Re: [中學] 資優班考題2題
※ 引述《stu2005131 (自由幻夢)》之銘言：
: 1. 設x.y.z為3個不全為零的實數
: 求(xy+2yz)/(x^2+y^2+z^2)的最大值
y = r cosθ, x = r sinθ cosφ, z = r sinθ sinφ
(xy+2yz)/(x^2+y^2+z^2)=(cosφ+2sinφ)sinθcosθ
=(√5 / 2)sin(φ+ω)sin2θ
max = √5 / 2
: 2. 設x=[(11^(1/2)+7^(1/2)]^12且y表示x的小數部分 求x(1-y)之值為?
Let (√11 + √7) = a
(√11 - √7) = b
a^2 = 18 + 2√77
b^2 = 18 - 2√77
a^2 + b^2 (Integer)
a^2 b^2 (Integer)
a^4 + b^4 = (a^2 + b^2)^2 - 2a^2 b^2 (Integer)
a^{12} + b^{12} = (a^4 + b^4)(a^8 - a^4 b^4 + b^8)
= (a^4 + b^4)[(a^4+b^4)^2-3a^4b^4] = K, (K is Integer)
but ab=4, and a>3+1=4, hence, b<1
hence, x=a^{12}=(K-1)+(1-b^{12})=(K-1)+y
y=x-K+1
x(1-y) = x (K-x) = a^{12}*b^{12}=4^{12}
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 27.147.57.77
※ 編輯: JohnMash 來自: 27.147.57.77 (07/29 12:52)
推
07/29 14:41, , 1F
07/29 14:41, 1F
推
07/29 18:55, , 2F
07/29 18:55, 2F
討論串 (同標題文章)