2012年四川省雅安市中考真题及答案

雅安市二○一二年初中毕业暨高中阶段教育学校招生考试
数 学 试 卷
本试卷分第Ⅰ卷（选择题）和第Ⅱ卷（非选择题）两部分，第Ⅰ卷1至2页，第Ⅱ卷3至8页.全卷满分120分，考试时间为120分钟．
第Ⅰ卷（选择题 共36分）
注意事项：
1．在答第Ⅰ卷之前，考生务必将自己的姓名、准考证号、考试科目等填涂在机读卡上．
2．每小题选出答案后，用2B 铅笔把机读卡上对应题目的答案标号涂黑，如需改动， 用橡皮擦干净后，再选涂其它答案，不能答在试卷上.
3．考试结束，由监考人员将试卷和机读卡一并收回．
一、选择题（本大题共12个小题，每小题3分，共36分）每小题的四个选项中，有且仅
有一个正确的.
1．9的平方根的是（ ）
（A ）3 （B ）3- （C ）3± （D ）6
2．如图1，已知O ⊙是ABC △的外接圆，110AOB = ∠，则C ∠的度数为（ ）
（A ）55 （B ）70 （C ）60 （D ）45
3．如果单项式212a x y -与313
b x y 是同类项，那么a b ，值分别为（ ） （A ）2，2 （B ）3-，2 （C ）2，3 （D ）3，2
4．如图2，已知12l l ∥，且1120= ∠
，则2∠=（ ）
（A ）40 （B ）50 （C ）60 （D ）70
5．计算222
()()a a b a b a b +-+等于（ ）
（A ）4a （B ）6a （C ）22a b （D ）22a b -
6．圆柱形水桶的底面周长为3.2πm ，高为0.6m ，它的侧面积是（ ）
（A ）1.536π2m （B ）1.92π2m （C ）0.96π2m （D ）2.56π2m
7．已知二次函数21y ax =-图象的开口向下，则直线1y ax =-经过的象限是（ ）
（A ）第一、二、三象限 （B ）第一、二、四象限
（C ）第一、三、四象限 （D ）第二、三、四象限
8．如图3是一个由多个相同小正方体堆积而成的几何体的俯视图，图中所示数字为该位置
小正方体的个数，则这个几何体的主视图是（ ）
9．由方程组213x m y m
+=⎧⎨-=⎩可得出x 与y 关系是（ ）
（A ）24x y += （B ）24x y -=
（C ）24x y +=- （D ）24x y -=-
10．某校图书管理员清理阅览室的课外书籍时，将其中甲、乙、丙三类书籍的有关数据制成
如图3不完整的统计图，已知甲类书有30本，则丙类书的本数是（ ）
（A ）90 （B ）144 （C ）200 （D ）80
11．在平面直角坐标系中，ABC △的三个顶点坐标分别是(45)(12)(42)A B C ，，
，，，，将ABC △向左平移5个单位后，A 的对应点A '的坐标是（ ）
（A ）(05)， （B ）(15)-， （C ）(95)， （D ）(10)-，
则这位选手得分的平均数和方差分别是（ ）
（A ）9.3，0.04 （B ）9.3，0.048 （C ）9.22，0.048 （D ）9.37，0.04
雅安市二○一二年初中毕业暨高中阶段教育学校招生考试
数 学 试 卷
第Ⅱ卷（非选择题 共84分）
注意事项：
1．用蓝、黑墨水的钢笔或圆珠笔直接答在试题中．
2．答卷前将密封线内的项目填写清楚，不要将答案或解答写在密封线内．
二、填空题（本大题共5个小题，每小题3分，共15分）请将答案直接写在相应题的横线
上.
13．若一元二次方程220x x m ++=无实数解，则m 的取值范围是 ．
14
．化简= ． 15．如图5，AB 是O ⊙直径，O 是圆心，BC 与O ⊙相切于B 点，CO 交O ⊙于点D ，且84BC CD ==，，那么O ⊙的半径是 ．
16．在一个暗盒中放有若干个红色球和3个黑色球（这些球除颜色外，无其它区别），从中随机取出1个球是红球的概率是25
.若在暗盒中增加1个黑球，则从中随机取出1个球是红球的概率是 ．
17．在ADC △和ADC △中，下列条件：①BD DC AB AC ==，；②B C =∠∠， BAD CAD =∠∠；③B C =∠∠，BD DC =；④ADB ADC =∠∠，BD DC =.能得出ABD ACD △≌△的序号是 ．
三、解答题（本大题共69分）解答要求写出必要的文字说明、演算步骤及推理过程
18．（本小题12分，每小题6分） ①计算：1
0120122sin3042-⎛⎫+++- ⎪⎝⎭ ．
②化简：
2
2
121 (1)
1
x x
x x
-+
+∙
-
．
19．（本小题6分）
解不等式组
213(2) 2 4.
x x
x
--⎧
⎨
-<
⎩
≥
20．（本小题7分）
用一根绳子环绕一个圆柱形油桶．若环绕油桶3周，则绳子还多4尺；若环绕油桶4周，则绳子又少了3尺.这根绳子有多长？环绕油桶一周需要多少尺？
21．（本小题10分）
如图6， ABCD 是平行四边形，P 是CD 上一点，且AP 和BP 分别平分DAB ∠和CBA ∠．
（1）求APB ∠的度数；
（2）如果5cm AD =，8cm AP =，求APB △的周长.
