2025-2026学年北京市朝阳区九年级数学中考二模模拟试卷（含答案详解与评分标准）

—PAGE1—2025-2026学年北京市朝阳区九年级数学中考二模模拟试卷（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分题型选择题填空题解答题合计分值10题×3分=30分6题×3分=18分6题=72分120分注意事项1．本试卷用于九年级数学中考二模阶段复习检测，重点考查基础知识、综合运用能力与规范表达能力。2．选择题请把正确选项填在答题卡相应位置；填空题只填写结果；解答题须写出必要的推理、计算过程。3．作图、证明和计算应书写清楚，答案区从题目后预留位置开始作答；参考答案与解析另起新页。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．下列各数中，最小的是（）A．B．C．D．2．某种微粒的直径约为米，将这个数用科学记数法表示为（）A．米B．米C．米D．米3．一个正多边形的每个外角都是，则这个正多边形的边数为（）A．6B．7C．8D．94．满足且的整数的个数是（）A．4B．5C．6D．75．一个不透明袋中装有3个红球和2个白球，这些球除颜色外完全相同。从袋中不放回地连续摸出2个球，则两个球都是红球的概率为（）A．B．C．D．6．将二次函数的图象向左平移1个单位，再向上平移4个单位，所得图象的顶点坐标是（）A．(3，1)B．(1，-7)C．(1，1)D．(2，1)7．在半径未知的圆中，弦，圆心到弦的距离为3，则这个圆的半径为（）A．4B．5C．6D．78．一组数据的平均数为5，则这组数据的中位数是（）A．4B．5C．6D．89．反比例函数的图象经过点，点也在该图象上，则的值为（）A．-6B．-3C．3D．610．关于二次函数，下列判断正确的是（）A．函数有最大值B．图象与轴交于和C．图象的对称轴是直线D．当时，随的增大而增大二、填空题（本大题共6小题，每小题3分，共18分）11．分解因式：__________。12．若是方程的两个根，则__________。13．在锐角三角形中，若，则__________。14．半径为6，圆心角为的扇形的弧长为__________。15．若一次函数的函数值随的增大而减小，则的取值范围是__________。16．在平面直角坐标系中，点，点，点在线段上，且，则点的坐标为__________。三、解答题（本大题共6小题，共72分。解答应写出文字说明、证明过程或演算步骤）17．（10分）计算与化简。（1）计算：；（2）先化简，再求值：，其中。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（10分）在矩形中，，，点是的中点。连接，过点作，垂足为。（1）求的长；（2）求点到直线的距离。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（12分）某校九年级为中考二模复习购买甲、乙两种专题资料。已知购买2本甲资料和3本乙资料共需84元，购买3本甲资料和2本乙资料共需91元。（1）求甲、乙两种资料每本的单价；（2）学校计划购买两种资料共100本，总费用不超过1800元，且甲资料的本数不少于乙资料本数的一半。设购买甲资料本，求的取值范围；（3）若每本甲资料可供3名学生专项训练，每本乙资料可供2名学生专项训练，在满足（2）的条件下，怎样购买可使可覆盖训练人数最多？最多可覆盖多少人？作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（12分）为了解九年级学生二模前计算能力专项训练情况，某校从全年级随机抽取40名学生，记录一次训练测试成绩（满分110分），并整理成下表。成绩x（分）70≤x<8080≤x<9090≤x<100100≤x≤110人数4101610（1）若全年级共有600名学生，估计成绩不低于90分的学生人数；（2）按各组中点估算这40名学生的平均成绩；（3）从样本中成绩不低于90分的学生里随机抽取2名参加解题展示，求恰好有1名学生来自“100≤x≤110”这一组的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（14分）在圆中，为直径，点在圆上，且，。过点作圆的切线，过点作，垂足为，连接。（1）求圆的半径；（2）证明：，并求的长；（3）求的长，并求的值。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（14分）在平面直角坐标系中，已知抛物线，直线。（1）求抛物线的顶点坐标和它与轴的交点坐标；（2）若直线经过抛物线与轴的交点，求的值，并求直线与抛物线的另一个交点坐标；（3）当时，设点在直线上，点在抛物线上，且。求线段长度的最大值；（4）若直线与抛物线有两个不同交点，求的取值范围。作答区：_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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