中等学校招生贵港市课改实验区统一考试数 学（考试时间120分钟，总分120分）一、细心填一填：本大题共10小题，每小题2分，共20分．请将答案填写在题中的横线上．1．3的相反数是 ． 2．计算：123．3．光的速度约为300000000米／秒，用科学记数法表示光的速度应记为 米／秒（保留一位有效数字）． 4．分解因式：322x x x ． 5．“明天的太阳从西方升起”这个事件属于 事件（用“必然”、“不可能”、“不确定”填空）． 6．如图，直线y x 是线段AB 的垂直平分线，若A 点的坐标是(02)，，则B 点的坐标是 ． 7．如图，在ABC △中，90B，D E ，分别是边AB AC ，的中点，410DEAC，，则AB ．8．如图，在O 中，弦AD 平行于弦BC ，若80AOC ，则DAB 度． 9．如图，将Rt ABC △绕点A 逆时针方向旋转90，则旋转后B 点的坐标是 ．10．观察下列各等式：1111212，1112323，1113434，… 根据你发现的规律，计算：2222122334(1)n n … （n 为正整数）．二、精心选一选：本大题共8小题，每小题3分，共24分．在每小题给出的四个选项中，只有一个是正确的，请将正确答案前的字母填入题后的括号内．每小题选对得3分，选错、不选或多选均得零分．11．下列计算中，正确的是（ ） Ａ．236x x x ·Ｂ．235a b ab Ｃ．321a aＤ．236()a a12．用下列同一种图形，不能密铺的是（ ） Ａ．三角形 Ｂ．正五边形 Ｃ．四边形Ｄ．正六边形xy Ay=xBO 第6题 ABCDE第7题ADC BO第8题第9题1 12 23 313．小明将两个全等且有一个角为60的直角三角形拼成如图所示的图形，其中两条较长直角边在同一直线上，则图中等腰三角形的个数是（ ）Ａ．4 Ｂ．3 Ｃ．2 Ｄ．1 14．已知正比例函数y kx 的图象与反比例函数6ky x的图象的一个交点坐标是(13)，，则另一个交点的坐标是（ ） Ａ．(13)， Ｂ．(31)， Ｃ．(12)， Ｄ．(23)， 15．某公司员工的月工资统计如下表，那么该公司员工月工资的平均数、中位数和众数分别是（ ）Ａ．1600，1500，1500 Ｂ．，1000，1000 Ｃ．1600，1500，1000 Ｄ．，1500，1000月工资（元） 3000 1000 人数（人）14516．由若干个小立方块搭成的几何体的三视图如图所示，则该几何体中小立方块的个数是（ ） Ａ．4 Ｂ．5Ｃ．6 Ｄ．717．在一个不透明的袋子中装有2个红球，3个白球，它们除颜色外其余均相同，随机从中摸出一球，记录下颜色后将它放回袋子中，充分摇匀后，再随机摸出一球，则两次都摸到红球的概率是（ ） Ａ．25Ｂ．23Ｃ．45Ｄ．42518．已知：如图，AD 是ABC △的角平分线，且:3:2AB AC ，则ABD △与ACD △的面积之比为（ ） Ａ．3:2 Ｂ．3:2Ｃ．2:3Ｄ．2:3三、解答题：本大题共8小题，满分76分．19．计算下列各题（本题满分11分，第（1）题5分，第（2）题6分） （1）01|31|sin 30(5tan 60)2（2）21224x x xA F CD H BM E G 主视图 左视图 俯视图 ABCD如图，图（1）是某中学九年级（一）班全体学生对三种蔬菜的喜欢人数的频数分布直方图． 解答下列问题：（1）九年级（一）班总人数为 人；（2）哪种蔬菜的喜欢人数频率最高？并求出该频率；（3）请根据频数分布直方图中的数据，补全图（2）中的扇形统计图； （4）根据上述统计的结果，请你为食堂的进货提出一条合理化的建议．21．（本题满分8分）如图所示，四边形ABCD 是平行四边形，E F ，分别在AD CB ，的延长线上，且DE BF ，连接FE 分别交AB CD ，于点H G ，．（1）观察图中有几对全等三角形，并把它们写出来； （2）请你选择（1）中的其中一对全等三角形给予证明．加油站聪明的你如果还有时间，请在上图中连接AF CE ，，你将发现图中出现了更多的全等三角形．请在下面的横线上再写出两对与（1）不同的全等三角形（不用证明）．你将可以获得奖励分，每写正确一对全等三角形奖励1分，共2分（加分后全卷得分不超过120分）． 1． ，2． ．菠菜 大白菜 空心菜 蔬菜10 203040喜欢人数 1218 30图（1） 空心菜图（2）A HF B CG ED我市某初中举行“八荣八耻”知识抢答赛，总共50道抢答题．抢答规定：抢答对1题得3分，抢答错1题扣1分，不抢答得0分．小军参加了抢答比赛，只抢答了其中的20道题，要使最后得分不少于50分，问小军至少要答对几道题？ 23．（本题满分9分）如图所示，AB 是O 的直径，AD 是弦，DBC A =∠∠． （1）求证：BC 与O 相切；（2）若OC 是BD 的垂直平分线，垂足为E ，6BD ，4CE =，求AD 的长．24．（本题满分10分）如图所示，图（1）是一座抛物线型拱桥在建造过程中装模时的设计示意图，拱高为30m ，支柱3350m A B ，5根支柱1122334455A B A B A B A B A B ，，，，之间的距离均为15m ，1515B B A A ∥，将抛物线放在图（2）所示的直角坐标系中． （1）直接写出图（2）中点135B B B ，，的坐标； （2）求图（2）中抛物线的函数表达式； （3）求图（1）中支柱2244A B A B ，的长度．30m图(1)图(2)如图所示，在一笔直的公路MN 的同一旁有两个新开发区A B ，，已知10AB千米，直线AB 与公路MN 的夹角30AON ∠，新开发区B 到公路MN 的距离3BC千米．（1）求新开发区A 到公路MN 的距离； （2）现要在MN 上某点P 处向新开发区A B ，修两条公路PA PB ，，使点P 到新开发区A B ，的距离之和最短．请你用尺规作图在图中找出点P 的位置（不用证明，不写作法，保留作图痕迹），并求出此时PA PB 的值．26．（本题满分12分）如图，已知直线l 的函数表达式为483yx ，且l 与x 轴，y 轴分别交于AB ，两点，动点Q 从B 点开始在线段BA 上以每秒2个单位长度的速度向点A 移动，同时动点P 从A 点开始在线段AO 上以每秒1个单位长度的速度向点O 移动，设点Q P ，移动的时间为t 秒．