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Probability, MET1 2017 VCAA 5

For Jac to log on to a computer successfully, Jac must type the correct password. Unfortunately, Jac has forgotten the password. If Jac types the wrong password, Jac can make another attempt. The probability of success on any attempt is . Assume that the result of each attempt is independent of the result of any other attempt. A maximum of three attempts can be made.

  1. What is the probability that Jac does not log on to the computer successfully?   (1 mark)

  2. Calculate the probability that Jac logs on to the computer successfully. Express your answer in the form , where and are positive integers.   (1 mark)

  3. Calculate the probability that Jac logs on to the computer successfully on the second or on the third attempt. Express your answer in the form , where and are positive integers.   (2 marks)

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a. 

 

b. 

 

c. 

Calculus, MET1 2017 VCAA 3

Let  

  1. Show that  .   (1 mark)

  2. Sketch the graph of   on the axes below. Label the axis intercepts and any stationary points with their coordinates.   (3 marks)

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  2.  
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a. 

 

b.  


 

 

Harder Ext1 Topics, EXT2 2017 HSC 15b

Consider the curve  , for    and  , where is a positive constant.

  1. Show that the equation of the tangent to the curve at the point    is given by  .  (2 marks)
  2. The tangent to the curve at the point meets the and axes at   and  respectively. Show that  , where is the origin.  (3 marks)

 

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(i)   

 

 

 

(ii) 

 

 

 

 
 
 
 
 
 

Financial Maths, STD2 F1 SM-Bank 12

A golf shop is having a Boxing Day sale.

  1. What is the percentage discount on a putter which is reduced from $120.00 to $102.00?  (1 mark)

  2. The same discount applies storewide. What discount amount is applicable to a box of golf balls whose original price was $25.00?  (1 mark)

  3. What is the sale price of a golf bag which originally cost $160.00  (1 mark)

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i.    
   
   

 

ii.   
   

 

iii.   
   
   

Financial Maths, STD2 F1 SM-Bank 6

Michelle intends to keep a car purchased for $17 000 for 15 years. At the end of this time its value will be $3500.

  1. By what amount, in dollars, would the car’s value depreciate annually if Michelle used the flat rate method of depreciation?  (1 mark)

  2. Determine the annual flat rate of depreciation correct to one decimal place.  (1 mark) 

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i.   
   

 

 

ii.   

Financial Maths, STD2 F1 SM-Bank 5

Khan paid $900 for a printer.

This price includes 10% GST (goods and services tax).

  1. Determine the price of the printer before GST was added.

     

    Write your answer correct to the nearest cent.  (2 marks)

  2. Khan is able to depreciate the full $900 purchase price of his printer for taxation purposes.

     

    Under flat rate depreciation the printer will be valued at $300 after five years.

     

    Calculate the annual depreciation in dollars.  (1 mark)

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i.   

COMMENT: Reverse GST questions regularly cause problems for many students.
 
 

 

ii.   

Financial Maths, STD2 F1 SM-Bank 3

A company purchased a machine for $60 000.

For taxation purposes the machine is depreciated over time using the straight line depreciation method.

The machine is depreciated at a flat rate of 10% of the purchase price each year.

  1. By how many dollars will the machine depreciate annually?  (1 mark)

  2. Calculate the value of the machine after three years.  (1 mark)

  3. After how many years will the machine be $12 000 in value?  (1 mark)

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i.   
   

 

ii.   

 
 

 

iii.   

 

Financial Maths, STD2 F1 SM-Bank 10

Hugo is a professional bike rider.

The value of his bike will be depreciated over time using the flat rate method of depreciation.

The graph below shows his bike’s initial purchase price and its value at the end of each year for a period of three years.
 

  1. What was the initial purchase price of the bike?  (1 mark)

  2. Use calculations to show that the bike depreciates in value by $1500 each year.  (1 mark)

  3. Assume that the bike’s value continues to depreciate by $1500 each year. Determine its value five years after it was purchased.  (1 mark)

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i.   
 

ii.   

