2026届南京市九年级数学中考压轴冲刺原创仿真模拟试卷（含答案详解与评分标准）第889套

第页2026届南京市九年级数学中考压轴冲刺原创仿真模拟试卷（含答案详解与评分标准）第889套考试卷头试卷名称2026届南京市九年级数学中考压轴冲刺原创仿真模拟试卷（含答案详解与评分标准）第889套考试节点2026届南京市九年级数学中考压轴冲刺限时综合训练适用范围南京市九年级学生中考考前综合复习、薄弱题型修复与规范作答训练考试时间100分钟满分120分注意事项：1.本卷为九年级数学中考压轴冲刺原创仿真模拟试卷，题目难度按基础、综合、压轴梯度设置，满分120分。2.答题前请填写姓名、班级和准考证号；选择题用规定符号填涂，非选择题写出必要的文字说明、证明过程或演算步骤。3.作图题可先用铅笔作图，确认后用黑色签字笔描清；计算结果需化简，含单位的结果应写明单位。4.解答题请在对应位置作答，答题过程应条理清楚；只写答案而无关键过程的题目，按评分标准酌情给分。答题要求：本卷共23题。选择题每题只有一个正确选项；填空题只填写最终结果；解答题需写出主要推理依据和计算步骤。题型题号每题分值小计选择题1-62分12分填空题7-162分20分解答题17-23见题号标注88分合计1-23—120分一、选择题（本大题共6小题，每小题2分，共12分。每小题给出的四个选项中，只有一项是符合题意的）1.（2分）计算(-2)^2+√9-|-5|的结果是（）A.-4B.2C.6D.122.（2分）若x=3是方程ax-2=7的解，则a的值为（）A.1B.2C.3D.43.（2分）等腰三角形两腰均为5，底边为6，则该三角形的面积为（）A.10B.12C.15D.184.（2分）一组数据2，3，4，4，7的平均数、中位数、众数中，数值相等的是（）A.只有平均数和中位数B.只有平均数和众数C.只有中位数和众数D.三者都相等5.（2分）在圆O中，弦AB所对的圆心角∠AOB=80°，点C在优弧AB上，则∠ACB的度数为（）A.20°B.40°C.80°D.100°6.（2分）抛物线y=(x-1)^2+2的顶点坐标与最小值分别为（）A.(1，2)，2B.(-1，2)，2C.(1，-2)，-2D.(2，1)，1二、填空题（本大题共10小题，每小题2分，共20分。请把答案填写在题中横线上）7.（2分）分解因式：x^2-9=____________。8.（2分）不等式2x-1≤5的解集是____________。9.（2分）在Rt△ABC中，∠C=90°，若tanA=3/4，则cosA=____________。10.（2分）袋中有3个红球、2个白球、1个黄球，这些球除颜色外完全相同，任取1个球，取到红球的概率为____________。11.（2分）一个矩形面积为48，长比宽多2，则该矩形的周长为____________。12.（2分）数列a_n=n^2-n+1，则a_10=____________。13.（2分）若x+y=7，xy=10，则x^2+y^2=____________。14.（2分）半径为6、圆心角为60°的扇形弧长为____________。15.（2分）两个相似三角形的相似比为2:3，较大三角形面积为27，则较小三角形面积为____________。16.（2分）直线y=-x+6与两坐标轴围成的三角形的内切圆半径为____________。三、解答题（本大题共7小题，共88分。解答应写出必要的文字说明、证明过程或演算步骤）17.（8分）（1）化简：[(a^2-4)/(a^2+4a+4)]÷[(a-2)/(a+2)]，并写出a的取值范围；

（2）解方程：x^2-4x-5=0。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（8分）某校九年级进行一次“定点投篮”训练，随机抽取20名学生的成绩，记分如下表。

分数：678910

人数：24752

（1）求这20名学生成绩的平均数、中位数和众数；

（2）若从成绩不低于9分的学生中随机抽取2人参加展示，求抽到的2人都为10分的概率。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（10分）在平行四边形ABCD中，E、F分别为BC、AD的中点，连接AE、CF。

（1）求证：四边形AECF是平行四边形；

（2）若AB=6，AD=8，∠BAD=60°，求四边形AECF的面积。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（10分）某公园路灯杆AB垂直于水平地面，B为灯杆底端。小明在同一直线上的C、D两点观测灯杆顶端A，C点较靠近B，测得∠ACB=60°，∠ADB=30°，且CD=20m。

（1）求灯杆AB的高度；

（2）若在C点放置一块高度为1.6m的竖直提示牌，灯光照射下其在地面上的影长为多少米？（结果可保留根号）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（12分）在平面直角坐标系中，一次函数l经过A(0，4)、B(2，0)，抛物线y=x^2-2x与l交于点B和点P。

（1）求一次函数l的表达式；

（2）求点P的坐标和△POB的面积；

（3）设Q(t，-2t+4)是线段AB上一点，过Q作y轴的平行线交抛物线于R(t，t^2-2t)，其中0≤t≤2，求线段QR的最大值。____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（18分）在平面直角坐标系中，圆O的圆心为原点，半径为5，A(-5，0)、B(5，0)，点C(3，4)在圆O上。过C作圆O的切线，交x轴于点T；过B作x轴的垂线，交该切线于点E。

