高一数学秋季学期期末考试
高一数学（普通高中卷）
本试卷分第Ⅰ卷（选择题）和第Ⅱ卷（非选择题）两部分．满分150分．考试时间120分钟．
说明：可以使用计算器，但未注明精确度的计算问题不得采取近似计算，建议根据题型特点把握好使用计算器的时机．相信你一定会有出色的表现！
第Ⅰ卷
本卷共12小题，每小题5分，共60分．在每小题给出的四个选项中，只有一项是符合题目要求的．请把符合题目要求的选项的字母填入答题卷的答题卡中．
一、选择题：
1．已知集合A＝｛a，b，c｝，那么
（A）a
A （B）a∈A （C）｛a｝∈A （D）｛a｝
A
2．已知集合A＝｛1，2｝，集合B满足A
B＝｛1，2｝，则集合B的个数是
（A）1 （B）2 （C）3 （D）4
3．如果命题“非P”为假，命题“P且q”为真，那么
（A）q为真 （B）q为假 （C）p或q为假 （D）q不一定为真
4．如果（x，y）在映射f作用下的象是（2x－y，x－2y），则（1，2）的象是
（A）（0，－1） （B）（4，1） （C）（0，－3） （D）（0，1）
5．已知三个命题：①方程x2－x＋2＝0的判别式小于或等于零；②若｜x｜≥0，则x≥0；
③5＞2且3＜7．其中真命题是
（A）①和② （B）①和③ （C）②和③ （D）只有①
6．如果函数y＝mx＋2与y＝3x－n的图象关于直线y＝x对称，则
（A）m＝
，n＝6 （B）m＝，n＝－6
（C）m＝3，n＝－2 （D）m＝3，n＝6
7．已知lgm＝b－2lgn，那么m等于
（A）
（B）
（C）b－2n （D）
8．若函数y＝
在
上为增函数，则a的取值范围是
（A）
（B） （C）
（D）
9．在数列｛an｝中，已知前n项和Sn＝7n2－8n，则a100的值为
（A）69200 （B）1400 （C）1415 （D）1385
10．设p：3是1和5的等差中项，q：4是2和5的等比中项，下列说法正确的是
（A）“p或q”为真 （B）“p且q”为真 （C）“非p”为真 （D）“非q”为假
11．若｛an｝是等比数列，且an＞0，a2a4＋2a3a5＋a4a6＝25，则a3＋a5的值为
（A）5 （B）10 （C）15 （D）20
12．若数列｛an｝是公差为
的等差数列，它的前100项和为145，则a1＋a3＋a5＋…＋a99的值是
（A）60 （B）72.5 （C）85 （D）120
第Ⅱ卷（本卷共10小题，共90分）
二、填空题：本大题共4小题；每小题4分，共16分．请将答案填写在答题卷中的横线上．
13．函数f（x）＝
的定义域为▲．
14．若函数y＝x＋1，则f－1（2）＝▲．
15．设等比数列｛an｝的前n项和为Sn，若S3＋S6＝2S9，则数列的公比q的值是▲．
16．已知函数f（x）满足f（x）＝
则
＝▲．
答题卷
高一数学（普通高中卷）
| 题号 | 一 | 二 | 三 | 总分 | |||||
| 1～12 | 13～16 | 17 | 18 | 19 | 20 | 21 | 22 | ||
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一、选择题答题卡：（每小题5分，共60分）
| 题号 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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二、填空题：（每小题4分，共16分）
13． ； 14． ； 15． ； 16． ．
三、解答题：本大题共6小题；共74分．解答应写出文字说明、证明过程或演算步骤．
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17．（本小题满分12分）
已知全集U＝｛x｜x
－7x＋10≥0｝，A＝｛x｜｜x－4｜＞2｝，B＝｛x｜
≥0｝.
