VCAA 2011 Exam 1 Q9 in Networks

[billyjackson768]

I got this one right according to my answer sheet, but looking at it now I have got no idea why... I actually so much as crossed off the point that VCAA stated that made the question correct and circled the wrong answer in my question booklet. I actually think I may have educated guessed it, but changed my guess at the last second or something.

Anyways, does anyone have a worked solution for this. VCAA didn't exactly explain why all the other statements were not true. I hate these questions where you need to go through such a long list of options to be sure an answer is correct.

Hopefully the exam doesn't have a lot more of those difficult questions like that this year. You know with spesh kids being allowed in and everything.

[Tonychet2]

1 and 2 can easily be ruled as wrong,

 the quickest way to do it is draw the prototypical euler path, that is A - B - C - D , in a square with a line going from A to D (diagonal)

[Will T]

Okay, I'll talk you through why each statement is false.
The first statement: In the network, there could be three vertices with odd degrees.
This is false, as this condition renders the network disconnected, and therefore an Euler path/circuit can not exist.
The second statement: The path could have passed an isolated vertex. Once again this just makes no sense.
The third statement: The path could have included vertex Q more than once. I drew a sample network and was able to show that it could pass Q more than once, although after some experience with Euler paths/circuits it should be known that this is true.
The fourth statement: The sum of the degrees of P and Q could equal seven. This is false as for an Euler path to exist, P and Q must either both be even or odd vertices. And in any case, the sum of two odd numbers and two even numbers is always an even number and therefore not 7.
The fifth statement: The sum of degrees of all the vertices in the network could be equal to seven. This is false for the same reason as the statement above is false. For example, if P and Q are even vertices, then summing even numbers will always yield an even number. If we consider the alternative case, P and Q are both odd vertices, then summing two odds makes an even, and then summing evens will still yield an even.

Hence, the only statement that is true is actually the third statement, and so the answer is B.
I hope this helped

[Tonychet2]

Okay, I'll talk you through why each statement is false.
The first statement: In the network, there could be three vertices with odd degrees.
This is false, as this condition renders the network disconnected, and therefore an Euler path/circuit can not exist.
The second statement: The path could have passed an isolated vertex. Once again this just makes no sense.
The third statement: The path could have included vertex Q more than once. I drew a sample network and was able to show that it could pass Q more than once, although after some experience with Euler paths/circuits it should be known that this is true.
The fourth statement: The sum of the degrees of P and Q could equal seven. This is false as for an Euler path to exist, P and Q must either both be even or odd vertices. And in any case, the sum of two odd numbers and two even numbers is always an even number and therefore not 7.
The fifth statement: The sum of degrees of all the vertices in the network could be equal to seven. This is false for the same reason as the statement above is false. For example, if P and Q are even vertices, then summing even numbers will always yield an even number. If we consider the alternative case, P and Q are both odd vertices, then summing two odds makes an even, and then summing evens will still yield an even.

Hence, the only statement that is true is actually the third statement, and so the answer is B.
I hope this helped

just to paraphase the reason why 4th is false is because for any euler path it must start at the odd vertice and end at the odd vertice if there are 2 odds, if the path is all evens, then even + even = even lol

[billyjackson768]

Thanks! Much clearer. I usually rely on notes for these questions, so when I has a lot of different options to it where you sort of need to apply it all at once it gets difficult. Did cross out 1 and 2 right away though so I guess having an educated guess can actually pay off.