# 台大二階筆試-數學考古2018
## 2018台大電機
### 1.
#### $w=\frac{-1+\sqrt{3}i}{2},w^{86}+w^{87}+...w^{365}=?,(x+yi)^{4}=-2+2\sqrt{3}i,x>0,y>0,x=?,y=?$
#### ans:
#### $1+w+w^2=0,$原式=$w^2=\frac{-1-\sqrt{3}i}{2}$
#### $(x+yi)^{4}=(\sqrt{2})^4(cos120^{\circ}+isin120^{\circ})\longrightarrow x=\frac{\sqrt{6}}{2}、y=\frac{\sqrt{2}}{2}$
### 2.
#### $ln \alpha+\alpha-6=0,e^{\beta}+\beta-6=0,\alpha+\beta=?,ln \alpha+e^{\beta}=?$
#### ans:
#### $ln \alpha+e^{ln\alpha}=6,\beta+e^{\beta}=6\longrightarrow ln\alpha=\beta$
#### $ln\alpha+\alpha=\alpha+\beta=6,ln \alpha+e^{\beta}=6$
### 3.
#### $2x^2-3x-1=0,$兩根為$cot\alpha、cot\beta,cot(\alpha+\beta)=?$
#### $2sin^2(\alpha+\beta)-3sin(\alpha+\beta)cos(\alpha+\beta)+cos^2(\alpha+\beta)=?$
#### ans:
#### $cot\alpha+cot\beta=\frac{sin(\alpha+\beta)}{sin\alpha sin\beta}=\frac{3}{2},cot\alpha cot\beta=\frac{cos\alpha cos\beta}{sin\alpha sin\beta}=\frac{-1}{2}$
#### $cot(\alpha+\beta)=\frac{cos(\alpha+\beta)}{sin(\alpha+\beta)}=\frac{\frac{-1}{2}sin\alpha sin\beta-sin\alpha sin\beta}{\frac{3}{2}sin\alpha sin\beta}=-1$
#### 另一答案為$1+\frac{3}{2}+\frac{1}{2}=3$
### 4.
#### $\overrightarrow{u}=(2,1),\overrightarrow{v}=(-1,2),4x\overrightarrow{u}-\overrightarrow{v}\perp x\overrightarrow{u}+3\overrightarrow{v},x=?$
#### $(k,1),(2,4),(0,2k-4)$共線,k=?
#### ans:
#### $x=\pm\frac{\sqrt{3}}{2},k=1、5$
### 5.
#### $\left\{\begin{matrix} a_1x+b_1y=c_1\\ a_2x+b_2y=c_2\end{matrix}\right.$恰一解$(\alpha,\beta)$,$\left\{\begin{matrix} a_1x+3b_1y=2c_1\\ a_2x+3b_2y=2c_2\end{matrix}\right.$求其解(以$\alpha、\beta$表示)
#### ans:
#### $x=2\alpha,y=\frac{2}{3}\beta$
### 6.
#### 染病者0.5%、真陽檢測為陽0.95、真陰檢測為陰0.95，求一人為陽則染病機率為?
#### ans:
#### $\frac{0.005\times0.95}{0.995\times0.05+0.005\times0.95}=\frac{19}{218}$
### 7.
#### 投硬幣，正面A給B1元，反之亦然，A有6元、B有3元，玩到有人輸光為止，P(n)為A有n元贏所有錢機率,P(0)=?、P(9)=?、P(n),P(n-1),P(n+1)關係式?、求A贏所有錢機率?
#### ans:
#### $P(0)=0,P(9)=1,P(n)=\frac{1}{2}P(n-1)+\frac{1}{2}P(n+1)$
#### $P(n)=\alpha+\beta n\longrightarrow P(n)=\frac{1}{9}n,P(6)=\frac{2}{3}$