浙江省五校2007年高三第二次联合考试数学(理科)试题
参考公式:
如果事件 A , B 互斥,那么P( A+ B )= P( A)+ P( B) , P( A+ B)= P( A)P( B)
如果事件A在一次试验中发生的概念是p,那么n次独立重复试验中恰好发生 k次的概率:
球的表面积公式:S=, 其中 R 表示球的半径
球的体积公式V=,其中R表示球的半径
第I卷
海淀2007高三数学查漏补缺题
07年高考数学北京卷应该是在06年北京卷成功的基础上,稳定的发展,复习中要对各区题目(尤其东城、西城、海淀)文科理科中重点板块不仅要明确知识点,而且还要掌握结构特点,要用联系的思想看知识间的综合,用运动的观点看能力的要求. 高考数学试题是以思维能力考查为主体的,试题展现数学关系常常选取不同展示形式(图表、图象、曲线图、表格、符号、等等)之一,同学们要善于利用数学信息的多种表述分析问题,联系已有知识方法,提高分析问题、解决问题能力.
一、函数与导数
1.〖理科〗 已知函数f (x)=6lnx―ax2―8x+b (a,b为常数),且x =3为f (x)的一个极值点.
(Ⅰ) 求a;
(Ⅱ) 求函数f (x)的单调区间;
(Ⅲ) 若y = f (x)的图象与x轴正半轴有且只有3个交点,求实数b的取值范围.
解: (Ⅰ) ∵ f ′ (x) =―2ax―8, ∴ f ′ (3) =2―
(Ⅱ) 函数f (x)的定义域为(0,+∞).
由(Ⅰ) 知f (x) =6lnx+x2―8x+b.
∴ f ′ (x) =+2x―8=.
由f ′ (x)>0可得x>3或x<1,由f ′ (x)<0可得1<x<3.
∴函数f (x)的单调递增区间为(0,1)和(3,+∞),单调递减区间为(1,3).
(注:单调区间应分开写,不能用“È”连接)
(Ⅲ) 由(Ⅱ)可知函数f (x)在(0,1)单调递增,在(1,3)单调递减,在(3,+∞)单调递增.
且当x=1或x=3时,f ′ (x)=0.
∴ f (x)的极大值为f (1)=6ln1+1―8+b=b―7,
f (x)的极小值为f (3)=6ln3+9―24+b=6ln3+b―15.
∵当x充分接近0时,f (x)<0,当x充分大时,f (x)>0,
∴要使f (x)的图象与x轴正半轴有且仅有三个不同的交点,只需
则7<b<15―6ln3
2.〖理科、文科〗设函数,其图象在点处的切线的斜率分别为.
(Ⅰ)求证:;
(Ⅱ)若函数的递增区间为,求的取值范围;
(Ⅲ)若当时(k是与无关的常数),恒有,试求k的最小值.
(Ⅰ)证明:,由题意及导数的几何意义得
, (1)
, (2)
又,可得,即,故
由(1)得,代入,再由,得
, (3)
将代入(2)得,即方程有实根.
故其判别式得 ,或, (4)
由(3),(4)得;
(Ⅱ)解:由的判别式,
知方程有两个不等实根,设为,
又由知,为方程()的一个实根,则由根与系数的关系得
,
当或时,,当时,,
故函数的递增区间为,由题设知,
因此,由(Ⅰ)知得
的取值范围为;
(Ⅲ)解:由,即,即,
因为,则,整理得,
设,可以看作是关于的一次函数,
由题意对于恒成立,
故 即得或,
由题意,,
故,因此的最小值为.
二、数列
3.对于数列{an}, {cn}数列,其中cn=an+1― an (nÎN*).
(Ⅰ) 若数列{an}的通项公式 (nÎN*),求{cn}的通项公式;
(Ⅱ) 若数列{an}的首项是1,且满足cn― an=2n.
(1) 求证:数列为等差数列;
(2) (理) 若(nÎN*),求证:.
(文) 求数列{an}的前n项和Sn.
证明:(Ⅰ)依题意cn=an+1― an,
∴ cn=.
(Ⅱ)(1)由cn― an=2n得an+1― an― an=2n,即an+1=2an+2n.
∴,即.
∵a1=1,,∴是以为首项、为公差的等差数列.
(2)(理)由(1)知an=n?2n-1.
∴ .
