2026届高二数学高二下册期中模拟试卷（重庆专用版·原创命题C卷含答案详解与评分标准）

2026届高二数学高二下册期中模拟试卷（重庆专用版·原创命题C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分题号范围：1—26范围：重庆专用版注意事项与答题要求：1.本卷为高二下册期中阶段测评卷，满分120分，考试时间120分钟；请在规定位置填写学校、班级、姓名和考号。2.选择题共10小题，每题只有一个选项符合题意；填空题只填写最终结果；解答题应写出必要的文字说明、运算步骤和结论。3.本卷知识覆盖导数及其应用、计数原理、二项式定理、概率与统计等内容，题目难度由基础到综合递进。4.书写应规范清晰，作图或列表辅助时应标明关键量；不得在试题区写出与答案无关的涂改信息。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.已知函数f(x)=x³-3x²+2，则f′(1)的值为（）。A.-6B.-3C.0D.32.5名同学排成一排，其中甲、乙两人不相邻的不同排法共有（）。A.48种B.60种C.72种D.96种3.二项式(1+2x)^5的展开式中x³项的系数为（）。A.40B.60C.80D.1604.随机变量X的分布列为P(X=0)=0.2，P(X=1)=0.5，P(X=2)=0.3，则E(X)=（）。A.0.8B.1.0C.1.1D.1.35.已知f(x)=lnx-ax（x>0）在x=2处取得极大值，则实数a的值为（）。A.1/4B.1/2C.1D.26.连续投掷一枚质地均匀的骰子3次，至少出现一次“6点”的概率为（）。A.1/216B.5/36C.91/216D.125/2167.若C(n,0)+C(n,1)+…+C(n,n)=64，则C(n,2)=（）。A.10B.15C.21D.288.函数f(x)=x³-3ax在x=-1处有极值，则a的值为（）。A.-1B.0C.1D.39.方程x1+x2+x3+x4=8的非负整数解中，满足x1≥2的解的个数为（）。A.56B.70C.84D.12010.关于x的方程x³-3x+m=0有三个互不相同的实数根，则实数m的取值范围是（）。A.m<-2B.m>2C.-2<m<2D.m=±2选择题答题栏12345678910二、填空题（本大题共6小题，每小题3分，共18分。请将答案填写在题中横线上。）11.函数y=xlnx（x>0）的导数y′=________。12.若C(n,2)=28，则正整数n=________。13.某盒中有8件合格品和2件次品，从中任取3件，取出的3件全为合格品的概率为________。14.展开式(x+1/x)^6中x²项的系数为________。15.若X服从正态分布N(70,5²)，且P(-1<Z<1)=0.6826，则P(65<X<75)=________。16.函数f(x)=x³-3x²+1的单调递减区间为________。填空题答题栏111213141516三、解答题（本大题共10小题，共72分。解答应写出必要的文字说明、证明过程或演算步骤。）17.（7分）已知函数f(x)=x³-3x²+2x。

（1）求f′(x)；

（2）求曲线y=f(x)在点x=1处的切线方程；

（3）求函数f(x)的单调区间。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（7分）重庆某校进行班级展示，6名代表A、B、C、D、E、F需排成一排上台。若A与B不能相邻，且C不能站在两端，求满足条件的不同排法数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（7分）某学习小组共有12名成员，其中男生7名、女生5名。现从中随机选出3名同学参加数学阅读交流，设选出的女生人数为X。

（1）写出X的分布列；

（2）求E(X)；

（3）在已知至少选出1名女生的条件下，求恰好选出2名女生的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（7分）某校高二年级抽取40名学生，调查“坚持整理错题”与“期中模拟达标”之间的关系，得到如下列联表。请根据表中数据作答。学生类型达标未达标合计坚持整理错题20525未坚持整理错题6915合计261440（1）分别估计两类学生的达标率；（2）依据K²=n(ad-bc)²/[(a+b)(c+d)(a+c)(b+d)]，判断在0.05水平下两变量是否有关联。临界值：K²≥3.841可认为在0.05水平下有关联，K²≥6.635可认为在0.01水平下有关联。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（7分）已知二项式(x-2/x)^6的展开式。

（1）求含x²的项；

（2）求常数项；

（3）求展开式各项系数之和。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（7分）某印刷店制作一张宣传页，正文印刷区域面积固定为600cm²，上、下边距各5cm，左、右边距各3cm。设正文区域宽为xcm（x>0）。

（1）用x表示整张宣传页面积S(x)；

（2）求整张宣传页面积的最小值，并给出此时正文区域的宽。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（7分）已知函数f(x)=lnx-ax（x>0）。

（1）当a=1/2时，求f(x)的单调区间和极值；

（2）若f(x)≤0对任意x>0恒成立，求实数a的取值范围。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（7分）某批螺钉的单件合格概率为0.9，各件是否合格相互独立。随机抽检4件，设合格件数为X。

（1）写出X的分布列，并求P(X≥3)；

（2）若每件合格记收益2元，每件不合格记损失3元，记4件抽检的收益为Y，求E(Y)。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________25.（8分）已知函数f(x)=x³-3ax+2（a为实数）。

（1）当a=1时，求f(x)的极值；

（2）讨论方程f(x)=0的实数根个数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________26.（8分）已知函数F_a(x)=x-alnx（x>0）。

