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3 Feb 2011: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

15 Feb 2011: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

20 Feb 2011: 5. Let f be an entire function such that for some a C and r > 0, f takes no values in the discD(a, r). ... Show that |f (z)| as |z| ,and deduce that |f (z)| takes a minimum at some a C.

13 Nov 2011: Explain geometrically why this is thecase. 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is,

1 Feb 2011: Xn be independent Poisson random variables, with Xi having parameteriθ for some θ > 0. ... b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T , and.

10 Jan 2011: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... b) Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible.

12 Oct 2011: about. (ii) m n a b (n m) [(a = 1) (b = 1) (ab 6= n)] ,. 5. The sum of some (not necessarily distinct) natural numbers is 100. ... 10. Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}?

12 Oct 2011: Show that L = K(α) = {a bα |a, b K} for some α F with α2 K. ... ii) Show that if L/K is a finite extension with Q K, for which there exist only finitelymany intermediate subfields K K′ L, then L = K(α) for some α

24 Oct 2011: Show that they have the samesplitting field if and only if b = cnar for some c K and r N with gcd(r, n) = 1. ... for some divisor d > 1 of n.

2 Nov 2011: 2. Find the convergents to the fraction 5744. Prove that if x and y are integers such that 57x 44y = 1 , then x = 17 44k and y = 57k 22 for some ... 3. Which of these are true for all natural numbers a , b and c?