2017北京市中考试题

D./r/n线段/r/nPD/r/nD./r/n线段/r/nPD/r/n的长度/r/nC./r/n线段/r/nPC/r/n的长度/r/n）/r/nD./r/nx/r/n丰/r/n4/r/n2017/r/n年北京市高级中等学校招生考试/r/n数学试卷/r/n一、选择题（本题共/r/n30/r/n分，每小题/r/n3/r/n分）/r/n1/r/n•如图所示，点/r/nP/r/n到直线/r/nl/r/n的距离是（/r/n）/r/n线段/r/nPA/r/n的长度/r/nB./r/n线段/r/nPB/r/n的长度/r/nx/r/n2/r/n•若代数式/r/n有意义，则实数/r/nx/r/n的取值范围是（/r/nx/r/n-/r/n4/r/nA./r/nx/r/n二/r/n0/r/nB./r/nx/r/n二/r/n4/r/nC./r/nx/r/n丰/r/n0/r/n3./r/n右图是某个几何题的展开图，该几何体是（）/r/nA./r/n三棱柱/r/nB./r/n圆锥/r/nC./r/n四棱柱/r/nD./r/n圆柱/r/n4./r/n实数/r/na,b,c,d/r/n在数轴上的对应点的位置如图所示，则正确的结论是/r/n（/r/n）/r/nA./r/nA./r/na/r/n>-4/r/nB./r/nbd/r/n>/r/n0/r/nC./r/n\a\/r/n>/r/nb/r/nD./r/nb/r/n+/r/nc/r/n>/r/n0/r/n5./r/n下列图形中，是轴对称图形但不是中心对称图形的是（/r/n）/r/n6./r/n若正多边形的一个内角是/r/n150°，/r/n则该正多边形的边数是（）/r/nA./r/n6/r/nB./r/n12/r/nC./r/n16/r/nD.18/r/n4/r/n1/r/na/r/n2/r/n7./r/n如果/r/na/r/n2/r/n+/r/n2a/r/n-/r/n1/r/n=/r/n0/r/n,那么代数式/r/na/r/ng/r/n-的值是（/r/n）/r/na/r/n丿/r/na/r/n一/r/n2/r/nA./r/n-3/r/nB./r/n-1/r/nC./r/n1/r/nD./r/n.3/r/n8./r/n下面的统计图反映了我国与“一带一路”沿线部分地区的贸易情况./r/n2011-2016/r/n年我国与东南亚地区和东欧地区的贸易额统计图/r/nA/r/n贸胡颔亿美元/r/n（以上数据摘自《“一带一路”贸易合作大数据报告（/r/n2017/r/n）》）根据统计图提供的信息，下列推理不合理的是（）/r/n与/r/n2015/r/n年相比，/r/n2016/r/n年我国与东欧地区的贸易额有所增长/r/n2011-2016/r/n年，我国与东南亚地区的贸易额逐年增长/r/n2011-2016/r/n年，我国与东南亚地区的贸易额的平均值超过/r/n4200/r/n亿美元/r/n2016/r/n年我国与东南亚地区的贸易额比我国与东欧地区的贸易额的/r/n3/r/n倍还多/r/n

/r/n9/r/n.小苏和小林在右图所示的跑道上进行/r/n4/r/nX50/r/n米折返跑.在整个过程中，跑步者距起跑线的距离/r/ny/r/n（单位：/r/nm/r/n）与跑步时间/r/nt/r/n（单位：/r/ns/r/n）的对应关系如下图所示•下列叙述正确的是（/r/n）/r/n小苏跑全程的平均速度大于小林跑全程的平均速度/r/n小苏前/r/n15/r/ns/r/n跑过的路程大于小林前/r/n15/r/ns/r/n跑过的路程/r/n小林在跑最后/r/n100/r/nm/r/n的过程中，与小苏相遇/r/n2/r/n次/r/n10./r/n下图显示了用计算机模拟随机投掷一枚图钉的某次实验的结果./r/n下面有三个推断：/r/n当投掷次数是/r/n500/r/n时，计算机记录“钉尖向上”的次数是/r/n308，/r/n所以“钉尖向上”的概率是/r/n0.616；/r/n随着实验次数的增加，“钉尖向上”的频率总在/r/n0.618/r/n附近摆动，显示出一定的稳定性，可以估计“钉尖向/r/n得到的，写出一中由/r/n得到的，写出一中由/r/nAOCD/r/n得到/r/nAAOB/r/n的过程：/r/n上”的概率是/r/n0.618；/r/n若再次用计算机模拟实验，则当投掷次数为/r/n1000/r/n时，“钉尖向上”的概率一定是/r/n0.620./r/n其中合理