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PAPER TITLE: PURE MATHEMATICS

EXAM DATE: FRIDAY 14, JUNE 2013

COURSE CODE: M208/J

 

Question 1                                                                          

Sketch the graph of the function f defined by

 

indicating clearly the main features

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Question 2

Let ∼ be the relation defined on Z by x ∼ y if x² −y² is divisible by 3.

 Prove that ∼ is an equivalence relation

Question 3

The group table for a group G is shown below.

0	e	a	b	c	d	f
e	e	a	b	c	d	f
a	a	f	d	e	c	b
b	b	d	e	f	a	c
c	c	e	f	d	b	a
d	d	c	a	b	f	e
f	f	b	c	a	e	D

 

(a) Show that G is the cyclic group generated by the element a, and find another generator of G.

(b) Find a subgroup of G of order 3.

(c) Give brief reasons why G is not isomorphic to either Z₄ or S(Δ).

Question 4

 The permutations p = (14) (253) and q = (12)(45) are elements of S₅.

(a) Write down the order and parity of each of p and q.

(b) Find the subgroup H of S₅ generated by p, giving the elements of H in cycle form.

(c) Find s = qop0q⁻¹.

(d) Determine, in cycle form, an element of S₅, other than q, that conjugates p to s.

Question 5

This question concerns the system of linear equations

 

(a) Write down the augmented matrix for this system of linear equations.

(b) Find the row-reduced form of the matrix that you wrote down in part (a).

(c) Use your answer to part (b) to solve the above system of linear equations

Question 6

Let t be the linear transformation

t: R³ −→ R³

(x, y, z)−→ (x−y, y + z, x+ z).

(a) Find a basis for Im(t).

(b) Determine the dimension of Ker(t).

(c) State whether t is one-one and justify your answer

Question 7

Determine the solution set of the inequality

 

Question 8

Determine whether or not each of the following series converges, naming any result or test that you use.

1. 
2. 

Question 9

This question concerns the symmetry group G of the cross shown below, which has four identical arms. The vertices at the ends of the arms are labelled as shown.

     

Let g ∈ G be the rotation of the cross through an angle of π/2 anticlockwise about its centre and let h ∈ G be the reflection of the cross in the line through the midpoints of the edges 12 and 56.

(a) Write g, g² and h in cycle form, using the numbering of the locations of the vertices as shown above.

(b) Express the conjugate ghg⁻¹ of h by g in cycle form and describe ghg⁻1 geometrically.

(c) Are the symmetries k = (18)(27)(36)(45) and h conjugate in G? Justify your answer

Question 10

This question concerns the groups (², +) and (R, +) and the function ∅ defined by

∅: R² −→ R,

(x, y) −→ x−3y

(a) Prove that ∅ is a homomorphism.

(b) Determine the kernel and image of ∅.

(c) Identify the quotient group R²/Ker(φ) up to isomorphism, justifying your answer

Question 11

The function f is defined on the interval [0,4] by

 

(a) Sketch the graph of f.

(b) Determine the values of the Riemann sums L(f, P) and U(f, P) for the partition P of [0,4] where P ={[0,1],[1,3],[3,4]

Question 12

(a) Determine the Taylor polynomial T₂(x) at 3 for the function

 

(b) Show that T₂(x) approximates f(x) with an error at most 0.01 on the interval [3,3.5]

Question 13

This question concerns the set

K = {z ∈ C^∗ :|z| =1}; that is, the elements of C^∗ with modulus 1.

(a) Show that K is a subgroup of the group (C^∗, ×) and explain why it is a normal subgroup.

(b) Describe the elements of each of the cosets 3K and 2iK.

(c) Show that every coset of K in C^∗ can be written as rK where r ∈ R⁺.

(d) Write down the rule for the binary operation rK . sK where r, s∈R⁺.

 (e) Show that the function

∅ :(C∗/K, ×)−→ (R⁺,×)

rK −→ r is an isomorphism

Question 14

This question concerns the matrix

A = 

(a) Show that (1,1,0) is an eigenvector of A and find the corresponding eigenvalue.

(b) Use the characteristic equation of A to check the eigenvalue that you obtained in part (a), and find the remaining eigenvalues of A.

(c) Find the eigenspaces of A.

(d) Find an orthonormal eigenvector basis of A.

(e) Write down an orthogonal matrix P and a diagonal matrix D such that P^TAP = D.

Question 15

1. Determine whether or not each of the following sequences {a_n} converges, naming any result or rule you use. If a sequence does converge, then find its limit
2. 
3. 
4.                                                                                                                                                                                     b) Prove that the polynomial                                                                                                                                            p(x)=x³ −3x² +1 has at least three zeros

Question 16

The set of matrices

G =

forms a group under matrix multiplication.

(You are NOT asked to show that G is a group.)

1. Show that the following equation defines a group action of G on the plane R².

 

The remainder of this question refers to the group action defined above.

2. (i) Find the orbit of each of

                (1,0); (0,1); (1,2).

 (ii) Give a geometric description of all the orbits of the action.

3. Find the stabiliser of each of

                (1,0); (1,2).

(d) Find Fix

Question 17

1. (i) The function f is defined by

 

Sketch the graph of f and determine whether or not f is diferentiable at 1. If f is diferentiable at 1, then evaluate the derivative f’(1). Name any results or rules that you use

(ii) Prove the inequality

                 for x ∈ [1,16]

(b) Use the ε−δ definition of continuity to prove that the function f(x)=x² −x is continuous at the point c = 3.

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