\((1)\)已知两条直线\({{l}_{1}}:kx+(1-k)y-3=0\)和\({{l}_{2}}:\left( k-1 \right)x+2y-2=0\)互相垂直，则\(k=\)__________．

\((2)\)如图，已知正方体\(ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}\)的棱长为\(2\)，\(M\)、\(N\)分别是棱\(AD\)和\(C{{C}_{1}}\)的中点，则异面直线\({{A}_{1}}M\)与\(BN\)所成角度为_________．

\((3)\)已知二次函数\(f\left( x \right)=a{{x}^{2}}+bx+c\)，任意的\(x\in R\)有\(f\left( 1-x \right)=f\left( 1 \right)+f\left( x \right)\)，若\(f\left( x \right)\)在区间\(\left[ 2m,m+1 \right]\)上不单调，则\(m\)的取值范围为_________________．

\((4)\)已知点\(A\)为圆\(O:{{x}^{2}}+{{y}^{2}}=5\)与圆\(C:{{x}^{2}}+{{(y-2)}^{2}}=1\)在第一象限内的交点，过\(A\)的直线\(l\)被圆\(O\)和圆\(C\)所截得的弦分别为\(NA\)和\(MA(M,N\)不重合\()\)，若\(\left| NA \right|=\left| MA \right|\)，则直线\(l\)的方程为______________\(.\)          

已知函数\(\therefore 2 < a < 3\)，\(\therefore 2 < a < 3\)．

\((\)Ⅰ\()\)若曲线\({{x}_{1}}+{{x}_{2}}=a,{{x}_{1}}{{x}_{2}}=3-a\)在点\((1,f(1))\)处的切线与直线\(=-\dfrac{1}{2}{{a}^{2}}+a-3+(3-a)\ln (3-a)\)垂直，求\(h(a)=-\dfrac{1}{2}{{a}^{2}}+a-3+(3-a)\ln (3-a),a\in (2,3)\)的值；

\((\)Ⅱ\()\)设\({{h}^{/}}(a)=-a-\ln (3-a)\)有两个极值点\({{h}^{/\!/}}(a)=-1+\dfrac{1}{3-a}=\dfrac{a-2}{3-a} > 0\)，且\({{h}^{/}}(a)\)，求证：\((2,3)\) ．

\(14.\)两圆\(x\)\({\,\!}^{2}+\)\(y\)\({\,\!}^{2}+4\)\(y\)\(=0\)，\(x\)\({\,\!}^{2}+\)\(y\)\({\,\!}^{2}+2(\)\(a\)\(-1)\)\(x\)\(+2\)\(y\)\(+\)\(a\)\({\,\!}^{2}=0\)在交点处的切线互相垂直，那么实数\(a\)的值为__________．

设曲线\(y= \dfrac{x+1}{x-1} \)在点\(\left(3,2\right) \)处的切线与直线\(ax+y+3=0\)垂直，则\(a=\)  (    )

\((1)\)若数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和\({{S}_{n}}={{n}^{2}}+n+1\)，则\({{a}_{n}}=\) ________________．

\((2)\)若椭圆\( \dfrac{x^{2}}{4}+ \dfrac{y^{2}}{k}=1\)的离心率\(e\)\(= \dfrac{2}{3}\)，则实数\(k\)的取值是______________________．

\((3)\)某观测站在城\(A\)南偏西\(20^{\circ}\)方向的\(C\)处，由城\(A\)出发的一条公路，走向是南偏东\(40^{\circ}\)，在\(C\)处测得公路距\(C\)处\(31\)千米的\(B\)处有一人正沿公路向城\(A\)走去，走了\(20\)千米后到达\(D\)处，此时\(C\)、\(D\)间的距离为\(21\)千米，问这人还要走千米可到达城\(A .\)

\((4)\)过点\(P(2,4)\)作两条互相垂直的直线分别交\(x\)轴、\(y\)轴于点\(A\)、\(B\)，则线段\(AB\)的中点\(M\)的轨迹方程为____________________．