\((1)\)在平面直角坐标系中,已知函数\(y=\log \;a(x-3)+2 (a > 0且a\neq 1) \)的图像过定点\(p \),角\(a \)的始边与\(x \)轴的非负半轴重合,终边过点\(p \),则\(3{\sin }^{2}α+\cos 2α \)的值为 .
\((2)\)设\({{S}_{n}}\)为数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和,已知\({{a}_{1}}=2\) ,对任意\(p,q\in N^{*}\),都有\({{a}_{p+q}}={{a}_{p}}+{{a}_{q}}\), 则\(f\left( n \right)=\dfrac{{{S}_{n}}+60}{n+1}(n\in N^{*})\)的最小值为
\((3)\)某几何体的三视图如图,则该几何体的外接球表面积 .
\((4)\Delta ABC\)中,\(AB=4\),\(\angle ABC=\dfrac{\pi }{4},\angle ACB=\dfrac{\pi }{6}\),\(O\)为外心,\(D\),\(E\)分别为\(AB\),\(AC\)上定点,满足\(\overrightarrow{AD}=\overrightarrow{DB},\overrightarrow{AE}=2\overrightarrow{EC}\),\(BE\),\(CD\)交于\(G\)点,则\(\overrightarrow{AO}\cdot \overrightarrow{AG}=\)______________