2026届武汉市九年级数学中考冲刺模拟试卷（含答案详解与评分标准）

2026届武汉市九年级数学中考冲刺模拟试卷（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：中考冲刺适用年级：九年级注意事项：1．本卷为武汉市九年级数学中考冲刺阶段综合检测卷；全卷共三大题22小题，满分120分，考试时间120分钟。2．答题前请将学校、班级、姓名、考号填写清楚；选择题请将答案填入作答栏，解答题写出必要的文字说明、计算过程或证明过程。3．可使用三角板、圆规、铅笔等常规作图工具；结果含根号时应化为最简形式，涉及实际问题时应写出符合题意的单位或范围。4．本卷兼顾基础运算、图形推理、统计概率、函数综合与实际应用，适合作为考前限时自测；答题时应先审清条件，再按题号顺序规范书写。答题提示：选择题重在快速判断与排除，填空题重在结果准确，解答题重在过程完整。涉及函数最值时需关注自变量范围，涉及几何证明时需先标明已知关系与要证结论，书写答案须规范。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分一、选择题（本大题共10小题，每小题3分，共30分）每小题给出的四个选项中，只有一项符合题目要求。请把正确选项填写在下表相应题号下。题号12345678910答案1．计算的结果是（）A．1B．−1C．7D．−72．2025年武汉市轨道交通全年客运量约为人次。把这个数还原成普通记数法为（）A．60200000B．602000000C．6020000000D．602000000003．不等式的解集是（）A．x≤1B．x≤4C．x≥4D．x≥14．在图形变换中，将一个图形绕某点旋转后与原图形重合，则这个图形关于该点成（）A．轴对称B．中心对称C．平移对称D．相似对称5．某冲刺小组5名同学一次限时训练得分为76，82，82，88，92，则这组数据的中位数和众数分别是（）A．82，82B．82，88C．84，82D．88，826．一个不透明袋中有3个红球和2个白球，这些球除颜色外完全相同。从袋中任取1个球，取到红球的概率是（）A．1/5B．2/5C．3/5D．4/57．一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8．在Rt△ABC中，∠C=90°，AC=6，BC=8，则sinA的值为（）A．3/5B．4/5C．3/4D．4/39．点A，B，C在同一圆上，若∠ACB=35°，则劣弧AB所对圆心角的度数为（）A．35°B．55°C．70°D．110°10．抛物线的顶点坐标和最大值分别是（）A．(1，5)，5B．(−1，5)，5C．(1，−5)，−5D．(−1，−5)，−5二、填空题（本大题共6小题，每小题3分，共18分）请把答案填写在题中横线处。11．分解因式：=__________________________。12．方程的解为__________________________。13．若一个正多边形的每个内角都是135°，则这个正多边形的边数为__________________________。14．某校九年级9名同学一次数学小测的平均分为80分，加入第10名同学的98分后，10名同学的平均分为__________________________分。15．若点在二次函数的图象上，则=__________________________。16．在Rt△ABC中，∠C=90°，∠A=30°，AB=12，则BC=__________________________。三、解答题（本大题共6小题，共72分）解答应写出文字说明、证明过程或计算步骤。17．（8分）先化简，再求值：，其中。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（10分）为了了解九年级学生在中考冲刺阶段整理数学错题的方式，某校随机抽取40名学生进行问卷调查。调查项目如下：A．只看订正；B．重做错题；C．归类整理；D．向同伴讲题。根据调查结果，制成下表。项目人数百分比扇形圆心角A1230%108°B1640%144°C820%72°D410%36°（1）补全表中数据的计算依据，并说明样本容量；（2）若该校九年级共有600名学生，估计选择“归类整理”的学生人数；（3）从选择C的8名学生中随机抽取2人作为经验分享代表，若其中有5名男生、3名女生，求抽到的2人恰好为1男1女的概率。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（10分）在平行四边形ABCD中，E，F分别是AB，CD的中点，连接DE，BF。（1）求证：四边形DEBF是平行四边形；（2）若AB=10，AD=6，∠A=60°，求四边形DEBF的周长。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（12分）某文具店销售一款中考冲刺错题本，每本进价20元。市场调查发现，当售价为30元时，月销售量为600本；售价每提高1元，月销售量减少20本。设每本售价为x元，月销售利润为y元。