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Quadratic, EXT1 2016 HSC 14c

The point   lies on the parabola with the equation  .

The tangent to the parabola at meets the directrix at .

The normal to the parabola at meets the vertical line through at the point , as shown in the diagram.

 

ext1-2016-hsc-q14

  1. Show that the point has coordinates .  (1 mark)
  2. Show that the locus of lies on another parabola .  (3 marks)
  3. State the focal length of the parabola .  (1 mark)

It can be shown that the minimum distance between and occurs when the normal to at is also the normal to at .  (Do NOT prove this.)

  1. Find the values of so that the distance between and is a minimum.  (2 marks)
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i.    

 

ii.   

♦♦ Mean mark 32%.
MARKER’S COMMENT: Normal equation is found in the Reference Sheet. Save time by using it!

 

 

 
 
 
 

 

 

iii.  

♦♦♦ Mean mark 15%.
 

 

 

iv.  

♦♦♦ Mean mark 11%.

 

 

Binomial, EXT1 2016 HSC 14b

Consider the expansion of  , where is a positive integer.

  1. Show that  .  (1 mark)
  2. Show that  .  (1 mark)
  3. Hence, or otherwise, show that  .  (2 marks)
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i.    

 

ii.  

 

iii. 

♦♦♦ Mean mark 14%.
 

 

 

 

Plane Geometry, EXT1 2016 HSC 13c

The circle centred at has a diameter . From the point outside the circle the line segments and are drawn meeting the circle at and respectively, as shown in the diagram. The chords and meet at . The line segment produced meets the diameter at .

ext1-2016-hsc-q13_1

Copy or trace the diagram into your writing booklet.

  1. Show that is a cyclic quadrilateral.  (2 marks)
  2. Hence, or otherwise, prove that is perpendicular to .  (2 marks)
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i.  

 


 

ii.  

♦♦♦ Mean mark 14%.

Mechanics, EXT2* M1 2016 HSC 13b

The trajectory of a projectile fired with speed    at an angle    to the horizontal is represented by the parametric equations

  and   ,

where is the time in seconds.

  1. Prove that the greatest height reached by the projectile is  .  (2 marks)

A ball is thrown from a point above the horizontal ground. It is thrown with speed at an angle of to the horizontal. At its highest point the ball hits a wall, as shown in the diagram.
 

     ext1-2016-hsc-q13
 

  1. Show that the ball hits the wall at a height of above the ground.  (2 marks)

The ball then rebounds horizontally from the wall with speed . You may assume that the acceleration due to gravity is .

  1. How long does it take the ball to reach the ground after it rebounds from the wall?  (2 marks)

  2. How far from the wall is the ball when it hits the ground?  (1 mark)

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i.   
 

 

 

 

ii.  

 

♦♦ Mean mark part (iii) 35%.
iii.   ext1-hsc-2016-13bi

 

MARKER’S COMMENT: Many students struggled to solve: .

 

 

iv.  

♦ Mean mark 43%.

Mechanics, EXT2* M1 2016 HSC 13a

The tide can be modelled using simple harmonic motion.

At a particular location, the high tide is 9 metres and the low tide is 1 metre.

At this location the tide completes 2 full periods every 25 hours.

Let be the time in hours after the first high tide today.

  1. Explain why the tide can be modelled by the function  .  (2 marks)

  2. The first high tide tomorrow is at 2 am.

     

    What is the earliest time tomorrow at which the tide is increasing at the fastest rate?  (2 marks)

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ii.   

 

 

 
 

 

Polynomials, EXT1 2016 HSC 12c

The graphs of    and    meet at the point where  α, as shown.

ext1-2016-hsc-q12_1

  1. Show that the tangents to the curves at  α  are perpendicular.  (2 marks)
  2. Use one application of Newton’s method with    to find an approximate value for α. Give your answer correct to two decimal places.  (2 marks)
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Calculus, EXT1 C1 2016 HSC 12b

In a chemical reaction, a compound is formed from a compound . The mass in grams of and are and respectively, where is the time in seconds after the start of the chemical reaction.

Throughout the reaction the sum of the two masses is 500 g. At any time , the rate at which the mass of compound is increasing is proportional to the mass of compound .

At the start of the chemical reaction,  and  .

  1.  Show that  .  (3 marks)

  2.  Show that    satisfies the equation in part (i), and find the value of .  (2 marks)

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i.  

♦ Mean mark (part i) 43%.

 

 

 

 

ii.   
 

 

 

 

Calculus, EXT1 C1 2016 HSC 12a

The diagram shows a conical soap dispenser of radius 5 cm and height 20 cm.
 

     ext1-2016-hsc-q12
 

At any time seconds, the top surface of the soap in the container is a circle of radius cm and its height is cm.

The volume of the soap is given by  .

