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PAPER TITLE: FURTHER PURE MATHEMATICS
EXAM DATE: FRIDAY 8, JUNE 2018
COURSE CODE: M303/H
Question 1
(a) (i) Use the Euclidean algorithm to find hcf(147,336).
(ii) Hence find integers s and t such that
147s + 336t = hcf(147,336).
(b) Prove by induction that
for all positive integers n.
ANSWERS(Purchase full paper to get all the solutions):
1ai)
hcf(147,336)using Euclidean algorithm.
336 = 2(147) + 42
147 = 4(42) + 21
42 = 2(21) + 0
21 = 1(21) + 0
Therefore, the hcf = 21
1aii)
147s + 336t = hcf(147,336)
21 = 1(147) + (-4(42))
42 = (336 - 2(147))
21 = 1(147) + (-4((1(336) - 2(147))))
21 = 1(147) +8(147) + (-4(336))
21 = 9(147) + (-4(336))
9(147) + (-4(336)) = 21 (1)
147s + 336t = hcf(147,336) (2)
By comparing (1) and (2)
s = 9, and t = -4.
1b)
1×2+2×3+3×4+···+ n(n + 1) =
By mathematical induction.
For n = 1
L.H.S: n(n + 1) = 1(1+1) = 2
R.H.S:
For n =1
L.H.S = R.H.S = 2, it is true.
For n = 2
L.H.S: n(n + 1) = 2+2(2+1) =8
L.H.S = R.H.S = 8, it is true
Assume, that n = k
1×2+2×3+3×4+···+ k(k + 1) = (1)
Show that n = k+1
1×2+2×3+3×4+···+ k(k + 1) + k+1(k+2) = (2)
Add k+1(k+2) to both side of (1)
1×2+2×3+3×4+···+ k(k + 1) + k+1(k+2) = (3)
Simplifying the R.H.S of (3)
Hence, the R.H.S of (3) = R.H.S of (2) =
The L.H.S of (3) = L.H.S of (2) = 1×2+2×3+3×4+···+ k(k + 1) + k+1(k+2)
Therefore,
1×2+2×3+3×4+···+ k(k + 1) + k+1(k+2) =
1×2+2×3+3×4+···+ n(n + 1) + n+1(n+2) = .
Question 2
(a) Let G = Z3 × Z4.
(i) Write down the order of G.
(ii) Write down the order of (3,0).
(iii) Write down an element of order 6.
(iv) Is G cyclic? Justify your answer.
(b) Consider the following two groups, G and H, both of which have order 360.
G = Z24 × Z2 × Z25
H = Z12 × Z20 × Z5
(iii) Is either of G or H isomorphic to D600? Briefly justify your answer.
Question 3
(a) Given 60 = 22 ×3×5, list all the divisors of 60.
Hence, or otherwise, find
(i) τ(60),
(ii) σ(60),
(iii) ∅(60).
(b) Calculate the following Legendre symbols (note that 179 and 223 are prime)
(i) (2/223),
(ii) (179/223).
(c) With the help of the Rational Root Test, prove that
f(x)=x3 −x2 +3x−2
is irreducible over Q
Question 4
Let
A = {(a, b) ∈ R2 :a2 + b2 ≤ 2},B={(a, b) ∈ R2 : |b|<|a|}
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) Determine whether (1,0) is a interior point of C for the Euclidean metric d(2)
(c) Write down in set notation the closure, interior and boundary of C for the Euclidean metric d(2) on the plane.
(d) Determine whether the closure of C for the Euclidean metric on the plane is compact
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 ∈ 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 −{(a, b):} by
d(x, y) = max{2|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((0,1),(1,0)).
(b) Write down a d-disconnection of X.
(c) Show that the sequence,is a d-convergent sequence in X
(d) by considering the sequence, show that X is not d-complete.
(e) Deduce that X is not d-sequentially compact.
Question 7
Let G be a group of order 539 = 11×72.
(ii) Show that G has a unique Sylow 11-subgroup and a unique Sylow 7-subgroup.
(iii) Deduce that G is abelian.
Dic10 ={a,b : a4 = e, a2 = b10, aba3b = e}
. Express abab3 in the form aibj, for i =0 ,1,...,3 and j =0 ,1,...,9.
Let p be a prime that divides m.
(i) Show that G has a normal subgroup of order p.
(ii) By using the Second Principle of Mathematical Induction on the order of |G|, the Correspondence Theorem and part (i) (or otherwise), prove that G has a subgroup of order m.
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}. (You may assume that this norm is multiplicative.)
(a) Show that the only elements r ∈ Z[i] for which N(r) = 5 are 1±2i, −1±2i,2±i and −2±i.
Explain why these elements cannot all be associates of each other.
(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 + 2i is an hcf of 3 + i and 5 in Z[i].
Let I ={5,3+i} be the ideal in Z[i] generated by the elements 5 and 3+i. Since Z[i] is a principal ideal domain, the ideal I is principal.
(d) Show that {1+2i,−1−2i,−2+i,2−i} is the set of all elements d ∈ Z[i] such that I = {d}.
(e) With the help of the previous parts or otherwise, show that I is a maximal ideal in Z[i].
(f) By part (d), I is a principal ideal in Z[i] generated by 2−i, that is, I ={2−i}. Using the results of the previous parts or otherwise, show that I = {α + βi : α,β ∈ Z,α+2β ≡ 0 (mod 5) }.
QUESTION 9
(a) Define a distance function d: N×N→ R as follows:
(i) Write down d(1,5).
(ii) Prove that d is a metric on N by showing that
(1) d satisfies (M1).
(2) d satisfies (M2).
(3) d satisfies (M3).?
(b) Let
A = {f ∈ C[0,1] : |f(x)−f(y)|≤ |x-y| for each x, y ∈ [0,1]}
B ={f ∈ C[0,1] : |f(x)| ≤1 for 0≤ x ≤ 1}.
iii) Hence or otherwise show that A∩B is dmax-sequentially compact. (You may assume that A is an equicontinuous set of functions
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Last updated: Sep 02, 2021 03:07 PM
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