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PAPER TITLE: PURE MATHEMATICS

EXAM DATE: TUESDAY 9, OCTOBER 2007

COUSRE CODE: M208/C

Question 1

Sketch the graph of the function f defined by

 

Your sketch should identify:

(a) any asymptotes to the graph;

(b) any points where the graph crosses the axes

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

The sketch of the graph

1. The vertical asymptotes of the graph is obtained by eating the denomination to zero

 

2x+1 = 0

2x=-1

x=-

x = -0.5

therefore, Va = -0.5

the horizontal asymptotes(Ha) equal the degree of the numerator and the denominator = 1

2. the point is indicated on the graph

Question 2

Let ∼ be the relation defined on Z by

x ∼ y if x² −y² is divisible by 2.

Prove that ∼ is an equivalence relation.

Question 3

The set {1,3,5,9,11,13} forms a group G under multiplication modulo 14. (You are NOT asked to prove this statement.)

(a) Show that G is cyclic.

(b) Find all the distinct subgroups of G.

(c) Find an isomorphism ∅ that maps (G,×₁₄) to (Z₆,+₆).

Question 4

The permutations p = (1 5 3 2), q = (2 5 4) and r = (1 5 3)are elements of S₅.

1. Write down each of the following as a permutation in cycle form:

p⁰q, p⁰q⁰p⁻¹,p ².

(b) State the order and parity of p⁰q and p².

(c) Find a permutation in cycle form which conjugates q to r.

Question 5

Find the matrix of the linear transformation

T: R²→R²

(x, y)−→ (2x + y, x−y)

with respect to

(a) the standard basis for both the domain and the codomain.

(b) the basis {(1,−3), (−1,4)} for the domain and the standard basis for the codomain.

(c) the basis {(1,−3),(−1,4)} for both the domain and the codomain.

Question 6

This question concerns the matrix

 

(a) Find the eigenvalues of A.

(b) For each eigenvalue of A, find a corresponding eigenvector.

(c) Write down a matrix P and a diagonal matrix D such that P⁻¹AP = D.

Question 7

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

1. 
2. 

Question 8

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

1. 
2. 

Question 9

This question concerns the symmetry group G of the regular octagon shown below

Let g ∈ G be the anticlockwise rotation of the octagon through an angle of 3π/4 about its Centre and let h ∈ G be the reflection of the octagon in the line through the vertices at locations 2 and 6.

(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 = (14)(23)(58)(67) and l = (15)(26)(37)(48) conjugate in G? Justify your answer.

Question 10

The set

            G =

forms a group under matrix multiplication. (You are NOT asked to prove this.)

This question concerns the function ∅ defined by

            ∅: G→(R*,×)

           

(a) Prove that ∅ is a homomorphism.

(b) Find Ker(∅).

(c) Determine Im(∅) and hence explain why

G/Ker(∅)  (,×).

Question 11

1. Prove that

 

2. Use Stirling’s Formula to prove that

 

Question 12

Prove that the following limit exists and determine its value.

           

Question 13

The group G ={e,a,b,c,d,f,g,h,i,j,k,l,m,n,o,p} is defined by the following group table. (You are NOT expected to show that G is a group.)

(a) Find H, the cyclic subgroup generated by the element b.

(b) Show that K = {e,d,i,m} is a subgroup of G.

(c) Show that H is a normal subgroup of G, and that K is not a normal subgroup of G.

(d) Write down the elements of the quotient group G/H.

(e) Determine a group that is isomorphic to G/H, justifying your answer

Question 14

Consider the following subset of R⁴.

            S ={(a,b,−3a +2b,4a−3b):a, b ∈ R }

1. Show that S is a subspace of R⁴.

(b) Show that {(1,1,−1,1),(1,0,−3,4)} is a basis for S and write down the dimension of S.

(c) Find an orthogonal basis for S that contains the vector (1,1,−1,1).

(d) Find the coordinates of (0,1,2,−3) with respect to the basis you found in part (c).

Question 15

1. For each of the following functions f, sketch the graph of f and determine whether f is continuous at 0.

 

          

(b) Consider the equation 3−x = .

(i) Show that this equation has a solution in [0, 1].

(ii) Show that this equation has only one real solution

Question 16

This question concerns the group (,×) and the plane R².

(a) Show that the following equation defines a group action of  on ².

r ∧ (x, y) = (x, ry)

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

1. (i) Find the orbit of each of

(1,0); (0,1); (2,1).

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

(c) Find the stabiliser of each of (1,0); (2,1).

(d) Find Fix(3).

Question 17

1. Calculate the Taylor polynomial T₂(x) at 1 for the function

                

 Show that T₂(x) approximates f(x) with an error at most  on the interval [1,1.5]

2. (i) Determine the interval of convergence of the power series

                     

     (ii) Hence, or otherwise, write down the radius of convergence of the series

                     

            briefly explaining your answer.

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