  4. Find  , expressing your answer in the form  , where and are real numbers.   (3 marks)

  5. Find the exact values of and if the block is to have maximum volume.   (2 marks)

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a.  

 

b.  

 

c.   
   
 

 

d.  

 

Calculus, MET2 2012 VCAA 21 MC

The volume,  cm³, of water in a container is given by    where cm is the depth of water in the container at time minutes. Water is draining from the container at a constant rate of 300 cm³/min. The rate of decrease of , in cm/min, when   is

A.  

B.  

C.  

D.  

E.  

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CORE*, FUR2 2014 VCAA 1

The adult membership fee for a cricket club is $150.

Junior members are offered a discount of $30 off the adult membership fee.

  1. Write down the discount for junior members as a percentage of the adult membership fee.   (1 mark)

Adult members of the cricket club pay $15 per match in addition to the membership fee of $150.

  1. If an adult member played 12 matches, what is the total this member would pay to the cricket club?   (1 mark)

If a member does not pay the membership fee by the due date, the club will charge simple interest at the rate of 5% per month until the fee is paid.

Michael paid the $150 membership fee exactly two months after the due date.

  1. Calculate, in dollars, the interest that Michael will be charged.   (1 mark)

The cricket club received a statement of the transactions in its savings account for the month of January 2014.

The statement is shown below.

     BUSINESS, FUR2 2014 VCAA 1

    1. Calculate the amount of the withdrawal on 17 January 2014.   (1 mark)

    2. Interest for this account is calculated on the minimum balance for the month and added to the account on the last day of the month.
    3. What is the annual rate of interest for this account?  Write your answer, correct to one decimal place.   (1 mark)

  1.  

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a.  

  
b.  

 

 

c.   
   
   

 

d.i.  


    

d.ii.  

 

  

Algebra, STD2 A1 SM-Bank 9

The volume of a sphere is given by    where    is the radius of the sphere.

If the volume of a sphere is  , find the radius, to 1 decimal place.  (3 marks)

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Probability, MET2 2013 VCAA 9 MC

Harry is a soccer player who practises penalty kicks many times each day.

Each time Harry takes a penalty kick, the probability that he scores a goal is 0.7, independent of any other penalty kick.

One day Harry took 20 penalty kicks.

Given that he scored at least 12 goals, the probability that Harry scored exactly 15 goals is closest to

A.   

B.   

C.   

D.   

E.   

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Polynomials, EXT2 2015 HSC 16b

Let    be a positive integer.

  1. By considering  , show that
  3.  
  4.  
  5. Let  ,  for    (2 marks)

  6.  
  7. Show that
    1.   (2 marks)

  9.  
  10. By considering the roots of  , find the value of
    1.   (3 marks)

    2.  
  11. Prove that
    1.   (2 marks)

 

 

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  4.  
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(i)   

 

 

(ii)  

 

♦♦♦ Mean mark 8%!
STRATEGY: Note that finding the correct roots gained 2 full marks in this part.

 

(iii) 

 

 

 

♦♦ Mean mark 14%.

 

(iv) 

 
 
 

Harder Ext 1 Topics, EXT2 2015 HSC 16a

  1. A table has rows and columns, creating cells as shown.
  2. Counters are to be placed randomly on the table so that there is one counter in each cell. There are identical black counters and identical white counters.
  4. Show that the probability that there is exactly one black counter in each column is   (2 marks)

  5. The table is extended to have rows and columns. There are counters, where are identical black counters and the remainder are identical white counters. The counters are placed randomly on the table with one counter in each cell.

  6. Let be the probability that each column contains exactly one black counter.

  7. Show that   (2 marks)
  10. Find   (2 marks)
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(i) 

 

 

 

(ii) 

♦♦ Mean mark 16%.

 

 

♦♦♦ Mean mark 5%.
COMMENT: Lowest State mean mark on the 2015 exam.
(iii)  
   
   
   
   

Proof, EXT2 P1 2015 HSC 15c

For positive real numbers and , .     (Do NOT prove this.)

  1. Prove  ,  for positive real numbers    and    (1 mark)

  2. Prove  ,  for positive real numbers    and    (2 marks)

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i.   

 

ii.   

♦♦ Mean mark 29%. 

