甘肃省兰州市中考真题

甘肃省兰州市中考真题
甘肃省兰州市中考真题

兰州市2007年初中毕业生学业考试

数学试卷

一、选择题(本题共12个小题,每小题4分,共计48分.在每小题给出的4个选项中,只有一项是符合题目要求的.)

1.下列方程中是一元二次方程的是( ) A.210x +=

B.21y x += C.210x += D.

2

1

1x x

+=

2.把R t ABC △各边的长度都扩大3倍得R t A B C '''△,那么锐角A ,A '的余弦值的关系

为( ) A.cos cos A A '=

B.cos 3cos A A '=

C.3cos cos A A '= D.不能确定

3.下列函数中,自变量x 的取值范围是2x >的函数是( ) A.2y x =

-

B.12y x

=

- C.12

y x =

- D.121

y x =

-

4.下列说法正确的是( )

A.在同一年出生的367名学生中,至少有两人的生日是同一天

B.彩票中奖的机会是1%,买100张一定会中奖

C.天气预报说明天下雨的概率是50%,所以明天将有一半的时间在下雨 D.抛一枚图钉钉尖着地和钉尖朝上的概率一样大

5.下列四个图形中,既是轴对称图形,又是中心对称图形的是( )

A.①② B.①③ C.②③ D.①②③ 6.顺次连接等腰梯形四边中点所得到的四边形是( )

A.等腰梯形 B.直角梯形 C.矩形 D.菱形 7.“圆柱与球的组合体”如图1所示,则它的三视图是( )

主视图 左视图 俯视图 主视图 左视图 俯视图 主视图 左视图 俯视图 主视图 左视图 俯视图

A. B. C. D. 8.二次函数2

y ax bx c =++图象如图2所示,则点()A ac bc ,在( ) A.第一象限

B.第二象限

C.第三象限

D.第四象限

① ② ③ ④

9.如图3,正方形A B C D 内接于O ,点E 在劣弧A D 上,则B E C ∠等于( ) A.45

B.60

C.30

D.55

10.在同一时刻的阳光下,小明的影子比小强的影子长,那么在同一路灯下( ) A.小明的影子比小强的影子长

B.小明的影子比小强的影子短 C.小明的影子和小强的影子一样长

D.无法判断谁的影子长

11.已知120k k <<,则函数1y k x =和2k y x

=

的图象大致是( )

12.如图4,小丽自己动手做了一顶圆锥形的圣诞帽,母线长是30cm ,底面半径是10cm ,

她想在帽子上缠一根漂亮的丝带,从A 出发绕帽子侧面一周,至少需要丝带( ) A.603cm

B.

303cm 2

C.303cm D.30cm

二、填空题(本题共8个小题,每小题4分,共32分,请把答案填在题中的横线上.) 13.右图是一个小熊的头像,图中反映出圆与圆的四种位置关系,但是其中有一种位置关系没有反映出来,请你写出这种位置关系,它是________.

14.老师给出了一个函数,甲、乙、丙三位学生分别指出了这个函数的一个性质,甲:第一象限内有它的图象;乙:第三象限内有它

的图象;丙:在每个象限内,y 随x 的增大而减小.请你写一个满足上述性质的函数解析式________________.

15.兰州市政府为解决老百姓看病难的问题,决定下调药品的价格,某种药品经过两次降价,由每盒72元调至56元.若每次平均降价的百分率为x ,由题意可列方程为________. 16.将抛物线2

2y x =先沿x 轴方向向左平移2个单位,再沿y 轴方向向下平移3个单位,所得抛物线的解析式是________.

17.赵亮同学想利用影长测量学校旗杆的高度,如图5,他在某一时刻立1米长的标杆测得其影长为1.2米,同时旗杆的投影一部分在地面上,另一部分在某一建筑的墙上,分别测得其长度为9.6米和2米,则学校旗杆的高度为________米.

y

x

O

2

图 1

A

E

D

C B

O

图 3

y x

O y x

O y x

O y

x

O A .

B .

C .

D.

A

4

18.抛物线2222y ax ax a =+++的一部分如图6所示,那么该抛物线在y 轴右侧与x 轴交点的坐标是________.

19.如图7,P A P B ,切O 于A B ,两点,若60APB = ∠,O 的半径为3,则阴影部分的面积为________.

20.下列是三种化合物的结构式及分子式,请按其规律,写出后一种化合物的分子式...为________.

三、作图题(本题满分5分,要求尺规作图,不写作法和证明,但保留作图痕迹) 21.在R t ABC △中,90ACB = ∠,30CAB = ∠,用两种方法把它分成两个三角形,且要求一个三角形是等腰三角形.

四、解答题(本大题共9道题,共计65分,解答时写出必要的文字说明、证明过程或演算步骤)

22.(本题满分5分)计算:200.25(cos 60)(31)tan 60-?---+

23.(本题满分6分)

阅读材料:为解方程2

2

2

(1)5(1)40x x ---+=,我们可以将2

1x -看作一个整体,然后设

A

P

B O

图7

y

x

O 3-

图 6

A B

C

A

B

C 2米

9.6米 图5

H H H H C CH 4

H H H C

C 2H 6

H H H C H H H C

C 3H 8

H H C C H H H

21x y -=……①,那么原方程可化为2

540y y -+=,解得11y =,24y =,当1y =时,

211x -=,2

2x ∴=,2x ∴=±;当4y =时,214x -=,25x ∴=,5x ∴=±,故

原方程的解为12x =,22x =-,35x =,45x =-.

