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Proof, EXT2 P1 2015 HSC 15b

Suppose that    and    is a positive integer.

  1. Show that    (2 marks) 

  2. Hence, or otherwise, show that 
     
            (2 marks)

  3. Hence, explain why
     
          (1 mark)

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i.   
 
 

 

♦♦ Mean mark 21%.
STRATEGY: The conversion of the middle term from a fraction into a logarithm should flag the need for integration of each term.
ii.   
 
 
 

 

 

iii.   
 
♦♦ Mean mark 29%.
 

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.    

 

Proof, EXT2 P1 2007 HSC 7a

  1. Show that    for     (2 marks)

  2. Let  .

     

    Show that the graph of    is concave up for     (2 marks)

  3. By considering the first two derivatives of  ,show that    for     (2 marks)

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i.   
 
MARKER’S COMMENT: A geometric proof using arc length and a right angled triangle caused problems as few students could deal with the case  .

 

 

 

ii.
 
 
 

 

 

iii. 

 


 

 

Proof, EXT2 P1 2014 HSC 16b

Suppose is a positive integer. 

  1. Show that 
     
        .  (3 marks)

  2. Use integration to deduce that
     
    .  (2 marks)

  3. Explain why   .  (1 mark)

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i.   

♦♦ Mean mark 12%.
STRATEGY: Applying the GP formula and simplifying the middle term is worth 2 full marks in this question.
 

 

 

 

 

ii.   

♦ Mean mark 43%.
 
 
 
 

 

 

iii. 

♦♦ Mean mark 28%.