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Proof, EXT2 P1 2023 HSC 16b

  1. Prove that  , for  .  (2 marks)

  2. Using part (i), or otherwise, prove that for all positive integers ,

  (3 marks)

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i.  

ii. 

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i.    


 

♦ Mean mark (i) 39%.

ii.   

 
 
   
 
 

 

 
 
 
 
 
♦♦ Mean mark (ii) 28%.

Proof, EXT2 P1 2022 HSC 16c

It is given that for positive numbers with arithmetic mean ,
 

                   (Do NOT prove this.)

Suppose a rectangular prism has dimensions    and surface area .

  1. Show that  .  (2 marks)

  2. Using part (i), show that when the rectangular prism with surface area is a cube, it has maximum volume.  (2 marks)

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i.   


♦♦♦ Mean mark (i) 26%.
 
 
 
 

 

ii.   


 


♦♦♦ Mean mark (ii) 26%.

Proof, EXT2 P1 2021 HSC 15a

For all non-negative real numbers and .  (Do NOT prove this.)

  1. Using this fact, show that for all non-negative real numbers , and ,
  2.     .  (2 marks)

  3. Using part (i), or otherwise, show that for all non-negative real numbers , and ,  
  4.     .  (2 marks)

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i.   

♦ Mean mark part (i) 50%.


 

 

 

ii.   

♦♦♦ Mean mark part (ii) 26%.

 

 

Proof, EXT2 P1 2006 HSC 8a

Suppose  

  1. Show that    (2 marks)

  2. Hence show that    (1 mark)

  3. By integrating the expressions in the inequality in part (ii) with respect to    from    to  , show that
     
          (2 marks)

  4. Hence show that for  
     
          (1 mark)

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i.   
 
 
 
 
 
 

 

ii.   

 

 

iii.  
 
 
 

 

iv.