2.
\((1)\)已知复数\(z=x-2+yi\)的模是\(2\sqrt{2}\),则点\((x,y)\)的轨迹方程是_________.
\((2)\)如果函数\(f(x)\)在区间\(D\)上是凸函数,那么对于区间\(D\)内的任意\(x_{1}\),\(x_{2}\),\(…\),\(x_{n}\),都有\(\dfrac{f\mathrm{(}x_{1}\mathrm{)}{+}f\mathrm{(}x_{2}\mathrm{)}{+}\mathrm{{…}}{+}f\mathrm{(}x_{n}\mathrm{)}}{n}\leqslant f\left( \dfrac{x_{1}{+}x_{2}{+}\mathrm{{…}}{+}x_{n}}{n} \right).\)若\(y=\sin x\)在区间\((0,π)\)内是凸函数,则在\(\triangle ABC\)中,\(\sin A+\sin B+\sin C\)的最大值是_____.
\((3)\)甲、乙、丙三位教师分别在哈尔滨、长春、沈阳的三所中学里教不同的学科\(A\)、\(B\)、\(C\),已知:
\(①\)甲不在哈尔滨工作,乙不在长春工作;\(②\)在哈尔滨工作的教师不教\(C\)学科;
\(③\)在长春工作的教师教\(A\)学科;\(④\)乙不教\(B\)学科\(.\)可以判断乙教的学科是______________.
\((4)\)将正整数对作如下分组,第\(1\)组为\(\left\{ \left( 1{,}2 \right){,}\left( 2{,}1 \right) \right\}\),第\(2\)组为\(\left\{ \left( 1{,}3 \right){,}\left( 3{,}1 \right) \right\}\),第\(3\)组为\(\left\{ \left( 1{,}4 \right){,}\left( 2{,}3 \right){,}\left( 3{,}2 \right){,}\left( 4{,}1 \right) \right\}\),第\(4\)组为\(\left\{ \left( 1{,}5 \right){,}\left( 2{,}4 \right)\left( 4{,}2 \right)\left( 5{,}1 \right) \right\}{⋅⋅⋅⋅⋅⋅}\)则第\(30\)组第\(16\)个数对为__________.