2007年南京数学中考试题

南京市2007年初中毕业学业考试数学注意事项：1．本试卷1至2页为选择题，共24分，3页6页为非选择题，共96分，全卷满分120分．考试时间120分钟，选择题答在答题卡上，非选择题答在答卷纸上．2．答选择题前考生务必将自己的考试证号，考试科目用2Ｂ铅笔填涂在答题卡上，每小题选出答案后，用2Ｂ铅笔把答题卡上对应题目的答案标号涂黑．如需改动，用橡皮擦干净后，再选涂其他答案．不能答在试卷上．3．答非选择题前考生务必将答纸密封线内的项目及桌号填写清楚．用铅笔或圆珠笔（蓝色或黑色）答在答卷纸上，不能答在试卷上．下列各题所用的四个选项中，有且只有一个是正确的．一、选择题（每小题2分，共24分）1．计算12-+的值是（）Ａ．3-Ｂ．1-Ｃ．1Ｄ．32．2007年5月2日，南京夫子庙、中山陵、玄武湖、雨花台四大景区共接待游客约518 000人，这个数可用科学记数法表示为（）Ａ．40.51810⨯Ｂ．55.1810⨯Ｃ．651.810⨯Ｄ．351810⨯3．计算3x x÷的结果是（）Ａ．4xＢ．3xＣ．2xＤ．34．14的算术平方根是（）Ａ．12-Ｂ．12Ｃ．12±Ｄ．1165．不等式组2110xx>-⎧⎨-⎩，≤的解集是（）Ａ．12x>-Ｂ．12x<-Ｃ．1x≤Ｄ．112x-<≤6．反比例函数2kyx=-（k为常数，0k≠）的图象位于（）Ａ．第一、二象限Ｂ．第一、三象限Ｃ．第二、四角限Ｄ．第三、四象限7．如图，一个可以自由转动的转盘被等分成6个扇形区域，并涂上了相应的颜色，转动转盘，转盘停止后，指针指向黄色区域的概率是（）Ａ．16Ｂ．13Ｃ．12Ｄ．23（第7题）8．下列轴对称图形中，对称轴条数最少的是（ ）Ａ．等边三角形 Ｂ．正方形 Ｃ．正六边形 Ｄ．圆9．下列四个几何体中，已知某个几何体的主视图、左视图、俯视图分别为长方形、长方形、圆，则该几何体是（ ） Ａ．球体 Ｂ．长方体 Ｃ．圆锥体 Ｄ．圆柱体 10．如果a ∠是等腰直角三角形的一个锐角，则tan α的值是（ ） Ａ．12Ｂ．2Ｃ．111．下列各数中，与 ）Ａ．2+Ｂ．2-Ｃ．2-+12．如图，在平面直角坐标系中，点P 在第一象限，P 与x 轴相切于点Q ，与y 轴交于(02)M ，，(08)N ，两点，则点P 的坐标是（ ） Ａ．(53)， Ｂ．(35)， Ｃ．(54)， Ｄ．(45)，二、填空题（每小题3分，共12分） 13．如果40a ∠= ，那么a ∠的补角等于．14．已知5筐苹果的质量分别为（单位：kg ）；5249505351，，，，，则这5筐苹果的平均质量为 kg ．15．如图，O 是A B C △的外接圆，30C ∠= ，2cm A B =，则O 的半径为cm ．16．已知点()P x y ，位于第二象限，并且4y x +≤，x y ，为整数，写出一个．．符合上述条件的点P 的坐标： ．三、（每小题6分，共18分） 17．解方程组42 5.x y x y +=⎧⎨-=⎩，18．计算：221111a a a a a a -÷----．（第15题）19．某养鸡场分3次用鸡蛋孵化出小鸡，每次孵化所用的鸡蛋数、每次的孵化率（孵化率=100%⨯孵化出的小鸡数孵化所用的鸡蛋数）分别如图1，图2所示：（1）求该养鸡场这3次孵化出的小鸡总数和平均孵化率；（2）如果要孵化出2000只小鸡，根据上面的计算结果，估计该养鸡场要用多少个鸡蛋？ 四、（第20题8分，第21题6分，第22题7分，共21分） 20．两组邻边分别相等的四边形我们称它为筝形．如图，在筝形A B C D 中，AB AD =，BC D C =，A C ，B D 相交于点O ， （1）求证：①A B C A D C △≌△； ②O B O D =，A C B D ⊥；（2）如果6A C =，4B D =，求筝形A B C D 的面积．21．将A B C D ，，，四人随机分成甲、乙两组参加羽毛球比赛，每组两人． （1）A 在甲组的概率是多少？ （2）A B ，都在甲组的概率是多少？22．如图，A B ，两地之间有一座山，汽车原来从A 地到B 地须经C 地沿折线A C B --行驶，现开通隧道后，汽车直接沿直线A B 行驶．已知10km A C =，30A ∠=，45B ∠=，则隧道开通后，汽车从A 地到B 地比原来少走多少千米？（结果精确到0.1km ）1.41≈1.73≈）ＣＡＢ3045图1孵化出用的鸡蛋数统计图批次第3次第2次 第1次 图2孵化率统计图五、（每小题7分，共14分）23．某市为了鼓励居民节约用水，采用分段计费的方法按月计算每户家庭的水费，月用水量不超过203m 时，按2元／3m 计费；月用水量超过203m 时，其中的203m 仍按2元／3m 收费，超过部分按2.6元／3m 计费．设每户家庭用用水量为3m x 时，应交水费y 元． （1）分别求出020x ≤≤和20x >时y 与x 的函数表达式； （2）小明家第二季度交纳水费的情况如下：24．如图，A 是半径为12cm 的O 上的定点，动点P 从A 出发，以2πcm /s 的速度沿圆周逆时针运动，当点P 回到A 地立即停止运动．（1）如果90POA ∠= ，求点P 运动的时间；（2）如果点B 是O A 延长线上的一点，A B O A =，那么当点P 运动的时间为2s 时，判断直线B P 与O 的位置关系，并说明理由．六、（每小题7分，共14分）25．某农场去年种植了10亩地的南瓜，亩产量为2000kg ，根据市场需要，今年该农场扩大了种植面积，并且全部种植了高产的新品种南瓜，已知南瓜种植面积的增长率是亩产量的增长率的2倍，今年南瓜的总产量为60 000kg ，求南瓜亩产量的增长率．26．在梯形A B C D 中，A D B C ∥，6A B D C A D ===，60ABC ∠=，点E F ，分别在线段A D D C ，上（点E 与点A D ，不重合），且120BEF ∠=，设A E x =，DF y =．（1）求y 与x 的函数表达式； （2）当x 为何值时，y 有最大值，最大值是多少？七、（本题10分）27．在平面内，先将一个多边形以点O 为位似中心放大或缩小，使所得多边形与原多边形对应线段的比为k ，并且原多边形上的任一点P ，它的对应点P '在线段O P 或其延长线上；接着将所得多边形以点O 为旋转中心，逆时针旋转一个角度θ，这种经过和旋转的图形变换叫做旋转相似变换，记为()O k θ，，其中点O 叫做旋转相似中心，k 叫做相似比，θ叫做旋转角．Ａ ＥＤＦＣＢ（1）填空：①如图1，将A B C △以点A 为旋转相似中心，放大为原来的2倍，再逆时针旋转60 ，得到A D E △，这个旋转相似变换记为A （，）；②如图2，A B C △是边长为1cm的等边三角形，将它作旋转相似变换)A，得到AD E △，则线段B D 的长为 cm ；（2）如图3，分别以锐角三角形ABC 的三边A B ，B C ，C A 为边向外作正方形A D E B ，B F GC ，C H IA ，点1O ，2O ，3O 分别是这三个正方形的对角线交点，试分别利用12AO O △与A B I △，C IB △与2C AO △之间的关系，运用旋转相似变换的知识说明线段12O O 与2A O 之间的关系．八、（本题7分）28．已知直线l 及l 外一点A ，分别按下列要求写出画法，并保留两图痕迹． （1）在图1中，只用圆规．．．．在直线l 上画出两点B C ，，使得点A B C ，，是一个等腰三角形的三个顶点；（2）在图2中，只用圆规．．．．在直线l 外画出一点P ，使得点A P ，所在直线与直线l 平行．南京市2007年初中毕业生学业考试 数学试题参考答案及评分标准二、填空题（每小题3分，共12分）Ａ Ｉ图1Ｉ图2ＣB DE图1 ＡBＣDE图2ＥＤＢＦＧＣＨＡＩ3O1O2O图313．140 14．51 15．2 16．(13)-，，(12)-，，(11)-，，(21)-，，(22)-，，(31)-，六个中任意写出一个即可三、（每小题6分，共24分）17．（本题6分） 解：①＋②，得39x =．