2009年中考呼和浩特数学试卷

2009年中考呼和浩特数学试题
一、选择题（本大题包括10个小题，每小题3分，共30分）
1．－2的倒数是（ ）
A ．－ 1 2
B ． 1
2 C ．2 D ．－2
2．已知△ABC 的一个外角为50º，则△ABC 一定是（ ）
A ．锐角三角形
B ．钝角三角形
C ．直角三角形
D ．锐角三角形或钝角三角形
3．有一个正方体，6个面上分别标有1～6这6个整数，投掷这个正方体一次，则出现向上
一面的数字为偶数的概率是（ ）
A ． 1 3
B ． 1 6
C ． 1 2
D ． 1
4 4．如图，AB 是⊙O 的直径，点C 在⊙O 上，CD ⊥AB ，DE ∥BC ，则图中与△ABC 相似的
三角形的个数为（ ）
A ．4个
B ．3个
C ．2个
D ．1个 5．用配方法解方程3x 2－6x ＋1＝0，则方程可变形为（ ）
A ．(x －3)2＝ 1 3
B ．3(x －1)2＝ 1
3
C ．(3x －1)2＝1
D ．(x －1)2＝ 2
3 6．为了解我市参加中考的15000名学生的视力情况，抽查了1000名学生的视力进行统计分
析．下面四个判断正确的是（ ）
A ．15000名学生是总体
B ．1000名学生的视力是总体的一个样本
C ．每名学生是总体的一个个体
D ．以上调查是普查 7．半径为R 的圆内接正三角形的面积是（ ）
A ．3
2R 2 B ．πR 2 C ．332R 2 D ．334R 2 8．在等腰△ABC 中，AB ＝AC ，中线BD 将这个三角形的周长分为15和12两个部分，则这
个等腰三角形的底边长为（ ）
A ．7
B ．11
C ．7或11
D ．7或10 9．右图哪一个是左边正方体的展开图（ ）
10．下列命题中，正确命题的个数为（ ）
①若样本数据3、6、a 、4、2的平均数是4，则其方差为2 ②“相等的角是对顶角”的逆命题 ③对角线互相垂直的四边形是菱形
④若抛物线y ＝(3x －1)2＋k 上有点(2，y 1)、(2，y 2)、(－5，y 3)，则y 3＞y 2＞y 1 A ．1个 B ．2个 C ．3个 D ．4个
二、填空题（本大题包括6个小题，每小题3分，共18分）
11．某种生物孢子的直径为0.00063m ，用科学记数法表示为 m ． 12．把45ab 2－20a 分解因式的结果是 ．
13．初三（1）班有48名学生，春游前，班长把全班学生对春游地点的意向绘制成了扇形统
10987
6
5
432
1
A B
C D C
A
B 地面
墙
α
计图，其中“想去野生动物园的学生数”的扇形圆心角为120º．请你计算想去其他地
点的学生有 人．
14．若|x －2y ＋1|＋|2x －y －5|＝0，则x ＋y ＝ ．
15．如图，四边形ABCD 中，∠ABC ＝120º，AB ⊥AD ，BC ⊥CD ，
AB ＝4，CD ＝53，则该四边形的面积是 ． 16．10个人围成一个圆圈做游戏．游戏的规则是：每个人心里都想好一个数，并把自己想好的数如实地告诉与他相邻的两个人，然后
每个人将与他相邻的两个人告诉他的数的平均数报出来．若报出
来的数如图所示，则报3的人心里想的数是 ．
三、解答题（本大题包括9个小题，共72分）
17．(1)(5分)计算：2009
1)1(45sin 68)12(−+−+−−o ；
(2)(5分)先化简再求值：⎝⎛⎠⎞a － a 2
－b 2
＋1 a ÷ b －1 a × 1 a ＋b
，其中a ＝－ 1 2，b ＝－2．
18．(5分)要想使人安全地攀上斜靠在墙上的梯子的顶端，梯子与地面所成的角α一般满足
50º≤α≤75º．如图，现有一个6m 长的梯子，梯子底端与墙角的距离围3m ． (1)求梯子顶端B 距墙角C 的距离(精确到0.1m )；
(2)计算此时梯子与地面所成的角α，并判断人能否安全使用这个梯子． (参考数据：2≈1.414，3≈1.732)
A
B C G
19．(7分)如图，正方形ABCD 的边CD 在正方形ECGF 的边CE 上，连接BE 、DG ．
(1)求证：BE ＝DG ；
(2)图中是否存在通过旋转能够互相重合的两个三角形？若存在，说出旋转过程；若不
存在，请说明理由．
20．(7分)试确定a 的取值范围，使以下不等式组只有一个整数解．
⎩
⎨⎧x ＋ x ＋1
4＞1，
1.5a － 1 2(x ＋1)＞ 1
2(a －x )＋0.5(2x －1)．
21．(7分)在直角坐标系中直接画出函数y ＝|x |的图象．若一次函数y ＝kx ＋b 的图象分别
过点A (－1，1)、B (2，2)，请你依据这两个函数的图象写出方程组⎩⎨⎧y ＝|x |
y ＝kx ＋b 的解．
22．(9分)某商场服装部为了调动营业员的积极性，决定实行目标管理，即确定一个月销售
目标，根据目标完成的情况对营业员进行适当的奖惩．为了确定一个适当的目标，商场统计了每个营业员在某月的销售额，并整理得到如下统计图(单位：万元)．请分析统计数据完成下列问题．
(1)月销售额在哪个值的人数最多？中间的月销售额是多少？平均月销售额是多少？ (
23．(8分)如图，反比例函数y ＝ m
x (x ＞0)的图象与一次函数y ＝－ 1 2x ＋ 5
2的图象交于A 、
B 两点，点
C 的坐标为(1， 1
2)，连接AC ，AC ∥y 轴．
(1)求反比例函数的解析式及点B 的坐标；
(2)现有一个直角三角板，让它的直角顶点P 在反比例函数图象上A 、B 之间的部分滑
动(不与A 、B 重合)，两直角边始终分别平行于x 轴、y 轴，且与线段AB 交于M 、N 两点，试判断P 点在滑动过程中△PMN 是否与△CBA 总相似？简要说明判断理由．
12
13
14
15
16
18
20
22
2628
3032
3435
24．(8分)如图，在直角梯形ABCD 中，AD ∥BC ，∠ABC ＝90º，AB ＝12cm ，AD ＝8cm ，
BC ＝22cm ，AB 为⊙O 的直径，动点P 从点A 开始沿AD 边向点D 以1cm/s 的速度运动，动点Q 从点C 开始沿CB 边向点B 以2cm/s 的速度运动，P 、Q 分别从点A 、C 同时出发，当其中一点到达端点时，另一个动点也随之停止运动．设运动时间为t (s )． (1)当t 为何值时，四边形PQCD 为平行四边形？(2)当t 为何值时，PQ 与⊙O 相切？
