題目列表(包括答案和解析)
給出下列四個結(jié)論:①函數(shù)在其定義域內(nèi)是增函數(shù);②函數(shù)的最小正周期是2π;③函數(shù)的圖象關(guān)于直線對稱;④函數(shù)是偶函數(shù).其中正確結(jié)論的序號是 .
(本題14分)設(shè)
(1)當時,求在處的切線方程;
(2)當時,求的極值;
(3)當時,求的最小值。
(本題12分)定義在R上的函數(shù),已知在上有最小值3。
(1)求的單調(diào)區(qū)間;
(2)求在上的最大值。
第二節(jié):以下是五個應(yīng)聘者的信息
。請閱讀
A至F職位招聘的信息
,選出符合他們要求的最佳選項。選項中有一項是多余選項。(答案寫在答題卷上)
Elizabeth Nagata.
Elizabeth once studied in a secretarial program when she was at university. She has worked in Sumitomo Corporation as an executive assistant for eight years.
James Florian.
James once worked as a salesperson and served as the director of new home sales in both Forest Lakes and Glenmore for three years. He can speak excellent German and English, and he is very good at dealing with people.
Arthur Smith.
Arthur, a 22-year-old boy, has just graduated from college. He’s got a diploma in Library Technician Program. Wishing to have some free time, he doesn’t want to work full time.
Gillian Anderson.
Gillian graduated from the University of Manchester and got a Master’s Degree in Library Science five years ago. As soon as she left school, she worked in St. Bride Printing Library as a librarian. Now she is skilled in dealing with almost everything in the library system.
.Jerome Fingelhamm.
Jerome, aged 32, majored in computer science when he was at university. He has worked in a software company for six years and is an expert at both software and hardware. He got a driver’s license when he was eighteen years old.
Sales Executive
We are looking for intelligent sales staff for our ever-growing team. Applicants must have at least one year sales experience, knowledge of English language publications and media, good English and social skills.
B. Executive Assistant Vancouver
Island Regional Library (VIRL) has an opportunity for an executive assistant. A diploma in Business Administration or a secretarial program is needed. A minimum (
最小值
) of five years’ experience as an executive assistant to an executive director is required.
C. Reference Specialist
You’re responsible for providing support for reference services in the library system. The successful applicant must have a Master’s Degree in Library Science and a minimum of three years’ experience as a professional librarian.
D. Library Technician
We need a part-time library technician. Your position is responsible for library cataloging (
編目錄
), binding, ordering publications, assisting in legal research. Both technical and reference services are required. The successful applicant should have a Library Technician diploma (
文憑
).
E. Circulation Director(
發(fā)行總監(jiān)
)
We are looking for a circulation director. The successful applicant must have good managing skills, including the ability to assign tasks to ensure the work of the circulation department is done to the highest standard. A sales experience of at least ten years is needed.
F. Computer & Network Assistant
This position is responsible for the activities related to the use of automated (
自動化的
) systems. Duties include installing (
安裝
), repairing and updating software and hardware. A driver’s license is necessary.
以下是五個應(yīng)聘者的信息
。請閱讀
A至F職位招聘的信息
,選出符合他們要求的最佳選項。選項中有一項是多余選項。(答案寫在答題卷上)
1.Elizabeth Nagata.
Elizabeth
once studied in a secretarial program when she was at university. She has
worked in Sumitomo Corporation as an executive assistant for eight years.
2.James Florian.
James once
worked as a salesperson and served as the director of new home sales in both
Forest Lakes and Glenmore for three years. He can speak excellent German and
English, and he is very good at dealing with people.
3.Arthur Smith.
Arthur, a
22-year-old boy, has just graduated from college. He’s got a diploma in Library
Technician Program. Wishing to have some free time, he doesn’t want to work
full time.
4.Gillian Anderson.
Gillian
graduated from the University of Manchester and got a Master’s Degree in
Library Science five years ago. As soon as she left school, she worked in St.
Bride Printing Library as a librarian. Now she is skilled in dealing with
almost everything in the library system.
5..Jerome Fingelhamm.
Jerome, aged
32, majored in computer science when he was at university. He has worked in a
software company for six years and is an expert at both software and hardware.
He got a driver’s license when he was eighteen years old.
A.
|
B.Executive Assistant Vancouver
|
C.Reference Specialist
|
D.Library Technician |
We need a
part-time library technician. Your position is responsible for library
cataloging (
編目錄), binding, ordering publications, assisting in legal research. Both
technical and reference services are required. The successful applicant should
have a Library Technician diploma (文憑).
E.
