3.
\((1)\)已知:\(\dfrac{\pi }{2} < \beta < \alpha < \dfrac{3\pi }{4}\),求\(\alpha -\beta \)的范围为__________.
\((2)\)设等差数列\(\left\{ {{a}_{n}} \right\}\)、\(\left\{ {{b}_{n}} \right\}\)的前\(n\)项和分别为\({{S}_{n}}\)、\({{T}_{n}}\),若对任意自然数\(n\)都有\(\dfrac{{{S}_{n}}}{{{T}_{n}}}=\dfrac{2n+3}{n-4}\),则\(\dfrac{{{a}_{9}}}{{{b}_{5}}+{{b}_{7}}}+\dfrac{{{a}_{3}}}{{{b}_{8}}+{{b}_{4}}}\)的值为________.
\((3)\)已知椭圆\(C:\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)\),直线\(l\)为圆\(O:{{x}^{2}}+{{y}^{2}}={{b}^{2}}\)的一条切线,若直线\(l\)的倾斜角为\(\dfrac{\pi }{3}\),且恰好经过椭圆的右焦点,则椭圆的离心率为________.
\((4)\)在正方体\(ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}\)中\((\)如图\()\),已知点\(P\)在直线\(B{{C}_{1}}\)上运动\(.\)则下列四个命题:
\(①\)三棱锥\(A-{{D}_{1}}PD\)的体积不变;
\(②\)直线\(AP\)与平面\(AC{{D}_{1}}\)所成角的大小不变;
\(③\)二面角\(P-A{{D}_{1}}-C\)的大小不变;
\(④M\)是正方形\({{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}\)所在平面内到点\(D\)和\({{C}_{1}}\)距离相等的点,则\(M\)点的轨迹是直线\({{A}_{1}}{{D}_{1}}\).
其中正确命题的编号是_______\(.(\)写出所有正确命题的编号\()\)