4.
填空题
\((1)\)已知等差数列\(\{ \)
\(a_{n}\)\(\}\)前\(9\)项的和为\(27\),
\(a\)\({\,\!}_{10}=8\),则
\(a\)\({\,\!}_{200}\)等于________.
\((2)\)过两直线\(x- \sqrt{3}y+1=0\)和\( \sqrt{3}x+y- \sqrt{3}=0\)的交点,并且与原点的最短距离为\( \dfrac{1}{2}\)的直线的方程为________.
\((3)\)设
\(m\) , \(n\)是不同的直线,\(α\),\(β\),\(γ\)是不同的平面,有以下四个命题:
\((1)\left. \begin{matrix} \alpha /\!/\beta \\ \alpha /\!/\gamma \\\end{matrix} \right\}\Rightarrow \beta /\!/\gamma \);\((2)\left. \begin{matrix} \alpha \bot \beta \\ m/\!/\alpha \\\end{matrix} \right\}\Rightarrow m/\!/\beta \)
\((3)\left. \begin{matrix} m\bot \alpha \\ m/\!/\beta \\\end{matrix} \right\}\Rightarrow \alpha \bot \beta \);\((4)\left. \begin{matrix} m/\!/n \\ n\subset \alpha \\\end{matrix} \right\}\Rightarrow m/\!/\alpha \),
其中假命题有 .
\((4)\)已知定义在\(R\)上的单调函数\(f(x)\)满足对任意的\(x_{1}\),\(x_{2}\),都有\(f(x_{1}+x_{2})=f(x_{1})+f(x_{2})\)成立\(.\)若正实数\(a\),\(b\)满足\(f(a)+f(4b-1)=0\),则\( \dfrac{1}{a}+ \dfrac{2}{b} \)的最小值为 .