Circulation Director(
發(fā)行總監(jiān))
We are
looking for a circulation director. The successful applicant must have good
managing skills, including the ability to assign tasks to ensure the work of
the circulation department is done to the highest standard. A sales experience
of at least ten years is needed.
F. Computer
& Network Assistant
This
position is responsible for the activities related to the use of automated (
自動化的) systems. Duties include
installing (安裝), repairing and updating software and
hardware. A driver’s license is necessary.
1.解析:,故選A。
2.解析:∵
,
故選B。
3.解析:由,得,此時,所以,,故選C。
4.解析:顯然,若與共線,則與共線;若與共線,則,即,得,∴與共線,∴與共線是與共線的充要條件,故選C。
5.解析:設(shè)公差為,由題意得,;,解得或,故選C。
6.解析:∵雙曲線的右焦點到一條漸近線的距離等于焦距的,∴,又∵,∴,∴,∴雙曲線的離心率是。故選B.
7.解析:∵、為正實數(shù),∴,∴;由均值不等式得恒成立,,故②不恒成立,又因為函數(shù)在是增函數(shù),∴,故恒成立的不等式是①③④。故選C.
8.解析:∵,∴在區(qū)間上恒成立,即在區(qū)間上恒成立,∴,故選D。
9.解析:∵
,此函數(shù)的最小值為,故選C。
10.解析:如圖,∵正三角形的邊長為,∴,∴,又∵,∴,故選D。
11.解析:∵在區(qū)間上是增函數(shù)且,∴其反函數(shù)在區(qū)間上是增函數(shù),∴,故選A
12.解析:如圖,①當或時,圓面被分成2塊,涂色方法有20種;②當或時,圓面被分成3塊,涂色方法有60種;
③當時,圓面被分成4塊,涂色方法有120種,所以m的取值范圍是,故選A。
13.解析:做出表示的平面區(qū)域如圖,當直線經(jīng)過點時,取得最大值5。
14.解析:∵,∴時,,又時,滿足上式,因此,,
∴。
15.解析:設(shè)正四面體的棱長為,連,取的中點,連,∵為的中點,∴∥,∴或其補角為與所成角,∵,,∴,∴,又∵,∴,∴與所成角的余弦值為。
16.解析:∵,∴,∵點為的準線與軸的交點,由向量的加法法則及拋物線的對稱性可知,點為拋物線上關(guān)于軸對稱的兩點且做出圖形如右圖,其中為點到準線的距離,四邊形為菱形,∴,∴,∴,∴,∴,∴向量與的夾角為。
17.(10分)解析:(Ⅰ)由正弦定理得,,,…2分
∴,,………4分
(Ⅱ)∵,,∴,∴,………………………6分
又∵,∴,∴,………………………8分
∴!10分
18.解析:(Ⅰ)∵,∴;……………………理3文4分
(Ⅱ)∵三科會考不合格的概率均為,∴學(xué)生甲不能拿到高中畢業(yè)證的概率;……………………理6文8分
(Ⅲ)∵每科得A,B的概率分別為,∴學(xué)生甲被評為三好學(xué)生的概率為!12分
(理)∵,,,。……………………9分
∴的分布列如下表:
0
1
2
3
∴的數(shù)學(xué)期望。……………………12分
19.(12分)解析:(Ⅰ)時,
,,
由得, 或 ………3分
+
0
-
0
+
遞增
極大值
遞減
極小值
遞增
, ………………………6分
(Ⅱ)在定義域上是增函數(shù),
對恒成立,即
………………………9分
又(當且僅當時,)
………………………4分
20.解析:(Ⅰ)∵∥,,∴,∵底面,∴,∴平面,∴,又∵平面,∴,∴平面,∴!4分
(Ⅱ)∵平面,∴,,∴為二面角的平面角,………………………6分
,,∴,又∵平面,,∴,∴二面角的正切值的大小為。………………………8分
(Ⅲ)過點做∥,交于點,∵平面,∴為在平面內(nèi)的射影,∴為與平面所成的角,………………………10分
∵,∴,又∵∥,∴和與平面所成的角相等,∴與平面所成角的正切值為。………………………12分
解法2:如圖建立空間直角坐標系,(Ⅰ)∵,,∴點的坐標分別是,,,∴,,設(shè),∵平面,∴,∴,取,∴,∴!4分
(Ⅱ)設(shè)二面角的大小為,∵平面的法向量是,平面的法向量是,∴,∴,∴二面角的正切值的大小為!8分
(Ⅲ)設(shè)與平面所成角的大小為,∵平面的法向量是,,∴,∴,∴與平面所成角的正切值為。………………………12分
21.(Ⅰ) 解析:如圖,設(shè)右準線與軸的交點為,過點
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