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Proof, EXT2 P1 2023 SPEC2 1 MC

Consider the following statement.

'If my football team plays badly, then they are not training enough.'

Which one of the following statements is the contrapositive of the statement above?

  1. If my football team plays badly, then they need more training.
  2. If they are training enough, then my football team does not play badly.
  3. If my football team doesn't play badly, then they are training enough.
  4. If they are training enough, then my football team will most likely win.
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Proof, EXT2 P1 2022 HSC 3 MC

Let be three points in three-dimensional space with .

Consider the following statement.

If is on the line , then there exists a real number such that  .

Which of the following is the contrapositive of this statement?

  1. If for all real numbers , then is on the line .
  2. If for all real numbers , then is not on the line .
  3. If there exists a real number such that  , then is on the line .
  4. If there exists a real number such that  , then is not on the line .
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Proof, EXT2 P1 2021 HSC 9 MC

Four cards have either RED or BLACK on one side and either WIN or TRY AGAIN on the other side.

Sam places the four cards on the table as shown below. 
 


 

A statement is made: ‘If a card is RED, then it has WIN written on the other side’.

Sam wants to check if the statement is true by turning over the minimum number of cards.

Which cards should Sam turn over?

  1.  1 and 4
  2.  3 and 4
  3.  1. 2 and 4
  4.  1, 3 and 4
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♦ Mean mark 48%.


 

Proof, EXT2 P1 2021 HSC 4 MC

Consider the statement:

‘For all integers , if is a multiple of 6, then is a multiple of 2’.

Which of the following is the contrapositive of the statement?

  1. There exists an integer such that is a multiple of 6 and not a multiple of 2.
  2. There exists an integer such that is a multiple of 2 and not a multiple of 6.
  3. For all integers , if is not a multiple of 2, then is not a multiple of 6.
  4. For all integers , if is not a multiple of 6, then is not a multiple of 2.
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Proof, EXT2 P1 2020 HSC 15a

In the set of integers, let be the proposition:

'If    is divisible by 3, then    divisible by 3.'

  1. Prove that the proposition is true.  (2 marks)

  2. Write down the contrapositive of the proposition .  (1 mark)

  3. Write down the converse of the proposition and state, with reasons, whether this converse is true or false.  (3 marks)

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i.     

 
 
 
 

 

 

ii.    


 

♦♦ Mean mark part (iii) 36%.

iii.  


 

 

 
 
 

 

 
 
 
 

 

Proof, EXT2 P1 2020 HSC 7 MC

Consider the proposition:

'If    is not prime, then is not prime'. 

Given that each of the following statements is true, which statement disproves the proposition?

  1. is prime
  2. is divisible by 9
  3. is prime
  4. is divisible by 23
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Proof, EXT2 P1 SM-Bank 5 MC

Four cards are placed on a table with a letter on one face and a shape on the other.
 

     

 
You are given the rule: "if N is on a card then a circle is on the other side."

Which cards need to be turned over to check if this rule holds?

  1. N and G
  2. G and triangle
  3. circle and N
  4. N and triangle
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Proof, EXT2 P1 SM-Bank 1 MC

Consider the following statement.

"If you have no treasure, I have no kingdom."

Which of the following is logically equivalent to this statement?

  1.  If I have no kingdom then you have no treasure.
  2.  If you have treasure then I have a kingdom.
  3.  If you have no kingdom then I have no treasure.
  4.  If I have a kingdom then you have treasure.
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