  1.  Explain why  .  (1 mark)

  2.  Show that  .  (1 mark)

The dispenser has a leak which causes soap to drip from the container. The area of the circle formed by the top surface of the soap is decreasing at a constant rate of  ².
 

  1.  Show that  .  (2 marks)

  2.  What is the rate of change of the volume of the soap, with respect to time, when ?  (2 marks)

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  4. ³
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i.   ext1-hsc-2016-12a

 

 

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Statistics, EXT1 S1 2016 HSC 11f

A darts player calculates that when she aims for the bullseye the probability of her hitting the bullseye is    with each throw.

  1. Find the probability that she hits the bullseye with exactly one of her first three throws.  (1 mark)

  2. Find the probability that she hits the bullseye with at least two of her first six throws.  (2 marks)

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ii.  

Calculus, 2ADV C3 2016 HSC 16b

Some yabbies are introduced into a small dam. The size of the population, , of yabbies can be modelled by the function

where is the time in months after the yabbies are introduced into the dam.

  1. Show that the rate of growth of the size of the population is
  2. .  (2 marks)

  3. Find the range of the function , justifying your answer.  (2 marks)

  4. Show that the rate of growth of the size of the population can be written as
  5. .  (1 mark)

  6. Hence, find the size of the population when it is growing at its fastest rate.  (2 marks)

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i.  
 
   
   

 

ii.  

♦♦♦ Mean mark (ii) 21%.

 

iii. 

♦♦♦ Mean mark (iii) 18%.

 
 

 

iv.  

♦♦♦ Mean mark (iv) 14%.

 

hsc-2016-16bi

Calculus, 2ADV C4 2016 HSC 16a

A particle moves in a straight line. Its velocity at time seconds is given by

  1. Find the initial velocity.  (1 mark)

  2. Find the acceleration of the particle when the particle is stationary.  (2 marks)

  3. By considering the behaviour of for large , sketch a graph of against for  , showing any intercepts.  (2 marks)

  4. Find the exact distance travelled by the particle in the first 7 seconds.  (3 marks)

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  3.  
    hsc-2016-16ai
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i.  

Mean mark 93%.
COMMENT: Great example of low hanging fruit late in the exam for students who progress efficiently.

 

ii.  
 

 

   

   

 

iii. 

♦ Mean mark 47%.

 hsc-2016-16ai

 

♦♦ Mean mark 26%.

iv.  

Plane Geometry, 2UA 2016 HSC 15c

Maryam wishes to estimate the height, metres, of a tower, , using a square, with side length metre.

She places the point on the horizontal ground and ensures that the point lies on the line joining to the top of the tower The point is the intersection of the line joining and and the side The point is the foot of the perpendicular from to the ground. Let have length metres and have length metres.

hsc-2016-15c

Copy or trace the diagram into your writing booklet.

  1. Show that and are similar.  (2 marks)
  2. Show that and are similar.  (2 marks)
  3. Find in terms of and .  (2 marks)
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ii.  

♦♦ Mean mark 27%.

 

 

 

 

iii.  

♦♦ Mean mark 27%.

 
 

 

 


   
 

Probability, 2ADV S1 2016 HSC 15b

An eight- sided die is marked with numbers  1, 2, … , 8. A game is played by rolling the die until an 8 appears on the uppermost face. At this point the game ends.

  1. Using a tree diagram, or otherwise, explain why the probability of the game ending before the fourth roll is

      

    .  (2 marks)

  2. What is the smallest value of for which the probability of the game ending before the th roll is more than  ?  (3 marks)

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ii. 

♦♦ Mean mark 29%.
ALGEBRA: Note that dividing by reverses the < sign as it is dividing by a negative number.
 

Calculus, 2ADV C3 2016 HSC 14c

A farmer wishes to make a rectangular enclosure of area 720 m². She uses an existing straight boundary as one side of the enclosure. She uses wire fencing for the remaining three sides and also to divide the enclosure into four equal rectangular areas of width m as shown.
 

hsc-2016-14c
 

  1. Show that the total length, m, of the wire fencing is given by
     
          .  (1 mark)

  2. Find the minimum length of wire fencing required, showing why this is the minimum length.  (3 marks)

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ii. 
 

 

 

Financial Maths, 2ADV M1 2016 HSC 14b

A gardener develops an eco-friendly spray that will kill harmful insects on fruit trees without contaminating the fruit. A trial is to be conducted with 100 000 insects. The gardener expects the spray to kill 35% of the insects each day and that exactly 5000 new insects will be produced each day.

The number of insects expected at the end of the th day of the trial is

  1. Show that  .  (2 marks)

  2. Show that  .  (1 mark)

  3. Find the expected insect population at the end of the fourteenth day, correct to the nearest 100.  (1 mark)

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Mechanics, EXT2 M1 2016 HSC 15b

A particle is initially at rest at the point which is metres to the right of

The particle then moves in a straight line towards

For the acceleration of the particle is given by    where is the distance from and is a positive constant.

