haihongyuan.com
海量文库 文档专家
全站搜索:
您现在的位置:首页 > 初中教育 > 学科竞赛学科竞赛

初二竞赛试题

发布时间:2013-11-13 08:01:34  

资中二中初二数学竞赛试卷

1、若a、b为实数,满足(A)-1

111ba

,则?的值是 ??

aba?bab

1

(B)0 (C)

2

( )

(D)1

2、下面4种说法:

(1)一个有理数与一个无理数的和一定是无理数 (2)一个有理数与一个无理数的积一定是无理数 (3)两个无理数的和一定是无理数 (4)两个无理数的积一定是无理数

其中,正确的说法种数为 ( ) (A)1 (B)2 (C)3 (D)4

3、已知一次函数y=kx+b,其中kb>0.则所有符合条件的一次函数的图象一定通过

( )

(A)第一、二象限 (B)第二、三象限 (C)第三、四象限 (D)第一、四象限

4、在凸四边形ABCD中,E、F、G、H分别为AB、BC、CD、DA的中点,EG与FH相交于

O,设四边形AEOH、BFOE、CGOF的面积分别为3、4、5,则四边形DHOG的面积为 ( ) (A)

2

(B)

15 4

(C)4 (D)6

( )

5、x=20072008,则x除以10的余数是

(A)1 (B)3 6、已知a、b、c为互不相同的有理数,满足(b?

(C)7 (D)9

2)2?(a?2)(c?2).则符合条件的a、

( )

(D)4组

b、c的组数共有 (A)0组 (B)1组 (C)2组 二、填空题(本大题满分28分,每小题7分)

1、关于x的不等式:|2x?1|?6的所有非负整数解的和为_________. 2、已知x?

12?3

,y?

12?,则x3+12xy+y3=___________.

3、用三种边长相等的正多边形地砖铺地,其顶点拼在一起,刚好能完全铺满地面.已知正多

边形的边数为x、y、z,则111??的值为____________. xyz

4、如图,在梯形ABCD中,AB∥DC,AB=AC,DA=DB,∠ADB=90o,

则∠ACD的度数等于____________.

三、(本大题满分20分)

如图,在梯形ABCD中,AD∥BC,AC⊥DB,AC=5,∠DBC=30o.

(1)求对角线BD的长度;(2)求梯形ABCD的面积.

四、(本大题满分25分)

设实数x满足:

五、(本大题满分25分)

已知正整数a、b、c满足:a<b<c,且ab+bc+ca=abc.求所有符合条件的a、b、c.

3x?14x?26x?313???.求2|x?1|?|x?4|的最小值. 23510

参考答案及评分细则

一、选择题(本题满分42分,每小题7分)

1、D 2、A 3、B 4、C 5、A 6、A

二、填空题(本大题满分28分,每小题7分)

1、6 2、64 3、

三、(本大题20分)

解一:(1)如图,过A作AE//DB交CB延长线于E.

∵AC⊥DB,AE//DB

∴AC⊥AE,∠AEC=∠DBC=30o

∴∠EAC=90o,即△EAC为直角三角形 ························································ 5分 ∴EC=2AC=10

∴AE=EC?AC221 4、30o 2?2?52?5

∵AD//BC且AE//DB,∴四边形AEBD为平行四边形.

∴DB=AE=5.

(2)记梯形ABCD的面积为S,过A作AF⊥BC于F,则

△AFE为直角三角形.

∵∠AEF=30o

∴AF=5351AE=,即梯形ABCD的高AF=. ································· 15分 222

∵四边形AEBD为平行四边形,∴AD=EB

∴S=111525(AD?BC)?AF?EC?AF??10??. ·············· 20分 22222

解二:(1)同解一.

(2)∵ABCD的对角线互相垂直,∴它的面积=1对角线之积. ·················· 15分 2

∴S=1153253AC?DB??10??. ················································ 20分

2222

四、(本大题25分)

解:原不等式两边同乘以30,得:15(3x-1)-10(4x-2)≥6(6x-3)-39 化简得:-31x≥-62

解得:x≤2····································································································· 5分 记y=2|x?1|?|x?4|

(1)当x≤-4时,y=-2(x-1)-(x+4) =-3x-2

所以,y的最小值都为(-3)×(-4)-2=10,此时x =-4 ··························· 10分

(2)当-4≤x≤1时,y=-2(x-1)+(x+4) =-x+6

所以,y的最小值为5,此时x =1 ································································· 15分

(3)当1≤x≤2时,y=2(x-1)+(x+4) =3x+2

所以,y的最小值为5,此时x =1 ································································· 20分 综上所述,2|x?1|?|x?4|的最小值为5,在x =1时取到.···················· 25分

五、(本大题25分)

解:由1≤a<b<c知abc=ab+bc+ca<3bc

所以a<3,故a=1或者a=2. ····································································· 5分

(1)当a=1时,有b+bc+c=bc

即b+c=0,这与b、c为正整数矛盾. ·························································· 10分

(2)当a=2时,有2b+bc+2c=2bc,即bc-2b-2c=0

所以(b-2)(c-2)=4 ····················································································· 15分 又因为2<b<c,故0<b-2<c-2

于是b-2=1,c-2=4.即b=3,c=6 ························································· 20分 所以,符合条件的正整数仅有一组:a=2,b=3,c=6.······························· 25分

网站首页网站地图 站长统计
All rights reserved Powered by 海文库
copyright ©right 2010-2011。
文档资料库内容来自网络,如有侵犯请联系客服。zhit326@126.com