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Mechanics, EXT2* M1 2017 HSC 13c

A golfer hits a golf ball with initial speed at an angle to the horizontal. The golf ball is hit from one side of a lake and must have a horizontal range of 100 m or more to avoid landing in the lake.
 

     

Neglecting the effects of air resistance, the equations describing the motion of the ball are

,

where is the time in seconds after the ball is hit and is the acceleration due to gravity in . Do NOT prove these equations.

  1. Show that the horizontal range of the golf ball is
     
          metres.  (2 marks)

  2. Show that if    then the horizontal range of the ball is less than 100 m.  (1 mark)

It is now given that    and that the horizontal range of the ball is 100 m or more.

  1. Show that  .  (2 marks)

  2. Find the greatest height the ball can achieve.  (2 marks)

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i.   

 

 
 

 

ii.   

♦ Mean mark 44%.

 

 

iii.   

 

iv.   

♦♦ Mean mark 35%.
 

 

 
 

 

 
 
 
 

Binomial, EXT1 2017 HSC 13b

Let be a positive EVEN integer.

  1. Show that
  2. .  (2 marks)

  3. Hence show that
  4. .  (1 mark)

  5. Hence show that
  6. .  (2 marks)

 

 

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(i)   

 

(ii)   

♦ Mean mark part (iii) 48%.

 

(iii)   

Plane Geometry, 2UA 2017 HSC 16c

In the triangle , the point is the mid-point of . The point lies on and

.

The line that passes through the point and is parallel to meets produced at .

Copy or trace this diagram into your writing booklet.

  1. Prove that is isosceles.  (3 marks)
  2. The point is chosen on so that is parallel to .
     
    Show that bisects .  (2 marks)

 

 

 

 

 

 

 

 

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(i)  

♦♦♦ Mean mark 21%.

 

 

 

(ii)  

♦♦ Mean mark 31%.

Calculus, 2ADV C3 2017 HSC 16a

John’s home is at point and his school is at point . A straight river runs nearby.

The point on the river closest to is point , which is 5 km from .

The point on the river closest to is point , which is 7 km from .

The distance from to is 9 km.

To get some exercise, John cycles from home directly to point on the river, km from , before cycling directly to school at , as shown in the diagram.
 


  

The total distance John cycles from home to school is km.

  1. Show that  .   (1 mark)

  2. Show that if  , then  .   (3 marks)

  3. Find the value of that makes  .   (2 marks)

  4. Explain why this value of gives a minimum for .   (1 mark)

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i.  

 
 

 

ii. 

♦ Mean mark 41%.

 

 

iii. 

♦♦ Mean mark 29%.
♦♦♦ Mean mark 9%.

 

iv.  

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%.

 

 

 

 

 

 

Financial Maths, 2ADV M1 2017 HSC 15b

Anita opens a savings account. At the start of each month she deposits into the savings account. At the end of each month, after interest is added into the savings account, the bank withdraws $2500 from the savings account as a loan repayment. Let be the amount in the savings account after the th  withdrawal.

The savings account pays interest of 4.2% per annum compounded monthly.

  1. Show that after the second withdrawal the amount in the savings account is given by
    .  (2 marks)

  2. Find the value of    so that the amount in the savings account is $80 000 after the last withdrawal of the fourth year.  (3 marks)

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i.  
   
 
   
   
   
♦ Mean mark part (ii) 48%.

 

ii.  

   
 

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%.
 

Algebra, 2UG 2017 HSC 30d

In an investigation, students used different numbers of identical small solar panels to power model cars. The cars were then tested and their speed measured in km/h. The results are summarised in the table.
 


 

The equation of the least-squares line of best fit, relating the speed and the number of solar panels, has been calculated to be
 


 

  1. What would be the speed of a car powered by 5 solar panels, based on this equation?  (1 mark)
  2. Calculate the correlation coefficient, , between the number of solar panels and the speed of a car.  (2 marks)

 

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(i)   
   

 

(ii)  

♦♦ Mean mark 23%.
 

Measurement, STD2 M6 2017 HSC 30c

The diagram shows the location of three schools. School is 5 km due north of school , school is 13 km from school and is 135°.
 


 

  1. Calculate the shortest distance from school to school , to the nearest kilometre.  (2 marks)

  2. Determine the bearing of school from school , to the nearest degree.  (3 marks)

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i.   

 
 
 

 

ii.   

♦♦ Mean mark 31%.
 

 

Statistics, STD2 S1 2017 HSC 30a

A set of data has a lower quartile () of 10 and an upper quartile () of 16.

What is the maximum possible range for this set of data if there are no outliers?  (2 marks)

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♦♦ Mean mark 34%.

 
 
 
 

 

 

Statistics, STD2 S5 2017 HSC 29d

All the students in a class of 30 did a test.

The marks, out of 10, are shown in the dot plot.
 


