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PAPER TITLE: PURE MATHEMATICS

EXAM DATE: TUESDAY 7, OCTOBER 2008

COUSRE CODE: M208/D

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

1. Write down the converse of the following statement for positive integers n.

If n is even, then n² −n is even.

2. Of the statement in part (a) and its converse, one is true, and one is false. Prove the true statement, and give a counterexample for the false statement

Question 3

In each of the following cases, determine whether the given set and binary operation form a group.

(a) ({1,2,4,8,10},×₁₂)

(b) ({Z∈ C:Re(Z)=0 },+).

Question 4

The group table for a group G is given below.

	e	a	b	c	d	f	g	h
e	e	a	b	c	d	f	g	h
a	a	e	c	b	f	d	h	g
b	b	c	a	e	g	h	f	d
c	c	b	e	a	h	g	d	f
d	d	f	h	g	e	a	c	b
f	f	d	g	h	a	e	b	c
g	g	h	d	f	b	c	e	a
h	h	g	f	d	c	b	a	e

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

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

(c) Show that H is a normal subgroup of G.

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

Question 5

The position vectors of the points A and B in the plane are

a = (−1,3) and b = (7 ,−1), respectively.

(a) Draw a sketch showing the points A and B, and the line L through A and B.

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

(c) Find a point C on L whose position vector is perpendicular to L.

Question 6

Let t be the linear transformation

            t:R³ −→ R³

(x,y,z)−→ (2x + y +3z,3x−y +2z,4x−3y + z).

(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

Find the solution set of 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 converges, Find its limit

1. 
2. 

Question 9 

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

Let g ∈ G be the clockwise rotation of the hexagon through π/3 about its Centre and let h ∈ G be the reflection of the hexagon in the line through the vertices at locations 2 and 5.

1. Write g,g² and h in cycle form, using the numbering of the locations of the vertices as shown above.
2. Express the conjugate ghg⁻¹ of h by g in cycle form and describe ghg⁻¹ geometrically.
3. Are the symmetries k = (1 4)(2 5)(3 6) and l = (1 6)(2 5)(3 4) conjugate in G? Justify your answer.

Question 10

Let

∅:(Z₈,+₈) −→ (Z₅*,×₅) be the homomorphism given by

∅ (n)=2ⁿ (mod 5).

(a) Find the image under ∅ of each element of Z₈.

(b) Write down Ker(∅), Im(∅) and the cosets of Ker(∅) in Z₈

Question 11

The function f is defined on the interval [0,1] 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 [0,1] where P =={[0,],[ , ],[ ,1]}.

Question 12

Determine the interval of convergence of the power series

 

Question 13

The permutations p = (1 4) (2 3 5) and q = (1 2 4 6) are elements of S₆.

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

                                     p²,q⁰p, q⁻¹,q⁰p⁰q⁻¹.

     (ii) State the order of each of p, q, p² and q⁰p.

2. (i) Find the subgroup H of S₆ generated by p.

  (ii) Determine the conjugate subgroup qHq⁻¹.

  (iii)Explain why s = (2 5) (1 6 3) is conjugate to p in S₆, and find all the elements of S₆ which conjugate p to s.

Question 14

This question concerns the matrix

                            A =

(a) Show that (0,1,0) 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

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

1. 
2. 
3. 
4. 

Question 16

The elements of the symmetry group of the equilateral triangle, S(), are the identity e, rotations a and b and reflections r, s, t as shown below.

An equilateral triangle is divided into six identical right-angled triangles numbered 1 to 6, as shown below.

1. Draw up a group action table showing the action of S() on this set of six triangles.

Each of the six small triangles is now to be coloured black or white to give a coloured equilateral triangle; examples of possible triangles are shown below.

2. Use the Counting Theorem to determine the number of different coloured equilateral triangles that are possible, where two triangles are regarded as the same if a rotation or reflection takes one to the other. Justify your answer

Question 17

1. For each of the following functions f, sketch the graph of f and determine whether f is diferentiable at 0. If f is diferentiable at 0, then evaluate the derivative f’(0). Name any results or rules you use.

                                           

                                          

2. Use the ε–δ definition of continuity to prove that

                                     f(x)=x² −3x

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