25．(10分)某超市经销一种销售成本为每件40元的商品．据市场调查分析，如果按每件50
元销售，一周能售出500件；若销售单价每涨1元，每周销量就减少10件．设销售单价为x 元(x ≥50)，一周的销售量为y 件．
(1)写出y 与x 的函数关系式(标明x 的取值范围)；
(2)设一周的销售利润为S ，写出S 与x 的函数关系式，并确定当单价在什么范围内变
化时，利润随着单价的增大而增大？
(3)在超市对该种商品投入不超过10000元的情况下，使得一周销售例如达到8000元，
销售单价应定为多少？
2009年呼和浩特市中考试卷
数学参考答案及评分标准
一、选择题（本大题共10个小题，每小题3分，共30分）
1 2 3 4 5 6 7 8 9 10
A B C A D
B D
C
D B
二、填空题（本大题共6个小题，每小题3分，共18分）
11．4
6.3
10−× 12．5(32)(32)a b b +−
13．
32 14．6 15．2
16．2− 三、解答题（本大题9个小题，共
72分）
17
．解：（1）
)
1
20091
6sin 45(1)−−+−°+−
1−−·······················································································3分 11++−−
=0·······························································································································5分
（2）22111a b b a a a a b ⎛⎞−+−−÷×⎜⎟
+⎝⎠
=2221a a b a −+−1
1a b a b
××
−+ =
(1)(1)1
1b b a a b a b +−×
−+·
=1b a b
++····································································································································3分
将122a b =−=−代入得：上式=12
552−=−·····································································5分
18．解：（1）在Rt ACB △中，
5.2m BC ===
········································································································2分 （2）在Rt ACB △中，31
cos 62
AC AB α=
== 60α∴=°
············································································5分 506075<°<°Q °
∴可以安全使用.··································································6分 19．（1）证明：∵正方形ABCD 和正方形ECGF
90BC CD CE CG BCE DCG ∴==∠=∠=，，°
B
C
A
α
·······················································3分
在BCE △和DCG △中，
BC CD BCE DCG CE CG =⎧⎪
∠=∠⎨⎪=⎩
(SAS)BCE DCG ∴△≌△
BE DG ∴=·····················································································································5分 （2）存在．BCE △绕点C 顺时针旋转90°得到DCG △（或将DCG △逆时针旋转90°得
到BCE △）····································································································································7分 20．解：解不等式①：414x x ++> 3
5
x ∴>
····································································································································2分 解不等式②：1111
1.50.52222
a x a x x −−>−+−
即a x <····································································································································5分
由数轴上解集表示可得：
当12a <≤，只有一个整数解······························································································8分
21．解：画出图象得4分
由图象可知，方程y x
y kx b
⎧=⎪⎨
=+⎪⎩的解为
21
21x x y y ==−⎧⎧⎨⎨==⎩⎩
或·······································6分 （画出函数y x =的图象得3分，画出y kx b =+
22．①销售额为18万元的人数最多，中间的月销售额为20万元，平均月销售额为22万元