的是（）/r/nTOC\o"1-5"\h\z/r/nA./r/n①/r/nB./r/n②/r/nC./r/n①②/r/nD./r/n①③/r/n二、填空题（本题共/r/n18/r/n分，每题/r/n3/r/n分）/r/n11./r/n写出一个比/r/n3/r/n大且比/r/n4/r/n小的无理数：/r/n./r/n12./r/n某活动小组购买了/r/n4/r/n个篮球和/r/n5/r/n个足球，一共花费了/r/n435/r/n元，其中篮球的单价比足球的单价多/r/n3/r/n元，求篮球/r/n的单价和足球的单价•设篮球的单价为/r/nx/r/n元，足球的单价为/r/ny/r/n元，依题意，可列方程组为/r/n./r/n13./r/n如图，在/r/nA/r/nABC/r/n中，/r/nM/r/n、/r/nN/r/n分别为/r/nAC/r/n,/r/nBC/r/n的中点.若/r/nS/r/n=/r/n1/r/n，则/r/nS/r/nN/r/nCMN/r/n四边形/r/nABNM/r/n14./r/n如图，/r/nAB/r/n为/r/ne/r/nO/r/n的直径，/r/nC/r/n、/r/nD/r/n为/r/ne/r/nO/r/n上的点，/r/nAD/r/n二/r/nCD/r/n.若/r/nZ/r/nCAB/r/n=/r/n40/r/n0/r/n，则/r/nZ/r/nCAD/r/n二/r/n15/r/n•如图，在平面直角坐标系/r/nxOy/r/n中，/r/nAAOB/r/n可以看作是/r/nAOCD/r/n经过若干次图形的变化（平移、轴对称、旋转）/r/n16./r/n下图是“作已知直角三角形的外接圆”的尺规作图过程/r/n已知：/r/nRt/r/nA/r/nABC,/r/nZ/r/nC/r/n二/r/n90/r/n0/r/n，求作/r/nRtAABC/r/n的外接圆./r/n作法：如图．/r/n分别以点/r/nA/r/n和点/r/nB/r/n为圆心，大于/r/n2/r/nAB/r/n的长为半径作弧，两弧相交于/r/nP/r/n,/r/nQ/r/n两点;/r/n作直线/r/nPQ/r/n，交/r/nAB/r/n于点/r/nO/r/n；/r/n以/r/nO/r/n为圆心，/r/nOA/r/n为半径作/r/ne/r/nO/r/n./r/ne/r/nO/r/n即为所求作的圆./r/n请回答：该尺规作图的依据/r/n三、解答题(本题共/r/n72/r/n分，第/r/n17/r/n题-/r/n26/r/n题，每小题/r/n5/r/n分，第/r/n27/r/n题/r/n7/r/n分，第/r/n28/r/n题/r/n7/r/n分，第/r/n29/r/n

/r/n题/r/n8/r/n分）解答应写出文字说明、证明过程或演算步骤./r/n计算：/r/n4cos30/r/no/r/n+/r/nC/r/n-迈/r/n）/r/n-、；/r/n!2/r/n+/r/n|-/r/n2/r/n./r/n2/r/n（x/r/n+/r/n1/r/n）/r/n>/r/n5/r/nx/r/n—/r/n7/r/n解不等式组：/r/n］/r/nx/r/n+/r/n10/r/n>/r/n2/r/nx/r/nI/r/n3/r/n如图，在/r/nAABC/r/n中，/r/nAB/r/n二/r/nAC/r/n,/r/nZ/r/nA/r/n二/r/n36/r/no/r/n,/r/nBD/r/n平分/r/nZ/r/nABC/r/n交/r/nAC/r/n于点/r/nD/r/n./r/n求证：/r/nAD=BC/r/n./r/n20./r/n数学家吴文俊院士非常重视古代数学家贾宪提出的“从长方形对角线上任一点作两条分别平行于两邻边的直线，则所容两长方形面积相等（如图所示）”这一推论，他从这一推论出发，利用“出入相补”原理复原了《海岛算经》九题古证./r/nD/r/n£/r/nG/r/nB/r/nF/r/nD/r/n£/r/nG/r/nB/r/nF/r/n（以上材料来源于《古证复原的原理》、《吴文俊与中国数学》和《古代世界数学泰斗刘徽》）请根据上图完成这个推论的证明过程．/r/nTOC\o"1-5"\h\z/r/n/r/n证明：/r/nS/r/n=/r/nS/r/n—/r/n（/r/ns/r/n+/r/nS/r/n）/r/n，/r/nS=S/r/n—/r/n（/r/n+/r/n）./r/n矩形/r/nNFGD/r/nA/r/nADC/r/nA/r/nANF/r/nA/r/nFGC/r/n矩形/r/nEBMF/r/nA/r/nABC/r/n易知，/r/nS/r/n=/r/nS/r/n，/r/n=/r/n，/r/n=/r/n．