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23 Mar 2009: d : R{0} N. such that for all a R and b 6= 0, b R we can write. ... N by. N(a b. 2. )= a2 2b2. (=. a b22) (a, b Z).

26 Oct 2009: Is it Lipschitz equivalent. to the uniform norm? (b) Let R[0, 1] denote the vector space of all integrable functions on [0, 1]. ... Usingthe fact that all norms on a finite-dimensional space are Lipschitz equivalent, deduce that α iscontinuous.

13 Oct 2009: with xn yn zn = wn. State explicitly what is meant by ‘sufficiently large’; that is, give some number N(depending on n) such that the result holds for all primes p ... Suppose that (i) |A| = 3 for each A A; and (ii) |A B| = 1for all distinct A, B A.

23 Feb 2009: 1. Which of the following propositions are tautologies, that is, are true for all valuations?(a) (A B) (B A). ... Show thatthe set of propositions that all believe is consistent and deductively closed.

15 Oct 2009: k=1 uk(x) to F (x) is uniform on [a, b]. Proof. ... It would suffice to assume e.g. that |fn(x)| 1for all n and x [a, b].

21 Jan 2009: Show thatf (x) = 0 for all x. 8. Let f : [a, b] R be bounded. ... iv) If f′(x) > 0 for all x (a, b) then f is strictly increasing.

5 Mar 2009: p. Show that the sumof residues of g at all its singularities is zero. ... 2π0. log |f (ρeiθ|dθ. Now suppose that w1,. , wn are all the zeros of f in the open disc B = {|z| < ρ}, repeatedaccording to multiplicities, but f never vanishes on

20 Nov 2009: 1+. xnfor all n 1. Show. that (xn)n=1 converges, and determine its limit. ... 13. Is there an uncountable family S PN such that AB is finite for all distinct A,B S?

29 Oct 2009: Is it Lipschitz equivalent. to the uniform norm? (b) Let R[0, 1] denote the vector space of all integrable functions on [0, 1]. ... Usingthe fact that all norms on a finite-dimensional space are Lipschitz equivalent, deduce that α iscontinuous.

9 Mar 2009: 3) Let a, b be positive integers. Let. x =1. a 1b+ 1. ... b) Find all C2(R2) functions with |Du| bounded and u constant in R2.

20 Feb 2009: αN has area (N 2)π. αj. Showthat, for every N > 3 and every α with 0 < α < (1 2N )π, there is a regular N -gon with all anglesequal to ... 15. The quaternions Q consist of all 2 2 complex matrices.

28 Jan 2009: The spherical circlewith centre C and radius r is the set of all points on the sphere at distance r from C. ... Show that two triangles that have all three angles the same are congruent.

30 Oct 2009: n 2 orf (xj )p(xj ) = (1)j1‖f p‖ for all j = 1, 2,. , ... c) Set pn = (2In)1p̃n. Check that pn is a polynomial of degree 2n, and that supx[a,b] |pn(x) f (x)| < 2 for all sufficiently large n.

10 Oct 2009: 9. Which of these are true for all natural numbers a, b and c?(i) (a,b) (a,c) = (a2,bc). ... 12. In the sequence 41, 43, 47, 53, 61,. , each difference is two more than the previous one.Are all the numbers in the sequence prime?

6 Feb 2009: 10. (i) Show that the set P (S) of all subsets of a given set S is a ring with respect to the operations ofsymmetric difference and intersection - the power-set ... 12. Let ̟ be a set of prime numbers. Write Z̟ for the collection of all rationals m/n

19 Feb 2009: Deduce that traces of all words in A and Band their inverses (i.e. ... of all elements of the group 〈A, B〉 generated by Aand B) are determined by the three numbers {tr(A), tr(B), tr(AB)}.

12 Feb 2009: Extend f to all of R as a continuous function such that the extendedfunction agrees with f on [a, b] and is identical to zero outside [0, 1]. ... x, 0) = 0 for all. x. Let F (t) = 11 g(x, t) dx.

20 Feb 2009: Show. that, for every N > 3 and every α with 0 < α < (1 2N )π, there is a regular N-gon with all anglesequal to α. ... 15. The quaternions Q consist of all 2 2 complex matrices.

30 Jan 2009: The spherical circlewith centre C and radius r is the set of all points on the sphere at distance r from C. ... Show that two triangles that have all three angles the same are congruent.

23 Feb 2009: iv) Show that in all other cases, |Sn(g)| = n. 3. ... Describe the set Z(A) of all Möbius transformations thatcommute with A, and hence describe the set {B(z) | B Z(A)}.