2026届杭州市九年级数学中考一模模拟试卷（含答案详解与评分标准）

第1页2026届杭州市九年级数学中考一模模拟试卷（含答案详解与评分标准）学校：________________班级：________________姓名：________________考号：________________考试时间：120分钟满分：120分考试节点：中考一模适用对象：2026届九年级注意事项：1．本试卷共三大题，22小题，满分120分，考试时间120分钟。2．答题前请填写学校、班级、姓名和考号。3．选择题用规定方式作答；填空题和解答题写在相应作答区。4．解答题应写出必要的文字说明、证明过程或演算步骤。本卷用于九年级数学中考一模考前检测。题型选择题填空题解答题总分分值30分18分72分120分一、选择题（本大题共10小题，每小题3分，共30分。每小题只有一个选项符合题意）1．计算的结果是（）A．B．C．D．2．一次函数的图象不经过的象限是（）A．第一象限B．第二象限C．第三象限D．第四象限3．分式方程的解为（）A．B．C．D．无解4．某学习小组6名同学一次数学限时训练成绩为88，92，92，95，97，100，则这组数据的中位数和众数分别是（）A．92，92B．，92C．95，92D．，955．如图形关系用文字描述：直线，一条截线与直线a形成的一个内错角为，则与它相等的内错角度数为（）A．B．C．D．6．一个不透明袋中有3个红球、2个白球，除颜色外完全相同。从袋中不放回地随机摸出2个球，则两球颜色相同的概率是（）A．B．C．D．7．二次函数的顶点坐标为（）A．B．C．D．8．从圆外一点P向圆O作两条切线PA、PB，切点分别为A、B。若，则的度数是（）A．B．C．D．9．关于x的一元二次方程有两个不相等的实数根，则m的取值范围是（）A．B．且C．且D．10．点A、点B同在反比例函数上，直线AB与坐标轴围成的三角形中，三角形AOB的面积为（）A．B．C．D．二、填空题（本大题共6小题，每小题3分，共18分）11．因式分解：____________。12．不等式组：，的解集为____________。13．若a，b是方程的两个根，则____________。14．某坡道与水平地面的夹角为，铅直高度为12m，则坡道的长为____________m。15．一个正多边形的每个外角都是，则这个正多边形的边数为____________。16．二次函数的最大值为____________。三、解答题（本大题共6小题，共72分。解答应写出必要的文字说明、证明过程或演算步骤）17．（8分）计算与解方程。（1）计算：；（2）解方程：。作答区：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（10分）为了解2026届九年级学生中考一模数学阶段复习情况，某校从九年级学生中随机抽取40名学生的数学模拟测试成绩，整理如下表：成绩x（分）70≤x<8080≤x<9090≤x<100100≤x<110110≤x≤120人数6121453（1）本次抽样的样本容量为多少？成绩所在频数最大的组是哪一组？（2）用每组组中值估计这40名学生的平均成绩。（3）若该校九年级共有200名学生，请估计成绩不低于90分的学生人数。（4）在样本中成绩不低于110分的3名学生中有1名为甲，现随机抽取2名参加讲评展示，求甲被抽到的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（10分）某校准备购买一批中考一模复习用错题本。商店规定：一次购买不少于40本且不超过80本时，每本价格为元，其中x为购买本数。设购买x本的总费用为y元。（1）求y关于x的函数表达式，并写出x的取值范围；（2）若购买50本，需要多少元？（3）若预算不超过840元，在满足优惠条件的前提下，最多能购买多少本？作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（12分）如图形关系用文字描述：在中，，，。点D在斜边AB上，过点D分别作于E，于F，且。（1）证明：；（2）求AD的长；（3）求矩形CEDF的面积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（14分）某校为中考一模考前心理活动周制作一座抛物线形拱门。以地面AB所在直线为x轴，拱门中心所在铅直线为y轴建立平面直角坐标系，拱门与地面的交点为、，最高点为，单位为米。（1）求拱门所在抛物线的函数表达式；（2）求距中心2m处拱门的高度；（3）若一块矩形宣传牌宽1.5m、高2.5m，要求其底边在地面上且关于y轴对称放置，判断它能否完全放入拱门内，并说明理由；（4）若矩形宣传牌高为3.6m，仍关于y轴对称放置，求它能完全放入拱门内时的最大宽度。作答区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（18分）如图形关系用文字描述：在平面直角坐标系中，抛物线经过、、三点。点M为第一象限内抛物线上的动点，设M的横坐标为t。（1）求抛物线的函数表达式及顶点坐标；（2）用含t的式子表示的面积S，并求S的最大值；（3）点D在x轴上，且。若点M在点D右侧且，求点M的坐标，并求四边形ACMD的面积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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参考答案与解析评分说明：选择题、填空题按答案给分；解答题按过程分、计算分、结论分分步给分。若学生使用其他正确方法，过程清楚、结果正确，可参照相应步骤给分；仅写结果但缺少必要过程时，解答题不得给满分。一、选择题答案与关键理由1．C因为，，，所以原式=。（3分）2．C一次函数中且，图象经过第一、第二、第四象限，不经过第三象限。（3分）3．C两边同乘，得，解得；经检验不是增根。（3分）4．B将数据按从小到大排列后为88，92，92，95，97，100，中位数为，众数为92。（3分）5．B两直线平行，内错角相等，因此所求角为。（3分）6．B同色包括两红或两白，概率为。（3分）7．A配方得，顶点坐标为。（3分）8．C两切线所成角与对应圆心角互补，故。（3分）9．B一元二次要求；两个不相等实根要求，得，故且。（3分）10．B两点均满足。三角形AOB面积为。（3分）选择题评分补充：第1题重在准确处理平方根、绝对值和负整数指数，若只写选项不给理由，客观题仍按选项给分；第2题考查一次函数图象与系数的对应关系，要从斜率和截距判断象限；第3题属于分式方程，必须注意分母不为零；第4题考查中位数和众数，中位数应取第3、4个数据的平均数；第5题考查平行线内错角；第6题考查不放回概率，需把两红和两白两种情形相加；第7题考查配方法求顶点；第8题考查切线夹角与圆心角互补；第9题同时考查一元二次方程定义和判别式；第10题考查反比例函数点坐标与坐标法求面积。二、填空题答案与解析11．。利用完全平方公式。（3分）12．。由得；由得，取公共部分。（3分）13．。由根与系数关系得，，所以。（3分）14．24。设坡道长为l，则，所以。（3分）15．8。正多边形外角和为，边数为。（3分）16．5。配方得，故最大值为5。（3分）填空题评分补充：第11题写成平方形式才完整；第12题若只写两个不等式的分别解而未取交集，应酌情扣分；第13题不能先求根再代入也可用根与系数关系，关键是分母乘积不能写错符号；第14题应识别直角三角形中的正弦关系；第15题利用外角和固定为360度；第16题要将二次函数化为顶点式或利用开口方向判断