2027届江西中考数学暑期衔接开学摸底卷

2027届江西中考数学暑期衔接开学摸底卷（压轴题原创可编辑资料·PAGE原创模拟/可编辑WORD2027届江西中考数学暑期衔接开学摸底卷（压轴题分层·答案详解·评分标准）120分钟原创训练｜基础覆盖｜综合应用｜压轴题过程评分适用对象

2027届江西九年级学生、数学教师、家长与暑期衔接班交付内容

1套120分学生卷、答案详解、分步评分、压轴题诊断、错题复盘与二刷题使用时点

2026年暑假、新九年级开学与第一轮诊断文件属性

A4版式｜可编辑｜可打印｜含填写/作答与复盘模块下载后按“使用导览→主体内容→评分/验收→复盘改进”顺序使用本资料为原创训练/管理工具，不声称为官方文件；涉及具体考试、项目标准、平台规则或企业制度时，请以最新正式要求为准。

STARTHERE/使用导览从这里开始，10分钟完成使用部署。先确认使用对象、交付结果和分发方式，再进入正文。适用对象｜2027届江西九年级学生、数学教师、家长与暑期衔接班会得到什么｜1套120分学生卷、答案详解、分步评分、压轴题诊断、错题复盘与二刷题最短使用路径｜独立限时完成→按评分细则订正→48小时二刷并复盘打印与分发｜主体内容与答案/复盘页建议分开；空白表格可按需多份复印或转为在线表单。3步启动01｜独立限时完成02｜按评分细则订正03｜48小时二刷并复盘使用边界本资料为原创训练材料，不对应任何学校内部卷或官方预测；请结合所在地区最新考试说明、学校教学进度与学生基础调整。内容清单学生卷与作答区答案解析与评分细则学情诊断与错题复盘二刷/阶段补弱计划

