2025-2026学年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分适用范围：九年级中考一模阶段定位：考前检测注意事项：1．本试卷共三大题，22小题，满分120分，考试时间120分钟。请在规定位置填写学校、班级、姓名和考号。2．选择题每题只有一个正确选项；填空题只写最终结果；解答题应写出必要的文字说明、计算过程或推理过程。3．本卷用于中考一模前的阶段复习检测，重点考查基础知识、核心方法、综合运用与规范表达。4．作图题可先用铅笔画图，确认后再描清；所有结果应符合题目条件，并注意单位与取值范围。题型选择题填空题解答题分值10题×3分=30分6题×3分=18分6题共72分一、选择题（本大题共10小题，每小题3分，共30分）1．若，则的值是（）A．-2B．2C．10D．-102．将用科学记数法表示，正确的是（）A．3.45×10⁻⁶B．3.45×10⁻⁷C．34.5×10⁻⁸D．0.345×10⁻⁶3．不等式组的解集是（）A．x<4B．x≥-2C．-2≤x<4D．-2<x≤44．一次函数的图象经过第一、三、四象限，则的取值范围是（）A．k<1B．1<k<2C．k=2D．k>25．关于的一元二次方程有两个相等的实数根，则的值是（）A．-4B．0C．4D．86．在△ABC中，AB=AC，∠A=40°，则∠B的度数为（）A．40°B．60°C．70°D．80°7．PA为⊙O的切线，A为切点，OA=5，OP=13，则PA的长为（）A．8B．10C．12D．138．一个不透明袋中有2个红球、3个白球、4个黑球，这些球除颜色外完全相同。随机摸出1个球，摸到“不是黑球”的概率是（）A．2/9B．1/3C．4/9D．5/99．点P(x₁，y₁)，Q(x₂，y₂)都在反比例函数的图象上，且x₁<0<x₂，则下列关系正确的是（）A．y₁>y₂B．y₁=y₂C．y₁<y₂D．无法确定10．关于二次函数，下列说法正确的是（）A．对称轴是直线x=-2B．函数的最小值为1C．当x>2时，y随x增大而增大D．图象与y轴交于点(0，-1)二、填空题（本大题共6小题，每小题3分，共18分）11．计算的结果为__________．12．若是方程的一个根，则的值为__________．13．矩形ABCD中，AB=8，BC=6，则它的外接圆半径为__________．14．一组数据9，10，10，11，13，a的平均数为11，则这组数据的中位数为__________．15．某同学在操场上测得一棵树的影长为12m，太阳光线与地面所成角为30°，则树高为__________m．16．点P(t，t²-2t-3)在第四象限，则的取值范围是__________．三、解答题（本大题共6小题，共72分）17．（本题10分）计算与化简。（1）计算：（2）先化简，再从-1，0，1，2中选择一个合适的数代入求值：作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（本题10分）中考一模后，学校为九年级学生购买甲、乙两种专项突破练习册。甲种练习册每本18元，乙种练习册每本24元，第一次共购买80本，总费用为1740元。（1）第一次购买甲、乙两种练习册各多少本？（2）第二次购买时，甲种练习册按九折销售，乙种练习册按八五折销售。学校计划共购买90本，甲种不少于30本，乙种不少于40本，总预算不超过1700元。在满足条件的情况下，乙种练习册最多可购买多少本？作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（本题12分）如图，在△ABC中，AB=AC=10，BC=12，D为BC的中点，点E在AC上，AE=4。过点E作EF∥BC，交AD于点F。（1）证明：AD⊥BC。（2）求EF的长，并求△AEF的面积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（本题12分）为了解九年级学生在中考一模前数学复习中的主要错题来源，某校随机调查了100名学生。调查结果分为“选择题”“填空题”“解答题”“综合题”四类，其中“综合题”在扇形统计图中所占圆心角为108°，其余数据如下表。错题来源选择题填空题解答题综合题人数18m32n（1）求m，n的值。