 

 

iii. 

 
 

Mechanics, EXT2 2017 HSC 14c

A smooth double cone with semi-vertical angle    is rotating about its axis with constant angular velocity .

Two particles, each of mass , are sitting on the cone as it rotates, as shown in the diagram.

Particle 1 is inside the cone at vertical distance above the apex, , and moves in a horizontal circle of radius .

Particle 2 is attached to the apex by a light inextensible string so that it sits on the cone at vertical distance below the apex. Particle 2 also moves in a horizontal circle of radius .

The acceleration due to gravity is .

  1. The normal reaction force on Particle 1 is .
    By resolving into vertical and horizontal components, or otherwise, show that  .  (2 marks)
  2. The normal reaction force on Particle 2 is and the tension in the string is  .

  3. By considering horizontal and vertical forces, or otherwise, show that

  4. .  (2 marks)
  6. Show that  .  (2 marks)

 

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(i)  

 

(ii)  

 

 

 

 

(iii) 

♦♦ Mean mark 34%.

Calculus, EXT2 C1 2017 HSC 14a

It is given that  .

  1. Find and so that  .  (1 mark)

  2. Hence, or otherwise, show that for any real number
     
         .  (2 marks)

  3. Find the limiting value as    of
     
         .  (1 mark)

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i. 

 

 


 

ii. 

 

iii. 

 
 

Measurement, STD2 M1 SM-Bank 8

A cannon ball is made out of steel and has a diameter of 23 cm.

  1. Find the volume of the sphere in cubic centimetres (correct to 1 decimal place).  (2 marks)

  2. It is known that the mass of the steel used is 8.2 tonnes/m³. Use this information to find the mass of the cannon ball to the nearest gram.  (2 marks)

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  1. ³
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i.   

 
 
  ³

 

ii.   ³³

³
  ³

 

 

 

Measurement, STD2 M1 SM-Bank 3

A 250-watt television is turned on for an average of 4 hours per day during off-peak periods for a week.

If the television is not running at any other time and electricity is charged at $0.36/kWh during off-peak, how much does it cost to run the television for a week?  (2 marks)

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Mechanics, EXT2 M1 2017 HSC 13c

A particle is projected upwards from ground level with initial velocity  , where is the acceleration due to gravity and is a positive constant. The particle moves through the air with speed    and experiences a resistive force.

The acceleration of the particle is given by  . Do NOT prove this.

The particle reaches a maximum height, , before returning to the ground.

Using  , or otherwise, show that    metres.  (4 marks)

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Harder Ext1 Topics, EXT2 2017 HSC 16a

  Let  , where  .

  1. Show that  , for any integer .  (1 mark)
  3. Let  , where is a positive integer.
  4. By summing the series, prove that  

    .  (3 marks)

  6. Deduce, from parts (i) and (ii), that.  (2 marks)

  7. Show that  
     is independent of .  (1 mark)

 

 

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(i)   

 
 

 

(ii)   

♦♦ Mean mark 34%.

 
 
 
 
 
♦ Mean mark part (iii) 48%.

 

(iii)   
   
   
   

 

 
 
 
 

♦♦♦ Mean mark part (iv) 28%.

 

(iv)   

 

Functions, EXT1′ F2 2017 HSC 12d

Let be a polynomial.

  1. Given that    is a factor of , show that
     
    .  (2 marks)

  2. Given that the polynomial    has a factor  , find the value of .  (2 marks)

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i.  
 
   

 

 

 

ii.  
 
   
   

 

 

CORE, FUR2 2017 VCAA 2

The back-to-back stem plot below displays the wingspan, in millimetres, of 32 moths and their place of capture (forest or grassland).

 

  1. Which variable, wingspan or place of capture, is a categorical variable?  (1 mark)

  2. Write down the modal wingspan, in millimetres, of the moths captured in the forest.  (1 mark)

  3. Use the information in the back-to-back stem plot to complete the table below.  (2 marks)

     

     

  4. Show that the moth captured in the forest that had a wingspan of 52 mm is an outlier.  (2 marks)

  5. The back-to-back stem plot suggests that wingspan is associated with place of capture.
  6. Explain why, quoting the values of an appropriate statistic.  (2 marks)

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  3.    