（1）求出点AB ，的坐标； （2）当t 为何值时，APQ △与AOB △相似？（3）求出（2）中当APQ △与AOB △相似时，线段PQ 所在直线的函数表达式．O P AQBy x2006年中等学校招生贵港市课改实验区统一考试数学参考答案及评分标准一、1．32．3 3．8310 4．2(1)x x5．不可能6．(20)，7．68．409．(22)， 10．21nn 二、11．Ｄ12．Ｂ 13．Ｂ14．Ａ15．Ｃ16．Ｂ17．Ｄ18．Ｂ三、19．解：（1）原式1131122··············································· 4分31； ···························································· 5分（2）原式12222x x x x···················································· 1分2222x xx x ································································· 3分222xx x ································································· 5分12x． ······································································· 6分20．解：（1）60···························································································· 2分 （2）喜欢空心菜的人数频率最高，30100%50%60； ····································· 4分 （3）如图：················································· 6分 （4）建议食堂购买菠菜、大白菜、空心菜时按2:3:5进货． ································ 8分 （只要学生回答合理，均可给分） 21．解：（1）2对． EDG FBH △≌△；·············································································· 2分空心菜 菠菜大白菜EAH FCG △≌△．·············································································· 4分 （2）选证EDG FBH △≌△． ∵四边形ABCD 是平行四边形， AE CF ∴∥，DC AB ∥，E F ∴， EGD AHG ． ··············································································· 6分 AHG FHB ∵， EGD FHB ∴． ············································································· 7分 DE =BF ∵，EDG FBH ∴△≌△． ·········································································· 8分 （其他方法合理即可给分）加油站：1．EDC FBA △≌△，2．EAF FCE △≌△（EGC FHA △≌△）． 22．解：设小军答对x 道题，依题意得： ··························································· 1分 3(20)50xx ≥ ························································································· 4分 解得：1172x ≥ ···························································································· 6分x 为正整数x 的最小正整数为18 ··················································································· 7分 答：小军至少要答对18道题． ········································································ 8分 23．（1）证明：AB 是直径90D································································································· 1分 90AABD ······················································································· 2分 又DBC A90DBC ABD ，即90ABC OB BC ········································· 3分 OB 是半径BC 与O 相切． ······················································································· 4分 （2）解：OC 垂直平分BD ，132BEBD················································· 5分 BEOC ，90BEO BEC ，90EOB OBE ．又90OBE EBC OBC ．