2025-2026学年高二数学高二下册期末模拟试卷（重庆专用版·考前适应卷含答案详解与评分标准）

2025-2026学年高二数学高二下册期末模拟试卷（重庆专用版·考前适应卷，含答案详解与评分标准）重庆专用版·考前适应卷学校班级姓名考号考试时间：120分钟满分：120分注意事项与答题要求：1.本卷共26题，选择题1—10题共30分，填空题11—16题共18分，解答题17—26题共72分。2.作答前请将学校、班级、姓名、考号填写清楚；选择题请在答题栏中填写选项，填空题请在横线上填写结果。3.解答题应写出必要的文字说明、计算过程或推理过程；结果应化简，图形关系要说明依据。4.全卷按高二下册期末考前适应要求设置，重在函数导数、数列、圆锥曲线、概率统计、空间向量等内容的综合运用。选择题答题栏（请将1—10题答案填写在相应题号下方）：题号12345678910答案一、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。1.函数f(x)=x³-3x+2在x=1处的导数值为（）。A.-3B.0C.3D.62.已知i为虚数单位，复数z=(1+i)²/(1-i)，则z等于（）。A.1+iB.-1+iC.1-iD.-1-i3.向量a=(2,-1)，b=(1,m)。若a⊥b，则实数m的值为（）。A.-2B.-1C.1D.24.双曲线x²/4-y²/5=1的离心率为（）。A.3/2B.√5/2C.√5/3D.2/35.某同学进行5次独立投篮训练，每次命中的概率均为0.4。设命中次数为X，则P(X=2)=（）。A.0.2304B.0.3456C.0.4000D.0.50006.等差数列{aₙ}中，a₁=3，公差d=2，若前n项和Sₙ=80，则n等于（）。A.6B.7C.8D.97.函数f(x)=x³-3x的单调递增区间是（）。A.(-1,1)B.(-∞,1)C.(-1,+∞)D.(-∞,-1)与(1,+∞)8.直线y=kx+1与抛物线y²=4x相切，则k的值为（）。A.-1B.0C.1D.29.某校高二年级调查50名学生参加数学社团情况：参加社团18人，其中女生8人；未参加社团32人，其中女生18人。从这50名学生中任取1人，已知取到的是女生，则其参加数学社团的概率为（）。A.4/13B.8/25C.9/25D.5/1310.若函数f(x)=lnx-ax（x>0）有唯一极大值，且该极大值为-1，则a的值为（）。A.1B.1/eC.eD.e²二、填空题：本题共6小题，每小题3分，共18分。11.函数f(x)=eˣcosx，则f′(0)=__________。12.过点P(2,1)作曲线y=x²的两条切线，则这两条切线斜率之和为__________。13.(1+2x)⁶展开式中x³的系数为__________。14.椭圆x²/9+y²/5=1的焦距为__________。15.随机变量X的分布列为P(X=0)=a，P(X=1)=2a，P(X=2)=3a，则E(X)=__________。16.若函数f(x)=x³-3ax在R上单调递增，则实数a的取值范围为__________。三、解答题：本题共10小题，共72分。解答应写出文字说明、证明过程或演算步骤。17.（6分）已知数列{aₙ}满足a₁=2，aₙ₊₁=aₙ+2n+1（n∈N*）。

（1）求aₙ的通项公式；

（2）求前n项和Sₙ。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（6分）重庆某校园气象小组调试3个独立温度传感器，每个传感器在一次检测中读数合格的概率均为0.8。设X为3个传感器中读数合格的个数。

（1）写出X的分布列；

（2）求P(X≥2)与E(X)。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（6分）已知函数f(x)=x³-3x²-9x+1。

（1）求f(x)的单调区间与极值；

（2）求曲线y=f(x)在x=0处的切线方程。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（7分）在空间直角坐标系中，A(0,0,0)，B(4,0,0)，C(0,3,0)，D(0,0,6)。

（1）求直线BD与平面ABC所成角的正弦值；

（2）求点A到平面BCD的距离。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

21.（7分）为分析高二学生期末复习时长与一次数学测评成绩的关系，某学习小组记录5名同学的数据如下：复习时长x（小时）12345测评成绩y（分）5863677280（1）求y关于x的线性回归方程ŷ=bx+a；

（2）预测复习6小时的测评成绩；

（3）求x=4时的残差。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（7分）已知椭圆C：x²/4+y²=1。

（1）求椭圆C的离心率；

（2）过点M(0,1)的直线l：y=kx+1与椭圆C相交于P、Q两点。若|PQ|=2，求k的值。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（8分）某劳动实践小组从6件同型号检测器材中随机抽取3件，其中4件合格、2件不合格。设抽到的合格件数为X。

（1）求X的分布列；

（2）若每抽到1件合格器材记10分，每抽到1件不合格器材记-5分，记总分为Y，求E(Y)；

（3）求总分不低于15分的概率。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（8分）已知函数fₐ(x)=eˣ-ax-1（x∈R）。

