2026届深圳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届深圳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分题量：22题阶段：中考冲刺检测注意事项：1．本试卷用于2026届九年级数学中考冲刺阶段考前检测，满分120分，考试时间120分钟。2．答题前，请将学校、班级、姓名和考号填写清楚；客观题答案写在规定位置，主观题写出必要步骤。3．作图题可先用铅笔作图，确认后再加深；计算题结果需化为最简形式，并注意单位和取值范围。4．全卷分为选择题、填空题和解答题三部分。选择题10题共30分，填空题6题共18分，解答题6题共72分。题型选择题填空题解答题总分阅卷人分值301872120一、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，请将正确选项填在括号内。1．计算的结果是（）A．1B．2C．3D．72．国家游泳中心某检测仪可记录到米级别的微小位移，把这个数用科学记数法表示为（）A．7.2×10⁻⁶B．7.2×10⁻⁷C．0.72×10⁻⁶D．72×10⁻⁸3．点P（2，－3）关于y轴的对称点坐标是（）A．（2，3）B．（－2，3）C．（3，－2）D．（－2，－3）4．在△ABC中，∠A＝42°，∠B＝68°，则∠C的外角度数为（）A．70°B．90°C．100°D．110°5．同时抛掷两枚质地均匀的骰子，两个点数之和大于9的概率是（）A．1/9B．1/6C．1/4D．1/36．若，则化简的结果是（）A．a＋bB．a－bC．abD．a²＋b²7．抛物线的顶点坐标是（）A．（－2，－3）B．（2，－3）C．（2，1）D．（4，1）8．某小组7名同学一周完成数学限时训练的次数分别为8，6，9，10，7，8，9，则这组数据的中位数是（）A．7B．8C．9D．109．从圆外一点P作⊙O的两条切线PA、PB，A、B为切点，若∠APB＝50°，则∠AOB的度数为（）A．100°B．120°C．130°D．140°10．若△ABC∽△DEF，且对应边AB：DE＝2：3，则△ABC与△DEF的面积比为（）A．4：9B．2：3C．3：2D．9：4二、填空题（本大题共6小题，每小题3分，共18分）请把答案直接填写在题中横线上。11．分解因式：________________。12．不等式的解集是________________。13．一个多边形的内角和是900°，则这个多边形的边数是________________。14．一次函数图象经过点（1，2）和（3，8），则该一次函数表达式为________________。15．一个不透明袋中有3个红球和2个蓝球，这些球除颜色外完全相同。一次摸出2个球且不放回，两个都是红球的概率是________________。16．半径为6cm、圆心角为120°的扇形弧长是________________cm。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．（8分）完成下列计算与解不等式组。（1）计算：；（2）解不等式组：作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（10分）解下列方程或方程组，并写出检验过程。（1）解方程组：（2）解分式方程：。作答区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（10分）某校九年级数学备课组为了解学生中考冲刺阶段每日课外数学训练时间，随机抽取200名学生进行问卷调查，按训练时间分为A、B、C、D四组，统计结果如下表。组别训练时间t（分钟）人数A0＜t≤2040B20＜t≤4080C40＜t≤6060Dt＞6020（1）求C组的频率，并求D组在扇形统计图中对应圆心角的度数；（2）若该校九年级共有1200名学生，请估计每日课外数学训练时间超过40分钟的学生人数；（3）备课组从A组志愿者2名男生、1名女生和D组志愿者1名男生、2名女生中各随机抽取1名学生访谈，求抽到的2名学生恰好一男一女的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（12分）在Rt△ABC中，∠C＝90°，AC＝6，BC＝8。点E在AC上，且AE＝2，过点E作EF∥AB，交BC于点F。（1）证明△CEF∽△CAB；（2）求EF、CF的长；（3）求△CEF的面积。作答区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（14分）某校为中考冲刺复习印制数学专题资料。