/r/nA/r/nADC/r/nA/r/nABC/r/n可得/r/nS/r/n=/r/nS/r/n．/r/n矩形/r/nNFGD/r/n矩形/r/nEBMF/r/n21./r/n关于/r/nx/r/n的一元二次方程/r/nx/r/n2/r/n—/r/n(/r/nk/r/n+/r/n3/r/n)/r/nx/r/n+/r/n2k/r/n+/r/n2/r/n=/r/n0/r/n./r/n求证：方程总有两个实数根；/r/n若方程有一根小于/r/n1,/r/n求/r/nk/r/n的取值范围./r/n22./r/n如图，在四边形/r/nABCD/r/n中，/r/nBD/r/n为一条对角线，/r/nAD/r/n//BC,/r/nAD/r/n=/r/n2/r/nBC/r/n,/r/nZABD/r/n=/r/n90/r/n0/r/n,/r/nE/r/n为/r/nAD/r/n的中点,/r/n连接/r/nBE/r/n./r/n求证：四边形/r/nBCDE/r/n为菱形；/r/n连接/r/nAC/r/n，若/r/nAC/r/n平分/r/n/BAD/r/n,/r/nBC/r/n=/r/n1/r/n，求/r/nAC/r/n的长./r/n23./r/n如图，在平面直角坐标系/r/nxOy/r/n中，函数/r/ny/r/n=/r/n-/r/n(/r/nx/r/n>/r/n0/r/n)/r/n的图象与直线/r/ny/r/n=/r/nx/r/n—/r/n2/r/n交于点/r/nA/r/n(/r/n3,/r/nm/r/n)/r/n./r/nx/r/n求/r/n—m/r/n的值；/r/n已知点/r/nP/r/n(/r/nn,n/r/n)(/r/nn/r/n>/r/n0/r/n)/r/n，过点/r/nP/r/n作平行于/r/nx/r/n轴的直线，交直线/r/ny/r/n=/r/nx/r/n—/r/n2/r/n于点/r/nM/r/n，过点/r/nP/r/n作平行于/r/ny/r/n轴的直线，交函数/r/ny/r/n=/r/n-/r/n(/r/nx/r/n>/r/n0/r/n)/r/n的图象于点/r/nN/r/n./r/nx/r/n当/r/nn/r/n=/r/n1/r/n时，判断线段/r/nPM/r/n与/r/nPN/r/n的数量关系，并说明理由；/r/n若/r/nPN/r/n>/r/nPM/r/n，结合函数的图象，直接写出/r/nn/r/n的取值范围./r/n

/r/n24./r/n如图，/r/nAB/r/n是/r/ne/r/nO/r/n的一条弦，/r/nE/r/n是/r/nAB/r/n的中点，过点/r/nE/r/n作/r/nEC/r/n丄/r/nOA/r/n于点/r/nC/r/n，过点/r/nB/r/n作/r/ne/r/nO/r/n的切线交/r/nCE/r/n的延长线于点/r/nD/r/n./r/n求证：/r/nDB/r/n=/r/nDE/r/n；/r/n若/r/nAB/r/n=/r/n12,/r/nBD/r/n=/r/n5/r/n，求/r/ne/r/nO/r/n的半径./r/n25./r/n某工厂甲、乙两个部门各有员工/r/n400/r/n人，为了解这两个部门员工的生产技能情况，进行了抽样调查，过程如下，请补充完整./r/n收集数据从甲、乙两个部门各随机抽取/r/n20/r/n名员工，进行了生产技能测试，测试成绩(百分制)如下：/r/n甲/r/n78/r/n86/r/n74/r/n81/r/n75/r/n76/r/n87/r/n70/r/n75/r/n90/r/n75/r/n79/r/n81/r/n70/r/n74/r/n80/r/n86/r/n69/r/n83/r/n77/r/n乙/r/n93/r/n73/r/n88/r/n81/r/n72/r/n81/r/n94/r/n83/r/n77/r/n83/r/n80/r/n81/r/n70/r/n81/r/n73/r/n78/r/n82/r/n80/r/n70/r/n40/r/n整理、描述数据/r/n

