# A few problems to have a go at...

Watch
Announcements

Page 1 of 1

Go to first unread

Skip to page:

Four problems that I hope you'll like. A level knowledge (inc FM where appropriate) is required.

**Problem 1**
Spoiler:

Determine whether or not there is a rational number that satisfies

Hint 1: a rational number is any number that can be expressed in the form where and are coprime (i.e they have a gcd of 1).

Hint 2: Use complex numbers.

Hint 3: An algebraic integer is a complex root of a polynomial with integer coefficients that has leading coefficient (that is the non-zero coefficient of the highest power) 1. The sum, difference and product of two algebraic integers is also an algebraic integer. If a number is both rational and an algebraic integer then it is an integer.

Show

Determine whether or not there is a rational number that satisfies

Hint 1: a rational number is any number that can be expressed in the form where and are coprime (i.e they have a gcd of 1).

Hint 2: Use complex numbers.

Hint 3: An algebraic integer is a complex root of a polynomial with integer coefficients that has leading coefficient (that is the non-zero coefficient of the highest power) 1. The sum, difference and product of two algebraic integers is also an algebraic integer. If a number is both rational and an algebraic integer then it is an integer.

**Problem 2****Problem 3****Problem 4**
2

reply

Report

#2

I have a funny feeling none of these are actually appropriate for "A level students"...

0

reply

(Original post by

I have a funny feeling none of these are actually appropriate for "A level students"...

**DFranklin**)I have a funny feeling none of these are actually appropriate for "A level students"...

Whenever I've done this in the past I've always received a decent response. We'll see what happens

0

reply

Report

#4

**DFranklin**)

I have a funny feeling none of these are actually appropriate for "A level students"...

0

reply

Report

#5

(Original post by

Well, definitely inappropriate for an A-level paper

Whenever I've done this in the past I've always received a decent response. We'll see what happens

**Indeterminate**)Well, definitely inappropriate for an A-level paper

Whenever I've done this in the past I've always received a decent response. We'll see what happens

1

reply

(Original post by

Having done A-Level maths + FM just last summer, I can safely say I have never seen anything like this unless I missed something...

**RDKGames**)Having done A-Level maths + FM just last summer, I can safely say I have never seen anything like this unless I missed something...

Well here's a start. For problem 4 consider the quadratic in the denominator and try a substitution.

(Original post by

Since there's a set of 10 problems I posted only yesterday with little response (and although some of them were hard, some were reasonably accessible) I'm not expecting a great deal...

**DFranklin**)Since there's a set of 10 problems I posted only yesterday with little response (and although some of them were hard, some were reasonably accessible) I'm not expecting a great deal...

Well, erm, at least we try.

0

reply

Report

#7

I

I'll have a crack at it tomorrow

Edit: Couldn't hack it

*think*I may be able to do 4...I'll have a crack at it tomorrow

Edit: Couldn't hack it

0

reply

Report

#8

Can you confirm whether or not Problem 2 is just f(x) = x. If so, do you want us to prove it?

0

reply

Report

#9

(Original post by

Can you confirm whether or not Problem 2 is just f(x) = x. If so, do you want us to prove it?

**BobBobson**)Can you confirm whether or not Problem 2 is just f(x) = x. If so, do you want us to prove it?

0

reply

**BobBobson**)

Can you confirm whether or not Problem 2 is just f(x) = x. If so, do you want us to prove it?

(Original post by

I think f(x) = 1+x works as well. Plug a,b = 0 in to get f(0)=0 or 1. Then let a=0, and see what you get. Also, with functional equations substitute back in to see if what you get works.

**A02**)I think f(x) = 1+x works as well. Plug a,b = 0 in to get f(0)=0 or 1. Then let a=0, and see what you get. Also, with functional equations substitute back in to see if what you get works.

0

reply

Report

#11

(Original post by

If you think you've found one (or more than one) solution, can't you find any others? Or can you show that there aren't any others?

**Indeterminate**)If you think you've found one (or more than one) solution, can't you find any others? Or can you show that there aren't any others?

I think this is right:

0

reply

Report

#12

Problem 4 is just ugly. Problem 3 isn't so bad, on the other hand (doubt an A-Level student would manage it, though):

Use to get

Now use and to get

Now use on our integral (pretty nifty identity, try proving it) to get:

Use to get

Now use and to get

Now use on our integral (pretty nifty identity, try proving it) to get:

3

reply

Report

#13

"An algebraic integer is a root of some polynomial with leading coefficient (that is the non-zero coefficient of the highest power) 1."

Really?

Really?

0

reply

(Original post by

Problem 4 is just ugly. Problem 3 isn't so bad, on the other hand (doubt an A-Level student would manage it, though):

Use to get

Now use and to get

Now use on our integral (pretty nifty identity, try proving it) to get:

**Zacken**)Problem 4 is just ugly. Problem 3 isn't so bad, on the other hand (doubt an A-Level student would manage it, though):

Use to get

Now use and to get

Now use on our integral (pretty nifty identity, try proving it) to get:

(Original post by

"An algebraic integer is a root of some polynomial with leading coefficient (that is the non-zero coefficient of the highest power) 1." Really?

**Zacken**)"An algebraic integer is a root of some polynomial with leading coefficient (that is the non-zero coefficient of the highest power) 1." Really?

0

reply

Report

#15

(Original post by

I expected you to chip in

**Indeterminate**)I expected you to chip in

When do your lectures start?

0

reply

(Original post by

Might as well before lectures start (tomorrow!!). ;lol:

When do your lectures start?

**Zacken**)Might as well before lectures start (tomorrow!!). ;lol:

When do your lectures start?

Day 1 for me too, but I know a few who haven't got any lectures until Monday.

But yeah looking forward to it of course!

0

reply

X

Page 1 of 1

Go to first unread

Skip to page:

### Quick Reply

Back

to top

to top