（2）若该校九年级共有480名学生，请估计主要错题来源为“解答题”或“综合题”的学生人数。（3）从2名“解答题”代表和2名“综合题”代表中随机选取2名学生参加经验交流，求恰好选到1名“解答题”代表和1名“综合题”代表的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（本题14分）如图，一次函数的图象为直线l，与反比例函数的图象交于点A(2，a)和点B。直线l的表达式为：反比例函数的表达式为：（1）求a，k的值，并写出点B的坐标。（2）求△AOB的面积。（3）点P是x轴上一动点，求PA+PB的最小值及此时点P的坐标。（4）结合图象，直接写出不等式x+2>k/x的解集。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（本题14分）如图，抛物线经过A(-1，0)，B(3，0)，C(0，-3)三点，点D为抛物线顶点。（1）求抛物线的表达式，并写出顶点D的坐标。（2）点P在抛物线上，且点P的横坐标为t，其中0<t<3。求△PBC面积的最大值，并求此时点P的坐标。（3）在抛物线的对称轴上是否存在点M，使△MAC为等腰三角形？若存在，请写出所有满足条件的点M的坐标；若不存在，请说明理由。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析一、选择题答案与关键理由题号12345678910答案ABCDCCCDCC分值33333333331．a²-2b=4-6=-2，故选A。2．小数点向右移动7位得到3.45，所以原数为3.45×10⁻⁷，故选B。3．由2x-1<7得x<4，由x+3≥1得x≥-2，合并得-2≤x<4，故选C。4．图象经过第一、三、四象限，需要斜率为正且截距为负，即k-1>0且2-k<0，得k>2，故选D。5．两个相等的实数根要求判别式为0，即16-4c=0，得c=4，故选C。6．AB=AC，所以∠B=∠C；∠B=(180°-40°)÷2=70°，故选C。7．PA为切线，OA⊥PA，在Rt△OAP中，PA=√(13²-5²)=12，故选C。8．总球数9个，不是黑球的有5个，概率为5/9，故选D。9．反比例函数y=6/x中，当x<0时y<0，当x>0时y>0，所以y₁<y₂，故选C。10．y=x²-4x+1=(x-2)²-3，对称轴x=2；当x>2时，函数随x增大而增大，故选C。二、填空题答案与解析题号111213141516答案11510.54√30<t<3分值33333311．√12=2√3，所以(√12-√3)/√3=(2√3-√3)/√3=1。12．把x=1代入2x²+mx-3=0，得2+m-3=0，m=1。13．矩形对角线长为√(8²+6²)=10，外接圆半径等于对角线的一半，为5。14．由平均数为11可得9+10+10+11+13+a=66，所以a=13；按从小到大排列为9，10，10，11，13，13，中位数为(10+11)/2=10.5。15．设树高为h，则tan30°=h/12，h=12×√3/3=4√3。16．第四象限要求t>0且t²-2t-3<0；由(t-3)(t+1)<0得-1<t<3，合并得0<t<3。三、解答题答案详解与评分标准17．（10分）（1）原式=3+1-2×1/2+4=3+1-1+4=7。评分标准：绝对值、零次幂、特殊角三角函数值、算术平方根各1分；合并结果1分，共5分。（2）原式=原式中x≠1，x≠-2，因此可取x=0。代入得(0-2)/(0-1)=2。评分标准：正确因式分解2分；正确约分1分；说明不能取的数1分；代入正确并求值1分，共5分。18．（10分）（1）设第一次购买甲种练习册x本，乙种练习册y本。由题意得：解得x=30，y=50。答：第一次购买甲种练习册30本，乙种练习册50本。（2）设第二次购买乙种练习册b本，则甲种练习册为90-b本。由题意得40≤b≤60，且即1458+4.2b≤1700，解得b≤57+13/21。因为b为整数，所以b的最大值为57。答：乙种练习册最多可购买57本。评分标准：第（1）问设未知数1分，列方程组2分，解得数量并作答2分；第（2）问设元并表示甲种数量1分，列不等式2分，结合整数与条件得到最大值2分，共10分。19．（12分）（1）因为AB=AC，D为BC的中点，所以AD是等腰三角形底边上的中线。等