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Proof, EXT2 P1 2024 HSC 3 MC

Consider the statement:

'If a polygon is a square, then it is a rectangle.'

Which of the following is the converse of the statement above?

  1. If a polygon is a rectangle, then it is a square.
  2. If a polygon is a rectangle, then it is not a square.
  3. If a polygon is not a rectangle, then it is not a square.
  4. If a polygon is not a square, then it is not a rectangle.
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\(A\)

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\(\text{Statement:}\  P \Rightarrow \ Q\)

\(\text{Converse of statement:}\  Q \Rightarrow \ P\)

\(\Rightarrow A\)

Proof, EXT2 P1 2023 HSC 2 MC

Consider the following statement.

'If an animal is a herbivore, then it does not eat meat.'

Which of the following is the converse of this statement?

  1. If an animal is a herbivore, then it eats meat.
  2. If an animal is not a herbivore, then it eats meat.
  3. If an animal eats meat, then it is not a herbivore.
  4. If an animal does not eat meat, then it is a herbivore.
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\(D\)

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\(\text{Statement:}\ \ P \Rightarrow Q \)

\(\text{Converse of statement:}\ \ Q \Rightarrow P \)

\(\Rightarrow D\)

Proof, EXT2 P1 2022 HSC 7 MC

Consider the statement `P`.

`P` : For all integers `n \geq 1`, if `n` is a prime number then `(n(n+1))/(2)` is a prime number.

Which of the following is true about this statement and its converse?

  1. The statement `P` and its converse are both true.
  2. The statement `P` and its converse are both false.
  3. The statement `P` is true and its converse is false.
  4. The statement `P` is false and its converse is true.
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`D`

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`text{Statement:}\ \ ∀n in ZZ^+, text{if}\ n\ text{is prime}\ \ =>\ \ (n(n+1))/(2)\ text{is prime.}`

`text{Converse:}\ \ ∀n in ZZ^+, text{if}\ (n(n+1))/(2)\ text{is prime}\ \ =>\ \ n\ text{is prime.}`

`text{Consider prime}\ n=3:`

`(n(n+1))/(2)=(3xx4)/2=6\ \ text{(not prime → Statement is false)}`

`text{Consider}\ \ (n(n+1))/(2):`

`(n(n+1))/(2)\ text{is a prime}\ iff\ n=2\ \ text{(prime)}`

`:.\ text{Converse is true}`

`=>D`


♦♦ Mean mark 37%.

Proof, EXT2 P1 2021 HSC 12b

Consider Statement A.

Statement A: ‘If `n^2` is even, then `n` is even.’

  1. What is the converse of Statement A?.  (1 mark)

  2. Show that the converse of Statement A is true. (1 mark)

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  1. `text{See Worked Solution}`
  2. `text{See Worked Solution}`
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i.     `text{Converse}`

`text{If} \ n \ text{is even, then} \ n^2 \ text{is even.}`
 

ii.    `text{If} \ n \ text{is even:}`

`n` `= 2p, \ p  ∈  ZZ`
`n^2` `= (2 p)^2`
  `=4 p^2`
  `= 2 (2p^2)`
  `=2q, \ q  ∈  ZZ`

 

`:. \ text{If} \ n \ text{is even, then} \ n^2 \ text{is even.}`

Proof, EXT2 P1 2020 HSC 15a

In the set of integers, let `P` be the proposition:

'If  `k + 1`  is divisible by 3, then  `k^3 + 1`  divisible by 3.'

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

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

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

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  1. `text{See Worked Solutions}`
  2. `text{See Worked Solutions}`
  3. `text{See Worked Solutions}`
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i.     `text{Let} \ \ k + 1 = 3N, \ N∈ Z`

`=>  k = 3N – 1`

`k^3 + 1` `= (3N -1)^3 + 1`
  `= (3N)^3 + 3(3N)^2 (-1) + 3(3N)(-1)^2 + (-1)^3 + 1`
  `= 27N^3 – 27N^2 + 9N – 1 + 1`
  `= 3 (9N^3 – 9N^2 + 3N)`
  `= 3Q \ , \ Q ∈ Z`

 
`therefore \ text{If} \ \ k+ 1 \ \ text{is divisible by 3}, text{then} \ \ k^3 + 1 \ \ text{is divisible by 3.}`
 

ii.    `text{Contrapositive}`

`text{If} \ \ k^3 + 1 \ \ text{is not divisible by 3, then}\ \ k + 1\ \ text{is not divisible by 3.}`
 

♦♦ Mean mark part (iii) 36%.

iii.   `text{Converse:}`

`text{If} \ \ k^3 + 1\ \ text{is divisible by 3, then}\ \ k + 1\ \ text{is divisible by 3.}`

`text(Contrapositive of converse:)`

`text{If}\ \ k + 1\ \ text{is not divisible by 3, then}\ \ k^3 + 1\ \ text{is not divisible by 3.}`
 
`text(i.e.)\ \ k + 1 \ \ text{is not divisible by 3 when}\ \ k + 1 = 3Q + 1\ \ text{or}\ \ k + 1 = 3Q + 2, text{where}\ Q ∈ Z`
 

`text{If} \ \ k + 1` `= 3Q + 1\ \ => \ k=3Q`
`k^3 + 1` `= (3Q)^3 + 1`
  `= 27Q^3 + 1`
  `= 3(9Q^3) + 1`
  `= 3M + 1 \ \ (text{not divisible by 3,}\ M ∈ Z)`

 

`text{If} \ \ k + 1` `= 3Q + 2\ \ => \ k=3Q+1`
`k^3 + 1` `= (3Q + 1)^3 + 1`
  `= (3Q)^3 + 3(3Q)^2 + 3(3Q) + 1 + 1`
  `= 27Q^3 + 27Q^2 + 9Q + 2`
  `= 3(9Q^3 + 9Q^2 + 3Q) + 2`
  `= 3M + 2 \ (text{not divisible by 3,}\ M ∈ Z) `

 

`therefore \ text{By contrapositive, if}\ \ k^3 + 1\ \ text {is divisible by 3, k + 1 is divisible by 3.}`