资料按套销售，若每套售价20元，每天可售100套；在此基础上，每降价1元，每天可多售10套。已知每套资料的成本为8元，设每套降价x元，且0≤x≤10，每天利润为W元。（1）求W关于x的函数表达式；（2）若某天利润为1200元，求当天每套资料的售价；（3）在“每天售出不少于120套”的条件下，求该校当天可获得的最大利润，并说明此时每套资料的售价。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（18分）如图形情境可由坐标条件确定：平面直角坐标系中，抛物线过点A（0，3）和B（3，0），且二次项系数为－1。设该抛物线与x轴的另一个交点为C，顶点为P。点D在第一象限内的抛物线上，点D的横坐标为t（0＜t＜3），过D作x轴的垂线，交直线AB于点E。（1）求抛物线的解析式，并写出点C、点P的坐标；（2）求△BCD面积的最大值及此时点D的坐标；（3）用含t的式子表示△ADE的面积；当△ADE面积为1时，求t的所有取值；并求△ADE面积的最大值。作答区：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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参考答案与解析说明：选择题每小题3分，填空题每小题3分；解答题按步骤给分，过程合理、结论正确可得相应分。题号12345678910答案CBDDBABBCA1．答案C。绝对值、算术平方根和乘方分别为3、4、4，故结果为3。2．答案B。把0.00000072的小数点向右移动7位得7.2，因此为7.2×10⁻⁷。3．答案D。关于y轴对称时横坐标变为相反数，纵坐标不变，所以为（－2，－3）。4．答案D。∠C＝180°－42°－68°＝70°，其外角为180°－70°＝110°。5．答案B。两枚骰子共有36种等可能结果，点数和大于9的有（4，6）、（5，5）、（6，4）、（5，6）、（6，5）、（6，6）共6种，概率为1/6。6．答案A。利用平方差公式，分子可化为（a－b）（a＋b），再约去非零因式a－b，得a＋b。7．答案B。配方得y＝（x－2）²－3，顶点坐标为（2，－3）。8．答案B。数据从小到大排列为6，7，8，8，9，9，10，第4个数为8。9．答案C。两条切线所成角与对应圆心角互补，∠AOB＝180°－50°＝130°。10．答案A。相似三角形面积比等于相似比的平方，所以面积比为2²：3²＝4：9。1．考查实数的混合运算。零指数、平方根、负数乘方常同时出现，关键是先分别求出每一项，再按加减顺序合并，不能把（－2）²误算为－4。2．考查科学记数法。小于1的正数写成a×10ⁿ时，1≤a＜10，指数n为负数，负指数的绝对值等于小数点向右移动的位数。3．考查平面直角坐标系中的轴对称。关于y轴对称时，只改变横坐标符号，纵坐标保持不变；关于x轴对称则相反。4．考查三角形内角和和外角性质。可先求内角∠C，也可直接用外角等于不相邻两内角和，得到42°＋68°＝110°。5．考查古典概型。列举时要把有序结果区分开，如（4，6）和（6，4）是不同结果；漏列会导致概率偏小。6．考查因式分解与分式化简。约分前必须说明a≠b，从而a－b不是0；只有在分母非零时才能约去公因式。7．考查二次函数配方和顶点。把一般式转化为顶点式时，常数项要同时补减，顶点横坐标也可用－b÷2a求得。8．考查统计量的意义。求中位数必须先排序，样本容量为7时取第4个数；不能直接观察原顺序中的第4项。9．考查切线性质。连接OA、OB后，OA⊥PA，OB⊥PB，四边形AOBP中两个直角与∠APB共同决定∠AOB。10．考查相似三角形面积比。边长比是线性比，面积比是平方比；把2：3直接写成面积比是常见失分点。二、填空题答案与解析答案：。解析：平方差公式。答案：。解析：移项得，所以解集为。答案：7。解析：设边数为n，则（n－2）×180°＝900°，解得n＝7。答案：。解析：斜率，代入（1，2）得b＝－1。答案：。解析：不放回摸球，两个都是红球的概率为。答案：。解析：弧长。11．考查平方差公式，两个平方项相减时优先观察是否能写成“平方差”的结构，分解后的两个因式应互为共轭形式。12．考查一元一次不等式。移项后若出现未知数系数为负，化为标准形式时不等号方向要随除以负数而改变；本题也可保持－8≤x的形式。13．考查多边形内角和公式。内角和比三角形多几个180°就对应多几条边，代公式时要使用n－2而不是n。14．考查待定系数法。先由两点求斜率，再代入任一点求截距；最后可用另一点回代检验表达式是否一致。1