黑龙江省齐齐哈尔市二○○九年初中毕业学业考试数学试卷(word版含答案)

黑龙江省齐齐哈尔市二○○九年初中毕业学业考试
数 学 试 卷
考生注意：
1．考试时间120分钟
2．全卷共三道大题，总分120分
一、单项选择题（每题3分，满分30分）
1．1
7-
的绝对值是（ ） A ．17 B ．1
7
- C ．7 D ．7-
2．如图，为估计池塘岸边A B 、的距离，小方在池塘的一侧选取一点
O ，测得15OA =米，OB =10米，A B 、间的距离不可能是（ ）
A ．20米
B ．15米
C ．10米
D ．5米
3．下列运算正确的是（ ）
A
．3= B ．0
(π 3.14)1-= C ．1
122-⎛⎫=- ⎪⎝⎭
D
3=±
4．一组数据4，5，6，7，7，8的中位数和众数分别是（ ） A ．7，7 B ．7，6.5 C ．5.5，7 D ．6.5，7
5．如图，O ⊙是ABC △的外接圆，AD 是O ⊙的直径，若O ⊙的
半径为
3
2，2AC =，则sin B 的值是（ ） A ．23 B ．32 C ．34 D ．43
6．梯形ABCD 中，AD BC ∥，1AD =，4BC =，70C ∠=°，40B ∠=°，则AB 的长为（ ）
A ．2
B ．3
C ．4
D ．5
7．一宾馆有二人间、三人间、四人间三种客房供游客租住，某旅行团20人准备同时租用这三种客房共7间，如果每个房间都住满，租房方案有（ ） A ．4种 B ．3种 C ．2种 D ．1种
8．一个水池接有甲、乙、丙三个水管，先打开甲，一段时间后再打开乙，水池注满水后关闭甲，同时打开丙，直到水池中的水排空，水池中的水量3
(m )v 与时间(h)t 之间的函数关系如图，则关于三个水管每小时的水流量下列判断正确的是（ ） A ．乙>甲 B ．丙>甲
D ．丙>乙
O
A B
第2题图
第5题图 h 第8题图 D A B C O E
F
H
第10题图 第9题
9．已知二次函数2
(0)y ax bx c a =++≠的图象如图所示，则下列结论：0ac >①；②方程2
0ax bx c ++=的两根之和大于0；y ③随x 的增大而增大；④0a b c -+<，其中正确的个数（ ）
A ．4个
B ．3个
C ．2个
D ．1个
10．在矩形ABCD
中，1AB AD AF ==，平分DAB ∠，过C 点作CE BD ⊥于E ，延长AF EC 、交于点H ，下列结论中：AF FH =①；BO BF =②；CA CH =③；④
3BE ED =，正确的是（ ）
A ．②③
B ．③④
C ．①②④
D ．②③④ 二、填空题（每题3分，满分30分）
11．中国齐齐哈尔SOS 儿童村座落在齐齐哈尔市区西部，建成于1992年3月，是由国际SOS 儿童村资助，以家庭形式收养、教育孤儿的社会福利事业单位，占地面积为37000平方米，这个数用科学记数法表示为___________平方米． 12
．函数y =
x 的取值范围是_____________． 13．在英语句子“Wish you success!”（祝你成功！）中任选一个字母，这个字母为“s ”的概率是____________． 14．反比例函数(0)m
y m x
=
≠与一次函数(0)y kx b k =+≠的图象如图所示，请写出一条正确的结论：______________．
15．已知相交两圆的半径分别为5cm 和4cm ，公共弦长为6cm ，则这两个圆的圆心距是______________．
16．当x =_____________时，二次函数2
22y x x =+-有最小值．
17．如图，正方形ABCD 的边长为3cm ，以直线AB 为轴，将正方形旋转一周，所得几何体的主视图的面积是_____________．
18．已知102103m n ==，，则3210
m n
+=____________．
19．如图，边长为1的菱形ABCD 中，60DAB ∠=°．连结对角线AC ，以AC 为边作第二个菱形11ACC D ，使160D AC ∠=°；连结1AC ，再以
1AC 为边作第三个菱形122AC C D ，使2160D AC ∠=°；……，按此规律
所作的第n 个菱形的边长为___________．
20．用直角边分别为3和4的两个直角三角形拼成凸四边形，所得的四边形的周长是____________． 三、解答题（满分60分） 21．（本小题满分5分）
A
D
C
B
第17题图
C 1
D 1
D 2
C 2
D
A
B
第19题图
先化简：22222a b ab b a a ab a ⎛⎫
-+÷+ ⎪-⎝⎭
，当1b =-时，请你为a 任选一个适当的数代入求值．
22．（本小题满分6分）
如图，在平面直角坐标系中，ABC △的顶点坐标为(23)A -，、(32)B -，、(1,1)C -． （1）若将ABC △向右平移3个单位长度，再向上平移1个单位长度，请画出平移后的
111A B C △；
（2）画出111A B C △绕原点旋转180°后得到的222A B C △；
（3）A B C '''△与ABC △是中心对称图形，请写出对称中心的坐标：___________； （4）顺次连结12C C C C '、、、，所得到的图形是轴对称图形吗？
23．（本小题满分6分）
在直角边分别为5cm 和12cm 的直角三角形中作菱形，使菱形的一个内角恰好是三角形的一个角，其余顶点都在三角形的边上，求所作菱形的边长．
24．（本小题满分7分）
为了解某地区30万电视观众对新闻、动画、娱乐三类节目的喜爱情况，根据老年人、成年人、青少年各年龄段实际人口的比例3∶5∶2，随机抽取一定数量的观众进行调查，得到如下统计图．
节目 新闻 娱乐 动画 图二：成年人喜爱的节目统计图 新闻
娱乐 动画 108°
（1）上面所用的调查方法是_________（填“全面调查”或“抽样调查”）； （2）写出折线统计图中A 、B 所代表的值； A ：_____________；B ：_____________；
（3）求该地区喜爱娱乐类节目的成年人的人数．
25．（本小题满分8分）
邮递员小王从县城出发，骑自行车到A 村投递，途中遇到县城中学的学生李明从A 村步行返校．小王在A 村完成投递工作后，返回县城途中又遇到李明，便用自行车载上李明，一起到达县城，结果小王比预计时间晚到1分钟．二人与县城间的距离s （千米）和小王从县城出发后所用的时间t （分）之间的函数关系如图，假设二人之间交流的时间忽略不计． （1）小王和李明第一次相遇时，距县城多少千米？请直接写出答案． （2）求小王从县城出发到返回县城所用的时间． （3）李明从A 村到县城共用多少时间？
26．（本小题满分8分）
如图1，在四边形ABCD 中，AB CD =，E F 、分别是BC AD 、的中点，连结EF 并延长，分别与BA CD 、的延长线交于点M N 、，则BME CNE ∠=∠（不需证明）．
（温馨提示：在图1中，连结BD ，取BD 的中点H ，连结HE HF 、，根据三角形中位线定理，证明HE HF =，从而12∠=∠，再利用平行线性质，可证得BME CNE ∠=∠．） 问题一：如图2，在四边形ADBC 中，AB 与CD 相交于点O ，AB CD =，E F 、分别是BC AD 、的中点，连结EF ，分别交DC AB 、于点M N 、，判断OMN △的形状，请直接写出结论．
