在\(\Delta ABC\)中，角\(A,B,C\)的对边分别为\(a,b,c,\cos C=\dfrac{3}{10}\)．

\((1)\)若\(\overrightarrow{CA}\bullet \overrightarrow{CB}=\dfrac{9}{2}\)，求\(\Delta ABC\)的面积；

\((2)\)设向量\( \overset{⇀}{x}=(2\sin ⁡B,− \sqrt{3}), \overset{⇀}{y}=(\cos ⁡2B,1−2{\sin }^{2} \dfrac{B}{2}) \)，且\( \overset{⇀}{x}/\!/ \overset{⇀}{y} \)，求角\(B\)的值．

如图，在同一个平面内，向量\(\overrightarrow{{OA}}{,}\overrightarrow{{OB}}{,}\overrightarrow{{OC}}\)的模分别为\(1{,}1{,}\sqrt{2}{,}\overrightarrow{{OA}}\)与\(\overrightarrow{{OC}}\)的夹角为\(\alpha\)，且\(\tan\alpha{=}7{,}\overrightarrow{{OB}}\)与\(\overrightarrow{{OC}}\)的夹角为\({45}^{∘} \)。若\(\overrightarrow{{OC}}{=}m\overrightarrow{{OA}}{+}n\overrightarrow{{OB}}(m{,}n{∈}R)\)，则\(m{+}n{=}\) ______ ．

已知\(e_{1}\)，\(e_{2}\)是互相垂直的单位向量\(.\)若\( \sqrt{3}e_{1}-e_{2}\)与\(e_{1}+λe_{2}\)的夹角为\(60^{\circ}\)，则实数\(λ\)的值是________\(.\)　

已知向量\(\overrightarrow{AB}\)与\(\overrightarrow{AC}\)的夹角为\(120^{\circ}\)，且\(|\overrightarrow{AB}|=3\)，\(|\overrightarrow{AC}|=2.\)若\(\overrightarrow{AP}=λ\overrightarrow{AB}+\overrightarrow{AC}\)，且\(\overrightarrow{AP}⊥\overrightarrow{BC}\)，则实数\(λ\)的值为__________．

已知向量\(\overrightarrow{a}\)与向量\(\overrightarrow{b}\)的夹角为\(\dfrac{\pi }{3}\)，且\(\overrightarrow{a}\left(1, \sqrt{2}\right) \)，\(\overrightarrow{a}\bot \left( \overrightarrow{a}-2\overrightarrow{b} \right)\)，则\(\left| \overrightarrow{b} \right|=\)_______________．

\((1)\)求值：\({co}{{{s}}^{4}}{{15}^{0}}-{si}{{{n}}^{4}}{{15}^{0}}= \)________．

\((2)\)已知\({|}a{|=}1\)，\({|}b{|=}2\)，若\(a⊥(a+b)\)，则向量\(a\)与\(b\)的夹角为__________．

\((3)\)数列\(\{a_{n}\}\)满足\(a_{1}=0\)，\(a_{n+1}=\dfrac{{{a}_{n}}-\sqrt{3}}{\sqrt{3}{{a}_{n}}+1}\) \((n∈N^{*})\)，则\(a_{2015}=\)________．

\((4)\)已知数列\(\left\{ {{a}_{n}} \right\}\)是各项均不为零的等差数列，\({{S}_{n}}\)为其前\(n\)项和，且\({{a}_{n}}=\sqrt{{{S}_{2n-1}}}\left( n\in {{N}^{*}} \right).\)若不等式\(\dfrac{\lambda }{{{a}_{n}}}\leqslant \dfrac{n+8}{n}\)对任意\(n\in {{N}^{*}}\)恒成立，则实数\(\lambda \)的最大值为_____________．