∴ . ∴
=.
=
(文)由(1)得an==n?2n-1,
∴ Sn = a1+a2+…+an=1?20+2?21+…+n?2n-1, ①
∴ 2Sn=1?21+2?22+…+n?2n. ②
①―②得:― Sn=1+2+22+…+2n-1― n?2n =― n?2n,
∴ Sn= n?2n― 2n+1=(n― 1)?2n +1.
4.〖理科、文科〗 设数列的各项都是正数,记Sn为数列的前n项和,且对任意n∈N+,都有.
(Ⅰ)求证:=2Sn-an;
(Ⅱ)求数列的通项公式;
(Ⅲ)若(为非零常数,n∈N+),问是否存在整数,使得对任意 n∈N+,都有bn+1>bn.
(Ⅰ)证明:在已知式中,当n=1时,
∵a1>0 ∴a1=1……………………………………1分
当n≥2时, ①
②
①-②得,…………………………3分
∵an>0 ∴=
即=2Sn-an ∵a1=1适合上式
∴=2Sn-an(n∈N+)……………………5分
(Ⅱ)解:由(1)知=2Sn-an(n∈N+) ③
当n≥2时, =2Sn-1-an-1 ④
③-④得-=2(Sn-Sn-1)-an+an-1=2an-an+ an-1= an+ an-1
∵an+an-1>0 ∴an-an-1=1……………………8分
∴数列{an}是等差数列,首项为1,公差为1,可得an=n………………9分
(Ⅲ)解:∵
欲使
即成立 ⑤……………………11分
当n=2k-1,k=1,2,3,…时,⑤式即为 ⑥
依题意,⑥式对k=1,2,3…都成立,∴λ<1………………12分
当n=2k,k=1,2,3,…时,⑤式即为 ⑦
依题意,⑦式对k=1,2,3,…都成立,
∴……………………13分
∴
∴存在整数λ=-1,使得对任意n∈N+,都有>
2007 高三英语查漏补缺题
2007-05-16
单项填空
(注意每组题后粗体为答案)
第一组
1. --- Jack, you have earned so much in such a short time!
--- Yes, I have. But I am still _______, in terms of fortune, than you.
A. better off B. more worse off C. less well off D. less badly off
2. --- Who is singing and dancing in the gym now?
---_______ the students from the art department.
A. They are B. It is C. There are D. That is
3. ---The plane is due to take
off at
---Try to take another flight then.
A. What if B. As if C. Even if D. Only if
4. It was the early 2000s
______ mobile phones were widely used in
______ it was considered the patent of the rich.
A. that; in which B. when; before which
C. that; after which D. when; in where
5. He realized his mistakes and took measures to reduce the loss _____ it was too late.
A. until B. when C. before D. as
6. ―What about film last night ?
―It is too moving , I think , film for me .
A./; the B.a; a C.the ; / D.the ; a
7. We trust you ; only you can him to give up smoking .
A.persuade B.make C.advise D.let
8. The news that Chinese athletes won 19 gold medals pleased us greatly , ?
A.did they B.did it C.didn’t it D.didn’t they
9. ―How will you go home , Tom ? It is raining .
―I’m waiting for my brother’s car to me .
A.send …up B.pick…up C.give…up D.call…up
10. I was driven by a to learn and determined to be ahead of my classmates .
A.desire B.wish C.hope D.expect
11. ―My son failed the exam again ! Is he a little bit slow , Miss Li ?
―In my opinion , your son is than slow .
A.much lazy B.lazier rather C.quite lazy D.more lazy
12. Now we realize problem of pollution is.
A.so serious a B.how serious the C.such serious a D.what serious
13. ―Who broke the window ? Anybody you know ?
―Well, it be Jackson , but that’s just a guess .
A.must B.would C.might D.should
14. E-shopping , when properly ,can save us a lot of time and energy .
A.to do B.is done C.doing D.done
15. Not only to stay at home , but he was also forbidden watching TV.
A.that he was forced B.he was forced
C.was he forced D.was forced he
1-5 CBABC 6-10 DACBA 11-15 DBCDC
第二组
1.Human beings are different from animals_____they can use language as a tool to communicate.
A.in that B.for that C.in which D.on which
2.You have to be very careful with the dry wood. It ___________ very easily. A.is lighted B.lights C.will be lighting D.is lighting
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