  1. Prove that    (2 marks)

  2. Using the substitution   show that the time taken to reach a distance metres to the right of is given by
     
           (3 marks)

It can be shown that    (Do NOT prove this.)

  1. What is the limiting time taken for the particle to reach   (1 mark)

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ii.   
 
 

 

 

 

 

 

 

iii.  

 

♦♦ Mean mark 29%.
 
 

Calculus, EXT2 C1 2016 HSC 14b

Let    for  

  1. Using a suitable substitution, show that    (3 marks)

  2. Show that    (1 mark)

  3. Find    (3 marks)

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ii.   
   
   
   
   

 

♦ Mean mark part (iii) 49%.
iii.   
   
   
   
   
   
   
   

Integration, EXT2 2016 HSC 14a

  1. Show that    (1 mark)
  2. Using a graphical approach, or otherwise, explain why

     

    , for all positive integers   (1 mark)

  3. The diagram shows the region enclosed by and the -axis for

    ext2-hsc-2016-14a
     

     

    Using the method of cylindrical shells and the results in parts (i) and (ii), find the exact volume of the solid formed when is rotated about the -axis.  (3 marks)

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i.   
   
   

 

ii.  

ext2-hsc-2016-14a-answer

 

iii.   
   

 

 

 
 

 

 

Quadratic, 2UA 2016 HSC 13b

Consider the parabola  

  1. By completing the square, or otherwise, find the focal length of the parabola.  (2 marks)
  2. Find the coordinates of the focus.  (1 mark)
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ii. 

Calculus, 2ADV C3 2016 HSC 13a

Consider the function  

  1. Find the two stationary points and determine their nature.  (4 marks)

  2. Sketch the graph of the function, clearly showing the stationary points and the and intercepts.  (2 marks)

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ii.
 ext2-hsc-2016-13bi

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ii.  ext2-hsc-2016-13bi

Calculus, 2ADV C4 2016 HSC 12d

  1. Differentiate   . (1 mark)

  2. Hence find the exact value of  .  (2 marks)

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ii. 

Statistics, STD2 S5 2016 HSC 30d

The formula to calculate  -scores can be rearranged to give

σ

 

where     is the mean
   is the score
  σ  is the standard deviation
   is the -score
   
  1. In an examination, Aaron achieved a score of 88, which corresponds to a  -score of 2.4.

     

    Substitute these values into the rearranged formula above to form an equation.  (1 mark)

  2. In the same examination, Brock achieved a score of 52, which corresponds to a  -score of  –1.2.

     

    Using this information, form another equation and solve it simultaneously with the equation from part (i) to find the values of    and  σ.  (3 marks)

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  1. σ
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i.   σ
 

ii.   σ

♦♦ Mean mark part (ii) 32%.
COMMENT: Recognise this is a simultaneous equation problem (now outside Std2 syllabus) and not a normal distribution one.

σ
 

σ
σ
σ

 
σ

 

Measurement, STD2 M1 2016 HSC 30c

A school playground consists of part of a circle, with centre , and a rectangle as shown in the diagram. The radius of the circle is 45 m, the width of the rectangle is 20 m and is 100°.
 

What is the area of the whole playground, correct to the nearest square metre?  (5 marks)

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²²

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²

 

  ²

 

²

 

²²

FS Comm, 2UG 2016 HSC 30b

Michael was transferring some video files from his computer onto a USB stick. At some point during the transfer, he observed the information shown below. 

2ug-2016-hsc-q30_1

  1. Show that, at that time, approximately MB of data remained to be transferred.  (1 mark)
  2. Calculate the speed required to transfer MB in minutes. Give your answer in megabits per second (Mbps), correct to the nearest whole number. (Note that megabit = bits.)  (3 marks)

 

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(i)   

 

(ii)   

♦ Mean mark part (ii) 35%.

 
 

 

Functions, EXT1′ F2 2016 HSC 13d

Suppose    with and real,

  1. Deduce that if    then  cuts the -axis only once.  (2 marks)

  2. If  , what is the multiplicity of the root    (2 marks)

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ii.  

 
 

 

♦♦ Mean mark 32%.

 

Mechanics, EXT2 2016 HSC 13c

The ends of a string are attached to points and , with directly above The points and are   m apart.

An object of mass kg is fixed to the string at The object moves in a horizontal circle with centre and radius   m, as shown in the diagram.

ext2-hsc-2016-13c

The tensions in the string from the object to points and are and respectively. The object rotates with constant angular velocity You may assume that the acceleration due to gravity is  

  1. Show that    (3 marks)
  2. For what range of values of  is    (1 mark)
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i.    ext2-hsc-2016-13c-answer1

 

 

 
 

 

ii.  