（1）当a=1时，证明f₁(x)≥0；

（2）若fₐ(x)≥0对任意实数x恒成立，求实数a的值。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________25.（8分）已知抛物线C：y²=4x，点A(1,2)在C上。直线l：y=m(x-1)+2过点A，且与C的另一个交点为B（m≠0，m≠1）。

（1）用m表示点B的坐标；

（2）若△OAB的面积为2，其中O为坐标原点，求m的值。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________26.（9分）已知函数f(x)=x³-3ax+2（a∈R）。

（1）当a=1时，求曲线y=f(x)在x=2处的切线方程；

（2）若f(x)在区间[-1,2]上单调递增，求a的取值范围；

（3）若方程x³-3ax+2=0有三个不同实根，求a的取值范围。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________26题续答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析一、选择题答案题号12345678910答案BBDABCDCAA1.【答案】B【解析】f′(x)=3x²-3，代入x=1得f′(1)=0。A、C、D多把函数值或导数公式代入错误。2.【答案】B【解析】(1+i)²=2i，z=2i/(1-i)=2i(1+i)/2=i+i²=-1+i。A、C、D主要错在没有乘以共轭复数。3.【答案】D【解析】a⊥b等价于a·b=0，即2×1+(-1)m=0，所以m=2。4.【答案】A【解析】双曲线中a²=4，b²=5，c²=a²+b²=9，故c=3，a=2，e=c/a=3/2。5.【答案】B【解析】X~B(5,0.4)，P(X=2)=C₅²×0.4²×0.6³=10×0.16×0.216=0.3456。6.【答案】C【解析】Sₙ=n/2[2a₁+(n-1)d]=n/2[6+2n-2]=n(n+2)。令n(n+2)=80，得n=8。7.【答案】D【解析】f′(x)=3x²-3=3(x-1)(x+1)。当x<-1或x>1时f′(x)>0，函数递增。8.【答案】C【解析】将y=kx+1代入y²=4x，得k²x²+(2k-4)x+1=0。相切时判别式为0，即(2k-4)²-4k²=0，解得k=1。9.【答案】A【解析】女生总人数为8+18=26，其中参加社团的女生为8，所以所求概率为8/26=4/13。10.【答案】A【解析】f′(x)=1/x-a。要有唯一极大值，需a>0，极值点为x=1/a。极大值f(1/a)=ln(1/a)-1。由ln(1/a)-1=-1得ln(1/a)=0，所以a=1。B把极大值条件误看成0。二、填空题答案11.【答案】1【解析】f′(x)=eˣcosx-eˣsinx=eˣ(cosx-sinx)，故f′(0)=1。12.【答案】8【解析】设切点为(t,t²)，切线方程为y=2tx-t²。过P(2,1)，得1=4t-t²，即t²-4t+1=0。两条切线斜率分别为2t₁、2t₂，其和为2(t₁+t₂)=8。13.【答案】160【解析】x³项系数为C₆³·2³=20×8=160。14.【答案】4【解析】a²=9，b²=5，c²=4，c=2，焦距为2c=4。15.【答案】4/3【解析】由a+2a+3a=1得a=1/6。E(X)=0·a+1·2a+2·3a=8a=4/3。16.【答案】a≤0【解析】f′(x)=3x²-3a=3(x²-a)。若在R上单调递增，需要f′(x)≥0对一切x成立。由于x²最小值为0，所以a≤0。三、解答题答案、解析与评分标准17.（6分）参考答案：（1）由aₙ₊₁-aₙ=2n+1，得aₙ=a₁+Σₖ₌₁ⁿ⁻¹(2k+1)=2+(n-1)n+(n-1)=n²+1。（2）Sₙ=Σₖ₌₁ⁿ(k²+1)=n(n+1)(2n+1)/6+n。评分标准：写出累加关系得2分，得到通项公式得2分，正确使用平方和公式并整理得2分。易错点是把递推中的2n+1当成固定公差。18.（6分）参考答案：X~B(3,0.8)，P(X=k)=C₃ᵏ·0.8ᵏ·0.2³⁻ᵏ，k=0，1，2，3。分布列为：X=0，P=0.008；X=1，P=0.096；X=2，P=0.384；X=3，P=0.512。P(X≥2)=0.384+0.512=0.896，E(X)=3×0.8=2.4。评分标准：识别二项分布得1分，四个概率各0.5分，分布列完整得1分，P(X≥2)得1分，期望得1分。19.（6分）参考答案：f′(x)=3x²-6x-9=3(x+1)(x-3)。当x<-1或x>3时f′(x)>0；当-1<x<3时f′(x)<0。故f(x)在(-∞,-1)和(3,+∞)上单调递增，在(-1,3)上单调递减。f(-1)=6，为极大值；f(3)=-26，为极小值。x=0时f(0)=1，f′(0)=-9，切线方程为y-1=-9x，即y=-9x+1。评分标准：导数正确得1分，单调区间得2分，极值各1分，切线方程得1分。易错点是把驻点与极值点混同而不判断导数符号。20.（7分）参考答案：（1）平面ABC为z=0。向量BD=D-B=(-4,0,6)，|BD|=√52=