| A1. A is the 3 x 3 matrix a11 = a22 = -a33 = a13 = 1, a23 = 2, other elements zero. Find all 3 x 3 matrices B such that AB + BA = 0. | |
| A2. A, B are real square matrices such that A2001 = 0 and AB = A + B. Show that det B = 0. | |
| A3. a, b, c, d are reals such that the quadratic ax2 + (b + c)x + d + e = 0 has a root x ≥ 1. Show that the quartic ay4 + by3 + cy2 + dy + e = 0 also has a root y ≥ 1. | |
| A4. a1, a2, ... , an are points in Rn (so each ai has n coordinates). A is the matrix whose i, j element is ai·aj. Show that det A is non-negative, and the eigenvalues of A are all non-negative. | |
| A5. Every element of an n x n matrix is an even integer. Show that none of its eigenvalues are odd integers. | |
| A6. A is an n x n matrix. The main diagonal elements (aii) are all a + b, where a and b are reals, the elements in the diagonal immediately above the diagonal (the elements ai i+1) are all ab, and the elements in the diagonal immediately below the main diagonal are all 1. The other elements are all 0. Find det A. | |
| B1. The function f(x) has f ''(x) > 0 for all x > 0 and the graph is asymptotic to y = ax + b as x→∞. Show that f(x) - ax - b has negative derivative for all x > 0, and that f(x) > ax + b for all x > 0. | |
| B2. p, q are reals such that p > 0, q < 0, p + q < 1. The non-negative real sequence a1, a2, a3, ... satisfies an+2 ≤ p an+1 + q an for all n. Show that the sequence converges and find its limit. | |
| B3. Let g(x) = (1 + x) (1 + x2) ... (1 + x2001), and f(x) = x2000/g(x). Show that ∫x1 f(t) dt = f(x) for some x ∈ (0, 1). | |
| B4. A real-valued function f on the reals has a second derivative everywhere and f(x) + f ''(x) ≥ 0 for all x. Show that f(x) + f(x + π) ≥ 0 for all x. | |
| B5. f(x) is defined for x ≥ 1 and f(1) is non-zero. It satisfies f(x + 1) = 2001 f(x)2 + f(x) for all x. Find lim ( f(1)/f(2) + f(2)/f(3) + ... + f(n)/f(n+1) ). | |
| B6. f(x) is a differentiable function on [a, b] such that f(x)2 + f '(x)2 > 0 for all x in [a, b]. Show that the function has only finitely many distinct roots. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
8 July 2003
Last corrected/updated 23 Aug 03