2026年舟山高三数学高考三模冲刺卷：概率统计与实际应用建模（区县教研共同体第6套）含参考答案、逐题解析与评分细则

2026届舟山高三数学高考三模冲刺·概率统计与实际应用建模·区县教研共同体第6套数学试卷2026年舟山高三数学高考三模冲刺卷：概率统计与实际应用建模（区县教研共同体第6套）含参考答案、逐题解析与评分细则舟山/区县教研共同体2026届高三数学高考三模冲刺概率统计与实际应用建模专题强化卷（第6套）科目数学满分150分考试时间120分钟适用对象2026届高三三模冲刺试卷结构单项选择题8题、多项选择题4题、填空题4题、解答题6题命题重点概率统计、实际应用建模、函数导数、圆锥曲线、立体几何注意事项1.答卷前，考生务必将姓名、准考证号、学校或区县信息填写在答题卡指定位置。2.单项选择题每题只有一个正确选项；多项选择题每题至少有两个正确选项，全部选对得满分，部分选对且无错选得2分，有错选得0分。3.填空题只需填写最终结果；解答题必须写出必要的文字说明、推理过程和演算步骤，答案写在指定作答区。4.本卷重点考查概率统计与实际应用建模，同时覆盖函数与导数、圆锥曲线、立体几何、数列与三角等高考核心内容。一、单项选择题：本大题共8小题，每小题5分，共40分。每小题只有一个选项符合题意。1.舟山某校高三年级48名学生参加“海洋科普讲解”和“海洋数据整理”两项志愿活动，其中参加讲解活动的有28人，参加数据整理活动的有20人，两项都参加的有12人。从中随机抽取1名学生，抽到至少参加一项活动的学生的概率为A.13B.12C.32.已知集合A={x|log2x+1>A.(−1,3]B.(1,3]C.[0,1)D.[0,3]3.等比数列{an}的首项为2，公比为12，前n项和为A.3B.4C.5D.64.函数fx=xA.−1B.0C.1D.35.某校统计近几次冲刺复习中“专题微课投入时间x（小时）”与“对应训练正确题数y（题）”的关系，得到回归直线y=A.48.6B.52.6C.54.6D.56.66.在空间直角坐标系中，A(1,0,2)，B(3,−1,4)，C(0,2,1)，则向量AB与向量AC的数量积为A.−8B.−6C.0D.67.随机变量X服从二项分布B4,1A.116B.14C.58.椭圆x2A.13B.23C.5单项选择题答题栏12345678二、多项选择题：本大题共4小题，每小题5分，共20分。全部选对得5分，部分选对且无错选得2分，有错选得0分。9.已知随机变量X的数学期望EX=4，方差DA.EY=5B.D10.设fxA.f′x=1x−aB.当a≤11.双曲线C:x2a2−y2A.b:a=2:312.某区县教研共同体对高三学生“概率统计专题复习满意度”进行抽样调查并建立线性模型。下列说法正确的是A.当不同区县学生人数差异较大时，分层抽样通常比简单随机抽样更能保证样本结构B.样本相关系数r=0.92可直接说明复习投入是满意度提高的唯一原因C.带截距项的一元线性回归中，最小二乘残差之和为0D.只要样本容量足够大，就一定可以消除问卷设计造成的系统偏差多项选择题答题栏9101112三、填空题：本大题共4小题，每小题5分，共20分。13.某航线统计显示，随机一天出现7级以上大风的概率为0.30，既出现7级以上大风又发生班船延误的概率为0.12。已知当天出现7级以上大风，则发生班船延误的条件概率为__________。14.总体标准差为12，从该总体中抽取容量为36的简单随机样本，则样本均值的标准误为__________。15.曲线y=ex16.一个圆锥的底面半径为3，高为4，则其轴截面内切圆半径为__________。四、解答题：本大题共6小题，共70分。解答应写出文字说明、证明过程或演算步骤。17.（10分）在△ABC中，角A，B，C所对的边分别为a，b，c。已知sinA=2sinB（作答区）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（12分）舟山某区县教研共同体拟从8名学生中随机选3人组成“海岛研学数据展示组”。这8名学生中，来自定海的3人、来自普陀的3人、来自岱山的2人。设随机变量X表示所选3人来自的区县种数。（1）求PX=1与P（作答区）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（12分）如图形关系用文字描述：长方体ABCD−A₁B₁C₁D₁中，AB=2，AD=3，AA₁=2，M为棱CC₁的中点。以A为原点，AB，AD，AA₁所在直线分别为x轴、y轴、z轴建立空间直角坐标系。

（1）求直线A₁M与平面ABCD所成角的正弦值；

（2）求点D₁到平面A₁BM的距离；

（3）结合向量法说明本题中“图形位置不画图也可求解”的关键。（作答区）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（12分）某校高三数学组记录“概率统计专题训练周”中6个观测时段的相对时间t与学生平均建模得分y（分），数据如下表。约定t已中心化，回归直线记为y=t−3−2−1123y111315182124（1）求y关于t的最小二乘回归直线方程；（2）预测t=4时的平均建模得分；（3）若每提高1分需要配套0.8万元讲评与训练资源，固定组织成本为3万元，请估计（作答区）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（12分）已知椭圆E: x24+y（作答区）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（12分）某海岛观测站对一天内进港船舶是否需要二次核验进行建模。设抽检强度为x 0<x<1，每艘船被发现需要二次核验的概率为px=1−e（作答区）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________备用草稿区本区域不计分，可用于概率树图、回归计算、导数符号表、空间向量草算和答案复核。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

解答题附加作答页（一）用于第17—22题续答。请注明题号后作答。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

解答题附加作答页（二）用于第17—22题续答、检验或修正。请保持书写规范。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与详解本部分按题号逐题给出参考答案、关键解析与评分细则。选择题解析突出关键知识点与易错选项，解答题评分细则按主要步骤给分。一、参考答案速查表题号12345678答案CBCCCBCB题号910111213141516答案ABDABCDABDAC2/522