第一部分学生卷2027届江西中考数学暑期衔接开学诊断卷适用范围：江西地区2027届原创训练考试时间：120分钟满分：120分学校：________________班级：________________姓名：________________考号：________________注意事项：1.本卷共26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；填空题只填写最终结果；解答题应写出必要的文字说明、推理过程或演算步骤。3.作图、证明、计算均应规范书写。结果含根式、分式或π时，在没有特别说明的情况下可以保留准确值。4.本卷用于暑期—开学阶段诊断，注重基础覆盖、综合运用和压轴题过程表达；答案与解析从新页开始。题型题号题量分值选择题1—1010题30分填空题11—166题18分解答题17—2610题72分一、选择题（本大题共10小题，每小题3分，共30分）下列各题的四个选项中，只有一个选项符合题意。请将正确选项填入答题栏。题号12345678910答案1.计算(-2)²-√16+|-3|的结果是（）A.-3B.0C.3D.112.2026年江西某地春季监测到一种花粉颗粒的平均直径约为0.00002026米，将0.00002026用科学记数法表示为（）A.2.026×10⁻⁶B.2.026×10⁻⁵C.20.26×10⁻⁶D.0.2026×10⁻⁴3.下列因式分解正确的是（）A.x²+4y²=(x+2y)²B.x²-4y²=(x+2y)(x-2y)C.x²-2x=x(x+2)D.x²-1=(x-1)²4.一组数据9，9，10，12，15的中位数和众数分别是（）A.10，9B.9，10C.10，10D.11，95.如图形情境所述，在△ABC中，DE∥BC，D在AB上，E在AC上，且AD:AB=2:5，则S△ADE:S△ABC为（）A.2:5B.4:25C.3:5D.6:256.不等式x²-3x+2>0的解集是（）A.x<1或x>2B.1<x<2C.x≤1或x≥2D.x≠1且x≠27.一个不透明袋中有3个红球、2个白球，这些球除颜色外完全相同。从中不放回地随机摸出2个球，两球都是红球的概率为（）A.1/5B.3/10C.2/5D.3/58.反比例函数y=k/x的图象经过点(-2，3)，则下列点在该函数图象上的是（）A.(2，3)B.(3，2)C.(1，-6)D.(-1，-6)9.半径为6的圆中，圆心角为120°的扇形面积是（）A.6πB.9πC.12πD.18π10.将抛物线y=(x-1)²+2向左平移2个单位，再向下平移3个单位，所得抛物线的解析式是（）A.y=(x+1)²-1B.y=(x-3)²-1C.y=(x+1)²+5D.y=(x-3)²+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上。11.因式分解：a²+2a+1=________。12.方程|x-2|=3的解为________。13.一次函数y=2x+b的图象经过点(1，-1)，则b=________。14.某商品打八折后的售价为80元，则该商品原价为________元。15.直角三角形的两条直角边长分别为6和8，则它的内切圆半径为________。16.二次函数y=x²-4x+1的最小值为________。三、解答题（本大题共10小题，共72分）解答应写出文字说明、证明过程或演算步骤。17.（6分）先化简，再求值：[(x-1)/(x+2)-(x+2)/(x-1)]÷[(x+1)²/(x²+x-2)]，其中x=3。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（6分）某校九年级开展“每天运动不少于半小时”暑期阶段健康监测，随机抽取40名学生某天运动时间，整理如下表。运动时间/分钟30456075人数614128（1）求这40名学生运动时间的平均数；（2）写出这组数据的中位数和众数；（3）若全校九年级有600名学生，估计当天运动时间不少于60分钟的学生人数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（6分）某文具店购进A、B两种中考复习笔记本。已知购进40本A种、30本B种共用690元；购进20本A种、50本B种共用730元。（1）求A、B两种笔记本的进价；（2）若该店计划再购进两种笔记本共100本，且B种不少于20本，总进价不超过1050元。A种按进价提高20%出售，B种按进价提高15%出售，怎样进货可使全部售出后的利润最大？最大利润是多少？作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（6分）如图形情境所述，矩形ABCD中，AB=8，BC=6，点E为BC的中点，连接AE、AC，过点C作CF⊥AE，垂足为F。（1）求AE的长；（2）求△ACE的面积；（3）求线段CF的长。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（7分）在平面直角坐标系中，一次函数y=x+2与反比例函数y=k/x的图象交于点A(2，4)和点B。（1）求k的值；（2）求点B的坐标；（3）根据图象，直接写出不等式x+2≥k/x的解集。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（7分）如图形情境所述，AB是圆O的直径，点C在圆O上，∠ABC=30°，AC=6。过点C作圆O的切线，与AB的延长线交于点D。（1）求圆O的半径；（2）判断点D位于AB的哪一侧延长线上，并求CD的长；（3）说明你的计算依据。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（8分）一个盒中有三张形状、大小完全相同的卡片，卡片上分别写有数字-1，1，2。从盒中随机抽取一张记为m，不放回，再随机抽取一张记为n，得到点P(m，n)。（1）列出所有可能的有序数对(m，n)；（2）求点P在直线y=x+2上的概率；（3）求mn<0的概率；（4）若把“点P在第二象限或第四象限”记为事件M，判断事件M与mn<0的关系。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（8分）在平面直角坐标系中，抛物线y=-1/4x²+bx+c经过点A(0，3)、B(4，3)，并与x轴正半轴交于点C。（1）求抛物线的解析式；（2）求抛物线的顶点坐标；（3）在y轴右侧的抛物线上是否存在点P，使S△OCP=12？