问题二：如图3，在ABC △中，AC AB >，D 点在AC 上，AB CD =，E F 、分别是
BC AD 、的中点，
连结EF 并延长，与BA 的延长线交于点G ，若60EFC ∠=°，连结GD ，判断AGD △的形状并证明．
A
C D
F
E N
M O E B
C D
H A F N M 1 2 图1
图2 图3
A
B
C
D
F G
E
27．（本小题满分10分）
某电脑公司经销甲种型号电脑，受经济危机影响，电脑价格不断下降．今年三月份的电脑售价比去年同期每台降价1000元，如果卖出相同数量的电脑，去年销售额为10万元，今年销售额只有8万元．
（1）今年三月份甲种电脑每台售价多少元？
（2）为了增加收入，电脑公司决定再经销乙种型号电脑，已知甲种电脑每台进价为3500元，乙种电脑每台进价为3000元，公司预计用不多于5万元且不少于4.8万元的资金购进这两种电脑共15台，有几种进货方案？
（3）如果乙种电脑每台售价为3800元，为打开乙种电脑的销路，公司决定每售出一台乙种电脑，返还顾客现金a元，要使（2）中所有方案获利相同，a值应是多少？此时，哪种方案对公司更有利？
28．（本小题满分10分）
直线
3
6
4
y x
=-+与坐标轴分别交于A B
、两点，动点P Q
、同时从O点出发，同时到达A
点，运动停止．点Q沿线段OA运动，速度为每秒1个单位长度，点P沿路线O→B→A
运动．
（1）直接写出A B
、两点的坐标；
（2）设点Q的运动时间为t秒，OPQ
△的面积为S，求出S与t之间的函数关系式；
（3）当
48
5
S=时，求出点P的坐标，并直接写出以点O P Q
、、为顶点的平行四边形的第
四个顶点M的坐标．
2009年齐齐哈尔市初中毕业学业考试
数学试卷参考答案及评分标准
一、选择题
1．A 2．D 3．B 4．D 5．A 6．B 7．C 8．C 9．C 10．D
二、填空题（多答案题全部答对得3分，否则不得分，带单位的答案，不写单位扣1分） 11．4
3.710⨯ 12．x ≥0且1x ≠ 13．
2
7
14．正确即可 15
．(4± 16．1- 17．2
18cm 18．72 19
．1
n - 20．14或16或18
三、解答题
21．原式=22()()2()a b a b a ab b a a b a ⎛⎫
+-++÷ ⎪-⎝⎭
·
························································ 1分 =
2()
a b a a a b ++· ································ 1分 =1a b
+ ··········································· 1分 a 值正确(01)a a ≠≠±、给1分，计算结果正确给
22．画出平移后的图形 ···················· 2分，
画出旋转后的图形··························· 2分，
写出坐标（0，0）··························· 1分，
答出“是轴对称图形” ···················· 1分． 23． ·········································································· 1分， 边长
60
cm 17
·
······························································································· 1分， ··········································································· 1分， 边长
156
cm 25
·································································································· 1分 ··········································································· 1分， 边长
65
cm 18
·
······························································································· 1分．
24．抽样调查 ································································································· 1分
2040A B ==， ·
···················································································· 各1分， 5
300000150000352
⨯
=++ ·
········································································ 1分， 108
30%360
= ·
······························································································ 1分， 15000030%45000⨯= ·
················································································· 2分 25．（1）4千米 ·························································································· 2分， （2）解法一：
61180604
-=- ·
············································································· 1分 6
608414
+= ·
································································································· 1分 84+1=85 ········································································································ 1分 解法二：求出解析式，1
214
