2026届江苏省九年级数学一模能力诊断卷（含答案详解、评分标准与可打印作答区）

2026届江苏省九年级数学一模能力诊断卷（含答案详解、评分标准与可打印作答区）使用说明本资料用于阶段训练与自主复盘，建议先独立完成正文内容，再结合答案详解与评分标准完成二次订正。改稿重点1.正文抽取已掉成0页，先重建成稳定docx正文结构，避免封面图、表格壳或异常对象主导全文。2.补齐题面、答案、逐题解析、评分标准和学生作答区，避免正文过薄或交付断层导致系统处理失败。3.清理预览占位、空白页、只含封面目录的薄稿结构，确保正文首屏后立即进入实质内容。正文考试时间：120分钟满分：120分题号范围：1—23题适用对象：江苏省九年级一模模拟选择题1—8题，共24分填空题9—18题，共30分解答题19—23题，共66分答案页另起新页，可独立打印学段：初中范围：江苏省学科：数学聚焦：数学一模模拟核心考点第040组注意事项：1.本试卷为原创一模模拟试卷，满分120分，考试时间120分钟；请在规定时间内独立完成。2.选择题每小题只有一个正确选项；填空题只需写出最后结果，结果应化为最简。3.解答题应写出必要的计算过程、推理依据和结论；只写答案但无过程的，按评分标准酌情给分。4.需要作图或辅助线时，可在题旁空白处补充；作答区域的横线可用于书写过程。5.答题时注意单位、取值范围和图形位置关系，保持卷面整洁。选择题答题表：题号12345678答案一、选择题（本大题共8小题，每小题3分，共24分。每小题只有一个正确答案）1.计算−2+9−−12的结果是（）。A.3B.5C.4D.62.江苏某市一段河道水质监测中，某指标质量浓度为0.0000736克/升，用科学记数法表示为（）。A.7.36×10−6克/升B.7.36×10−5克/升C.0.736×10−4克/升D.73.6×10−6克/升3.若点P−2，m在一次函数y=3x+1的图像上，则m的值为（）。A.−5B.−4C.5D.74.在△ABC中，D、E分别在边AB、AC上，且DE∥BC，AD:DB=2:3，则△ADE与△ABC的面积比为（）。A.2:3B.2:5C.4:25D.9:255.一个不透明袋中有3个红球和2个白球，除颜色外完全相同。一次随机摸出2个球，摸出的2个球颜色相同的概率是（）。A.15B.25C.35D.456.抛物线y=−x2+4x−1的顶点坐标与最大值分别是（）。A.−2，−13，最大值−13B.2，−1，最大值−1C.−2，3，最大值3D.(2，3)，最大值37.半径为5的圆中，一条弦长为6，则圆心到这条弦的距离为（）。A.4B.3C.34D.278.关于x的一元二次方程x2−m+1x+m=0有两个不相等的实数根，则m应满足（）。A.m>1B.m<1C.m≠1D.任意实数二、填空题（本大题共10小题，每小题3分，共30分。请把答案填写在题中横线上）9.分解因式：x2−9=________________。10.不等式2x−5<x+1的解集是________________。11.若∠α的余角为36°，则∠α=________________。12.一组数据4，6，7，8，10的中位数是________________。13.反比例函数y=kx的图像经过点2，−3，则k=________________。14.半径为6、圆心角为60°的扇形面积是________________。15.平面直角坐标系中，A−1，2，B3，5，则AB=________________。16.方程x2−5x+6=0的两个根分别为x1、x2，则x12+x22=________________。17.边长为2的正六边形面积为________________。18.某学习小组用影长测量旗杆高度：同一时刻，长1.6m的标杆影长为2m，旗杆影长为15m，则旗杆高为________________m。三、解答题（本大题共5小题，共66分。解答应写出文字说明、证明过程或演算步骤）19.（10分）计算与解方程。本题考查二次根式化简、绝对值意义和分式方程的基本解法。请写出必要步骤，并注意分式方程的检验。（1）计算：12−32+1−3；（2）解分式方程：2x−1x+1−1=3x+1。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（10分）某校为了解九年级学生一模复习阶段每天自主学习数学的时长，随机抽取60名学生进行调查，按时长分为A、B、C、D四组，统计如下表。请根据样本数据完成统计推断，结果涉及人数时按最接近的整数给出。组别每天自主学习数学时长t人数频率A0≤t<30分钟60.10B30≤t<60分钟180.30C60≤t<90分钟24mDt≥90分钟120.20（1）求表中m的值；（2）若用扇形统计图表示上述数据，求C组所对应扇形的圆心角度数；（3）若该校九年级共有720名学生，估计每天自主学习数学时长不少于60分钟的学生人数；（4）从D组中随机选取4名代表，其中男生2名、女生2名。若再从这4名代表中随机选取2名交流学习方法，求恰好选到1名男生和1名女生的概率。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（12分）如图，在Rt△ABC中，∠C=90°，AC=6，BC=8，点D在AB上，且CD⊥AB；点E在AC上，CE=2，连接DE。