 

  1. Find the median test mark.  (1 mark)

  2. The mean test mark is 5.4. The standard deviation of the test marks is 4.22.

     

    Using the dot plot, calculate the percentage of the marks which lie within one standard deviation of the mean.  (2 marks)

  3. A student states that for any data set, 68% of the scores should lie within one standard deviation of the mean. With reference to the dot plot, explain why the student’s statement is NOT relevant in this context.  (1 mark)

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♦ Mean mark 50%.
i.   
   
   

 

ii.   

♦♦ Mean mark 34%.

 

iii.   

♦♦♦ Mean mark 13%.

Probability, STD2 S2 2017 HSC 29c

A group of Year 12 students was surveyed. The students were asked whether they live in the city or the country. They were also asked if they have ever waterskied.

The results are recorded in the table.
  


  

  1. A person is selected at random from the group surveyed. Calculate the probability that the person lives in the city and has never waterskied.  (2 marks)

  2. A newspaper article claimed that Year 12 students who live in the country are more likely to have waterskied than those who live in the city.

     

    Is this true, based on the survey results? Justify your answer with relevant calculations.  (2 marks)

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i.    
   
   
♦ Mean mark part (ii) 47%.

 

ii.  
   
   

 

 
 

 

Algebra, STD2 A4 2017 HSC 28e

A movie theatre has 200 seats. Each ticket currently costs $8.

The theatre owners are currently selling all 200 tickets for each session. They decide to increase the price of tickets to see if they can increase the income earned from each movie session.

It is assumed that for each one dollar increase in ticket price, there will be 10 fewer tickets sold.

A graph showing the relationship between an increase in ticket price and the income is shown below.
 


 

  1. What ticket price should be charged to maximise the income from a movie session?  (1 mark)

  2. What is the number of tickets sold when the income is maximised?  (1 mark)

  3. The cost to the theatre owners of running each session is $500 plus $2 per ticket sold.

     

    Calculate the profit earned by the theatre owners when the income earned from a session is maximised.  (2 marks)

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i.   

♦ Mean mark 50%.
 

 

ii.   

♦ Mean mark 45%.
 

 

 
 

 

iii.  
   

 

Financial Maths, STD2 F4 2017 HSC 28c

Michelle borrows $100 000. The interest rate charged is 12% per annum compounded monthly. The monthly payment is $1029 and the first repayment is made after one month.

What is the amount outstanding immediately after the SECOND monthly repayment is made?  (3 marks)

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♦ Mean mark 41%.

 

 

 

 

Probability, 2UG 2017 HSC 28b

Five people are in a team. Two of them are selected at random to attend a competition.

  1. How many different groups of two can be selected?  (1 mark)
  2. If Mary is one of the five people in the team, what is the probability that she is selected to attend the competition?  (1 mark)

 

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(i)  
   
♦♦ Mean mark part (ii) 33%.

 

(ii)  

 

Algebra, 2UG 2017 HSC 28a

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. Solve the following equations simultaneously, using either the substitution method or the elimination method.  (2 marks)

     

  3. The graphs of   and    are shown below.
  4.  


  5. What does the result from part (ii) mean in the context of the graph?  (1 mark)     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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(i)  
   

 

(ii)  
 

 

♦♦ Mean mark 31%.
MARKER’S COMMENT: An area that requires attention.

 

♦♦♦ Mean mark 20%.

 

(iii)  

Algebra, 2UG 2017 HSC 30b

The cost of a jewellery box varies directly with the cube of its height.

A jewellery box with a height of 10 cm costs $50.

Calculate the cost of a jewellery box with a height of 12 cm.  (2 marks)

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♦ Mean mark 49%.

 

 

Measurement, 2UG 2017 HSC 27d

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


 

  1. What is the longitude of island ?  (1 mark)
  2. What time is it on island when it is 10 am on island ?  (1 mark)
  3. A ship leaves island and travels west along the equator to island . It travels at a constant speed of 40 km/h.
    How long will the ship take to arrive at island ? Give your answer in days and hours to the nearest hour.  (3 marks)

 

 

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(i)   
   
   

 

(ii)  

♦ Mean marks of 40% and 45% for parts (i) and (ii) respectively.

 

 

(iii)   
   
   

♦ Mean mark part (iii) 38%.
 
 

Financial Maths, STD2 F5 2017 HSC 27c

A table of future value interest factors for an annuity of $1 is shown.
 


 

An annuity involves contributions of $12 000 per annum for 5 years. The interest rate is 4% per annum, compounded annually.

  1. Calculate the future value of this annuity.  (1 mark)

  2. Calculate the interest earned on this annuity.  (1 mark)

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i.  

 
♦♦ Mean mark part (ii) 22%.
COMMENT: A very poorly answered question dealing with a core concept in this area.

 

ii.  
   
   

Statistics, STD2 S1 2017 HSC 27a

Jamal surveyed eight households in his street. He asked them how many kilolitres (kL) of water they used in the last year. Here are the results.

  1. Calculate the mean of this set of data.  (1 mark)

  2. What is the standard deviation of this set of data, correct to one decimal place?  (1 mark)

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i.  
   
