2026届高二数学高二下册期末模拟试卷（河南专用版·考前适应卷含答案详解与评分标准）

2026届高二数学高二下册期末模拟试卷（河南专用版·考前适应卷，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分适用范围标签：河南专用版本卷为高二数学高二下册期末考前适应卷，重点覆盖导数及其应用、计数原理、二项式定理、概率与统计、离散型随机变量及分布等内容。考试说明与答题要求1.答题前填写学校、班级、姓名和考号；选择题答案填入答题栏，填空题答案写在相应横线上。2.解答题应写出必要的文字说明、计算过程或推理步骤；只写结果而无过程的，按评分标准酌情给分。3.本卷共26题，题型结构为：选择题1-10题，每题3分，共30分；填空题11-16题，每题3分，共18分；解答题17-26题，共72分。4.作图、列表、计算均应书写清楚；概率与统计题中可保留分数，也可按题目要求给出小数结果。题型题号题量分值小计选择题1-1010每题3分30分填空题11-166每题3分18分解答题17-2610见各题标注72分合计1-2626120分选择题答题栏题号12345678910答案填空题答题栏题号111213141516答案一、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)=x^3-3x+1，则曲线y=f(x)在点x=1处切线的斜率为A.-3B.0C.1D.32.在(1+x)^6的展开式中，x^2的系数为A.12B.15C.20D.303.设事件A、B相互独立，P(A)=0.6，P(B)=0.5，则P(A∪B)等于A.0.30B.0.50C.0.80D.1.104.随机变量X满足E(X)=10，D(X)=4，令Y=2X-3，则E(Y)与D(Y)分别为A.17，8B.17，16C.20，16D.23，45.函数f(x)=x^3-3x^2+4的单调递增区间为A.(-∞,0)∪(2,+∞)B.(0,2)C.(-∞,2)D.(0,+∞)6.一个袋中有5个红球、3个蓝球，从中不放回任取2个球，恰有1个红球的概率为A.5/14B.15/28C.3/7D.5/87.离散型随机变量X的分布列为P(X=0)=1/4，P(X=1)=1/2，P(X=2)=1/4，则D(X)为A.1/4B.1/2C.1D.3/28.某次数学测评成绩X近似服从正态分布N(100,σ^2)，且P(X>110)=0.1587。已知P(Z>1)=0.1587，则P(90<X<110)约为A.0.3413B.0.5000C.0.6826D.0.84139.5名志愿者排成一排参加展示，甲、乙两人不相邻的排法共有A.48种B.60种C.72种D.96种10.在第一象限内，直线x+2y=12与坐标轴围成的三角形中作内接矩形，矩形两边分别在两坐标轴上，则矩形面积的最大值为A.12B.16C.18D.24二、填空题（本大题共6小题，每小题3分，共18分）11.已知f(x)=lnx+x^2，则f'(1)=__________。12.在(2x-1/x)^6的展开式中，常数项为__________。13.某同学每次投篮命中率为0.8，连续投篮3次，恰好命中2次的概率为__________。14.经验回归方程为y=2x+1，当x=5时，y的预测值为__________。15.若f'(x)=x^2-4x+3，则函数f(x)的单调递增区间为__________。16.一组数据70，75，80，85，90的方差为__________。三、解答题（本大题共10小题，共72分。解答应写出必要步骤）17.（6分）已知函数f(x)=x^3-6x^2+9x+1。

（1）求函数f(x)的单调区间和极值；

（2）求曲线y=f(x)在x=0处的切线方程。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（6分）设(1+2x)^6=a0+a1x+a2x^2+…+a6x^6。

（1）求a2与a3；

（2）求a0+a2+a4+a6的值。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（7分）某学习小组从6份待检查作业中随机抽取3份，其中4份为合格作业、2份为需返修作业。记随机变量X为抽到的合格作业份数。

（1）求X的分布列；

（2）求E(X)与D(X)。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（7分）某班5名同学的阶段复习天数x与测评成绩y如下表。

复习天数x测评成绩y158262365471574（1）求x、y的平均数；（2）用最小二乘法求y关于x的经验回归方程；（3）预测复习6天时的测评成绩。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（7分）6名学生报名参加3个不同社团，每名学生只能参加1个社团，且每个社团至少有1人。若甲、乙两名学生不能参加同一个社团，求不同的分配方法数。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（7分）用薄板制作一个无盖长方体容器，底面为正方形，容积为32dm^3。设底面边长为xdm，高为hdm。

（1）用x表示材料面积S；

（2）求材料面积S的最小值，并给出此时底面边长和高。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（8分）某工厂产品由甲、乙两条生产线共同生产，甲线产量占60%，次品率为2%；乙线产量占40%，次品率为5%。从全部产品中随机抽取1件。

（1）求抽到次品的概率；

（2）若已知抽到的是次品，求它来自甲线的概率；

（3）若独立抽取3件产品，求恰有1件次品的概率（结果可精确到0.0001）。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（8分）某校高二数学月度测评成绩X近似服从正态分布N(80,6^2)。已知标准正态分布函数满足Φ(1)=0.8413，Φ(2)=0.9772。

