Maths Help

leavingcert2003

Some Help with the following would be appreciated (mixture of first year Analysis and Algebra):

A = ﴾ 2 –1 ﴿
﴾ 3 4 ﴿

A is a matrix


- Find the eigenvalues and eigenvectors of A.
- Write down an invertible matrix E, and a diagonal matrix D, such that A=EDE-1

Calculate An and hence or otherwise solve the recurrence relation

Xn+1 = 2Xn – Yn
Yn+1 = -3Xn + 4Yn

given that X0 = 2, Y0 = 1.

2. Use the cross product to find all the solutions of the system of linear equations

X1 + 3X2 + 4 X3 = 0
2X1 + 4X2 - 3X3 = 0

3. State the Well-ordering Axiom for Z (integers), and show that it fails for Q (rationals).

4.
- Use the Sieve Method to find all the primes in the interval [8,96]

- Prove that there are infinitely many primes of the form 4q + 3,
where q is a positive integer.

- Prove that 7^1/2 is irrational.

5.
- Prove that if a sequence is increasing and bounded above then it converges.

- The sequence (an ) is defined by a1 = 2, an+1 = (an + 6)1/2 (n is greater thatn or equal to 1).
(1) Calculate a few terms and guess the behaviour of the sequence.
(2) Given that the sequence converges, find its (positive) limit.
(3) Now show that part (a) applies, so that the sequence does indeed converge.

6.
-What does it mean to say that the function f is continuous at c in (a,b)? Use the limit theorems to show that f defined by

f(x) = x5 – 3x4 + 1

is continuous at 2.

- State the Intermediate Value Theorem and use it to show that the polynomial above (f(x) = x5 – 3x4 + 1) has a root in [-1,0] and another in [0,1].

- Outline a proof of the Intermediate Value Theorem.

7.
- The function g is defined on (1,3) by

g(x) = {px^2 + x + 3 if 1 < x £ 2
{3x + q if 2 < x < 3

- Show that g is continuous on (1,3) if and only if q = 4p – 1. What are the values of p and q if g is differentiable on (1,3)?

- Prove the Mean Value Theorem, assuming Rolle’s Theorem.
- The function f satisfies f’(x) > 0 for all x in (1,2). Show that f is increasing on (1,2).

8.
- Show using the definition

lnx = {Integral between x and 1} of: (1/t)dt (x > 0),

that ln(ab) = ln(a) + ln(b) (a,b > 0 )

Explain briefly why given c there exists d > 0 such that ln(d) = c, and hence define
the exponential ex.

- Calculate the lower Riemann sum L(h,p) where h(x) = 1/x and P is the partition of [1,3] into eight equal parts. What does the answer say about the value of e?

9.

- Show that the improper integral:

{Integral between ∞ and 1} of: dx/(x+1)1/2

diverges.

- Show also that:

1/(21/2) + 1/(31/2) +……….+ 1/(n1/2) ≥{Integral between n and 1}of: dx/(x + 1)1/2 (n≥2)

and deduce that the series
∑1/(n + 1)1/2 [sum starting value is: n = 1, sum ending value is: ∞]

diverges
10.

-Define the Euler ø function.
Calculate ø(m) when (1) m = 128, (2) m = 24.511.137

State and prove Euler’s theorem.

Use Euler’s Theorem (or otherwise) to find the remainder when 2986 is divided by 45.

Find the missing digit in the ISBN number 3-540-7?1-87-x.

ecksor

What exactly is that from? You're having problems with all of it?

silverside

can you not sit down with some of the guys in your class and go through it - this should all have been covered in one form or another in your notes/tutorials.

smiles

Looks like an old leaving cert exam paper.

RTFM.

(read the book)

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ecksor

Euler's totient function and Dirichlet's theorem are on the leaving cert syllabus??

GuanYin

I'm not overly sure this forum exists for people to have their homework done for them.

I'll give Leavingcert2003 a bit of time to explain what exactly he wants help with before I decide to close this or not.

Nowhere man

there is no way that that is on the leaving certificate course

silverside

nah looks like first year maths somewhere (UCD?)

leavingcert2003

First Year NUIG actually. The final exam is on Friday week and I dont have a clue of any of this as I had to miss a lot of lectures. I really need help with this. Please

silverside

You wont learn this off boards. Get the tutorial notes of one of your classmates and study them like crazy. Do all the past papers you can. If they are nice to you they may help you out by explaining them. What were you doing all year if you cant answer any of these?

smiles

Originally posted by leavingcert2003
First Year NUIG actually. The final exam is on Friday week and I dont have a clue of any of this as I had to miss a lot of lectures. I really need help with this. Please

Right, well being a lovely nice NUIG student (and freakily organised) I've got all the notes from last years Algebra & Analysis (Johnny Burns / John McDermott / Ted Hurley / McCloskey) and all the exam papers done out.

I can bring 'em up to galway (heading down tonight/tomorrow) if you want to "borrow" em.

Plus I give remarkably cheap grinds..... (average of 95% in those exams)

PM me if you want the note or help

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