本题中点D为斜边AB上的垂足，解题时可综合使用勾股定理、相似三角形和坐标法。（1）求AB与CD的长；（2）求证：△ACD∽△ABC，并求AD的长；（3）求tan∠CDE的值。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（16分）如图，抛物线y=−x2+bx+c与x轴交于A−1，0、B(3，0)两点，与y轴交于点C，顶点为D。图像仅用于辅助理解，计算时以题中点的坐标和函数关系为准。（1）求b、c的值，并写出抛物线的解析式；（2）求顶点D的坐标及对称轴；（3）点P在该抛物线第一象限部分上，设P的横坐标为t（0<t<3）。用含t的式子表示△PAB的面积S，并求S的最大值；（4）在抛物线的对称轴上是否存在点Q，使△QAB为直角三角形？若存在，求点Q的坐标；若不存在，请说明理由。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（18分）综合应用。如图，在矩形ABCD中，AB=12cm，BC=8cm。点P从A出发沿AB向B运动，速度为2cms；点Q从C出发沿CB向B运动，速度为1cms。两点同时出发，运动时间为ts（0<t≤6）。连接DP、PQ、AQ，设DP与AQ交于点R。运动过程中点P、Q均未超过相应边的端点，解题时应先明确t的取值范围，再建立函数或方程。（1）在以A为原点、AB所在直线为x轴、AD所在直线为y轴的平面直角坐标系中，写出P、Q的坐标，并用t表示DP2和PQ2；（2）求△DPQ的面积S关于t的函数关系式，并说明在0<t≤6内S的变化趋势；（3）当∠DPQ=90°时，求t的值和此时△DPQ的面积；（4）若R为AQ的中点，求t的值；并说明此时△DPQ的形状。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________参考答案与解析总评分原则：本卷满分120分。选择题按唯一正确选项给分；填空题只看最终结果，若结果等价且化简合理可给满分；解答题按步骤给分，关键结论正确但过程缺失时不超过该问分值的一半；运算中出现非关键性笔误且后续思路正确的，可在相应步骤内酌情扣分。涉及单位、取值范围、存在性讨论的问题，应写明单位或限制条件。一、选择题题号12345678答案CBACBDAC1.−2=2，9=3，−12=1，所以原式=2+3−1=4，选C。2.0.0000736=7.36×10−5，a×10n中1≤a<10，选B。3.将x=−2代入y=3x+1，得m=3×−2+1=−5，选A。本题体现函数图像上的点必须同时满足对应解析式。4.DE∥BC，△ADE∽△ABC，且AD:AB=2:2+3=2:5，面积比等于相似比平方，为4:25，选C。5.共C5,2=10种等可能结果；同色有C3,2+C2,2=3+1=4种，概率为410=25，选B。6.y=−x2+4x−1=−x−22+3，顶点为(2，3)，开口向下，最大值为3，选D。7.垂径定理：半弦长为3，圆心到弦距离d满足d2+32=52，得d=4，选A。8.判别式Δ=m+12−4m=m−12。两个不相等实数根要求Δ>0，故m≠1，选C。注意Δ=0时虽有实数根，但两个实数根相等，不符合题意。二、填空题题号答案9x+3x−310x<61154°12713−6146π1551613176318129.利用平方差公式：x2−9=x2−32=x+3x−3。10.2x−5<x+1，移项得x<6。11.互余的两个角和为90°，故∠α=90°−36°=54°。12.数据已按从小到大排列，中间一项为7，所以中位数为7。13.将2，−3代入y=kx，得−3=k2，故k=−6。14.扇形面积=60360×π×62=6π。15.AB=3+12+5−22=16+9=5。16.两根为2、3，或由根与系数关系得x1+x2=5，x1x2=6，所以x12+x22=x1+x22−2x1x2=25−12=13。17.正六边形可分成6个边长为2的等边三角形，面积为6×34×22=63。18.同一时刻物高与影长成正比，旗杆高h满足h15=1.62，得h=12。三、解答题19.参考解答与评分标准（10分）（1）12=23，所以12−32=32=3。又3>1，1−3=3−1。故原式=3+3−1=2+3。（2）原方程两边乘以x+1（x≠−1），得2x−1−x+1=3，即x−2=3，解得x=5。检验：当x=5时x+1≠0，所以x=5是原方程的解。评分点分值正确化简12并处理绝对值3分得出（1）的结果2+32分（2）正确去分母、求解并检验5分评分说明：第19题第（2）问若未写检验但求得x=5，最多扣1分；若去分母时未注明x≠−1但最后进行了检验，可不重复扣分。20.参考解答与评分标准（10分）（1）m=24÷60=0.40。（2）C组频率为0.40，圆心角为360°×0.40=144°。（3）不少于60分钟包括C、D两组，共24+1260=0.60，估计人数为720×0.60=432人。（4）设男生为M1、M2，女生为F1、F2。从4人中选2人的等可能结果有6种：M1M2、M1F1、M1F2、M2F1、M2F2、F1F2。其中恰好1男1女有4种