（1）求P(74≤X≤86)；

（2）求P(X≥92)；

（3）若参加测评的学生有500人，估计成绩在74分到92分之间的人数。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________25.（8分）已知函数f(x)=x^3-3ax+2，其中a>0。

（1）求f(x)的单调区间；

（2）若f(x)在区间[-1,2]上的最小值为-2，求a的值；

（3）在（2）的条件下，判断方程f(x)=0的实根个数，并说明理由。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________26.（8分）某数学活动设置8张题卡，其中5张基础题卡每张满分2分，3张综合题卡每张满分4分。学生随机不放回抽取2张题卡，记Y为抽到两张题卡的满分总和。

（1）求Y的分布列；

（2）求E(Y)与D(Y)；

（3）若活动入场需扣5积分，学生完成抽题后按Y获得积分，判断该规则从期望意义下是否有利于学生，并给出公平入场积分。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析（含评分标准）说明：客观题每题按结果给分；填空题若结果等价且书写清楚，均可得分。解答题按关键步骤给分，过程正确但因计算失误导致后续结果偏差的，按已完成的正确步骤给相应分。题号12345678910答案BBCBABBCCC题号111213141516答案3-1600.38411(-∞,1)∪(3,+∞)501.f'(x)=3x^2-3，代入x=1得f'(1)=0，切线斜率为0。A、D多由未代入或把导数符号判断错造成；C把函数值误当斜率。本题3分。2.由二项式定理，(1+x)^6中x^2项系数为C(6,2)=15，选B。C是C(6,3)，D是把部分项系数相加造成。本题3分。3.事件独立，所以P(A∩B)=0.6×0.5=0.3。P(A∪B)=P(A)+P(B)-P(A∩B)=0.6+0.5-0.3=0.8。本题3分。4.线性变换Y=2X-3满足E(Y)=2E(X)-3=17，D(Y)=2^2D(X)=16。方差受倍数平方影响，不受平移影响，选B。本题3分。5.f'(x)=3x^2-6x=3x(x-2)。当x<0或x>2时f'(x)>0，当0<x<2时f'(x)<0，故递增区间为(-∞,0)∪(2,+∞)。本题3分。6.从8个球中任取2个共有C(8,2)=28种；恰有1红1蓝共有C(5,1)C(3,1)=15种，概率为15/28。选B。本题3分。7.E(X)=0×1/4+1×1/2+2×1/4=1；E(X^2)=0+1/2+4×1/4=3/2，所以D(X)=E(X^2)-[E(X)]^2=1/2。选B。本题3分。8.P(X>110)=0.1587=P(Z>1)，可知(110-100)/σ=1，σ=10。于是P(90<X<110)=P(-1<Z<1)=2Φ(1)-1=0.6826。选C。本题3分。9.5人全排列为5!=120种。甲乙相邻时把甲乙看成一个整体，甲乙两人顺序2种，与另外3人共4个对象排列，得2×4!=48种。故不相邻为120-48=72种。选C。本题3分。10.设矩形右上角为(x,y)，则x+2y=12，面积S=xy=x(12-x)/2=6x-x^2/2。二次函数开口向下，顶点x=6，此时y=3，Smax=18。选C。本题3分。11.f'(x)=1/x+2x，故f'(1)=1+2=3。本题3分。12.通项为C(6,k)(2x)^(6-k)(-x^(-1))^k=C(6,k)2^(6-k)(-1)^kx^(6-2k)。常数项要求6-2k=0，k=3，故常数项为C(6,3)×2^3×(-1)^3=-160。本题3分。13.恰好命中2次服从二项分布，概率为C(3,2)×0.8^2×0.2=0.384。本题3分。14.将x=5代入y=2x+1，得y=2×5+1=11。本题3分。15.f'(x)=x^2-4x+3=(x-1)(x-3)。当x<1或x>3时f'(x)>0，函数递增，故填(-∞,1)∪(3,+∞)。本题3分。16.平均数为80，方差为[(70-80)^2+(75-80)^2+0^2+(85-80)^2+(90-80)^2]/5=(100+25+0+25+100)/5=50。本题3分。解答题详解与采分标准17.解：f'(x)=3x^2-12x+9=3(x-1)(x-3)。当x<1时f'(x)>0，当1<x<3时f'(x)<0，当x>3时f'(x)>0。故f(x)在(-∞,1)、(3,+∞)上递增，在(1,3)上递减；x=1处取得极大值f(1)=5，x=3处取得极小值f(3)=1。又f(0)=1，f'(0)=9，切线方程为y-1=9(x-0)，即y=9x+1。评分：求导正确1分；符号表或单调区间2分；极值2分；切线方程1分，共6分。18.解：（1）二项式通项为C(6,k)(2x)^k，因此a2=C(6,2)×2^2=60，a3=C(6,3)×2^3=160。（2）偶次项系数和可用代入法：令x=1得a0+a1+…+a6=3^6=729；令x=-1得a0-a