| A1. Let X be an n x n non-singular matrix with columns Xi. Y is the matrix with columns X2, X3, ... , Xn, 0. Show that YX-1 and X-1Y have rank n-1 and all eigenvalues 0. | |
| A2. f: [0,1] → R is continuous and ∫x1 f(t) dt ≥ (1-x2)/2 for all x. Show that ∫01 f(t)2 dt ≥ 1/3. | |
| A3. f: (0,∞) → R has continuous second derivative and limx→0 f '(x) = -∞, limx→0 f "(x) = ∞. Show that limx→0 f(x)/f '(x) = 0. | |
| A4. F: (1,∞) → R is defined by F(x) = ∫xx2 dt/ln t. Show that F is injective and find its range. | |
| A5. A, B are real n x n matrices. There are distinct real numbers t1, t2, ... , tn+1 such that A + tiB are all nilpotent. Show that A and B are nilpotent. | |
| A6. Given p > 1, show that we can find Kp > 0 such that if |x|p + |y|p = 2, then (x - y)2 ≤ Kp (4 - (x+y)2). | |
| B1. A is a real 3 x 3 matrix such that for each column vector u ∈ R3, Au and u are orthogonal. Show that the transpose of A is -A and that we can find a vector v such that Au = the vector product v x u for all u | |
| B2. The sequence b0, b1, b2, ... is defined by b0 = 1, bn = 2 + √bn-1 - 2 √(1 + √bn-1). Find ∑1∞ bn2n. | |
| B3. p(z) is a polynomial of degree n with complex coefficients. All its roots lie on the unit circle. Show that all roots of 2zp'(z) - np(z) also lie on the unit circle. | |
| B4. Show that for any ε > 0 we can find some real numbers λ1, ... , λn such that |x - ∑ λk x2k+1| ≤ &epsilon for |x| ≤ 1. Show that if f is an odd continuous function on [-1,1], then we can find some real numbers μ1, ... , μn such that |f(x) - ∑ μk x2k+1| < &epsilon for |x| ≤ 1. | |
| B5. Show that if |a0| < 1, then a0/2 + cos x + ∑2N an cos nx takes both positive and negative values. Show that ∑1100 cos(n3/2x) has at least 40 zeros in (0,1000). | |
| B6. f1, f2, f3, ... is a sequence of continuous functions on [0,1] such that ∫01 fm(x)fn(x) dx = 1 if m = n, 0 otherwise, and sup{ |fn(x)|: x ∈ [0,1] and n = 1, 2, 3, ... } < ∞. Show that there is no subsequence fnk such that limk→∞ fnk(x) exists for all x. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
IMC home
© John Scholes
jscholes@kalva.demon.co.uk
1 Dec 2003