解得3x =． ················································································································· 3分 把3x =代入②，得1y =． ·························································································· 5分∴原方程组的解是31x y =⎧⎨=⎩，． ························································································· 6分 18．（本题6分） 解：原式221111aa a a a a -=---- ···················································································· 2分 (1)(1)11(1)1aa a a a a a -+=---- ························································································· 4分 1111a a a +=---·············································································································· 5分1a a =-． ····················································································································· 6分19．（本题6分）解：（1）该养鸡场这3次孵化出的小鸡总数为4082.550786080120⨯+⨯+⨯=％％％（只）． ················································································································· 2分 这3次的平均孵化率为12010080405060⨯=++％％． ·················································· 4分（2）2000802500÷= ％ （个）．∴估计该养鸡场要用2 500个鸡蛋． ············································································· 6分 四、（第20题8分，第21题6分，第22题7分，满分21分）20．证明：（1）①在A B C △和A D C △中， AB AD =，BC D C =，AC AC =， ········································································ 2分A B C A D C ∴△≌△． ································································································· 3分②A B C A D C △≌△，E A O D A O ∴=∠∠． ································································································· 4分 AB AD = ，O B O D ∴=，A C B D ⊥． ························································································· 6分（2）筝形A B C D 的面积A B C =△的面积＋A C D △的面积 1122A CB O ACD O =⨯⨯+⨯⨯116422A CB D =⨯⨯=⨯⨯12=． ························································································································· 8分 21．（本题6分）解：所有可能出现的结果如下：总共有6（1）所有的结果中，满足A 在甲组的结果有3种，所以A 在甲组的概率是12， ·········· 2分（2）所有的结果中，满足A B ，都在甲组的结果有1种，所以A B ，都在甲组的概率是16．·············································································· 6分22．（本题7分） 解：过点C 作C D AB ⊥，垂足为D ．········································································· 1分 在R t C A D △中，30A = ∠，10km A C =，15km 2C D A C ∴==，cos 30AD AC ==． ···················································································· 3分在R t BC D △中，45B =∠，5km B D C D ∴==，sin 45C D B C ==． ··························································································· 5分5)km AB AD BD ∴=+=，105)AC BC AB ∴+-=+-555 1.415 1.73 3.4(km )=++⨯-⨯≈． ················································ 6分答：隧道开通后，汽车从A 地到B 地比原来少走约3.4km ． ········································· 7分 五、（每小题7分，共14分） 23．（本题7分）解：（1）当020x ≤≤时，y 与x 的函数表达式是2y x =； 当20x >时，y 与x 的函数表达式是220 2.6(20)y x =⨯+-，即 2.612y x =-； ········································································································ 3分 （2）因为小明家四、五月份的水费都不超过40元，六月份的水费超过40元，所以把30y =代入2y x =中，得15x =；把34y =代入2y x =中，得17x =；把42.6y =代入2.612y x =-中，得21x =． ······················································································ 5分 所以15172153++=． ······························································································· 6分 答：小明家这个季度共用水253m ． ············································································· 7分 24．