··························································································································································7分 ②目标应定为20万元，因为样本数据的中位数为20··································································9分 23．（1）由112C ⎛⎞
⎜⎟⎝⎠
，得(12)A ，，代入反比例函数m
y x
=中，得2m = ∴反比例函数解析式为：2
(0)y x x
=
>························································································2分 解方程组15222
y x y x ⎧=−+⎪⎪⎨
⎪=⎪⎩
由15222x x −+=化简得：2
540x x −+= (4)(1)0x x −−= 1241x x ==，
所以142B ⎛
⎞⎜⎟⎝⎠
，··································································································································5分 （2）无论P 点在AB 之间怎样滑动，PMN △与CAB △总能相似．因为B C 、两点纵坐标相等，所以BC x ∥轴．
又因为AC y ∥轴，所以CAB △为直角三角形．
同时PMN △也是直角三角形，AC PM BC PN ∥，∥．
∴PMN CAB △∽△．·······································································································8分 （在理由中只要能说出BC x ∥轴，90ACB ∠=°即可得分．） 24．（1）解：∵直角梯形ABCD ，AD BC ∥
PD QC ∴∥
∴当PD QC =时，四边形PQCD
为平行四边形．
由题意可知：2AP t CQ t ==，
82t t ∴−=
38t = 83
t =
∴当8
3
t s =时，四边形PQCD 为平行四边形．········································································3分
（2）解：设PQ 与O ⊙相切于点H ， 过点P 作PE BC ⊥，
垂足为E Q 直角梯形ABCD AD BC ，∥
PE AB ∴=
由题意可知：2AP BE t CQ t ===，
222BQ BC CQ t ∴=−=−
222223EQ BQ BE t t t =−=−−=− Q AB 为O ⊙的直径，90ABC DAB ∠=∠=° AD BC ∴、为O ⊙的切线
AP PH HQ BQ ∴==，
22222PQ PH HQ AP BQ t t t ∴=+=+=+−=−···························································5分
在Rt PEQ △中，2
2
2
PE EQ PQ +=
22212(223)(22)t t ∴+−=−
B
Q
B
Q
E
即：2
8881440t t −+=
211180t t −+= (2)(9)0t t −−=
1229t t ∴==，························································································································7分
因为P 在AD 边运动的时间为8
811
AD ==秒 而98t =>
9t ∴=（舍去）
∴当2t =秒时，PQ 与O ⊙相切．························································································8分
25．解：（1）50010(50)y x =−−
=100010(50100)x x −≤≤·······································································3分
（2）(40)(100010)S x x =−−
210140040000x x =−+− 210(70)9000x =−−+
当5070x ≤≤时，利润随着单价的增大而增大．····································································6分 （3）2
101400400008000x x −+−=
2101400480000x x −+= 214048000x x −+= (60)(80)0x x −−=
126080x x ==，····························································································································8分
当60x =时，成本=[]4050010(6050)1600010000×−−=>不符合要求，舍去． 当80x =时，成本=[]4050010(8050)800010000×−−=<符合要求．
∴销售单价应定为80元，才能使得一周销售利润达到8000元的同时，投入不超过10000
元．·······································································································································10分。