2026届北京市朝阳区九年级数学中考三模模拟试卷（含答案详解与评分标准）

2026届北京市朝阳区九年级数学中考三模模拟试卷（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分试卷类型：中考三模模拟交卷前请复查答题过程注意事项：1．本试卷为九年级数学中考三模模拟检测卷，侧重考前综合诊断、阶段复习效果检测与关键能力查漏补缺。2．全卷共三大题、22小题，满分120分，考试时间120分钟。选择题10题，每题3分，共30分；填空题6题，每题3分，共18分；解答题6题，共72分。3．请将选择题答案填涂或填写在规定位置；填空题只写结果；解答题应写出必要的文字说明、推理过程、计算步骤和作图依据。4．作答时可使用黑色字迹签字笔、铅笔、直尺、圆规和三角板。答案中涉及近似值时，除题目另有要求外，可保留根号或分数形式。5．本卷中所给图形均为示意图，除题中特别说明外，图形不一定按比例绘制。一、选择题（本题共10小题，每小题3分，共30分）每小题只有一个选项符合题意。请把正确选项填在题后括号内。1．-2的相反数是（）。A．-2B．2C．1/2D．-1/22．据一次三模考前适应性训练统计，某区共有约14200名九年级学生参加数学模拟检测。将14200用科学记数法表示为（）。A．1.42×10³B．1.42×10⁴C．14.2×10³D．0.142×10⁵3．若二次根式√(x-2)有意义，则x的取值范围是（）。A．x＞2B．x≥2C．x≤2D．x≠24．点P(3，-4)关于x轴对称的点的坐标是（）。A．(-3，-4)B．(-3，4)C．(3，4)D．(4，3)5．一个不透明袋中装有除颜色外完全相同的3个红球和2个蓝球。从中任意摸出1个球，摸到红球的概率是（）。A．2/5B．3/5C．1/2D．3/26．方程x²-4x+3=0的两个根是（）。A．-1和-3B．1和3C．-1和3D．1和-37．如图形变换复习中，把△ABC绕点O顺时针旋转90°得到△A′B′C′。下列说法一定正确的是（）。A．AB=A′B′B．AB⊥A′B′C．∠ABC=90°D．点A与点A′关于x轴对称8．在⊙O中，弦AB所对的圆心角∠AOB=80°，点C在圆上且与点O位于弦AB的同侧。若∠ACB为弦AB所对的圆周角，则∠ACB的度数为（）。A．20°B．40°C．80°D．160°9．一次函数y=kx+b的图象经过第二、第一、第四象限，则k与b的符号分别是（）。A．k＞0，b＞0B．k＞0，b＜0C．k＜0，b＞0D．k＜0，b＜010．二次函数y=x²-2x-3在-1≤x≤4上的最大值是（）。A．-4B．0C．3D．5二、填空题（本题共6小题，每小题3分，共18分）请将答案直接填写在题中横线上。11．计算：√9+(-1)⁰=____________。12．分解因式：x²-9=____________。13．不等式2x-5＜1的解集是____________。14．若反比例函数y=k/x的图象经过点(2，-3)，则k=____________。15．一个等腰三角形有两条边长分别为4和9，则它的周长为____________。16．某学习小组5名同学一次数学限时训练成绩分别为82，85，87，90，a，若这组数据的平均数是87，则a=____________。三、解答题（本题共6小题，共72分）解答应写出必要的计算过程、推理依据或作图说明。17．（本小题10分）计算与化简。（1）计算：√12-2√3+|1-√3|。（2）先化简，再求值：【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（本小题10分）三模复习期间，某学习小组整理48套数学错题资料。甲同学每小时整理的套数比乙同学多2套，甲单独完成这项工作所用时间比乙单独完成少4小时。求甲、乙两名同学每小时分别整理多少套资料。要求：根据题意列出方程，写明未知数的意义，并检验所得结果是否符合实际。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（本小题12分）为了解九年级学生在中考三模前的数学复习方式，某校随机调查了200名学生，并将主要复习方式分为四类：A．错题整理；B．专题训练；C．限时模拟；D．同伴讲评。统计结果如下表。复习方式A．错题整理B．专题训练C．限时模拟D．同伴讲评人数5648m24所占百分比28%24%12%（1）求m的值，并求“C．限时模拟”在扇形统计图中所对应扇形的圆心角度数；（2）若北京市朝阳区某次同类三模模拟检测约有4200名九年级学生参加，请估计选择“C．