（1）求AC、BC的长，并说明∠ACB=90°；

（2）求切线CT的表达式和点T的坐标；

（3）求证：EC=EB；

（4）求四边形OCEB的面积。____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（22分）在平面直角坐标系中，抛物线Γ：y=-x^2+6x与x轴交于O(0，0)、B两点，顶点为A。点P(t，-t^2+6t)为第一象限内抛物线上的动点，过P作PQ⊥x轴，垂足为Q。

（1）求点B、A的坐标及抛物线Γ的对称轴；

（2）若矩形OQPR的顶点R在y轴上，求该矩形面积的最大值及此时点P的坐标；

（3）当3<t<6时，直线AB与PQ交于点M，求PM的最大值；

（4）是否存在点P，使OP⊥AB？若存在，求t的值及△OPB的面积；若不存在，请说明理由。____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析（含评分标准）评分说明：本评分标准按中考数学规范作答要求设置。解答题中若学生采用其他正确方法，且逻辑清楚、计算准确，可参照相应步骤给分；若过程正确但计算失误，按关键步骤酌情扣分；若只写最终答案而缺少必要推理，按题目要求扣除过程分。一、选择题答案123456BCBDBA1.答案：B。解析：(-2)^2=4，√9=3，|-5|=5，所以4+3-5=2。评分标准：选对得2分。2.答案：C。解析：把x=3代入ax-2=7，得3a-2=7，3a=9，a=3。评分标准：选对得2分。3.答案：B。解析：等腰三角形底边为6，则底边上的高把底边平分为3和3，高h=√(5^2-3^2)=4，面积S=1/2×6×4=12。评分标准：选对得2分。4.答案：D。解析：平均数为(2+3+4+4+7)÷5=4；按从小到大排列后中位数为4；出现次数最多的是4，众数为4，故三者都相等。评分标准：选对得2分。5.答案：B。解析：同弧所对圆周角等于圆心角的一半。∠ACB=1/2∠AOB=40°。评分标准：选对得2分。6.答案：A。解析：y=(x-1)^2+2为顶点式，顶点为(1，2)，且二次项系数为正，最小值为2。评分标准：选对得2分。二、填空题答案78910111213141516(x+3)(x-3)x≤34/51/22891292π126-3√27.答案：(x+3)(x-3)。解析：平方差公式：x^2-9=x^2-3^2=(x+3)(x-3)。评分标准：填对得2分；化简等价形式正确得满分。8.答案：x≤3。解析：2x-1≤5，移项得2x≤6，故x≤3。评分标准：填对得2分；化简等价形式正确得满分。9.答案：4/5。解析：tanA=对边/邻边=3/4，可设两直角边为3k、4k，斜边为5k，所以cosA=4/5。评分标准：填对得2分；化简等价形式正确得满分。10.答案：1/2。解析：共有6个球，其中红球3个，所求概率为3/6=1/2。评分标准：填对得2分；化简等价形式正确得满分。11.答案：28。解析：设宽为x，则长为x+2，x(x+2)=48，得x=6，长为8，周长为2(6+8)=28。评分标准：填对得2分；化简等价形式正确得满分。12.答案：91。解析：a_10=10^2-10+1=91。评分标准：填对得2分；化简等价形式正确得满分。13.答案：29。解析：x^2+y^2=(x+y)^2-2xy=7^2-2×10=29。评分标准：填对得2分；化简等价形式正确得满分。14.答案：2π。解析：弧长l=(60/360)×2π×6=2π。评分标准：填对得2分；化简等价形式正确得满分。15.答案：12。解析：面积比等于相似比的平方，即4:9。较小三角形面积为27×4/9=12。评分标准：填对得2分；化简等价形式正确得满分。16.答案：6-3√2。解析：直线与两坐标轴围成直角等腰三角形，直角边均为6，斜边为6√2。内切圆半径r=面积/半周长=18/(6+3√2)=6-3√2。评分标准：填对得2分；化简等价形式正确得满分。

三、解答题答案与评分标准17.（1）a≠-2且a≠2。原式=[(a-2)(a+2)/(a+2)^2]×[(a+2)/(a-2)]=1。（2）x^2-4x-5=0，分解为(x-5)(x+1)=0，所以x=5或x=-1。评分标准：第（1）问4分，其中写出限制条件2分，正确化简2分；第（2）问4分，其中正确分解或使用公式2分，求出两个根2分。18.（1）平均数为(6×2+7×4+8×7+9×5+10×2)/20=161/20=8.05。按从小到大排列，第10个和第11个数据都为8，所以中位数为8；出现次数最多的是8，所以众数为8。（2）成绩不低于9分的学生有5+2=7人，其中10分学生2人。从7人中任取2人共有C(7,2)=21种等可能结果，2人都为10分只有C(2,2)=1种，概率为1/21。评分标准：第（1）问5分，平均数2分，中位数1.5分，众数1.5分；第（2）问3分，列出总情况2分，概率1分。19.（1）因为四边形ABCD是平行四边形，所以AD∥BC，AD=BC。又E、F分别为BC、AD的中点，所以EC=1/2BC，AF=1/2AD，且EC∥AF，从而EC=AF且EC∥AF。四边形AECF有一组对边平行且相等，因此四边形AECF是平行四边形。（2）平行四边形ABCD的面积为AB·AD·sin60°=6×8×(√3/2)=24√3。由（1）可知AECF的一组邻边可看作AF与AE，其中AF=1/2AD，且它的面积等于原平行四边形ABCD面积的一半。因此S_AECF=12√3。评分标准：第（1）问5分，写出中点关系2分，得到平行且相等2分，结论1分；第（2）问5分，求原平行四边