解答问题:(1)上述解题过程,在由原方程得到方程①的过程中,利用________法达到了解方程的目的,体现了转化的数学思想; (2)请利用以上知识解方程4260x x --=. 24.(本题满分6分)

将背面相同,正面分别标有数字1234,,

,的四张卡片洗匀后,背面朝上放在桌面上. (1)从中随机抽取一张卡片,求该卡片正面上的数字是偶数的概率; (2)先从中随机抽取一张卡片(不放回...

),将该卡片正面上的数字作为十位上的数字;再随机抽取一张,将该卡片正面上的数字作为个位上的数字,则组成的两位数恰好是4的倍数的概率是多少?请用树状图或列表法加以说明. 25.(本题满分7分) 如图8,已知A B 为O 的直径,弦C D A B ⊥,垂足为H . (1)求证:2AH AB AC = ;

(2)若过A 的直线与弦C D (不含端点)相交于点E ,与O 相交于点F ,求证:

2

A E A F A C = ;

(3)若过A 的直线与直线C D 相交于点P ,与O 相交于点Q ,判断2AP AQ AC = 是否成立(不必证明).

26.(本题满分7分)

兰州市某校对九年级学生进行“综合素质”评价,评价的结果为A (优)、B (良好)、C (合格)、D (不合格)四个等级.现从中抽测了若干名学生的“综合素质”等级作为样本进行数据处理,并作出图9所示的统计图,已知图中从左到右的四个长方形的高的比为:

14:9:6:1,评价结果为D 等级的有2人,请你回答以下问题:

(1)共抽测了多少人?

(2)样本中B 等级的频率是多少?C 等级的频率是多少? (3)如果要绘制扇形统计图,A D ,两个等级在扇形统计图中所占的圆心角分别是多少度? (4)该校九年级的毕业生共300人,假如“综合素质”等级为A 或B 的学生才能报考示范

E F

O

图8 D A

C B

H

性高中,请你计算该校大约有多少名学生可以报考示范性高中?

27.(本题满分7分)

某农场计划建一个养鸡场,为了节约材料,鸡场一边靠着原有的一堵墙(墙足够长),另外的部分用30米的竹篱笆围成,现有两种方案:①围成一个矩形(如下左图);②围成一个半圆形(如下右图).设矩形的面积为1S 平方米,宽为x 米,半圆形的面积为2S 平方米,半径为r 米,设你通过计算帮助农场主选择一个围成区域面积最大的方案(π3≈).

28.(本题满分8分)

兰州市城市规划期间,欲拆除黄河岸边的一根电线杆A B (如图10),已知距电线杆A B 水平距离14米处是河岸,即14B D =米,该河岸的坡面C D 的坡面C D F ∠的正切值为2,岸高C F 为2米,在坡顶C 处测得杆顶A 的仰角为30 ,D E ,之间是宽2米的人行道,请你通过计算说明在拆除电线杆A B 时,为确保安全,是否将此人行道封上?(在地面上以点

B 为圆心,以A B 长为半径的圆形区域为危险区域)

29.(本题满分9分)

如图11所示,在A B C △中,分别以A B A C B C ,,为边在B C 的同侧作等边ABD △,等边A C E △,等边B C F △.

(1)求证:四边形D A E F 是平行四边形;

(2)探究下列问题:(只填满足的条件,不需证明) ①当A B C △满足________条件时,四边形D A E F 是矩形;

②当A B C △满足________条件时,四边形D A E F 是菱形;

③当A B C △满足________条件时,以D A E F ,,,为顶点的四边形不存在.

O A B C D

等级

人数

图9

x

1S

2S

E D

F C

A

G

B

图10

30.(本题满分10分)

已知抛物线2y ax bx c =++的图象交x 轴于点0(0)A x ,和点(20)B ,,与y 轴的正半轴交于点C ,其对称轴是直线1x =-,tan 2B A C =∠,点A 关于y 轴的对称点为点D . (1)确定A C D ,,三点的坐标;

(2)求过B C D ,,三点的抛物线的解析式;

(3)若过点(03),且平行于x 轴的直线与(2)小题中所求抛物线交于M N ,两点,以M N 为一边,抛物线上任一点()P x y ,为顶点作平行四边形,若平行四边形的面积为S ,写出S 关于P 点纵坐标y 的函数解析式. (4)当

142

x <<时,(3)小题中平行四边形的面积是否有最大值,若有,请求出,若无,

请说明理由.

A D

E

F

B

C

图11

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中考英语单选精选300 题 1 .Roy made several kites, but _____ of them can fly high in the sky. A. neither B. none C. all D. most 2. Have you finished your work yet? No, not yet. I think it?ll take _____ ten minutes. A. another B. other C. others D. More 3 ---I?m sorry to have kept you waiting long. ---Never mind. I ____ here for only a few minutes. A. have been B. have come C. have arrived D. waited 4 .The teacher told the children, “it is very important___ careful ___fire.‖ A.to be; with B. is; with C. being; with D. to be; to 5 . She saw ____ boy playing with his dog A eight-year-old B.An eight-years-old C.An eight-year-old D.A eight-years-old 6 .The story sounds___ . A.interestingly B. interesting C. interested D. Danger 7. John told me to __ the pictures __ tomorrow. A. take; here B. bring; there C. taking; there D. bring; here 8. There __ a lot of __ in the kitchen and it was very__. A. was; smoke; hot B. were; smokes; hot C. was; smoke; hotter D. were; smoke; hotter 9. Do you __ drive on the left in England? A. have to B. must C. should D. May 10. It is very hot today, please keep the window__. A. opening B. open C. opens D. to open 11. --May I play with my dog? --I am afraid not. You can’t play with it __ you finish your homework A.when B. until C.after D. Because 12. I am afraid I won’t __ finish reading the book in two days A. can B. could C. be able to D. May 13. They have ___. A. enough food to eat B. food enough eat C. enough food eating D. food enough eating

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