若存在，求点P坐标；若不存在，说明理由。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________25.（10分）已知抛物线y=-x²+4x+5与x轴交于A、B两点（A在B的左侧），与y轴交于点C。点P(t，-t²+4t+5)在第一象限内的抛物线上，过点P作PQ∥y轴交直线BC于点Q。（1）求A、B、C三点坐标；（2）用含t的式子表示线段PQ的长；（3）求△PBC面积的最大值；（4）当S△PBC=15时，求点P的坐标。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________26.（8分）如图形情境所述，正方形ABCD的边长为6，点E在BC边上，点F在CD边上，BE=CF=t，且0<t<6。连接AE、BF，交于点P。（1）证明AE⊥BF；（2）用含t的式子表示AP:BP；（3）若AP:BP=2:1，求t的值、点P到AB和BC的距离，并求△ABP的面积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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，若学生方法不同但推理正确、结果准确，可参照相应采分点给分。1.答案：C。解析：(-2)²=4，√16=4，|-3|=3，所以4-4+3=3。A把平方或绝对值处理成负数，B漏掉绝对值项，D把减号误作加号。2.答案：B。解析：0.00002026的小数点向右移动5位得到2.026，所以0.00002026=2.026×10⁻⁵。C不是规范科学记数法，D虽数值相等但前面的数不在[1，10)范围。3.答案：B。解析：平方差公式为x²-(2y)²=(x+2y)(x-2y)。A把平方和误拆，C符号错误，D把x²-1误写为完全平方。4.答案：A。解析：按从小到大排列仍为9，9，10，12，15，第3个数为中位数10；出现次数最多的是9，所以众数为9。5.答案：B。解析：DE∥BC，△ADE∽△ABC，相似比AD:AB=2:5，面积比等于相似比的平方，即4:25。A把面积比误当长度比，D没有平方。6.答案：A。解析：x²-3x+2=(x-1)(x-2)，开口向上，大于0取两根外侧，所以x<1或x>2。B是小于0的解集，C把严格不等号写成含等号。7.答案：B。解析：不放回摸2个共有C(5,2)=10种组合，两球都是红球有C(3,2)=3种，概率为3/10。8.答案：C。解析：点(-2，3)代入得k=-6，函数为y=-6/x。点(1，-6)满足，其他选项代入均不满足。9.答案：C。解析：扇形面积=(120/360)×π×6²=12π。A把半径当直径，D把圆心角比例误算。10.答案：A。解析：y=(x-1)²+2左移2个单位得到y=(x+1)²+2，再下移3个单位得到y=(x+1)²-1。11.答案：(a+1)²。解析：a²+2a+1符合完全平方公式a²+2ab+b²=(a+b)²，其中b=1。12.答案：x=5或x=-1。解析：|x-2|=3表示x-2=3或x-2=-3，分别得x=5、x=-1。13.答案：-3。解析：把(1，-1)代入y=2x+b，得-1=2+b，所以b=-3。14.答案：100。解析：设原价为x元，八折即0.8x=80，解得x=100。15.答案：2。解析：斜边为√(6²+8²)=10，直角三角形内切圆半径r=(6+8-10)/2=2。16.答案：-3。解析：y=x²-4x+1=(x-2)²-3，二次项系数为正，所以最小值为-3。17.参考答案：原式=[((x-1)²-(x+2)²)/((x+2)(x-1))]÷[(x+1)²/((x+2)(x-1))]。因为(x-1)²-(x+2)²=[(x-1)-(x+2)][(x-1)+(x+2)]=-3(2x+1)，所以原式=-3(2x+1)/(x+1)²。把x=3代入，得原式=-3×7/16=-21/16。解析要点：先通分，再把除法转化为乘法，最后代入；x=3满足分式有意义条件。评分标准：正确写出公分母和通分结果2分；正确化简到-3(2x+1)/(x+1)²得2分；正确代入并得-21/16得1分；书写分式有意义条件或过程规范得1分。18.参考答案：（1）平均数=(30×6+45×14+60×12+75×8)/40=2130/40=53.25（分钟）。（2）把数据按从小到大排列，第20个数为45，第21个数为60，所以中位数为(45+60)/2=52.5（分钟）；出现次数最多的是45分钟，众数为45分钟。（3）样本中不少于60分钟的有12+8=20人，占20/40=1/2，估计全校为600×1/2=300人。解析要点：平均数要按人数加权；中位数因样本容量为偶数，取第20和第21个数据的平均。评分标准：平均数列式和结果2分；中位数、众数各1分；估计人数列式和结果2分。19.参考答案：（1）设A种进价为x元/本，B种进价为y元/本。由题意得40x+30y=690，20x+50y=730，解得x=9，y=11。故A种9元/本，B种11元/本。（2）设购进A种m本，则B种100-m本。条件为100-m≥20，9m+11(100-m)≤1050，得m≤80且m≥25。利润W=20%×9m+15%×11(100-m)=1.8m+1.65(100-m)=165+0.15m。因为W随m增大而增大，所以m取最大值80，B种为20本，最大利润W=165+12=177元。解析要点：方程组求单价，方案优化需同时使用数量限制和费用限制。评分标准：设未知数并列出方程组2分；解得两种进价1分；建立m的限制条件1分；写出利润函数并判断增减1分；给出方案和最大利润1分。20.参考答案：E为BC中点，BC=6，所以BE=EC=3。在Rt△ABE中，AB=8，BE=3，AE=√(8²+3²)=√73。（2）以CE为底，CE=3，A到直线BC的距离为AB=8，所以S△ACE=1/2×3×8=12。（3）因为CF⊥AE，且△ACE也可看成以AE为底、CF为高，所以12=1/2×√73×CF，得CF=24/√73，也可写为24√73/73。解析要点：矩形中相邻边垂直，中点给出CE长度；同一三角形面积的两种表示是求高的关键。评分标准：求出BE=CE=3得1分；AE=√73得2分；面积12得1分；由面积等式求出CF得2分。21.参考答案：（1）把A(2，4)代入y=k/x，得4=k/2，所以k=8。（2）交点满足x+2=8/x，即x²+2x-8=0，解得x=2或x=-4。x=2对应A，另一个交点B的横坐标为-4，