♦ Mean mark part (ii) 47%.
IMPORTANT: The population standard deviation is required here.

 

ii.  
   

Measurement, STD2 M1 2017 HSC 25 MC

In the circle, centre , the area of the quadrant is 100 cm².
 


 

What is the arc length , correct to one decimal place?

  1. 8.9 cm
  2. 11.3 cm
  3. 17.7 cm
  4. 25.1 cm
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♦ Mean mark 44%.

 

 
 
 

 

Algebra, STD2 A2 2017 HSC 20 MC

A pentagon is created using matches.

By adding more matches, a row of two pentagons is formed.

Continuing to add matches, a row of three pentagons can be formed.

Continuing this pattern, what is the maximum number of complete pentagons that can be formed if 100 matches in total are available?

A.    

B.    

C.    

D.    

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Probability, STD2 S2 2017 HSC 15 MC

The faces on a twenty-sided die are labelled  $0.05, $0.10, $0.15, … , $1.00.

The die is rolled once.

What is the probability that the amount showing on the upper face is more than 50 cents but less than 80 cents?

A.     

B.     

C.     

D.     

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♦ Mean mark 50%.

 

GEOMETRY, FUR1 SM-Bank 35 MC

Kim lives in Perth (32°S, 115°E). He wants to watch an ice hockey game being played in Toronto  (44°N, 80°W)  starting at 10.00 pm on Wednesday.

What is the time in Perth when the game starts?

A.   9.00 am on Wednesday

B.   7.40 pm on Wednesday 

C.   9.00 pm on Wednesday

D.   12.20 am on Thursday

E.   11.00 am on Thursday

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GEOMETRY, FUR2 SM-Bank 15

Osaka is at  34°N, 135°E, and Denver is at  40°N, 105°W. 

  1. Show that there is a 16-hour time difference between the two cities.
    (Ignore time zones.)    (1 mark)
  2. John lives in Denver and wants to ring a friend in Osaka. In Denver it is 9 pm Monday.     

     

    What time and day is it in Osaka then?   (1 mark)

  3. John’s friend in Osaka sent him a text message which happened to take 14 hours to reach him. It was sent at 10 am Thursday, Osaka time.

     

    What was the time and day in Denver when John received the text?    (1 mark)

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a.  
   

 

 

 

b. 

 

 

c. 

GEOMETRY, FUR2 SM-Bank 14

Pontianak has a longitude of 109°E, and Jarvis Island has a longitude of 160°W.

Both places lie on the Equator

  1. Find the shortest great circle distance between these two places, to the nearest kilometre. You may assume that the radius of the Earth is 6400 km.    (2 marks)
  2. The position of Rabaul is 4° to the south and 48° to the west of Jarvis Island. What is the latitude and longitude of Rabaul?     (1 mark)
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a.  
   

 

 

 

 
 
 

 

b.  
 
 
 
   
 
 
 
 

 

 

 

Calculus, MET1 SM-Bank 28

The function    has the rule  .

  1. Show that the graph of    cuts the -axis at  .   (1 mark)

  2. Sketch the graph    for    showing where the graph cuts each of the axes.   (2 marks)

  3. Find the area under the curve    between    and  .   (3 marks)

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  1.  
  2.  
       
  3. ²
Show Worked Solution

a.   

 

 

 

b.   2UA HSC 2006 7b

 

c. 
 
  ­­
 
 
  ²

Algebra, MET1 SM-Bank 23

The function    with rule    is drawn below.

Inverse Functions, EXT1 2004 HSC 5b

  1. Copy or trace this diagram into your writing booklet.
  2. On the same set of axes, sketch    where   is the inverse function of  .   (1 mark)

  3. Find the domain of the inverse  .   (1 mark)

  4. Find an expression for    in terms of  .   (2 marks)

  5. The graphs of    and    meet at exactly one point  .Let  α  be the -coordinate of  . Explain why  α  is a root of the equation
  6.      .   (1 mark)

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a.  

Inverse Functions, EXT1 2004 HSC 5b Answer

b.  

 

c. 

 

 

 

d.   

 

α

Calculus, MET1 SM-Bank 21

The rule for function   is  .  The diagram shows the graph  .

 Inverse Functions, EXT1 2010 HSC 3b

The graph has two points of inflection. 

  1. Find the coordinates of these points.   (2 marks)

  2. Explain why the domain of must be restricted if is to have an inverse function.    (1 mark)

  3. Find the rule for the inverse function if the domain of is restricted to     (2 marks)

  4. Find the domain for .    (1 mark)

  5. Sketch the curve  .   (1 mark)

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  5. Inverse Functions, EXT1 2010 HSC 3b Answer

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a.   
 
 
   
   

 

 
   
 
   
COMMENT: It is also valid to show that is an even function and if a P.I. exists at , there must be another P.I. at .
 

 


 

 

 

 

b.  
 
 
 

 

c.  

 

 

 

 

d.  

 

e. 

Inverse Functions, EXT1 2010 HSC 3b Answer