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PAPER TITLE: FURTHER PURE MATHEMATICS
EXAM DATE: WEDNESDAY 1, JUNE 2016
COURSE CODE: M303/D
Question 1
a) Prove by induction that
=
for all positive integers n.
b) (i) Use the Euclidean algorithm to find hcf(231,715).
(ii) Hence find integers s and t such that
231s + 715t = hcf(231,715).
ANSWERS(Purchase full paper to get all the solutions):
1a)
i=1nii+1 = nn+1 n+23
By mathematical induction.
For n = 1
L.H.S: = 2
R.H.S:
L.H.S = R.H.S = 2, it is true
For n = 2
L.H.S: = 8
L.H.S = R.H.S = 8, it is true
Assume, that n = k
(1)
Show that n = k+1
(2)
Add ( to both side of (1)
(3)
Simplifying the right hand of (3)
The right-hand side of (3) = right-hand side of (2)
The left-hand side of (3) = the left-hand side of (2)
Therefore,
→is true for all positive integer n.
1b)
hcf(231,715) using Euclidean algorithm.
715 = 3(231) + 22
231 = 10(22) + 11
22 = 2(11) + 0
11 = 1(11) + 0
Therefore, the hcf = 11
1bii)
11 = 1(231) + (-10(22))
22 = (715 - 3(231))
11 = 1(231) + (-10((715 - 3(231))))
11 = 1(231) +30(231) + (-10(715))
11 = 31(231) + (-10(715))
31(231) + (-10(715)) = 11 (1)
231s + 715t = hcf(231,715). (2)
By comparing (1) and (2)
s = 31, and t = -10.
Question 2
(a) Let G = Z2 × Z6.
(i) Write down the order of G.
(ii) Write down the maximum order of an element in G.
(iii) Find the number of elements of order 3 in G
(b) Let G be a group with proper subgroups H and K. Determine whether each of the following statements is true or false, and either give a proof or a counter-example.
(i) If H and K are normal in G, then H ∩ K is normal in G. You may assume that H ∩ K is a subgroup of G.
(ii) If G is abelian, then H is cyclic.
(c) Let ∅ be the homomorphism
∅: Z6 → Z6
x → x + x
Find Im(∅) and Ker( ∅).
Question 3
(a) Calculate the following Legendre symbols (note that 101 and 103 are prime)
(i) (3/103),
(ii) (101/103),
(iii) (303/103).
(b) Use the rational root test to show that
is irreducible over Q.
(c) Can 51 be expressed as the sum of two squares? Briefly justify your answer
Question 4
Let
A = {(a,b) ∈ R2 : a2 + b2 ≤ 4},
B = {(a,b) ∈ R2 : a2 + b2 >1} and
C = A∩B.
(a) Sketch on separate diagrams, or describe in words, each of A, B and C taking care to indicate which points lie in the sets.
(b) Write down in set notation the closure, interior and boundary of C for the following metric.
(i) the Euclidean metric d(2) for the plane
(ii) the discrete metric d0 for the plane.
(c) Show that C is not compact for the Euclidean metric
Question 5
Let α = − ; you may assume that
[Q() : Q]=[Q() : Q] = 2, ∉ Q() and ∉ Q().
(a) Express α3 in the form a+b for suitable a,b ∈ Q.
(b) With the help of (a), or otherwise, show that are in Q(α).
(c) Hence show that Q(α) = Q().
(d) Determine the degree [Q(α):Q].
(e) By considering α2 find the minimal polynomial for α over Q
Question 6
Define a metric, d, on X = R2 −{(x, y):y=2} by
d(x, y) = max{|x1 −y1|, |x2 −y2|}
where x =( x1, x2) and y = ( y1, y2) are points in R2. (You do not have to show that d is a metric.)
(a) Write down d((3,7),(1,2)).
(b) Write down a d-disconnection of X.
(c) Show that the sequence, is a d-convergent sequence in X
(d) show that R2 −{(x, y):y=2} is not d-complete.
(e) Deduce that X is not d-sequentially compact
Question 7
Let G be a group of order 32q, where q is prime and q>3.
(a) Show that G has a unique Sylow q-subgroup.
(b) Let q = 11.
(i) Find the possible number of Sylow 3-subgroups of G.
(ii) Use parts (a) and (b)(i) to deduce that G is abelian.
(iii) Hence classify all groups of order 99.
(c) Does the argument in part (b) generalise to show that all groups of order 32q, where q is prime and q>3, are necessarily abelian? Justify your answer.
(d) Give an example of a non-abelian group of order p2r, where p and r, are prime and p
Question 8
Let N(a + bi) = a2 + b2 be the usual Euclidean norm on the ring of Gaussian integers Z[i]={a + bi : a,b ∈ Z}.
(a) Show that the only elements r ∈ Z[i] for which N(r) = 2 are 1+i, 1-i,-1+i and −1-i.
(b) Let r be a common factor of s,t∈ Z[i]. Show that N(r) is a common factor of N(s) and N(t) in Z
(c) Hence show that 1 + i is an hcf of 3 + i and 6 in Z[i]. Briefly explain why 1−i, −1+i and −1−i are also hcfs of 3 + i and 6.
Let I ={6,3+i} be the ideal in Z[i] generated by the elements 6 and 3+i. Since Z[i] is a principal ideal domain, the ideal I is principal.
(d) Show that {1+i,1−i,−1+i,−1−i} is the set of all elements d ∈ Z[i] such that I = {d}.
(e) Using the results of parts (c) and (d) or otherwise, show that I ={α + βi; α,β ∈ Z,α ≡ β (mod 2)}.
(f) With the help of the previous parts or otherwise, show that I is a maximal ideal in Z[i].
Question 9
This question concerns the set C[0,1] with the max metric
(a) Let () be the sequence of functions where : [0 ,1]→ R is given by
(i) Write down (0) and (1).
(ii) Prove that () is - convergent to , where denotes the zero function
Now consider the set
S ={f ∈ C[0,1] : f(x) > 0 for each x ∈ [0,1]}
(iii) Show that ∈ S for each n ∈ N.
(iv) Hence, or otherwise, show that S is not -compact.
(b) Let
A = {f ∈ C[0,1] : |f(x)−f(y)|≤ for each x, y ∈ [0,1]}
a
B ={f ∈ C[0,1] : |f(x)| ≤ 2 for each x, y ∈ [0,1]}
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Last updated: Sep 02, 2021 02:16 PM
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