2025-2026学年高二数学高二下册期中模拟试卷（辽宁专用版·学生练习版含答案详解与评分标准）

第页2025-2026学年高二数学高二下册期中模拟试卷（辽宁专用版·学生练习版，含答案详解与评分标准）辽宁专用版高二数学高二下册期中模拟试卷考试时间：120分钟满分：120分学校：__________班级：__________姓名：__________考号：__________注意事项：1.本卷共26题，满分120分。选择题、填空题请按要求填写答案；解答题应写出必要的文字说明、运算步骤和结论。2.答题时保持卷面整洁，作图题可用铅笔先作辅助线，再用黑色签字笔确认主要结论。3.计算结果如含根式、对数或分数，请化为最简形式；概率结果可用分数表示。4.本卷按阶段测评要求命制，范围标签为辽宁专用版，适合作为高二下册期中前学生练习。考查范围提示：本卷突出导数及其应用、数列、计数原理、二项式定理、概率分布与统计图表等内容，兼顾基础运算、模型建立、分类讨论与综合推理。答题要求提示：选择题应先独立判断再填入答案栏；填空题写最终结果；解答题应在空白处分步书写，关键公式、代入过程和结论需完整呈现。书写规范提示：涉及概率、组合数、导数符号与数列求和时，请保留必要中间式，便于阅卷时判断思路。题型题号题量每题分值小计选择题1—10103分30分填空题11—1663分18分解答题17—2610见题注72分合计1—2626120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.（3分）已知函数f(x)=x³-3x²+2，则f'(2)=（）A.-2B.-1C.0D.22.（3分）二项式(1+2x)⁵的展开式中x²的系数为（）A.20B.30C.32D.403.（3分）等差数列{a_n}中，a₃=5，a₈=20，则前10项和S₁₀=（）A.120B.125C.130D.1354.（3分）同时掷两枚质地均匀的骰子，已知至少有一枚出现6点，则两枚点数和大于8的概率为（）A.5/11B.6/11C.7/11D.8/115.（3分）函数f(x)=lnx+x在x=1处的切线方程为（）A.y=2x-1B.y=xC.y=2x+1D.y=x+16.（3分）数列{a_n}满足a₁=2，a_{n+1}=a_n+2n，则a₅=（）A.12B.16C.20D.227.（3分）将字母A，A，B，C，D排成一列，要求两个A不相邻，不同排法共有（）A.24B.36C.48D.608.（3分）随机变量X的分布为P(X=2)=0.2，P(X=4)=0.5，P(X=6)=0.3，则D(X)=（）A.1.60B.1.80C.1.96D.2.049.（3分）函数f(x)=x³-3x+a的单调递减区间为（）A.(-∞,-1)B.(-1,1)C.(1,+∞)D.(-∞,+∞)10.（3分）从格点A(0,0)沿最短路径到B(5,3)，每步只能向右或向上，且不经过P(2,1)，不同路径条数为（）A.20B.24C.26D.30选择题答案栏：题号12345678910答案二、填空题（本大题共6小题，每小题3分，共18分）11.（3分）函数f(x)=x²lnx在x=1处的导数值为__________。12.（3分）若C(n,2)=28，则正整数n=__________。13.（3分）等比数列{b_n}中，b₁=3，公比q=2，则S₅=__________。14.（3分）(2x-1/x)⁶的展开式中的常数项为__________。15.（3分）甲、乙两人各射击一次，命中率分别为0.6和0.5，且两人射击相互独立，则恰有一人命中的概率为__________。16.（3分）函数f(x)=x³-3x+1在区间[-2,2]上的最大值为__________。三、解答题（本大题共10小题，共72分。解答应写出文字说明、运算过程和结论）17.（6分）已知函数f(x)=x³-3x²+4。

（1）求曲线y=f(x)在点x=1处的切线方程；

（2）求函数f(x)的单调区间及极值。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（6分）已知等差数列{a_n}满足a₂+a₅=16，a₄=10。

（1）求数列{a_n}的通项公式；

（2）求使前n项和S_n>180的最小正整数n。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（7分）完成下列两个计数问题。