     

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a.   

 

b.   

 

c.   

 

 

d.   
 

 

 

 

 

e.   

CORE, FUR2 2017 VCAA 1

The number of eggs counted in a sample of 12 clusters of moth eggs is recorded in the table below.
     

  1. From the information given, determine
  2.  i. the range   (1 mark)

  3. ii. the percentage of clusters in this sample that contain more than 170 eggs.   (1 mark)

In a large population of moths, the number of eggs per cluster is approximately normally distributed with a mean of 165 eggs and a standard deviation of 25 eggs.

  1. Using the 68–95–99.7% rule, determine
  2.  i. the percentage of clusters expected to contain more than 140 eggs.   (1 mark)

  3. ii. the number of clusters expected to have less than 215 eggs in a sample of 1000 clusters.   (1 mark)

  4. The standardised number of eggs in one cluster is given by  
  5. Determine the actual number of eggs in this cluster.   (1 mark)

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a.   

b.i.  

b.ii. 

c. 

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a.i. 

a.ii. 

 

 

b.i.   
   

 

b.ii.   
   

 

 

c.   
 
 
   

Functions, EXT1′ F1 2017 HSC 12a

Consider the function  .

  1.  Show that    is increasing for all .  (1 mark)

  2.  Show that    is an odd function.  (1 mark)

  3.  Describe the behaviour of    for large positive values of .  (1 mark)

  4.  Hence sketch the graph of  .  (1 mark)

  5.  Hence, or otherwise, sketch the graph of  .  (1 mark)

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  4.  
  5.  

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i. 


 

 
 

 


 

 
 

 

 

ii.  
     
   
   

 

 

iii. 

 

iv.  

 

v.  

Conics, EXT2 2017 HSC 11b

An asymptote to the hyperbola    makes an angle with the positive -axis, as shown.

Find the value of .  (2 marks)

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GRAPHS, FUR1 2017 VCAA 2 MC

The graph below shows the volume of water in a water tank between 7 am and 5 pm on one day.

Which one of the following statements is true?

  1. The volume of water in the tank decreases between 8 am and 11 am.
  2. The volume of water in the tank increases at the greatest rate between 4 pm and 5 pm.
  3. The volume of water in the tank is constant between 12 noon and 2 pm.
  4. The tank is filled with water at a constant rate of 100 L per hour.
  5. More water enters the tank during the first five hours than during the last five hours.
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NETWORKS, FUR1 2017 VCAA 2 MC

Two graphs, labelled Graph 1 and Graph 2, are shown below.
 

 
The sum of the degrees of the vertices of Graph 1 is

  1. two less than the sum of the degrees of the vertices of Graph 2.
  2. one less than the sum of the degrees of the vertices of Graph 2.
  3. equal to the sum of the degrees of the vertices of Graph 2.
  4. one more than the sum of the degrees of the vertices of Graph 2.
  5. two more than the sum of the degrees of the vertices of Graph 2.
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Algebra, STD2 A1 SM-Bank 5

Fried's formula is used to calculate the medicine dosages for children aged 1-2 years.
 


 

Ben is 1.5 years old and receives a daily dosage of 450 mg of a medicine.

According to Fried's formula, what would the appropriate adult daily dosage of the medicine be?  (2 marks)

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GEOMETRY, FUR1 2017 VCAA 1 MC

A wheel has five spokes equally spaced around a central hub, as shown in the diagram below.
 


 

The angle between two spokes is labelled on the diagram.

What is the angle ?

  1.  
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CORE, FUR1 2017 VCAA 19-20 MC

Shirley would like to purchase a new home. She will establish a loan for $225 000 with interest charged at the rate of 3.6% per annum, compounding monthly.

Each month, Shirley will pay only the interest charged for that month.

