2026届湖北省高三数学高考二模模拟试卷（含答案详解与评分标准）

2026届湖北省高三数学高考二模模拟试卷（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：150分题型单项选择题填空题解答题合计题量10题6题6题22题分值30分18分102分150分注意事项：本卷用于高考二模阶段复习检测。答题前请填写学校、班级、姓名和考号；选择题用规定符号填涂；填空题只写结果；解答题应写出必要的演算过程、推理依据和结论。选择题答案栏：题号12345678910答案填空题答案栏：题号111213141516答案一、单项选择题（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。）1．已知集合，，则等于（）。A．B．C．D．2．复数，则等于（）。A．B．C．D．3．二项式的展开式中，的系数为（）。A．B．C．D．4．已知单位向量与的夹角为，则等于（）。A．B．C．D．5．在等比数列中，，，且公比为正数，则等于（）。A．B．C．D．6．函数在处取得极小值，则该极小值为（）。A．B．C．D．7．一个盒中有3个红球、2个黑球，从中不放回地任取2个球，则取出的2个球颜色相同的概率为（）。A．B．C．D．8．正方体的棱长为2，则直线与平面所成角的正弦值为（）。A．B．C．D．9．椭圆的离心率为（）。A．B．C．D．10．设函数，方程有三个互不相同的实根，则实数的取值范围为（）。A．B．C．D．二、填空题（本题共6小题，每小题3分，共18分。请将结果填写在答题栏中。）11．函数的最小正周期为________。12．抛物线在点处的切线方程为________。13．二项式的展开式中，的系数为________。14．若一组二模阶段训练成绩为82，86，88，90，94，则这组数据的平均数为________。15．经过点，，的圆的方程为________。16．已知数列满足，，则=________。三、解答题（本题共6小题，共102分。解答应写出文字说明、证明过程或演算步骤。）17．（17分）已知函数。

（1）将化为的形式，其中，；

（2）求在区间上的最大值、最小值及取得相应值时的；

（3）求方程在区间内的所有解。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（17分）如图意设正方体的棱长为2，点为棱的中点。

（1）证明：；

（2）求直线与平面所成角的正弦值；

（3）若点在线段上，且，求使成立的。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（17分）某校高三年级在高考二模阶段复习检测后，从同一层次班级中随机抽取40名学生的数学成绩，按分数段整理如下表：分数段[70,80)[80,90)[90,100)[100,110)[110,120]人数4101484（1）用各组中点估计这40名学生的平均成绩与方差；

（2）以样本频率估计总体概率，若从该层次学生中独立抽取3人，记其中成绩不低于100分的人数为，求；

（3）从样本中成绩不低于100分的学生里任选2人参加复习经验交流，求至少有1人成绩在内的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（17分）已知函数，定义域为。

（1）当时，求的单调区间和最大值；

（2）若对任意恒成立，求实数的值；

（3）证明：对任意且，都有。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（17分）已知椭圆。

（1）求椭圆的焦点坐标和离心率；

（2）过点的非竖直直线与椭圆交于两点，求弦长关于的表达式；

（3）在所有过点的直线中，求使弦长最大的直线方程及最大弦长。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（17分）已知数列满足，。

（1）设，证明为等差数列，并求；

（2）求前项和；

（3）证明：对任意正整数，都有。作答区：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________