2008 年湖南省湘潭市中考数学真题及答案
一、选择题(本题共 8 个小题,每小题有且只有一个正确答案,请将正确答案的选项代号涂在答题卡相应
的位置上,每小题 3 分,满分 24 分)
1.55°角的余角是(
A. 55°
)
B.45°
C. 35°
D. 125°
2.如图,数轴上 A、B两点所表示的两数的(
)
A. 和为正数
B. 和为负数
A
-3
O
C. 积为正数
D. 积为负数
B
3.如图,已知 D、E分别是 ABC
的 AB、 AC边上的点,
DE BC 且
,
S
S
四边形
ADE
)
于(
A.1 : 9
C.1 : 8
B.1 : 3
D.1 : 2
4.已知样本数据 1,2,4,3,5,下列说法不正确...的是(
)
A.平均数是 3
C.极差是 4
5.已知 ABC
A. 3
5
B.中位数是 4
D.方差是 2
中,AC=4,BC=3,AB=5,则 sin A (
C. 5
3
B. 4
5
)
B
D. 3
4
那么 :AE AC 等
1
DBCE
A
D
E
第 3 题图
C
6.将五张分别印有北京 2008 年奥运会吉祥物 “贝贝,晶晶,欢欢,迎迎,妮妮”的卡片(卡片的形状、大
小一样,质地相同)放入盒中,从中随机抽取一张卡片印有“妮妮”的概率为(
A. 1
2
C. 1
4
B. 1
3
D. 1
5
)
第 6 题图
7.下列式子,正确的是(
)
A. 3
2
12
C.
2
3 2
8.下列命题是假.命题的是(
)
B. ( 2 1)( 2 1) 1
2
(
x
D.
2
xy
2
x
y
2
y
)
A. 若 x
y ,则 x+2008
三、解答题(本大题共 10 个小题,解答应写出文字说明、证明过程或演算步骤,请将解答过程写在答题卡
相应的位置上,满分 72 分)
17.(本题满分 6 分)
计算:
1
(3
)
0
(
11
)
2
.
18.(本题满分 6 分)
如图方格纸中每个小方格都是边长为 1 个单位的正方形,在建立平面直角坐标系后, ABC的顶点在格
点上,点 B的坐标为(5,-4),请你作出 A B C
并 写 出 B
的坐标.
与 ABC关于 y轴对称,
,使 A B C
19.(本题满分 6 分)
先化简,再求值:
x
x
1
2
x
x
1
x
1
x
2
2
2
,其中 x 满足 2
x
3
x
.
2 0
20.(本题满分 6 分)
如图,四边形 ABCD是矩形,E是 AB上一点,且 DE=AB,
过 C作 CF⊥DE,垂足为 F.
D
C
(1)猜想:AD与 CF的大小关系;
(2)请证明上面的结论.
A
F
B
E
21.(本题满分 6 分)
四川的强烈地震,牵动着花蕊小朋友的心. 花蕊小朋友用 280 元,买了每支 0.2 元的铅笔和每支 5 元的
钢笔一共 200 支,寄给灾区的小朋友,请你计算出她买的铅笔和钢笔的支数.
22.(本题满分 6 分)
阅读材料:
如 果 1x , 2x 是 一 元 二 次 方 程 2
ax
bx
x
的 两 根 , 那 么 有 1
c
0
x
2
,b
a
x x
1 2
c
a
.
,x
这是一元 二 次 方 程 根 与 系 数 的 关 系 , 我 们 利 用 它 可 以 用 来 解 题 , 例 1
x 是 方 程
2
2
x
6
x
的 两 根 , 求 2
x
1
3 0
x 的 值 .解 法 可 以 这 样 : 1
x
2
2
x
2
6 ,
x x 则
1 2
3 ,
2
x
1
2
x
2
(
x
1
x
2
2
)
2
x x
1 2
2
( 6)
2 ( 3)
42
. 请你根据以上解法解答下题:
,x
已知 1
x 是方程 2
x
2
4
x
的两根,求:
2 0
(1)
1
x
1
(2)
(
x
1
的值;
1
x
2
x 的值.
)
2
2
23.(本题满分 8 分)
某县七年级有 15000 名学生参加安全应急预案知识竞赛活动,为了了解本次知识竞赛的成绩分布情况,
从中抽取了 400 名学生的得分(得分取正整数,满分 100 分)进行统计:
分 组
49.5~59.5
59.5~69.5
69.5~79.5
79.5~89.5
89.5~100.5
合 计
频 率 分 布 表
频 数
频 率
20
32
124
144
400
0.08
0.20
0.36
1
请你根据不完整的频率分布表. 解答下列问题:
(1)补全频率分布表;
(2)补全频数分布直方图;
160
140
120
100
80
60
40
20
频数(人)
144
124
32
49.5 59.5 69.5 79.5
89.5
100.5
成绩(分)
(3)若将得分转化为等级,规定得分低于 59.5 分评为“D”,59.5~69.5 分评为“C”,
69.5~89.5 分评为“B”,89.5~100.5 分评为“A”,这次 15000 名学生中约有多少人评为“D”?如
果随机抽取一名参赛学生的成绩等级,则这名学生的成绩评为“A”、“B”、“C”、“D”哪一个等级
的可能性大?请说明理由.
24.(本题满分 8 分)
如图所示, O 的直径 AB=4,点 P是 AB延长线上的一点,过 P点作 O 的切线,切点为 C,连结 AC.
(1)若∠CPA=30°,求 PC的长;
(2)若点 P在 AB的延长线上运动,∠CPA的平分线交 AC于点 M. 你认为∠CMP的大小是否发生变化?