Part 1

After three years, the amount that Shirley will owe is

  1.    $73 362
  2.  $170 752
  3.  $225 000
  4.  $239 605
  5.  $245 865

 
Part 2

Let be the value of Shirley’s loan, in dollars, after months.

A recurrence relation that models the value of is

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Algebra, STD2 A4 SM-Bank 3

Temperature can be measured in degrees Celsius () or degrees Fahrenheit ().

The two temperature scales are related by the equation  .

  1. Calculate the temperature in degrees Fahrenheit when it is  −20 degrees Celsius.  (1 mark)

  2. The following two graphs are drawn on the axes below:
     
             and  
     

         

    Explain what happens at the point where the two graphs intersect.  (1 mark)

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  2.  

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i.  
   

 

ii.   

CORE, FUR1 2017 VCAA 1-3 MC

The boxplot below shows the distribution of the forearm circumference, in centimetres, of 252 people.
 

Part 1

The percentage of these 252 people with a forearm circumference of less than 30 cm is closest to

 

Part 2

The five-number summary for the forearm circumference of these 252 people is closest to

 

Part 3

The table below shows the forearm circumference, in centimetres, of a sample of 10 people selected from this group of 252 people.
 

 
The mean, , and the standard deviation, , of the forearm circumference for this sample of people are closest to

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Measurement, STD2 M2 SM-Bank 01

Island A and island B are both on the equator. Island B is west of island A. The longitude of island A is 5°E and the angle at the centre of Earth (O), between A and B, is 30°.
 


 

  1. What is the longitude of island B?  (1 mark)
  2. What time is it on island B when it is 10 am on island A?  (1 mark)
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(i)   
   
   

 

 

(ii)  

 

Statistics, EXT1 S1 2017 HSC 11g

The probability that a particular type of seedling produces red flowers is  .

Eight of these seedlings are planted.

  1. Write an expression for the probability that exactly three of the eight seedlings produce red flowers.  (1 mark)

  2. Write an expression for the probability that none of the eight seedlings produces red flowers.  (1 mark)

  3. Write an expression for the probability that at least one of the eight seedlings produces red flowers.  (1 mark)

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i.  

 

ii.   

 

iii.   

Calculus, EXT1* C1 2017 HSC 15c

Two particles move along the -axis.

When  , particle is at the origin and moving with velocity 3.

For  , particle has acceleration given by  .

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

When  , particle is also at the origin.

For  , particle has velocity given by  .

  1. When do the two particles have the same velocity?  (2 marks)

  2. Show that the two particles do not meet for  .  (3 marks)

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i.  
 
   
   

 

 

ii. 

 

iii.  
   
   

 

♦ Mean mark (iii) 39%.

 

 

 

 

 

 

Calculus, EXT1* C1 2017 HSC 14c

Carbon-14 is a radioactive substance that decays over time. The amount of carbon-14 present in a kangaroo bone is given by

where and are constants, and is the number of years since the kangaroo died.

  1. Show that satisfies  .  (1 mark)

  2. After 5730 years, half of the original amount of carbon-14 is present.

     

    Show that the value of , correct to 2 significant figures, is – 0.00012.  (2 marks)

  3. The amount of carbon-14 now present in a kangaroo bone is 90% of the original amount.

     

    Find the number of years since the kangaroo died. Give your answer correct to 2 significant  figures.  (2 marks)

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i.  
   
   

 

ii. 

 
 

 

iii. 

 
 
 

Calculus, 2ADV C4 2017 HSC 14b

  1. Find the exact value of
     
  2. .  (1 mark)

  3. Using Simpson’s rule with one application, find an approximation to the integral
  4.  

     
  5. leaving your answer in terms of and .  (2 marks)
     

  6. Using parts (i) and (ii), show that
     
  7. .  (1 mark)

 

 

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(i)  
(ii)  
(iii)  
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(i)  
   
   

 

(ii)  
                    
                     
                  
 
 
 

 

(iii)  

♦ Mean mark 49%.