（1）当a=2时，求F_a(x)的最小值；

（2）若对任意x>0，恒有x-alnx≥1，求实数a的值；

（3）利用（2）的结论证明：对任意x>0，lnx≤x-1，并指出等号成立条件。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析本部分覆盖第1—26题，给出参考答案、关键步骤和评分标准。阅卷时应依据过程完整性、运算准确性和结论规范性综合给分。一、选择题答案与解析12345678910BCCCBCBCCC1.选B。f′(x)=3x²-6x，代入x=1得f′(1)=-3。A项把二次项求导符号处理错误，C项漏写了-6x。2.选C。5人全排列为5!=120种；甲、乙相邻时可把二人看成一个整体，有2×4!=48种，所以不相邻为120-48=72种。A项只算了相邻数，D项多减或少减了内部顺序。3.选C。展开式通项为C(5,k)(2x)^k，令k=3，系数为C(5,3)×2³=80。B项少乘一个2，D项把次数误作4。4.选C。E(X)=0×0.2+1×0.5+2×0.3=1.1。A、B项常见于只取最大概率值或漏掉X=2的贡献。5.选B。f′(x)=1/x-a，极值点x=2满足f′(2)=0，因此a=1/2；且f′由正变负，确为极大值。6.选C。至少一次出现6点的对立事件是3次都不是6点，概率为(5/6)³=125/216，所以所求为1-125/216=91/216。D项是对立事件。7.选B。由二项式系数和2ⁿ=64，得n=6，所以C(n,2)=C(6,2)=15。C项对应n=7的情形。8.选C。f′(x)=3x²-3a，x=-1为极值点需f′(-1)=0，得a=1；此时f″(-1)=-6≠0，极值存在。9.选C。令y1=x1-2≥0，则y1+x2+x3+x4=6。非负整数解个数为C(6+4-1,4-1)=C(9,3)=84。10.选C。函数g(x)=x³-3x+m，g′(x)=3x²-3，极大值点为x=-1，极小值点为x=1。三个不同实根需g(-1)=m+2>0且g(1)=m-2<0，故-2<m<2。二、填空题答案与解析11.答案：lnx+1。利用乘积求导法则，(xlnx)′=1·lnx+x·(1/x)=lnx+1。12.答案：8。C(n,2)=n(n-1)/2=28，得n(n-1)=56，正整数解n=8。13.答案：7/15。从10件中任取3件的总数为C(10,3)，全为合格品的取法为C(8,3)，故概率为56/120=7/15。14.答案：15。(x+1/x)^6的通项为C(6,k)x^(6-2k)，令6-2k=2，得k=2，系数C(6,2)=15。15.答案：0.6826。标准化Z=(X-70)/5，65<X<75等价于-1<Z<1，故概率为0.6826。16.答案：(0,2)。f′(x)=3x²-6x=3x(x-2)，当0<x<2时f′(x)<0，所以函数单调递减区间为(0,2)。三、解答题答案、解析与评分标准17.参考答案：（1）f′(x)=3x²-6x+2。（2）f(1)=0，f′(1)=-1，切线方程为y-0=-(x-1)，即y=-x+1。（3）令f′(x)=0，得3x²-6x+2=0，x=1±√3/3。由于二次项系数为正，f′(x)>0在(-∞,1-√3/3)与(1+√3/3,+∞)上成立，f′(x)<0在(1-√3/3,1+√3/3)上成立。评分标准：求导正确2分；切点函数值与斜率各1分，切线方程1分；临界点1分，单调区间1分。18.参考答案：先不考虑A、B是否相邻，只要求C不在两端。6人全排列6!种，C在两端有2×5!种，所以C不在两端共有6!-2×5!=480种。再计算其中A、B相邻且C不在两端的排法：把A、B看成一个整体，连同C、D、E、F共5个对象，排列并考虑A、B内部顺序有2×5!=240种；其中C在两端时为2×(2×4!)=96种，所以A、B相邻且C不在两端为240-96=144种。所求排法数为480-144=336。评分标准：正确计算C不在两端2分；正确使用捆绑法2分；正确扣除C在两端的相邻情形1分；得出336并表述完整2分。19.参考答案：总取法数为C(12,3)=220。X可取0,1,2,3。P(X=0)=C(5,0)C(7,3)/220=35/220=7/44；P(X=1)=C(5,1)C(7,2)/220=105/220=21/44；P(X=2)=C(5,2)C(7,1)/220=70/220=7/22；P(X=3)=C(5,3)C(7,0)/220=10/220=1/22。因此E(X)=0×7/44+1×21/44+2×7/22+3×1/22=5/4。也可用超几何分布期望E(X)=3×5/12=5/4。在至少选出1名女生的条件下，P(X=2|X≥1)=P(X=2)/P(X≥1)=(70/220)/(185/220)=14/37。评分标准：列出取值1分；分布列每两项1分，共2分；期望2分；条件概率表达与结果各1分。20.参考答案：列联表可记为：坚持整理错题且达标a=20，坚持整理错题且未达标b=5；未坚持整理错题且达标c=6，未坚持整理错题且未达标d=9，n=40。（1）坚持整理错题学生达标率为20/(20+5)=80%；未坚持整理错题学生达标率为6/(6+9)=40%。（2）K²=40×(20×9-5×6)²/[(25)(15)(26)(14)]。其中20×9-5×6=150，故K²=900000/136500≈6.59。因为6.59>3.841，所以在0.05水平下可认为“坚持整理错题”与“期中模拟达标”有关联；又6.59<6.635，尚不能在0.01水平下作出同样判断。评分标准：达标率各1分；公式代