限时模拟”为主要复习方式的学生人数；（3）现从2名选择A方式的学生、1名选择C方式的学生和1名选择D方式的学生中随机抽取2人交流复习经验，求抽到的2人中恰好有1人选择A方式、1人选择非A方式的概率。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（本小题12分）如图，在矩形ABCD中，AB=8，AD=6，点E在AB上，AE=2，连接CE，过点D作DF⊥CE，垂足为F。（1）求线段CE的长；（2）求线段DF的长；（3）若点G是线段CE上一点，且DG=DE，求CG的长。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（本小题14分）在平面直角坐标系xOy中，抛物线y=x²-2x-3与x轴交于A，B两点（点A在点B的左侧），与y轴交于点C。点D(4，5)在该抛物线上，直线CD与抛物线在点C，D处相交。（1）求A，B，C的坐标和抛物线的顶点坐标；（2）求直线CD的函数表达式；（3）点P(t，t²-2t-3)在C，D之间的抛物线上，过点P作PQ∥y轴，交直线CD于点Q。求线段PQ关于t的表达式，并求PQ的最大值；（4）点M在x轴上，若△CDM的面积等于△ABC的面积，求点M的坐标。【作答区】____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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参考答案与解析本答案按题号逐题对应，评分标准用于阅卷参考。客观题只给出唯一正确答案，主观题若有其他正确解法，可参照相应步骤和结论给分。一、选择题12345678910BBBCBBABCD1．答案B。相反数是只有符号不同、绝对值相等的数，-2的相反数为2。2．答案B。14200=1.42×10000=1.42×10⁴，且1≤1.42＜10。3．答案B。二次根式有意义要求被开方数非负，所以x-2≥0，得x≥2。4．答案C。关于x轴对称时横坐标不变，纵坐标变为相反数，所以P′(3，4)。5．答案B。袋中共有5个球，其中红球3个，任意摸出1个球为红球的概率为3/5。6．答案B。x²-4x+3=(x-1)(x-3)，所以方程的两个根为1和3。7．答案A。旋转是全等变换，能保持对应线段长度相等，因此AB=A′B′。其余说法不一定成立。8．答案B。同弧所对圆周角等于圆心角的一半，所以∠ACB=1/2∠AOB=40°。9．答案C。一次函数图象经过第一、二、四象限，说明函数随x增大而减小，故k＜0；与y轴交于正半轴，故b＞0。10．答案D。y=x²-2x-3=(x-1)²-4，在区间[-1，4]上开口向上，最大值在端点处取得。y(-1)=0，y(4)=5，所以最大值为5。评分标准：每题3分，共30分。选项正确给3分；未选、多选或错选不给分。二、填空题11．4。解析：√9=3，(-1)⁰=1，所以原式=3+1=4。12．(x+3)(x-3)。解析：利用平方差公式a²-b²=(a+b)(a-b)。13．x＜3。解析：2x-5＜1，移项得2x＜6，所以x＜3。14．-6。解析：将点(2，-3)代入y=k/x，得-3=k/2，所以k=-6。15．22。解析：若腰长为4，则4+4＜9，不能构成三角形；若腰长为9，则周长为9+9+4=22。16．91。解析：5个数据平均数为87，所以总和为435。已知四个数之和为344，故a=435-344=91。评分标准：每题3分，共18分。结果正确给3分；结果形式等价给满分；只写中间过程未写最终结果，视完成程度给1—2分。三、解答题17．答案与解析。（1）√12=2√3，且√3＞1，所以|1-√3|=√3-1。原式=2√3-2√3+√3-1=√3-1。（2）因为x²-1=(x-1)(x+1)，所以当x=2时，原式=2/(2+1)=2/3。评分标准：第（1）问5分，其中化简√12给1分，判断绝对值给2分，计算结果给2分；第（2）问5分，其中因式分解和通分给2分，正确约分给2分，代入求值给1分。18．答案与解析。设乙同学每小时整理x套资料，则甲同学每小时整理(x+2)套资料。根据“甲单独完成所用时间比乙少4小时”，得两边同乘x(x+2)，得48(x+2)-48x=4x(x+2)，即96=4x²+8x。整理得x²+2x-24=0。解得x=4或x=-6。由于每小时整理套数应为正数，所以x=4。甲同学每小时整理x+2=6套。检验：当x=4时，x(x+2)≠0，且48/4-48/