若变化,请说明理由;若不变化,求出∠CMP的大小.
C
M
O
A
B
P
25.(本题满分 10 分)
我市花石镇组织 10 辆汽车装运完 A、B、C三种不同品质的湘莲共 100 吨到外地销售,按计划 10 辆汽
车都要装满,且每辆汽车只能装同一种湘莲,根据下表提供的信息,解答以下问题:
湘 莲 品 种
A
每辆汽车运载量(吨) 12
每吨湘莲获利(万元)
3
B
10
4
C
8
2
(1)设装运 A种湘莲的车辆数为 x,装运 B种湘莲的车辆数为 y,求 y与 x之间的函数关系式;
(2)如果装运每种湘莲的车辆数都不少于 2 辆,那么车辆的安排方案有几种?并写出每种安排方案;
(3)若要使此次销售获利最大,应采用哪种安排方案?并求出最大利润的值.
26.(本题满分 10 分)
已知抛物线
y
2
ax
bx
经过点 A(5,0)、B(6,-6)和原点.
c
(1)求抛物线的函数关系式;
(2)若过点 B的直线 y
kx b
与抛物线相交于点 C(2,m),请求出 OBC的面积 S的值.
(3)过点 C作平行于 x轴的直线交 y轴于点 D,在抛物线对称轴右侧位于直线 DC下方的抛物线上,任
取一点 P,过点 P作直线 PF平行于 y轴交 x轴于点 F,交直线 DC于点 E. 直线 PF与直线 DC及两坐标
轴围成矩形 OFED(如图),是否存在点 P,使得 OCD与 CPE相似?若存在,求出点 P的坐标;若不
存在,请说明理由.
y
D
C E
P
F
G
A
5
1
2
x
B
6
D
7
B
8
B
6
4
2
-2
-4
-6
5
A
湘潭市 2008 年初中毕业学业考试
数学参考答案及评分标准
一、选择题:
题 次
答 案
1
C
2
D
3
B
4
B
二、填空题:
9. 6
三、解答题:
10. 60
11. 10
12.90° 13. 2
x
14. 13
15.
10
4.2 10
16. 4
17、解:
1 (3
)
0
(
=1 1 2
······························································ 4 分
11
)
2
=0 ··································································································· 6 分
18、作图(略) ····························································································· 4 分
点 B 的坐标为(-5,-4)··········································································· 6 分
19、解:
2
x
x
x
1
2
x
1
2
x
1
2
x
=
x
1) (
(
x x
1
x
1)(
x
(
1)
x
2
1)
x
·································································· 3 分
x
2 3
x
2 0,
x
(
2)(
x
1) 0
x 或 2.
x
1,
················································································· 5 分
当 1x 时,
(
x
1)
2
分式
0,
2
x
1
2
x
1
2
x
无意义.
原式的值为 2.··················································································6 分
20、解:(1) AD CF . ··············································································· 2 分
(2)四边形 ABCD 是矩形,
AED
FDC DE AB CD
,
············································ 3 分
又
CF DE
,
CFD
A
90 ,
··················································4 分
FCD
······································································· 5 分
·················································································6 分
21、解:设买的铅笔为 x支,买的钢笔为 y支.···················································· 1 分
ADE
AD CF
根据题意得:
x
y
5
0.2
y
x
200
280
①
②
····················································· 3 分
解得
x
y
150
50
···················································································· 5 分
答:略································································································ 6 分
x
22、解: 1
x
2
4,
x x
1 2
2
········································································ 2 分
(1)
1
x
1
1
x
2
x
2
x
1
x x
1 2
4
2
······························································· 4 分
2
(2)
(
x
1
2
x
2
)
(
x
1
2
x
2
)
4
x x
1 2
2
4
4 2 8
··································· 6 分
23、解:(1)略 ····························································································· 3 分
(2)略·······························································································5 分
(3)15000 0.05 750
(人)······························································ 6 分
B 的频率为 0.2 0.31 0.51
··································································································· 8 分
,大于 A、C、D的频率,故这名学生评为 B等的可能性最大.
24、解:(1)连结 OC,
AB
4,
OC
2,
PC 为 O 的切线,
CPO
30 ,
PC
OC
tan 30
2
3
3
2 3.
············· 4 分
A
C
M
O
B
P
的大小没有变化 ································································· 5 分
··································································· 6 分
····································································· 7 分
(2) CMP
CMP
1
2
1 (
2
1 90
2
COP
COP
A MPA
1
CPO
2
CPO
)
45
················································································ 8 分
25、解(1)装 A种为 x辆,装 B种为 y辆,装 C种为 10-x-y辆,······················· 1 分
··············································· 2 分
由题意得:12
) 100
8(10
10
x
y
x
y
y
10 2
x
··················································································· 3 分
(2)10
x
y
10
x
(10 2 )
x
······················································· 4 分
x
故装 C种车也为 x 辆.
x
≥
10 2
x
2
≥
2
····················································· 5 分
解得 2
x
4.
x为整数,
x
2,3,4
··················································· 6 分
故车辆有 3 种安排方案,方案如下:
方案一:装 A种 2 辆车, 装 B种 6 辆车, 装 C种 2 辆车;
方案二:装 A种 3 辆车, 装 B种 4 辆车, 装 C种 3 辆车;··································· 7 分
方案三:装 A种 4 辆车, 装 B种 2 辆车, 装 C种 4 辆车.
(3)设销售利润为 W(万元),则
W=3 12
x
4 10 (10 2 ) 2 8
x
x
= 28
x
400
······················································································9 分
故 W 是 x是的一次函数,且 x增大时,W 减少.
故 2
x 时, maxW =400-28 2 344
(万元) ················································· 10 分