2026届河北省九年级数学中考三模模拟试卷（含答案详解与评分标准）

2026届九年级数学中考三模模拟试卷2026届河北省九年级数学中考三模模拟试卷（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分适用：2026届九年级节点：中考三模考前检测注意事项：1．本试卷用于九年级数学中考三模阶段复习检测，试题覆盖数与式、方程与不等式、函数、图形与几何、统计与概率及综合应用。2．请将选择题答案填入答题表，填空题直接写出最简结果；解答题应写出必要的文字说明、证明过程或演算步骤。3．作图、计算和证明应规范清楚；本卷满分120分，考试时间120分钟。题型选择题填空题解答题合计分值30分18分72分120分客观题作答表：题号12345678910答案得分一、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．2B．2.5C．3D．42．2026年三模质量分析中，某地九年级数学模拟参考人数约为69.8万人。将69.8万用科学记数法表示为（）A．6.98×10⁴B．6.98×10⁵C．0.698×10⁶D．69.8×10⁴3．若，则下列结论不一定成立的是（）A．a+2>b+2B．-3a<-3bC．a-b>0D．a²>b²4．一组用于三模错题统计的数据为2，3，5，x，8，若这组数据的中位数为5，则x的取值范围是（）A．x<3B．x=5C．x≥5D．3≤x≤55．不透明袋中有3个红球、2个白球，这些球除颜色外完全相同。从袋中不放回地随机摸出2个球，则至少摸到1个红球的概率是（）A．1/10B．3/5C．3/4D．9/106．一次函数的函数值随x的增大而减小，则m的取值范围是（）A．m<2B．m>2C．m≤2D．m≥27．若关于x的一元二次方程有两个相等的实数根，则c的值为（）A．-4B．4C．8D．168．在半径为6的圆O中，弦AB所对的圆心角为120°，则弦AB的长为（）A．6B．6√2C．6√3D．129．菱形ABCD的两条对角线AC、BD分别为8和6，则菱形ABCD的周长是（）A．14B．20C．24D．4810．二次函数在上的最大值是（）A．-4B．0C．5D．8二、填空题（本大题共6小题，每小题3分，共18分）题号111213141516答案11．计算：＝__________．12．设是方程的两个根，则＝__________．13．一个正多边形的每个内角都是144°，则这个正多边形的边数为__________．14．反比例函数的图象经过点，则当时，y＝__________．15．半径为6、圆心角为60°的扇形的弧长为__________．16．将抛物线向上平移m个单位后，所得抛物线的顶点恰好在x轴上，则m＝__________．三、解答题（本大题共6小题，共72分。解答应写出必要的计算、证明或说明过程）17．（本题10分）本题考查数式运算与分式化简，是三模考前基础能力检测。（1）计算：；（2）先化简，再求值：，其中．作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（本题12分）某校在九年级数学中考三模后，为了解学生对综合题的掌握情况，从全年级随机抽取40名学生的一次数学综合训练成绩（满分120分），按成绩x分成四组，统计如下表。组别成绩范围人数A80≤x<904B90≤x<1008C100≤x<11014D110≤x≤12014（1）求样本中成绩达到优秀（x≥110）的百分比，并估计全年级720名学生中达到优秀的人数；（2）求C组在扇形统计图中所对应扇形的圆心角度数；（3）若D组中有男生8人、女生6人，学校准备从D组随机抽取2名学生进行经验分享，求恰好抽到1名男生和1名女生的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（本题12分）如图形情境所述：在△ABC中，AB=AC=8，以AB为直径作圆O，圆O交BC于点D（点D不与点B重合），连接AD。已知∠C=30°。（1）求证：AD⊥BC，且D为BC的中点；（2）求AD和CD的长；（3）若以D为圆心、r为半径的圆与直线AC相切，求r的值。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（本题12分）某文具店为配合中考三模后的错题本整理活动，销售一种数学错题本。进价为50元/本，原售价为80元/本时，每周可销售200本。经调查，售价每降低2元，每周销量可增加20本。设每本降低2x元（x为整数），且0≤x≤15，每周销售利润为W元。（1）求W与x之间的函数关系式；（2）该文具店每周销售利润能否达到8300元？请说明理由；（3）求每周销售利润的最大值。若在获得最大利润的方案中优先提高销量，应确定每本售价为多少元？作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（本题12分）已知关于x的一元二次方程．（1）证明：无论m取何实数，该方程总有两个不相等的实数根；（2）设该方程的两个实数根为，若，求m的值；（3）若该方程的两个根均为正数，且m是不大于5的整数，求m的所有可能取值。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（本题14分）如图形情境所述：在平面直角坐标系中，抛物线与x轴交于A、B两点（A在B的左侧），与y轴交于点C，顶点为D。点P在抛物线上，横坐标为t，且．（1）求A、B、C、D四点的坐标；（2）连接PB、PC，求△PBC的面积S关于t的函数表达式，并求S的最大值；（3）当△PBC的面积为10时，求点P的坐标，并判断点P在抛物线对称轴的哪一侧；（4）过点P作PQ∥y轴交直线BC于点Q，是否存在点P，使PQ=PC？若存在，求点P的坐标；若不存在，请说明理由。作答区：_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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参考答案与解析说明：客观题每小题选对或填对得满分，选错、漏选、多选或结果不化简不得分；解答题按步骤给分，关键过程正确但个别计算误差不影响后续方法时，可根据评分标准酌情给后续方法分。一、选择题答案题号12345678910答案CBDCDABCBC1．C。2^{-1}=1/2，√9=3，|-1/2|=1/2，原式=3。2．B。69.8万=698000=6.98×10⁵，科学记数法中首数应满足1≤a<10。3．D。不等式两边同时加同一个数仍成立；同乘负数不等号方向改变；a-b>0成立；但a²>b²不一定成立，例如a=1，b=-2。4．C。五个数据按从小到大排列后的第三个数为中位数。要使中位数为5，x不能小于5，因此x≥5。5．D。至少1个红球的对立事件是