 
 
   
 

Proof, EXT2 P1 2015 HSC 15b

Suppose that    and    is a positive integer.

  1. Show that    (2 marks) 

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

  3. Hence, explain why
     
          (1 mark)

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i.   
 
 

 

♦♦ Mean mark 21%.
STRATEGY: The conversion of the middle term from a fraction into a logarithm should flag the need for integration of each term.
ii.   
 
 
 

 

 

iii.   
 
♦♦ Mean mark 29%.
 

Mechanics, EXT2 M1 2015 HSC 15a

A particle   of unit mass travels horizontally through a viscous medium. When  , the particle is at point    with initial speed  . The resistance on particle    due to the medium is  , where    is the velocity of the particle at time    and    is a positive constant.

When  , a second particle    of equal mass is projected vertically upwards from    with the same initial speed    through the same medium. It experiences both a gravitational force and a resistance due to the medium. The resistance on particle    is  , where    is the velocity of the particle    at time  . The acceleration due to gravity is  .

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

  2. By considering the velocity    of particle  , show that
     
           (3 marks)

  3. Show that the velocity    of particle    when particle    is at rest is given by
     
           (1 mark)

  4. Hence, if    is very large, explain why  
     
           (1 mark)

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i.   

 

ii.   

   
   
   
   
 

 

 

 

iii.   

 

 

iv.   

Mechanics, EXT2 2015 HSC 14c

A car of mass    is driven at speed    around a circular track of radius  . The track is banked at a constant angle    to the horizontal, where  .  At the speed    there is a tendency for the car to slide up the track. This is opposed by a frictional force  , where    is the normal reaction between the car and the track, and  . The acceleration due to gravity is  .

  1. Show that  
  2.   (3 mark)
  4. At the particular speed  , where  , there is still a tendency for the car to slide up the track.

  5.  

    Using the result from part (i), or otherwise, show that    (2 marks)

 

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(i)  

 

 
 

 

(ii)  

♦ Mean mark 46%. 

 

 

Functions, EXT1′ F2 2015 HSC 14b

The cubic equation    has roots    and  .

It is given that    and  .

  1. Show that     (1 mark)
  2. Find the value of     (2 marks)
  3. Find the value of     (2 marks)
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(i)   

 

(ii)   
 
 

 

COMMENT: Part (iii) proved challenging for many students with the State mean mark just over 50%.
(iii)  
 
 

 

Calculus, EXT2 C1 2015 HSC 14a

  1. Differentiate  , expressing the result in terms of    only.  (2 marks)

  2. Hence, or otherwise, deduce that
     
         ,  for   (2 marks) 

  3. Find    (1 mark)

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i.   

 

ii.  

 

iii.  
   
     
   

Harder Ext1 Topics, EXT2 2015 HSC 13c

A small spherical balloon is released and rises into the air. At time    seconds, it has radius   cm, surface area    and volume  .

As the balloon rises it expands, causing its surface area to increase at a rate of  . As the balloon expands it maintains a spherical shape.

  1. By considering the surface area, show that  
    1.   (2 marks)

  2. Show that  
    1.   (2 marks)

  3. When the balloon is released its volume is  ³. When the volume of the balloon reaches  ³  it will burst.
  4. How long after it is released will the balloon burst?  (2 marks)

 

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(i)   
 
 
   

 

(ii)  

­
­
­
­

 

(iii) 

 
 
 

Volumes, EXT2 2015 HSC 13b

Two quarter cylinders, each of radius  , intersect at right angles to form the shaded solid.

A horizontal slice    of the solid is taken at height    from the base. You may assume that    is a square, and is parallel to the base.

  1. Show that    (1 mark)
  2. Find the volume of the solid.  (2 marks)
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  2. ³

 

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(i)  

 

(ii)  

­
­
­
­ ³

Volumes, EXT2 2015 HSC 12d

The diagram shows the graph    for  . The shaded region is rotated about the line    to form a solid.

Use the method of cylindrical shells to find the volume of the solid.  (4 marks)

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³

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δδ

­
 
 

­
 
 
 
  ³

 

­
­
­
­
­
­
­
­
­ ³

Polynomials, EXT2 2015 HSC 12b

The polynomial    has roots    and    where  and    are real and  

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

 

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(i)  

 

 

(ii)