/r/n按如下分数段整理、描述这两组样本数据：/r/n成绩/r/nX/r/n人数/r/n部门/r/n40/r/n</r/nx/r/n</r/n49/r/n50/r/n</r/nx/r/n</r/n59/r/n60/r/n</r/nx/r/n</r/n69/r/n70/r/n</r/nx/r/n</r/n79/r/n80/r/n</r/nx/r/n</r/n89/r/n90/r/n</r/nx/r/n</r/n100/r/n甲/r/n0/r/n0/r/n1/r/n11/r/n7/r/n1/r/n乙/r/n（说明：成绩/r/n80/r/n分及以上为生产技能优秀，/r/n70/r/n--/r/n79/r/n分为生产技能良好，/r/n60/r/n--/r/n69/r/n分为生产技能合格，/r/n60/r/n分以下为生产技能不合格）/r/n分析数据两组样本数据的平均数、中位数、众数如下表所示：/r/n部门/r/n平均数/r/n中位数/r/n众数/r/n甲/r/n78.3/r/n77.5/r/n75/r/n乙/r/n78/r/n80.5/r/n81/r/n得出结论：/r/na/r/n.估计乙部门生产技能优秀的员工人数为/r/n；/r/nb/r/n.可以推断出/r/n部门员工的生产技能水平较高，理由为/r/n.（至少从两个不同的角度说/r/n明推断的合理性）/r/n26/r/n•如图，/r/nP/r/n是/r/nAB/r/n所对弦/r/nAB/r/n上一动点，过点/r/nP/r/n作/r/nPM/r/n丄/r/nAB/r/n交/r/nAB/r/n于点/r/nM/r/n,连接/r/nMB/r/n,过点/r/nP/r/n作/r/nPN/r/n丄/r/nMB/r/n于点/r/nN/r/n./r/n已知/r/nAB/r/n=/r/n6/r/ncm/r/n,设/r/nA/r/n、/r/nP/r/n两点间的距离为/r/nxcm/r/n,/r/nP/r/n、/r/nN/r/n两点间的距离为/r/ny/r/ncm/r/n./r/n（当点/r/nP/r/n与点/r/nA/r/n或点/r/nB/r/n重合时，/r/ny/r/n的值为/r/no）/r/n小东根据学习函数的经验，对函数/r/ny/r/n随自变量/r/nx/r/n的变化而变化的规律进行了探究./r/n下面是小东的探究过程，请补充完整：/r/n（/r/n1/r/n）通过取点、画图、测量，得到了/r/nX/r/n与/r/ny/r/n的几组值，如下表：/r/nx/r/n//r/ncm/r/n0/r/n1/r/n2/r/n3/r/n4/r/n5/r/n6/r/ny/r/n//r/ncm/r/n0/r/n2.0/r/n2.3/r/n2.1/r/n0.9/r/n0/r/n说明：补全表格时相关数值保留一位小数）/r/n2）/r/n建立平面直角坐标系，描出以补全后的表中各对对应值为坐标的点，画出该函数的图象/r/n（3）/r/n结合画出的函数图象，解决问题：当/r/nA/r/nPAN/r/n为等腰三角形时，/r/nAP/r/n的长度约为/r/ncm/r/n./r/n27/r/n•在平面直角坐标系/r/nxOy/r/n中，抛物线/r/nY/r/n二/r/nx/r/n-/r/n4/r/nx/r/n+/r/n3/r/n与/r/nx/r/n轴交于点/r/nA/r/n、/r/nB/r/n（点/r/nA/r/n在点/r/nB/r/n的左侧），与/r/nY/r/n轴交于点/r/nC/r/n./r/n（1）/r/n求直线/r/nBC/r/n的表达式；/r/n（2）/r/n垂直于/r/ny/r/n轴的直线/r/n//r/n与抛物线交于点/r/nP/r/n（/r/nx,y/r/n）/r/n,/r/nQ/r/n（/r/nx/r/n,/r/ny/r/n）/r/n，与直线/r/nBC/r/n交于点/r/nN/r/n（/r/nx/r/n,/r/ny/r/n）/r/n，若/r/nx/r/n</r/nx/r/n</r/nx/r/n，/r/n1/r/n2/r/n3/r/n结合函数的图象，求/r/nx/r/n+/r/nx/r/n+/r/nx/r/n的取值范围./r/n123/r/n28/r/n•在等腰直角/r/nAABC/r/n中，/r/nZACB/r/n=/r/n90/r/n0/r/n，/r/nP/r/n是线段/r/nBC/r/n上一动点（与点/r/nB/r/n、/r/nC/r/n不重合），连接/r/nAP/r/n,/r/n延长/r/nBC/r/n至/r/n点/r/nQ/r/n，使得/r/nCQ/r/n=/r/nCP/r/n，过点/r/nQ/r/n作/r/nQH/r/n丄/r/nAP/r/n于点/r/nH/r/n，交