114 臺南女中

[thepiano]

請參考附件

[thepiano]

第 4 題
以下均為向量
AD = AB + BC + CD
|AD|^2 = |AB|^2 + |BC|^2 + |CD|^2 + 2(AB˙BC + BC˙CD + CD˙AB)
5^2 = 2^2 + 3^2 + 4^2 + 2(AB˙BC + BC˙CD + CD˙AB)
AB˙BC + BC˙CD + CD˙AB = -2

AC˙BD = (AB + BC)(BC + CD) = AB˙BC + AB˙CD + |BC|^2 + BC˙CD = -2 + 9 = 7

[thepiano]

第 2 題
g(x) = f(x) - x^3 = ax^2 + bx + c
g(a) = g(b) = 0
a、b 是 g(x) = 0 之二根
a + b = -b/a，ab = c/a
c = a^2b = a^2 * [-a^2/(a + 1)] = -a^4/(a + 1) = (-a^3 + a^2 - a + 1) - [1/(a + 1)] 為整數
a = -2，c = 16，b = 4
a + b + c = 18

[thepiano]

填充第 12 題
f(x) = √3sin(kx) + cos(kx) = 2sin(kx + π/6)

(-1/4)π ≦ x ≦ (1/3)π
(-k/4)π ≦ kx ≦ (k/3)π
[(-3k + 2)/12]π ≦ kx + π/6 ≦ [(2k + 1)/6]π

當 [(-3k + 2)/12]π = (-1/2)π 時，是圖形最低點
當 [(2k + 1)/6]π = (1/2)π 時，是圖形最高點
分別往左、右延伸
故 (-3/2)π ＜ [(-3k + 2)/12]π ≦ -(1/2)π 且 (1/2)π ≦ [(2k + 1)/6]π ＜ (3/2)π 時
其圖形恰有一個最高點和一個最低點

8/3 ≦ k ＜20/3 且 1 ≦ k ＜4
8/3 ≦ k ＜ 4

[thepiano]

計算第 4 題
易知 P_1(a，c)，P_2(a^2 + bc，ac + cd)

(1) c = 0
OP_1 = 1，a^2 + c^2 = 1，a = ±1
(i) a = 1，P_n = (1，0)，OP_n = 1
(ii) a = -1，P_n = (±1，0)，OP_n = 1

(2) c ≠ 0
det(A) = 1，ad - bc = 1
OP_1 = 1，a^2 + c^2 = 1
OP_2 = 1，(a^2 + bc)^2 + (ac + cd)^2 = 1

a^4 + 2a^2bc + b^2c^2 + a^2c^2 + 2ac^2d + c^2d^2 = 1
a^4 + 2a^2(ad - 1) + (ad - 1)^2 + a^2(1 - a^2) + 2ad(1 - a^2) + (1 - a^2)d^2 = 1
d^2 = a^2

(i) d = a，a^2 - bc = 1 = a^2 + c^2，b = -c，矩陣 A 是旋轉矩陣，OP_n = 1
(ii) d = -a，P_1(a，c)，P_2(-1，0)，P_3(-a，-c)，P_4(1，0)，P_5(a，c)，P_6(-1，0)，...，OP_n = 1