（1）写出y关于x的函数解析式，并指出自变量x的实际取值范围；（2）当售价为多少元时，月销售利润最大？最大利润是多少元？（3）若店主希望月销售量不少于500本，在此条件下最大月利润是多少元？【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（14分）AB是⊙O的直径，AB=10，点C在⊙O上，AC=6，BC=8。过点C作⊙O的切线，交直线AB于点D，且D在A的外侧。（1）求sin∠ABC的值；（2）求证：∠DCA=∠ABC；（3）求CD的长。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（18分）在平面直角坐标系中，抛物线经过A(−1，0)，B(3，0)，C(0，−3)三点。直线BC与抛物线围成的弓形区域用于一道函数综合题训练。（1）求该抛物线的解析式和顶点坐标；（2）设点P在抛物线的C，B两点之间运动，P的横坐标为t（0<t<3）。过点P作y轴的平行线交直线BC于点Q，求线段PQ的最大值及此时点P的坐标；（3）在（2）的条件下，求△PBC面积的最大值；（4）设点R在y轴上，四边形CPQR为平行四边形，写出R的坐标（用t表示），并说明当△PBC面积最大时，点R与点C的位置关系。【作答区】____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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参考答案与解析题号12345678910答案ABBBACCBCA分值3333333333一、选择题答案详解与评分标准选择题评分采用客观给分方式，每小题3分。下面给出的关键理由用于核对学生思路，也可作为讲评时定位知识点的依据。若学生只在作答栏写出正确选项，该小题仍按满分计。1．A按运算顺序，先算乘方与绝对值：，故结果为1。选A。本题检查有理数运算中乘方与绝对值的先后顺序。评分：选对得3分。2．B科学记数法表示小数点向右移动8位，得到602000000。选B。科学记数法还原时要注意指数8对应8位移动。评分：选对得3分。3．B由得，所以。选B。不等式两边同除以正数时方向不变。评分：选对得3分。4．B一个图形绕某点旋转180°后能与原图形重合，是中心对称图形的判定特征。选B。本题区分轴对称与中心对称两个概念。评分：选对得3分。5．A数据从小到大已经排列：76，82，82，88，92，中间位置的数据是82，出现次数最多的数据也是82。选A。中位数看位置，众数看次数，两者不能混用。评分：选对得3分。6．C共有5个等可能结果，其中红球3个，所以取到红球的概率为3/5。选C。概率模型要求每个球除颜色外完全相同，保证结果等可能。评分：选对得3分。7．C一次函数的斜率为负，截距为正，图象过第一、第二、第四象限，不经过第三象限。选C。也可分别观察与坐标轴的交点进行判断。评分：选对得3分。8．BRt△ABC中，AB为斜边，AB=10；对∠A而言，对边BC=8，故sinA=8/10=4/5。选B。锐角三角函数要先找准研究角，再确定对边与斜边。评分：选对得3分。9．C同弧所对圆心角等于圆周角的2倍，因此劣弧AB所对圆心角为70°。选C。若题干强调同一圆或同弧，通常优先考虑圆周角定理。评分：选对得3分。10．A抛物线已写成顶点式，顶点为(1，5)。开口向下，所以函数最大值为5。选A。顶点式中括号内符号与横坐标相反，这是常见失分点。评分：选对得3分。二、填空题答案详解与评分标准填空题每小题3分，要求结果准确、形式规范。若结果等价但未化为常用最简形式，应结合题目要求酌情给分；本卷给出的答案均为标准写法。11．答案：(x+3)(x−3)。解析：。评分：结果完全正确得3分；若写成两因式乘积但顺序不同，仍为正确。12．答案：x=2或x=3。解析：，所以方程的两个根为2，3。评分：两个解均正确得3分，只写对一个得1分；若只写因式分解式未写解，最高得2分。13．答案：8。解析：正多边形每个外角为45°，边数为360°÷45°=8。评分：答案正确得3分；若只求出外角45°未写边数，可得1分。14．答案：81.8。解析：9名同学总分为720分，加入98分后总分818分，平均分为81.8分。评分：列式正确得1分，结果正确得3分；若写成分数形式818/10，未化成小数也可得满分。15．答案：0。解析：把x=2代入二次函数，m=4−6+2=0。评分：答案正确得3分；若代入过程正确但计算成非零值，按计算失误给1分。16．答案：6。解析：30°角所对直角边等于斜边的一半，故BC=6。评分：答案正确得3分；若用正弦关系列式求得同一结果，也为正确。三、解答题答案详解与评分标准以下评分标准按主要思路和关键步骤给分。学生若采用其他正确方法，只要推理严密、计算正确、结论符合题意，可参照相应步骤给分；若前一步计算错误但后续方法正确，可根据后续过程的独立价值酌情给分。书写中出现单位、范围或结论缺失的，应结合该问考查重点进行扣分。17．（8分）解：，。所以原式。当时，原式。评分标准：正确展开两个乘法式各2分；合并同类项得2分；代入正确得1分；结果正确得1分。容易失分点是符号处理，特别是减去第二个整式时，括号内每一项都要变号。先化简