两道牛津数学系入学加试题，找到原英文题目

282842712474

Q1. In this question we shall consider the function f (x) defined by f (x) = x^2-px + 3 where p is a constant. (i) Show that the function f (x) has one stationary value in the range 0 < x < 1 if0 < p < 1, and no stationary values in that range otherwise. In the remainder of the question we shall be interested in the smallest value attained by f (x) in the range 0 ≤ x ≤ 1. Of course, this value, which we shall call m, will depend on p. (ii) Show that if p > 1 then m = 4-2p. (iii) What is the value of m if p ≤ 0? (iv) Obtain a formula for m in terms of p, valid for 0 < p < 1. (v)Using the axes opposite, sketch the graph of m as a function of p in the range -2 ≤ p ≤ 2. Q2 Songs of the Martian classical period had just two notes (let us call them x and y) and were constructed according to rigorous rules: I. the sequence consisting of no notes was deemed to be a song (perhaps the most pleasant); II. a sequence starting with x, followed by two repetitions of an existing song and ending with y was also a song; III. the sequence of notes obtained by interchanging xs and ys in a song was also a song. All songs were constructed using those rules. (i) Write down four songs of length six (that is, songs with exactly six notes). (ii) Show that if there are k songs of length m then there are 2k songs of length 2m+2. Deduce that for each natural number there are 2^n songs of length 2^(n+1)-2. Songs of the Martian later period were constructed using also the rule: IV. if a song ended in y then the sequence of notes obtained by omitting that y was also a song. (iii)What lengths do songs of the later period have? That is, for which natural numbers n is there a song with exactly n notes? Justify your answer. 本文转自：http://pm.yhlive.com/thread-28-1-2.html [ 本帖最后由 282842712474 于 2008-2-1 11:16 编辑 ]

mathe

毕竟是入学考试，不难。不过不知道是本科还是研究生入学考试，怎么已经用到微积分了。 不过不懂stationary value的意义，查了下mathworld.wolfram.com才知道， 就是极值的意思。不过奇怪，应该是0

gxqcn

http://www.admissions.ox.ac.uk/interviews/tests/Math.pdf 或直接从论坛下载： Math.pdf (182.58 KB, 下载次数: 9)

2008-2-1 13:22 上传

点击文件名下载附件
THE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS, AND COMPUTER SCIENCE

mathe

我说呢， f (x) = x^2-px + 3 应该改为 f (x) = x^2-2px + 3 不然根本不成立