（本题7分）解：（1）当90POA = ∠时，点P 运动的路程为O 周长的14或34．设点P 运动的时间为s t ． 当点P 运动的路程为O 周长的14时，122124t π=π ．解得3t =； ·················································································································· 2分 当点P 运动的路程为O 周长的34时，322124t π=π ．解得9t =．∴当90POA =∠时，点P 运动的时间为3s 或9s ． ···················································· 4分 （2）如图，当点P 运动的时间为2s 时，直线B P 与O 相切． ··································· 5分理由如下：当点P 运动的时间为2s 时，点P 运动的路程为4cm π． 连接O P P A ，．O 的周长为24cm π，AP ∴的长为O 周长的16，60POA ∴=∠．BAPOO P O A = ，O AP ∴△是等边三角形． O P O A A P ∴==，60OAP = ∠， AB O A = ，AP AB ∴=． O A P A P B B =+ ∠∠∠，30APB B ∴==∠∠．90OPB OPA APB ∴=+=∠∠∠．O P B P ∴⊥．∴直线B P 与O 相切． ······························································································ 7分 六、（每小题7分，共14分）25．（本题7分）解：设南瓜亩产量的增长率为x ，则种植面积的增长率为2x ．····································· 1分 根据题意，得10(12)2000(1)60000x x ++= ． ············································································ 4分 解这个方程，得10.5x =，22x =-（不合题意，舍去）． ············································· 6分 答：南瓜亩产量的增长率为50％． ··············································································· 7分26．（本题7分） 解：（1）在梯形A B C D 中，A D B C ∥，6A B D C A D ===，60ABC = ∠，120A D ∴==∠∠，18012060AEB ABE ∴+=-=∠∠． 120BEF =∠，18012060AEB DEF ∴+=-=∠∠， ABE D EF ∴=∠∠．ABE D EF ∴△∽△． ································································································· 2分 A E A B D F D E ∴=． ············································································································ 3分 A E x = ，DF y =，66x yx∴=-． ·············································································································· 4分∴y 与x 的函数表达式是 211(6)66y x x x x =-=-+ ； ······················································································ 5分（2）216y x x =-+ 213(3)62x =--+． ····································································································· 6分∴当3x =时，y 有最大值，最大值为32．··································································· 7分七、（本题10分）27．解：（1）①2，60 ；···························································································· 2分 ②2； ·························································································································· 4分（2）12AO O △经过旋转相似变换45)A，得到A B I △，此时，线段12O O 变为线段B I ； ························································································· 6分C IB △经过旋转相似变换452C ⎛⎫ ⎪ ⎪⎝⎭，得到2C AO △，此时，线段B I 变为线段1AO ．···························································································· 8分12=，454590+=，122O O AO ∴=，122O O AO ⊥． ·················································································10分八、（本题7分）28．（1）画法一：以点A 为圆心，大于点A 到直线l 的距离长为半径画弧，与直线l 交于B C ，两点，则点B C ，即为所求． ······················································································· 1分画图正确． ··················································································································· 2分 画法二：在直线l 上任取一点B ，以点B 为圆心，A B 长为半径画弧，与直线l 交于点C ，则点B C ，即为所求． ·································································································· 1分画图正确． ··················································································································· 2分 （2）画法：在直线l 上任取B C ，两点，以点A 为圆心，B C 长为半径画弧，以点C 为圆心，A B 长为半径画弧，两弧交于点P ．则点P 即为所求．······························································ 5分AlA lAlP。