s t =-+ ······························································· 1分， 084s t ==， ·
···························································································· 1分， 84+1=85 ········································································································ 1分 （3）写出解析式1
520
s t =-
+ ·
········································································· 1分 620s t ==-， ·
····························································································· 1分 20+85=105 ····································································································· 1分
26．（1）等腰三角形 ······················································································ 1分 （2）判断出直角三角形 ··················································································· 1分 证明：如图连结BD ，取BD 的中点H ，连结HF HE 、， ····································· 1分 F 是AD 的中点，
HF AB ∴∥，1
2
HF AB =， 13∴∠=∠．
同理，1
2
HE CD HE CD =
∥，， 2EFC ∴∠=∠．
AB CD =，
∴HF HE =，
12∴∠=∠． ·
······························································································· 1分 60EFC ∠=°，
360EFC AFG ∴∠=∠=∠=°，
AGF ∴△是等边三角形． ·
··············································································· 2分 AF FD =， GF FD ∴=，
30FGD FDG ∴∠=∠=° 90AGD ∴∠=°
即AGD △是直角三角形． ··············································································· 2分 27．（1）解：设今年三月份甲种电脑每台售价x 元
A B
C
D F G H E
1
2 3
10000080000
1000x x =
+ ························································································· 1分 解得：4000x = ···························································································· 1分 经检验：4000x =是原方程的根， ···································································· 1分
所以甲种电脑今年每台售价4000元． （2）设购进甲种电脑x 台，
4800035003000(15)50000x x +-≤≤ ·
·························································· 2分 解得610x ≤≤ ···························································································· 1分
因为x 的正整数解为6，7，8，9，10，所以共有5种进货方案 ································· 1分 （3）设总获利为W 元，
(40003500)(38003000)(15)
(300)1200015W x a x a x a
=-+---=-+- ················································· 1分
当300a =时，（2）中所有方案获利相同． ························································ 1分 此时，购买甲种电脑6台，乙种电脑9
28．（1）A （8，0）B （0，6） ············ 1分
（2）86OA OB ==， 10AB ∴=
点Q 由O 到A 的时间是
8
81
=（秒） ∴点P 的速度是
610
28
+=（单位/秒） ·
1分 当P 在线段OB 上运动（或03t ≤≤）时，OQ t =，2S t = ·········································································································· 1分
当P 在线段BA 上运动（或38t <≤）时，6102162OQ t AP t t ==+-=-，, 如图，作PD OA ⊥于点D ，由
PD AP BO AB =
，得4865
t
PD -=， ······························ 1分 21324255
S OQ PD t t ∴=⨯=-+ ·
······································································ 1分 （自变量取值范围写对给1分，否则不给分．）
（3）82455P ⎛⎫ ⎪⎝⎭
， ···························································································· 1分
12382412241224555555I M M 2⎛⎫⎛⎫⎛⎫-- ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭
，，，，， ···················································· 3分
注：本卷中各题，若有其它正确的解法，可酌情给分．。