22．（本小题12分）
如图7，一次函数1y x =+与反比例函数k y x
=
的图象相交于点A （2，3）和点B ． （1）求反比例函数的解析式；
（2）求点B 的坐标；
（3）过点B 作BC x ⊥轴于C ，求ABC S △．
23．（本小题10分）
已知O ⊙的弦CD 与直径AB 垂直于F ，点E 在CD 上，且AE CE =．
求证：（1）2
CA CE CD =∙；
（2）已知53CA EA ==，，求sin EAF ∠.
24．（本小题12分）
在直角坐标系中，已知抛物线2y ax bx c =++与x 轴交于点A （1，0）和点B ，顶点为P ． （1） 若点P 的坐标为(1
4)-，，求此时抛物线的解析式； （2） 若点P 的坐标为(1
)k -，，0k <，点Q 是y 轴上一个动点，当k 为何值时，QB QP +取得最小值5；
（3） 试求满足（2）时动点Q 的坐标.
雅安市二○一二年初中毕业暨高中阶段教育学校招生考试
数学试题参考答案及评分意见
一、选择题（每题3分，共36分）
1.C
2.A
3.D
4.C
5.A
6.B
7.D
8.C
9.A 10.D
11.B 12.B
二、填空题（每题3分，共15分）
13.1m >
14.13
17.①②④ 三、解答题（共69分）
18.（12分）解：①原式=1+2+1+4 ················································································· 4分 =8 ····························································································· 6分
②原式=2
1(1)(1)(1)
x x x x x +-∙-+ ····························································· 4分 =1x x
- ····················································································· 6分 19.（6分）解：原不等式可化为21362x x x --⎧⎨>-⎩
≥ ······················································· 2分 即52
x x ⎧⎨>-⎩≤ ······································································· 4分 ∴不等式组解集为25x -<≤ ······························································ 6分
20.（7分）解：设这根绳子长为x 尺，环绕油桶一周需y 尺 ······································ 1分
由题意得方程组3443y x y x +=⎧⎨-=⎩
······························································· 3分 解得257
x y =⎧⎨=⎩ ········································································· 6分 答：这根绳子长为25尺，环绕油桶一周需7尺. ································· 7分
21.（10分）解：（1）如图1.
ABCD 是平行四边形，
AD CB AB CD ∴∥，∥.
180DAB CBA ∴+= ∠∠ ······························································ 2分
又AP 和BP 分别平分DAB ∠和CBA ∠，
1()902
PAB PBA DAB CBA ∴+=
+= ∠∠∠∠ ·························· 4分 在APB △中， 180()90APB PAB PBA ∴=-+= ∠∠∠ ·
·································· 5分 （2）AP 平分DAB ∠且AB CD ∥，
DAP PAB DPA ∴==∠∠∠.
ADP ∴△是等腰三角形.
5cm AD DP ∴==·········································································· 7分
同理5cm PC CB ==，即10cm AB DP PC =+=.
在Rt APB △中，10cm AB =，8cm AP =，
6cm BP ∴==.
APB ∴△的周长是6+8+10=24cm·················································· 10分
22.（12分）解：如图2.①将A 点坐标代入反比例函数k y x
=得6k =， ∴反比例函数的解析式为6y x
= ································································· 4分
②由题意得方程组：16y x y x =+⎧⎪⎨=⎪⎩
得：(1)6x x +=，即260x x +-=. (3)(2)0x x ∴+-=.
得1232x x =-=，······················································································· 8分
则B 点坐标为(32)--， ··············································································· 9分
③在ABC △中，以BC 为底边，则高为2(3)5--=，
则12552
ABC S =
⨯⨯=△ ············································································ 12分 23.（10分）解：（1）在CEA △和CAD △中
弦CD 垂直于直径AB ， AC AD ∴=
D C ∴=∠∠ ····························································································· 2分 又A
E EC =
CAE C ∴=∠∠ ······················································································· 3分 CEA CAD ∴△∽△ ∴CA CE CD CA
= ····························································································· 4分 即2
CA CD CE =∙ ···················································································· 5分
（2）如图3.253CA CD CE AC EC =∙== ，，， 253CD ∴=∙，253
CD =
························································
····
··········· 6分 又CF FD = ， 1125252236
CF CD ∴==⨯= ··································································· 7分 257366
EF CF CE =-=-= ··································································· 8分 ∴在Rt AFE △中，7
76sin 318
EF EAF AE ===∠ ································· 10分 24．（12分）解：（1）由题得抛物线的解析式为2
(1)4y a x =++ ····························· 2分
将A 点坐标代入，得1a =-，
∴抛物线解析式为：2(1)4y x =-++（或223y x x =--+） ·
··········· 4分
（2）作P 关于y 轴对称点(1
)P k '，，如图4. QP QP '∴= ······························································································ 5分
由题意知(30)B -， ······················································································ 6分
若QB QP +最小，即QB QP '+最小，则B ，Q ，P '三点共线，
即5P B '= ·································································································· 7分 又4AB =，
连结P A '，得P AB '△是直角三角形.
则P A '=3····································································································· 8分 3k ∴=- ···································································································· 9分
（3）由（2）知，BOQ BAP '△∽△，
BO OQ BA AP
∴
=' ··························································································· 10分 即343
OQ =. 94
OQ ∴= ································································································ 11分 则Q 点坐标为9(0)4-， ············································································ 12分 注：其它解法参照评分.。