部分題目參考解答
有錯還請不吝指教
1. \(\displaystyle 2\sqrt{3}\)
4.\(\displaystyle f(x)=x^3+6x^2+10x+4\)(感謝yosong老師指正)
5.\(\displaystyle \sqrt{23}\ or\ \sqrt{47}\)(感謝leilei老師指正)
7(1).\(\displaystyle ,\sqrt{3}-2,\sqrt{3}-1,\sqrt{3},\sqrt{3}+1,\sqrt{3}+2\) (感謝yosong老師指正)
8.由題目得\(\displaystyle (2k-1)^2=8m-7 \Rightarrow\ 4k^2-4k+8=8m\)
可得\(\displaystyle m=\frac{k^2-k+2}{2}\)
簡單討論一下k的奇偶情形，易知無論哪個k都能滿足存在一個\(\displaystyle m\in \mathbb{N}\)
9.設矩陣A的行向量分別為\(\displaystyle a_1,a_2 \cdots ,a_n\)，且\(a_k\)的分量分別為\(p_{k1},p_{k2},\cdots,p_{kn}\)
矩陣B的行向量分別為\(\displaystyle b_1,b_2 \cdots ,b_n\),且\(b_k\)的分量分別為\(q_{k1},q_{k2},\cdots,q_{kn}\)
若\(c_k\)為矩陣C的第k行向量，且\(c_k\)的分量分別為\(r_{k1},r_{k2},\cdots,r_{kn}\)
有\(\displaystyle c_k=q_{k1}(a_1)+q_{k2}(a_2)+\cdots +q_{kn}(a_n)\)
可知\(\displaystyle \Sigma_{i=1}^n\ r_{ki} =q_{k1}\times \displaystyle\Sigma_{i=1}^n\ p_{1i}+q_{k2}\times \displaystyle\Sigma_{i=1}^n\ p_{2i}+\cdots +q_{kn}\times \displaystyle\Sigma_{i=1}^n\ p_{ni}\)
因為AB皆為轉移矩陣，故\(\displaystyle \Sigma_{i=1}^n\ p_{ki}=1, \displaystyle\Sigma_{i=1}^n\ q_{ki}=1\)
所以\(\displaystyle \Sigma_{i=1}^n\ r_{ki}=1\)，C為轉移矩陣
10.給\(\displaystyle \overline{BC},\angle{ABC},\angle{ACB},\angle{ABO}\)或是\(\displaystyle \overline{BC},\angle{ABC},\angle{ACB},\angle{ACO}\)
11. p=6時，有f(p)有min 40
12.\(n(S)=6\)
13.幾何分布老梗題目 : \(\displaystyle E(X)= \frac{1}{p} , Var(X)=\frac{1-p}{p^2}\)
15.\(\displaystyle \frac{27}{65}\)
16.我算\(\displaystyle E(X)=500\times (\frac{4}{3})^{998}\times \frac{3005}{6}\)，醜到懷疑人生，不知道是否有誤
17.感覺分母應該是\(n^2\) ?
[ 本帖最後由 satsuki931000 於 2023-4-9 21:49 編輯 ]