求：（1）
UA；（2）A
B．
18．（本小题满分12分）
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用函数单调性的定义证明：f（x）＝
在区间（－
，－3）上是减函数．
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19．（本小题满分12分）
已知函数f（x）＝loga
，其中a＞0且a≠1．
（1）求f（x）的定义域；
（2）求函数f（x）的反函数．
20．（本小题满分12分）
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三个不同的实数a、b、c成等差数列，且a、c、b成等比数列，求a∶b∶c．
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21．（本小题满分12分）
已知一扇形的周长为c（c＞0），当扇形的弧长为何值时，它有最大面积？并求出面积的最大值．
22．（本小题满分14分）
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已知｛an｝是等差数列，其中a1＝1，S10＝100．
（1）求｛an｝的通项公式an；
（2）设an＝log2bn，证明数列｛bn｝是等比数列；
（3）求数列｛bn｝的前5项之和．
高一数学（普通高中卷）参考答案及评分标准
一、选择题：（每小题5分，共60分）
| 题号 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 答案 | B | D | A | C | B | A | D | A | D | A | A | A |
二、填空题：（每小题4分，共16分）
13．［－2，2］ 14．1 15．－
16．
三、解答题：
17．解：U＝｛x｜x≥5或x≤2｝，······································································································· 2分
A＝｛x｜x＞6或x＜2｝，········································································································ 4分
B＝｛x｜x＞5或x≤2｝，········································································································ 6分
（1）
UA＝｛x｜5≤x≤6或x＝2｝；·························································································· 9分
（2）AB｛x｜x＞6或x＜2｝．································································································ 12分
18．解：取任意的x1，x2∈（－，－3），且x1＜x2，则························································· 2分
f（x1）－f（x2）＝
－
＝
．··········································· 4分
∵x1＜x2，∴x2－x1＞0．·········································································································· 5分
又∵x1，x2∈（－，－3），∴x1＋3＜0，x2＋3＜0，············································ 7分
∴（x1＋3）（x2＋3）＞0．······································································································ 8分
∴f（x1）－f（x2）＞0．········································································································ 10分
根据定义知：f（x）在区间（－，－3）上是减函数．···································· 12分
19．解：（1）由题意可知
＞0，即x＞1或x＜－1，····················································· 3分
∴函数f（x）的定义域为（－∞，－1）∪（1，＋∞）；······················ 5分
（2）设u＝，x∈（－∞，－1）∪（1，＋∞），
则u∈（0，1）∪（1，＋∞）．··········································································· 6分
∴y＝logau，u∈（0，1）∪（1，＋∞）的值域为｛y｜y≠0｝．·········· 7分
即函数f（x）的值域为｛y｜y≠0｝．································································· 8分
由y＝loga可以解得x＝
．···························································· 10分
∴f（x）的反函数为f－1（x）＝（x≠0）．······································ 12分
20．解：∵a、b、c成等差数列，∴2b＝a＋c．①······································································· 3分
又∵a、c、b成等比数列，∴c2＝ab．②········································································ 6分
联立①，②解得a＝－2c或a＝c（舍去），b＝－
，············································ 9分
∴a∶b∶c＝（－2c）∶（－）∶c＝（－4）∶（－1）∶2．···················· 12分
21．解：设扇形的半径为R，弧长为l，面积为S
∵c＝2R＋l，∴R＝
（l＜c）．····················································································· 3分
则S＝Rl＝×·l＝
（cl－l 2）······································································· 5分
＝－（l 2－cl）＝－（l－）2＋
．·························································· 7分
∴当l＝时，Smax＝．··································································································· 10分
答：当扇形的弧长为时，扇形有最大面积，扇形面积的最大值是． 12分
22．解：（1）设等差数列｛an｝公差为d，
∵a1＝1，由S10＝10a1＋
·d＝100，················································ 2分
∴d＝2．·································································································································· 4分
∴an＝1＋（n－1）·2＝2n－1；················································································· 6分
（2）由an＝log2bn，∴bn＝
＝
．············································································ 7分
∵
＝4，b1＝21＝2，·················································································· 9分
∴｛bn｝是以2为首项，公比为4的等比数列；············································ 10分
（3）∴S5＝
＝682．··························································································· 14分