SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Proof, EXT2 P1 2021 HSC 5 MC

Which of the following statements is FALSE?

  1.                                              
  2.                                              
  3.                                
  4.                    
Show Answers Only

Show Worked Solution

♦ Mean mark 50%.


 

Proof, EXT2 P1 2012 HSC 16c

Let    be an integer where  . Integers from    to    inclusive are selected randomly one by one with repetition being possible. Let    be the probability that exactly    different integers are selected before one of them is selected for the second time, where  .

  1. Explain why  .   (2 marks)

  2. Suppose  . Show that  .   (2 marks)

  3. Show that if   then the integers    and    satisfy  .   (2 marks)

  4. Hence show that if    is not a perfect square, then    is greatest when    is the closest integer to  . 

     

    You may use part (iii) and also that    if  .   (2 marks)

Show Answers Only
Show Worked Solution

i.   

♦♦♦ Mean mark part (i) 7%.

 

 
 

♦ Mean mark part (ii) 38%.

 

ii.
 
 
 
 

 

♦♦♦ Mean mark part (iii) 10%.
iii.  
 
 
 
 

 

 

 

iv. 

 

♦♦♦ Mean mark part (iii) 1%.

 

Proof, EXT2 P1 2012 HSC 15a

  1. Prove that  , where    and  .   (1 mark)

  2. If  ,  show that  .   (2 marks)

  3. Let    and    be positive integers with  .
     
    Prove that   (2 marks)

  4. For integers  , prove that
     
        .   (1 mark)

     
Show Answers Only
Show Worked Solution
i.
 
 
 

 

ii. 

♦♦ Mean mark part (ii) 28%.

 

 
 
 

 

 

iii.  

♦ Mean mark part (iii) 39%.

 

 

 

iv. 

♦♦♦ Mean mark part (iv) just 2%!

 

 

Proof, EXT2 P1 2013 HSC 16a

  1. Find the minimum value of  , for    (2 marks)

  2. Hence, or otherwise, show that for  ,
     
          (1 mark)

  3. Hence, or otherwise, show that for    and  ,
     
          (2 marks)

Show Answers Only
Show Worked Solution
♦ Mean mark 43%.

STRATEGY: The best responses carefully considered the domain restrictions, notably
i.
 
   
   
 

 

 

 

♦ Mean mark 47%.

ii. 
 
 
 
 
 

 

 

♦♦ Mean mark 19%.

STRATEGY: The best responses carefully considered the domain restrictions, notably
iii.