（1）求(x²-1/x)⁶的展开式中x³的系数；

（2）从5本不同的数学书和4本不同的物理书中选出4本，要求至少选2本数学书，求不同选法数。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（7分）一个不透明盒中有3个红球、2个蓝球、1个白球，这6个球除颜色外完全相同。一次从盒中不放回地随机取出2个球，设X为取到的红球个数。

（1）写出X的分布列；

（2）求E(X)与D(X)。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（7分）某文创社团制作一批纪念徽章，单件售价为p元时，预计日销售量q=120-4p（10≤p≤25），单件成本为8元。设日利润为L(p)元。

（1）写出L(p)的表达式；

（2）利用导数求日利润最大时的售价，并求最大日利润。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（8分）已知函数f(x)=x³-3x²+ax+1。

（1）当a=0时，求f(x)的单调区间和极值；

（2）若f(x)在区间[1,3]上单调递增，求实数a的取值范围。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（8分）数列{a_n}满足a₁=1，a_{n+1}=2a_n+3（n∈N*）。

（1）证明数列{a_n+3}为等比数列；

（2）求a_n的通项公式；

（3）求前n项和S_n。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（8分）某校高二学生参加一次数学素养测评，随机抽取50名学生的成绩，得到如下分组统计表。成绩分组[60,70)[70,80)[80,90)[90,100]人数5152010（1）用各组中点值估计这50名学生成绩的平均数；（2）从成绩不低于80分的学生中随机抽取3人，设其中成绩不低于90分的人数为Y，求P(Y=1)；（3）若按成绩分组采用分层抽样方法从50人中抽取10人，则成绩在[90,100]的学生应抽取多少人？________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________25.（7分）已知函数f(x)=lnx-mx+1（x>0）。

（1）当m=1时，求曲线y=f(x)在点x=1处的切线方程；

（2）若对任意x>0，恒有f(x)≤0，求实数m的取值范围。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________26.（8分）已知函数f(x)=x³-3x+m。

