Math Is Fun Forum

Au101

Member

Registered: 2010-12-01

Posts: 353

A funny question

In a style I've never seen before. I'm not really sure where to begin with this one. I had a go, but by the time I got to part (v) I was just scratching my head really:

They don't seem to write questions like that anymore! Does anybody know what they're looking for here (no answer provided).

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: A funny question

Let's first look at A. What that notation means is we are looking at the set of all points P for which the line segment PQ has length 1. Do you know what that might be?

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The knowledge of some things as a function of age is a delta function.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: A funny question

Sure, unless I really have forgotten everything! If we were to show it on a graph we would have a circle centre (1,1) (i.e. Q) with radius 1. The set A will therefore be infinitely large, but some members of that set will be the point (1,2), the point (1,0), the point (2,1) and the point (0, 1).

Bob

Administrator

Registered: 2010-06-20

Posts: 10,156

Re: A funny question

hi Au101

Does anybody know what they're looking for here ?

Well it tests a lot of maths:

Do you understand set builder notation ?
Can you draw each locus correctly ?
Can you work out intersections and unions correctly ?

Your first answer is correct. 

Post the others if you want a second opinion on your answers.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: A funny question

Oh, sure, okay Thanks

Well, if we were to show B on a graph it would obviously be a straight line with equation x = y. So, the gradient is 1 and the y-intercept is zero. The set B will also be infinitely large, but some members of that set will be the point (0,0), the point (1,1) (i.e. Q), the point (2,2), etc.

Now, C doesn't seem that hard either. On a graph it will be a straight vertical line with x = 2. So it is also infinitely large, but includes the point (2,0), the point (2,1) and the point (2,2).

Now, D is less generous, but I believe, on a graph, the locus would be two lines parallel to x = y. One of the lines will be at a distance of 1 unit from x = y to the "left" (x < y) and the other will be at a distance of 1 unit from x = y to the "right" (x > y). Obviously another infinitely large set. Not 100% sure but I think a couple of points in this set would be:

So, it seems to me that: A and B will have two members in common, since the straight line that is the locus of B passes through the centre of the circle that is the locus of A.

B and C will have one member in common: the point (2,2).

A and C will have one member in common: the point (2,1).

There is no element common to all of A, B and C.

The two lines which form the locus of D will cross the line x = 2 at one point each, so the C and D will have two members in common.

B and D can have no members in common.

A and D will, I think, have four members in common - two on each of the lines which form the locus of D.

I don't think there will be an element common to all of A, C and D.

Now, how to draw that? Assuming I'm right!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,156

Re: A funny question

hi Au101,

Here's a diagram for all of those:

I don't think your points are right yet.

I'll just check.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: A funny question

Ooooh that's lovely, and now I see it all laid out nicely I can see how I was wrong about D in that A and D will only have two members in common - one on each of the lines, because of course, they must be tangents to the circle that is the locus of A. The points I suggest as elements of D came from choosing the point (0,0) (which is on the line x = y).

I proceeded as follows: The perpendicular distance between the point (0,0) and the line D1 is 1. Thus we can draw a right angle triangle - can we not? - with hypotenuse 1 and the other two sides of length a:

By inspection it is clear, then, that the point on D1 which is 1 unit from (0,0) is (-1/√2, 1/√2).

Bob

Administrator

Registered: 2010-06-20

Posts: 10,156

Re: A funny question

Thanks for that calculation.  That makes it very clear.

Now for E.

(1) Get A intersect B
(2) Get A intersect D
(3) Then form the union of those two answers.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: A funny question

Well indeed, now comes the tricky bit - I assumed I was supposed to convert all of this lovely (hard-earned ) information into a Venn diagram. And this is where Au101's brain goes: well forget that, I'm off for a cup of tea!

Er, this is what I have, assuming I've done this uploading an image thing right

On second thoughts, how does one upload an image

Let's try:

Last edited by Au101 (2015-01-05 07:34:19)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,156

Re: A funny question

For this question, you don't need a Venn diagram.  The answer should be a set containing members.  So just identify the members and list them like this

{answer 1, answer 2, .....}

Bob

Got to log off until later........

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: A funny question

Hmmmm, well, that's what I took "illustrate the following sets on the same diagram" to mean, but if you think we just need the elements of the set:

Then it should be easy enough to work them out, I get:

Bob

Administrator

Registered: 2010-06-20

Posts: 10,156

Re: A funny question

I think the diagram is the graph on post 6.

Just checking your co-ords.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,156

Re: A funny question

Co-ords good too.

B

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: A funny question

Thank you both very very much