91學年度指定科目考試數學(甲)非選擇題詳解

1. 令總球數$=$白球$+$黑球$=n$，則$P(2$白$)$ $=\displaystyle{\frac{C_{2}^{7}}{C_{2}^{n}}}$ $=\displaystyle{\frac{\displaystyle{\frac{7\times 6}{2}}}{\displaystyle{\frac{n\times (n-1)}{2}}}}$ $=\displaystyle{\frac{42}{n(n-1)}}$ $=\displaystyle{\frac{7}{22}}$ $\Rightarrow$ $n(n-1)=132$得$n=12$或$-11$(但負不合) $\Rightarrow$ $n=12$，故黑球個數為$12-7=5$(顆)
   
   
2. 令$f(x)=3{{x}^{4}}-4m{{x}^{3}}+1$ $\Rightarrow$ ${f}'(x)=12{{x}^{3}}-12m{{x}^{2}}$，解${f}'(x)=0$得$x=m$或$0$(重根) $\Rightarrow$解$f(m)>0$得$3{{m}^{4}}-4{{m}^{4}}+1>0$ $\Rightarrow$ ${{m}^{4}}<1$得$-1<m<1$