（1）当m=1时，求f(x)的极大值和极小值；

（2）讨论方程x³-3x+m=0在区间[0,2]内有两个不同实根时m的取值范围。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析评分标准总则：选择题、填空题按最终答案给分；解答题按关键步骤给分，方法正确但表达不完整时酌情给步骤分，计算结果与过程不一致时以过程合理性为主要依据。一、选择题答案与解析题号12345678910答案CDBCADBCBC1.C。f'(x)=3x²-6x，代入x=2得f'(2)=12-12=0。A、B、D均未按导数公式代入。2.D。x²项来自C(5,2)(2x)²，系数为C(5,2)·4=10×4=40。易误把2x中的2遗漏而得10。3.B。由a₈-a₃=5d=15得d=3，a₁=a₃-2d=-1，S₁₀=10(a₁+a₁+9d)/2=125。4.C。至少一枚为6的结果有11种；其中点数和大于8的为(6,3),(6,4),(6,5),(6,6),(3,6),(4,6),(5,6)，共7种，概率为7/11。5.A。f(1)=1，f'(x)=1/x+1，f'(1)=2，切线为y-1=2(x-1)，即y=2x-1。6.D。由递推式a₅=a₁+2(1+2+3+4)=2+20=22。7.B。5个字母的不同排列总数为5!/2=60；把两个A相邻看成一个整体，与B、C、D共4个对象排列有4!=24种，故不相邻为60-24=36。8.C。E(X)=2×0.2+4×0.5+6×0.3=4.2，D(X)=(2-4.2)²×0.2+(4-4.2)²×0.5+(6-4.2)²×0.3=1.96。9.B。f'(x)=3x²-3=3(x-1)(x+1)，当-1<x<1时f'(x)<0，因此函数在(-1,1)上单调递减。10.C。从A到B共需5步向右、3步向上，总路径数为C(8,3)=56；经过P的路径数为C(3,1)C(5,2)=3×10=30，所求为56-30=26。客观题讲评1：第1题与第5题都围绕导数意义展开，一个考查瞬时变化率，一个考查切线方程，答题时应先求导再代入切点。客观题讲评2：第2题与第14题同属二项式展开问题，关键是把通项写清楚；指数方程确定后，系数中的符号和常数倍要同步保留。客观题讲评3：第3题与第18题均考查数列基本量。已知项转化为首项与公差，是解决等差数列问题的常用入口。客观题讲评4：第4题属于条件概率，样本空间已被“至少有一枚6点”缩小，不能再用36个结果作分母。客观题讲评5：第7题采用反面计数更直接，先算全部不同排列，再扣除两个A相邻的情形，可有效避免分类不全。客观题讲评6：第8题计算方差前必须先求数学期望，方差中的每一项都是“偏差平方乘以概率”，不能把偏差直接相加。客观题讲评7：第9题的参数a不影响导函数的符号，因此单调区间与a的具体取值无关，这是本题的易错点。客观题讲评8：第10题使用最短路径模型，先确定总步数，再扣除经过指定点的路径，前后两段路径均需按组合数计算。二、填空题答案与解析题号111213141516答案1893-1600.5311.f'(x)=2xlnx+x，故f'(1)=1。12.C(n,2)=n(n-1)/2=28，得n(n-1)=56，正整数解为n=8。13.S₅=3(1-2⁵)/(1-2)=3×31=93。14.通项为C(6,k)(2x)^{6-k}(-1/x)^k，x的指数为6-2k。令6-2k=0，得k=3，常数项为C(6,3)·2³·(-1)³=-160。15.恰有一人命中概率为0.6×(1-0.5)+(1-0.6)×0.5=0.3+0.2=0.5。16.f'(x)=3x²-3，临界点为x=±1。比较f(-2)=-1，f(-1)=3，f(1)=-1，f(2)=3，最大值为3。填空题讲评1：第11题求导时要同时使用乘积求导法则和lnx的导数，代入x=1后ln1=0，结果只保留后一项。填空题讲评2：第12题的组合数方程应结合n为正整数这一条件，舍去不合题意的根，最终写出唯一正整数值。填空题讲评3：第13题使用等比数列求和公式时，公比q=2不等于1，符号处理可改写为3(2⁵-1)以减少计算失误。填空题讲评4：第15题的两个情形是“甲中乙不中”和“甲不中乙中”，互斥且覆盖恰有一人命中的全部可能。填空题讲评5：第16题在闭区间求最值，必须同时比较端点和临界点的函数值，只比较导数为零的点会漏掉端点。三、解答题答案、解析与评分标准17.（6分）f'(x)=3x²-6x=3x(x-2)。（1分）当x=1时，f(1)=1-3+4=2，切线斜率k=f'(1)=-3，切线方程为y-2=-3(x-1)，即y=-3x+5。（2分）由f'(x)>0得x<0或x>2；由f'(x)<0得0<x<2。（1分）所以f(x)在(-∞,0)、(2,+∞)上单调递增，在(0,2)上单调递减。（1分）x=0处取得极大值f(0)=4，x=2处取得极小值f(2)=0。（1分）讲评要点：本题考查导数的几何意义和函数单调性。切线方程必须同时写出切点坐标与斜率；求单调区间时应依据f'(x)的正负变化，不能只写临界点。18.（6分）设等差数列首项为a₁，公差为d。由a₂+a₅=16得2a₁+5d=16；由a₄=10得a₁+3d=10。（2分）两式联立，得d=4，a₁=-2。（1分）故a_n=a₁+(n-1)d=-2+4n-4=4n-6。（1分）S_n=n[2a₁+(n-1)d]/2=n[-4+4(n-1)]/2=2n²-4n。（1分）令2n²-4n>180，即n(n-2)>90。n=10时为80，n=11时为99，故最小正整数n=11。（1分）讲评要点：本题考查等差数列基本量运算。联立方程求出a₁与d后，再写通项和前n项和；比较最小正整数时应代入相邻两个整数验证。19.（7分）（1）展开式通项为C(6,k)(x²)^{6-k}(-1/x)^k=C(6,k)(-1)^kx^{12-3k}。（2分）令12-3k=3，得k=3，故x³的系数为C(6,3)(-1)³=-20。（1分）（2）