Premium Resources

PAPER TITLE: PURE MATHEMATICS

EXAM DATE: THURSDAY 12, OCTOBER 2006

COUSRE CODE: M208/H

Question 1            

Sketch the graph of the function f defined by

 

indicating clearly the main features

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

Sketch of the graph

The vertical asymptotes of the graph = 1

Question 2

Let

z =

(a) Express z in Cartesian form.

(b) Find the modulus and argument of z.

Question 3

(a) Show that the following groups are cyclic, and find a generator of each.

(i) G₁ =( {0,3,6,9},+₁₂)

(ii) G₂ =( {1,5,8,12},×₁₃)

(b) Find an isomorphism from G₁ to G₂, justifying your answer.

(c) Explain why the group G₃ =( {1,3,5,7},×8) is not isomorphic to G₁ or G₂

Question 4

The set G ={1,2,4,5,8,10,11,13,16,17,19,20} forms a group under multiplication modulo 21. (You are NOT asked to prove this result.)

(a) Show that H ={1,4,16} is a subgroup of G.

(b) Write down the left cosets of H in G.

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

(d) Write down a group from the course that is isomorphic to the quotient group G/H. Explain your answer.

Question 5

The position vectors of the points A and B are

a = (−1,2) and b = (3 ,−1), respectively.

(a) Draw a sketch showing the points A and B in the plane, and the line l through A and B.

(b) Find the position vector r of a general point on the line l.

(c) Find the point C on l whose position vector is perpendicular to l.

Question 6

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 equations

Question 7

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

1.  
2. 

Question 8

Determine whether the function , defined below, is continuous at 0.

Name any result or rule you use

Question 9

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

 

Let g ∈ G be the reflection of the pentagon in the vertical axis through the vertex at location 1 and let h∈ G be the anticlockwise rotation of the pentagon through 4π/5 about its Centre.

(a) Write g, h and h² 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 this conjugate geometrically.

(c) Are the symmetries (1 5)(2 4) and (1 4)(2 3) conjugate in G? Justify your answer.

Question 10

This question concerns the group (C,+) and the function ∅ defined by

∅: C → C

    z → z + z-

(a) Prove that ∅ is a homomorphism.

(b) Determine the kernel and image of ∅.

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

Question 11

Prove that the following limit exists and determine its value.

 

Question 12

Determine the interval of convergence of the power series.

 

Question 13

The permutations p = (2 3 7 6), q = (1 3 5 2 4) and r = (1 3 7) are elements of S₇.

1. (i) Find each of the following as a permutation in cycle form: p²,p⁰q, q⁻¹,r⁰q⁰r⁻¹.

(ii) State the orders of p, q, p² and p⁰q.

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

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

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

(iii) Write down a group from the course that is isomorphic to H’. Give a brief reason for your answer.

3. (i) Write down p⁰q as a composite of transpositions, and hence deduce the parity of (p⁰q)².

Question 14

This question concerns the matrix

A =

(a) Show that (1,-1,2) 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

Determine which of the following series are convergent. Name any result or test you use.

1.  
2. 
3. 
4. 

Question 16

The set of matrices

                            G =

forms a group under matrix multiplication.

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

a) 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.

b) (i) Find the orbit of each of

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

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

c) Find the stabiliser of each of

                (1,0); (0,1).

1. Find Fix  and Fix

Question 17

1. The function  is defined on the interval [0,2] by

 

1. Sketch the graph of
2. Determine the values of the Riemann sums L(f,P) andU(f,P) for the partition P of [0,2]where P = {[0, ],[

2. Let

                               =  

1. Evaluate .
2.  Prove that for n≥1

  =

3. Hence determine the values of  and .

(Purchase full paper by adding to cart) 

Last updated: Sep 02, 2021 03:08 PM

add to cart ← Back