♦ Mean mark 41%.

Harder Ext1 Topics, EXT2 2016 HSC 13b

The circle centred at has diameter . A point on produced is chosen so that is a tangent to the circle at and   The tangents to the circle at and meet at

ext2-hsc-2016-13b-i

Copy or trace the diagram into your writing booklet.

Prove that   (4 marks)

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ext2-hsc-2016-13b-i-answer

   

 

Conics, EXT2 2016 HSC 12d

  1. Show that the equation of the normal to the hyperbola , at is given by   (2 marks)
  2. The normal at meets the hyperbola again at
    Show that       (3 marks)
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ii.  

 

 
 

Functions, EXT1′ F1 2016 HSC 11d

The diagram shows the graph of  
 

ext2-hsc-2016-11d

 
Draw a separate half-page diagram for each of the following functions, showing all asymptotes and intercepts.

  1.    (2 marks)

  2.    (2 marks)

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i.   

ext2-hsc-2016-11d-answer2
ii.   

ext2-hsc-2016-11d-answer4
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i.    ext2-hsc-2016-11d-answer2

 

ii.    ext2-hsc-2016-11d-answer4

Volumes, EXT2 2016 HSC 9 MC

The diagram shows the dimensions of a polyhedron with parallel base and top. A slice taken at height parallel to the base is a rectangle.

ext2-hsc-2016-9a

What is a correct expression for the volume of the polyhedron?

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Probability, 2UG 2016 HSC 29a

Two unbiased coins are tossed.

  1. What is the probability that one coin shows heads and the other shows tails?  (1 mark)
  2. A game is played in which one player tosses the two coins. The rules are as follows:
  3. • If both coins show heads, the player wins
    • If both coins show tails, the player wins
  4. • If one coin shows heads and the other shows tails, the player loses .

  5.  

    What is the financial expectation of this game?  (2 marks)

 

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(i)   

 

(ii)   

♦ Mean mark part (ii) 43%.

 

Measurement, STD2 M1 2016 HSC 28e

A company makes large marshmallows. They are in the shape of a cylinder with diameter 5 cm and height 3 cm, as shown in the diagram.

2ug-2016-hsc-q28_4

  1. Find the volume of one of these large marshmallows, correct to one decimal place.  (2 marks)

A cake is to be made by stacking 24 of these large marshmallows and filling the gaps between them with chocolate. The diagrams show the cake and its top view. The shading shows the gaps to be filled with chocolate.
 

2ug-2016-hsc-q28_5

  1. What volume of chocolate will be required? Give your answer correct to the nearest whole number.  (3 marks)

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i.   
   
   
    ³

 

ii.    2ug-2016-hsc-q28-answer1

♦ Mean mark part (ii) 35%.


 

 

Measurement, STD2 M7 2016 HSC 28a

Jacob has a large jar of silver coins. He adds 20 gold coins into the jar. He then seals the jar and shakes it to ensure that the gold coins are mixed in thoroughly with the silver coins. Jacob then opens the jar and takes a handful of coins. In his hand he has 33 silver coins and 4 gold coins. 

  1. Based on Jacob’s handful, if a coin is selected at random from the jar, what is the probability that it is a gold coin?  (1 mark)

  2. Jacob returns the handful of coins to the jar. Estimate the total number of coins in the jar.  (2 marks)

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i.   
   
♦ Mean mark part (i) 28%.

 

ii. 

 

Financial Maths, STD2 F4 2016 HSC 27d

Marge borrowed $19 000 to buy a used car. Interest on the loan was charged at 4.8% pa at the end of each month. She made a repayment of $436 at the end of every month. The table below sets out her monthly repayment schedule for the first four months of the loan. 
 

2ug-2016-hsc-q27_1

  1. Some values in the table are missing. Write down the values for and .  (2 marks)

  2. Calculate the value of .  (2 marks)

  3. Marge repaid this loan over four years.

     

    What is the total amount that Marge repaid?  (1 mark)

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ii.   

 
♦♦ Mean mark part (iii) 28%.
COMMENT: Read carefully whether total paid or total interest paid is required.

 

iii.   

Financial Maths, STD2 F1 2016 HSC 26f

Theo is completing his tax return. He has a gross salary of $82 521 and income from a rental property totalling $10 920. He is claiming $13 420 in allowable deductions.

  1. Determine Theo’s taxable income.  (1 mark)

  2. Using the tax table below, calculate Theo’s tax payable.  (2 marks)

  1. In addition to the above tax, Theo must also pay a Medicare levy of $1600.42
  2. Theo has already paid $20 525 as Pay As You Go (PAYG) tax.
  3. Should Theo receive a tax refund or will he owe more tax? Justify your answer with calculations.  (2 marks)

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i.   

 

ii.   

 

iii.