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PAPER TITLE: PURE MATHEMATICS

EXAM DATE: TUESDAY 11, OCTOBER 2011

COUSRE CODE: M208/D

Question 1

Sketch the graph of the function f defined by

f(x)=

indicating clearly the main features

ANSWERS (Purcahse full paper to get all the solutions):

Sketch of the graph

The horizontal asymptotes (Ha) = 1

Question 2

(a) Write down the converse of the following statement for positive integers m and n. If 5 divides each of the integers m and n, then 5 divides 2n +3m.

(b) Of the statement in part (a) and its converse, one is true, and one is false.

Prove the true statement, and give a counter-example for the false statement

Question 3

For each of the following sets, with the binary operation given, determine whether or not it forms a group, justifying your answer.

(a) ({1,2,4,8,16}, ×₁₇)

(b) ({5^k: k ∈ Z}, ×

Question 4

The set G = {1,3,7,9,11,13,17,19} forms a group under multiplication modulo 20. (You are NOT asked to prove this result.)

(a) Show that H = {1,9} is a subgroup of G.

(b) Determine the left cosets of H in G.

(c) Explain why H is a normal subgroup of G.

(d) Determine a group from M208 to which the quotient group G/H is isomorphic.

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 system of linear equations above

 

Question 6

Let t be the linear transformation

t: R³ −→ R³

(x, y, z) −→ (x−2y +3z,3x + y +2z,2x−y +3z).

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

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

(c) State whether t is one-one, justifying your answer

Question 7

Solve the inequality

          

Question 8

Determine whether or not each of the following sequences {a_n} converges, naming any result or rule you use. If a sequence does converge, find its limit.

1.  
2. 

Question 9

The diagrams below show the eleven non-identity elements of S().

 

The conjugacy classes of S () are{e}, {a, a⁵}, {a², a⁴}, {a³}, {q¹, q³, q⁵}, {q², q4, q⁶}.

(a) Determine a normal subgroup of S () of order 2, a normal subgroup of order 3 and a normal subgroup of order 6. Justify your answers briefly.

(b) Show that H = {e, a³, q₃, q₆} is a subgroup of S () and explain briefly why it is not normal.

Question 10

The set L =  of lower triangular matrices forms a group under matrices multiplication. (You are NOT asked to prove this.)

This question concerns the function ∅ defined by

∅: L → (, x) is defined by

  → .

(a) Prove that ∅ is a homomorphism.

(b) Determine the kernel and image of ∅.

(c) Identify the quotient group L/Ker (∅) up to isomorphism, justifying your answer briefly

Question 11

Prove that the following limit exists, and determine its value

     

Question 12

Prove that

                            ≤π22 

Question 13

The permutations p = (1 3 7 5) and q = (1 2 6) (3 7) are elements of S₇.

1. (i) Find each of the following as a permutation in cycle form:

                p², q ⁻¹, p⁰q, p⁰q⁰p⁻¹.

(ii) State the order and parity of each of p, q, p², and p⁰q.

(iii)Write down a permutation in S₇ that is conjugate to p

2. (i) Find the subgroup H of S₇ generated by q.

(ii) Find the conjugate subgroup H’ = p⁰H⁰p⁻¹.

(iii)Write down a group from M208 that is isomorphic to H’. Justify your answer briefly.

Question 14

This question concerns the matrix

 

(a) Show that (−1,0,1) 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 the function f, defined below, is continuous at 0.

 

Name any results or rules you use

b) Determine the least upper bound and the greatest lower bound of the set

            E =

Question 16

Let X be the set of all 3×3 ‘chessboards’ in which each square is coloured either black or white. Some diferent elements of X are illustrated below

This question concerns the natural group action of S() on X. The elements of S() are the identity, rotations a, b, c and reflections r, s, t, u as shown below.

1. For each of the two chessboards shown below, draw each of the elements in its orbit, and list the elements of its stabilizer

2. Use the Counting Theorem to find the number of orbits of the group action, explaining your answer carefully

Question 17

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

 

Show that T₂(x) approximates f(x) with error less than  on the interval [3,3.25]

b (i) Use the ε−δ definition of continuity to prove that the function

 is continuous at point c =2.

(ii) Determine whether the function

 

is uniformly continuous on the interval [0,1], justifying your answer briefly

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