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L&E, 2ADV E1 2024 HSC 13

The graph shows the populations of two different animals, and , in a conservation park over time. The -axis is the size of the population and the -axis is the number of years since 1985 .

Population is modelled by the equation  .

Population is modelled by the equation  .
 

Complete the table using the information provided.   (3 marks)

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Calculus, 2ADV C3 2023 HSC 30

Let  .

  1. Find the coordinates of the stationary points of for  . You do NOT need to check the nature of the stationary points.  (3 marks)

  2. Without using any further calculus, sketch the graph of  , for  , showing stationary points and intercepts.  (2 marks)
     


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a.   

b.    
         

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a.   

 
 
 
Mean mark (a) 54%.


 


 

 
b.   
 

 

♦♦ Mean mark (b) 34%.

Calculus, 2ADV C3 2022 HSC 27

Let  .

It is given that  .

  1. Show that  .  (2 marks)

  2. Find any stationary points of    and determine their nature.  (2 marks)

  3. Sketch the curve  , showing any stationary points, points of inflection and intercepts with the axes.  (3 marks)

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  3.  
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a.   

 
   
   
   

 

b.   

 
 

 

 
   

 

 

c.   


 


Mean mark part (c) 54%.

Functions, 2ADV F2 SM-Bank 13

 

  1. Show that the function    is an odd function?  (1 mark) 

  2. Sketch  , labelling all intercepts and asymptotes.  (2 marks)

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i.   

 
 
 

 

 

ii.   


 

L&E, 2ADV E1 SM-Bank 2

The population of Indian Myna birds in a suburb can be described by the exponential function

where is the time in months.

  1.  What will be the population after 2 years?  (1 mark)

  2.  Draw a graph of the population.  (2 marks)

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  2.  

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i.   

 
 
 

 

ii.

Algebra, STD2 A4 2016 HSC 29b

The mass kg of a baby pig at age days is given by    where is a constant. The graph of this equation is shown.
 

2ug-2016-hsc-q29_1

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

  2. What is the daily growth rate of the pig’s mass? Write your answer as a percentage.   (1 mark)

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i.   

♦ Mean mark (i) 48%.
♦♦♦ Mean mark part (ii) 6%!

 
ii.   

Algebra, STD2 A4 2004 HSC 26a

  1. The number of bacteria in a culture grows from 100 to 114 in one hour.

     

    What is the percentage increase in the number of bacteria?   (1 mark)

  2. The bacteria continue to grow according to the formula  , where    is the number of bacteria after    hours.

     

    What is the number of bacteria after 15 hours?   (1 mark)

  1. Use the values of from    to    to draw a graph of  .

     

    Use about half a page for your graph and mark a scale on each axis.   (4 marks)

  2. Using your graph or otherwise, estimate the time in hours for the number of bacteria to reach 300.   (1 mark)

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i.   

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii. 

 
 

 

iii. 

 

iv. 

Algebra, STD2 A4 2012 HSC 30c

In 2010, the city of Thagoras modelled the predicted population of the city using the equation

.

That year, the city introduced a policy to slow its population growth. The new predicted population was modelled using the equation

.

In both equations, is the predicted population and is the number of years after 2010.  

The graph shows the two predicted populations.
 

  1. Use the graph to find the predicted population of Thagoras in 2030 if the population policy had NOT been introduced.   (1 mark)

  2. In each of the two equations given, the value of is 3 000 000.

     

    What does represent?   (1 mark)

  3. The guess-and-check method is to be used to find the value of , in  .

     

    (1) Explain, with or without calculations, why 1.05 is not a suitable first estimate for .  (1 mark)

     

    (2) With    and  , use the guess-and-check method and the equation    to estimate the value of to two decimal places. Show at least TWO estimate values for , including calculations and conclusions.  (2 marks)

  4. The city of Thagoras was aiming to have a population under 7 000 000 in 2050. Does the model indicate that the city will achieve this aim?

     

    Justify your answer with suitable calculations.  (2 marks)

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  1.  
  2.  
  3.  

     

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i.   

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii.    


 

♦ Mean mark part (ii) 48%

iii. (1) 

   
 

iii. (2) 

 

 

 

 

 

iv.