2026届湖北省高三数学高考冲刺模拟试卷（含答案详解与评分标准）

2026届湖北省高三数学高考冲刺模拟试卷2026届湖北省高三数学高考冲刺模拟试卷（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分适用对象2026届湖北省高三考试节点高考冲刺模拟注意事项：1．本试卷用于高考冲刺阶段综合检测，重点考查基础落实、主干方法、综合建模和规范表达。2．请将选择题答案填涂在答题卡相应位置；填空题和解答题写在规定区域，写出必要的文字说明、演算步骤或证明过程。3．全卷共22题，选择题10题30分，填空题6题18分，解答题6题102分，合计150分。一、选择题（本大题共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．已知复数，则等于A．B．C．D．2．设集合，，则为A．B．C．D．3．已知向量，，若，则实数的值为A．B．C．D．4．若，，则的值为A．B．C．D．5．函数在区间内既有极大值又有极小值，则的取值范围是A．B．C．D．6．袋中有4个红球和3个蓝球，除颜色外完全相同。从中不放回地任取2个球，则两球颜色不同的概率为A．B．C．D．7．数列满足，，则等于A．B．C．D．8．圆被直线截得的弦长为A．B．C．D．9．二项式的展开式中项的系数为A．B．C．D．10．若，且，则的最大值为A．B．C．D．二、填空题（本大题共6小题，每小题3分，共18分。）11．曲线在点处的切线方程为__________。12．等差数列中，，，则=__________。13．双曲线的一条渐近线斜率为，则其离心率为__________。14．随机变量，则=__________。15．方程的实数根个数为__________。16．已知，，与的夹角为，则=__________。三、解答题（本大题共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；（2）求时的最大值及取得最大值时的取值；（3）求方程在内的所有解。学生作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（17分）在冲刺阶段复习中，某数学小组整理出数列的递推模型：，。（1）设，证明数列为等比数列；（2）求及前项和；（3）求使成立的最小正整数。学生作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（17分）如图形关系所述，四棱锥的底面为边长为2的正方形，平面，且。（1）证明：平面；（2）求直线与底面所成角的正切值；（3）求三棱锥的体积。学生作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（17分）已知椭圆，其左右焦点分别为。（1）求椭圆的离心率；（2）直线与椭圆交于两点。求的面积关于的表达式；（3）若的面积为，求的所有取值。学生作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（17分）湖北省某校对高三学生进行一次高考冲刺模拟检测，随机抽取100名学生的数学成绩，按10分一组整理如下表：成绩区间[90,100)[100,110)[110,120)[120,130)[130,140)[140,150]频数61422282010（1）用组中值估计这100名学生的平均成绩；（2）以120分及以上记为“冲刺达标”，估计该校本次检测的冲刺达标率；（3）从成绩在的学生中按分层抽样抽取6人，再从这6人中随机选2人进行解题展示，求恰有1人来自的概率。学生作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（17分）已知函数，其中。（1）当时，求的单调区间，并判断方程的根的情况；（2）讨论方程在上的实数根个数；（3）当，设方程的两个根为，证明。学生作答区：_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________