2025-2026学年上海市浦东新区九年级数学中考一模模拟试卷（含答案详解与评分标准）

2025-2026学年上海市浦东新区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分注意事项：1．本卷用于九年级数学中考一模考前检测与阶段复习，全卷共三大题、22小题。2．选择题每题只有一个正确选项；填空题只填写最终结果；解答题需写出必要的计算过程、推理依据或作图说明。3．请将答案写在相应作答区内，书写清楚，步骤完整；不得在密封线外书写个人信息。4．作答时可使用规定范围内的直尺、圆规、三角尺和计算器，所有结论应由题设条件推出。题型选择题填空题解答题总分分值10题×3分=30分6题×3分=18分6题=72分120分一、选择题（本大题共10题，每题3分，共30分）请把唯一正确选项填在题后的括号内。1．化简的结果是（）。A．B．C．D．答案：________2．一元二次方程的较大实根是（）。A．1B．2C．3D．6答案：________3．二次函数的顶点坐标与开口方向分别是（）。A．顶点(1，-4)，开口向上B．顶点(-1，-4)，开口向上C．顶点(1，4)，开口向下D．顶点(-1，4)，开口向下答案：________4．反比例函数的图象经过点(2，-3)，当时，函数值y等于（）。A．-6B．-3C．3D．6答案：________5．已知AB是⊙O的直径，点C在圆上，，则的度数是（）。A．35°B．55°C．70°D．110°答案：________6．不透明袋中有3个红球和2个白球，每个球除颜色外完全相同。从袋中一次取出2个球，则取出的2个球都是红球的概率是（）。A．B．C．D．答案：________7．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．m<1B．m≤1C．m>1D．m≥1答案：________8．在△ABC中，D、E分别在AB、AC上，且DE∥BC。若AD∶DB=2∶3，则△ADE与△ABC的面积比是（）。A．2∶3B．4∶9C．4∶25D．2∶5答案：________9．抛物线经过点(-1，0)、(3，0)、(0，3)，并且开口向下。下列结论正确的是（）。A．对称轴是直线x=0B．当x=1时，y取得最大值4C．当x=0时，y取得最大值3D．图象与x轴只有一个交点答案：________10．在平面直角坐标系中，A(0，4)、B(3，0)，O为坐标原点。若以O为圆心的圆与直线AB相切，则该圆半径为（）。A．B．C．3D．4答案：________二、填空题（本大题共6题，每题3分，共18分）请将结果直接填写在横线上。11．计算：__________。答：________________________________12．函数的定义域是__________。答：________________________________13．若正比例函数的图象经过点(1，3)，则m=__________。答：________________________________14．一个圆锥的底面半径为3，母线长为5，则它的侧面积是__________。答：________________________________15．方程的两个实根之和为__________。答：________________________________16．若二次函数的函数值小于0，则x的取值范围是__________。答：________________________________三、解答题（本大题共6题，共72分）解答应写出文字说明、证明过程或演算步骤。17．（本题10分）计算与解方程。（1）计算：；（2）解方程：。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（本题10分）为了解九年级学生中考一模前数学阶段复习情况，某校从九年级随机抽取40名学生进行数学小测，满分120分。成绩分组统计如下表。成绩x（分）人数组中值频率60≤x<724660.1072≤x<8410780.2584≤x<9616900.4096≤x≤108101020.25（1）求样本中成绩不低于84分的学生人数，并写出相应频率；（2）用组中值估计这40名学生成绩的平均数；（3）若全年级共有500名九年级学生，请估计成绩不低于96分的人数。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（本题12分）如题意作图：AB为⊙O的直径，点C在⊙O上，且AB=10，AC=6，BC=8。过点C作⊙O的切线，该切线与BA的延长线交于点D。（1）证明：△ACD∽△CBD；（2）求线段CD、AD的长。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（本题12分）在平面直角坐标系中，已知抛物线与x轴交于A、B两点，与y轴交于C点，顶点为V。（1）求A、B、C、V的坐标；（2）求直线CV的解析式，并说明直线CV与抛物线的另一个交点；（3）点P在x轴上，且位于A、B之间。若△CPV的面积为10，求点P的坐标。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（本题14分）浦东新区某学校布置中考一模考前主题展板，需要搭建一个对称的抛物线形展架。以地面为x轴，展架的对称轴为y轴，建立平面直角坐标系。展架两端落地点为(-4，0)、(4，0)，最高点为(0，6)，单位为米。（1）求表示展架外轮廓的二次函数解析式；（2）若在离地3.75米的高度安装一条水平装饰条，求装饰条在展架内的最大长度；（3）若要在展架正中放入一个宽5米、高3.5米的矩形展示板，展示板底边在地面上，左右两边关于y轴对称，判断能否完全放入，并说明理由。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（本题14分）在平面直角坐标系中，抛物线与x轴交于A、B两点，与y轴交于C点，其中点B在点A的右侧。点P在抛物线第四象限部分上运动，设点P的横坐标为t，且。过点P作PQ∥y轴，交直线BC于点Q。（1）求点A、B、C的坐标及直线BC的解析式；（2）用含t的代数式表示线段PQ的长，并求△PBC面积的最大值；（3）若，求t的值及点P的坐标。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析评分说明：选择题每题3分，选对得3分，不选、多选或错选不得分；填空题每题3分，结果等价且书写清楚得3分；解答题按步骤给分，若方法正确但计算有小误差，可在相应步骤分内酌情扣分。一、选择题答案与解析1．A解析：，，所以原式=。评分：选A得3分。2．C解析：，两根为2和3，较大根为3。评分：选C得3分。3．A解析：函数已写成顶点式，顶点为(1，-4)，二次项系数为正，开口向上。评分：选A得3分。4．D解析：由点(2，-3)在图象上，得，所以。当时，。评分：选D得3分。5．C解析：同弧AC所对的圆周角是，圆心角是。圆心角等于同弧圆周角的2倍，所以。评分：选C得3分。6．B解析：共有种等可能结果，取出2个红球有种，概率为。评分：选B得3分。7．A解析：有两个不相等实数根需满足判别式大于0。，得。评分：选A得3分。8．C解析：AD∶AB=2∶5，且DE∥BC，所以△ADE∽△ABC，相似比为2∶5，面积比为。评分：选C得3分。9．B解析：由根为-1和3，可设。又过(0，3)，得，所以。抛物线为，顶点为(1，4)，最大值为4。评分：选B得3分。10．B解析：直线AB方程为。圆心O到直线AB的距离即切圆半径：。评分：选B得3分。二、填空题答案与解析11．4ab解析：。评分：写出4ab得3分。12．x≥1/2且x≠3解析：根号内需满足，分母需满足，所以且。评分：两个条件均正确得3分。13．5解析：点(1，3)在函数图象上，代入得，故。评分：写出5得3分。14．15π解析：圆锥侧面积公式为，所