EOB EBC·························································································· 6分 OBE BCE △∽△ OE BEBEEC································································································ 7分 223944BE OEEC ···················································································· 8分 OA OB BE DE ，OE 是ABD △的中位线， 922ADOE·························································································· 9分 24．（1）1(30)B ，0，3(030)B ，，5(300)B ，； ················································· 3分（2）设抛物线的表达式为(30)(30)y a x x ， ······································· 4分 把3(030)B ，代入得(030)(030)30y a ．130a∴．························································································· 6分 ∵所求抛物线的表达式为：1(30)(30)30y x x ． ································ 7分 （3）4B ∵点的横坐标为15， 4B ∴的纵坐标4145(1530)(1530)302y ． ······································ 8分 3350A B ∵，拱高为30，∴立柱44458520(m)22A B ． ···························································· 9分 由对称性知：224485(m)2A B A B ． ······················································ 10分 25．解：（1）330BC AOC，∠，6OB ····························································································· ·· 2分 过点A 作AE MN 于点E ，16AO AB OB ，8AE ．···················································································· 3分 即新开发区A 到公路的距离为8千米． ················································· 4分（2）画图正确． ·············································································· 6分过D 作DF AE 的延长线（点D 是点B 关于MN 的对称点），垂足为F ，则3EF CD BC11AF AE EF AE BC ······················································ ·········· ７分 过B 作BG AE 于GBGDFcos3053BGAB ······························································· 8分2222115319614ADAF DF ， ······················· 9分 连结PB ，则PBPD ，14PA PB PA PD AD（千米）． ····································· 10分 26．解：（1）由483y x ，令0x ，得8y；令0y，得6x． ············································································· 1分 A B ，∴的坐标分别是(60)(08)，，，． ··························································· 2分（2）由8BO ，6AO ，得10AB ．当移动的时间为t 时，AP t ，102AQt ． ·········································· 3分 QAPBAO ∵，∴当PA QAOABA时 APQ AOB △∽△102610t t∴， ···················································································· 4分 3011t ∴（秒）． ··················································································· 5分QAP BAO ∵，∴当PA AQAB AO时， AQP AOB △∽△，102106t t∴． ·················································································· 6分 5013t ∴（秒）． ··················································································· 7分3011t ∴秒或5013秒，经检验，它们都符合题意，此时AQP △与AOB △相似． 8分（3）当3011t 秒时，PQ OB ∥， PQ OA ⊥，3011PA，3611OP ∴，36011P ∴，． ···················································· 9分 ∴线段PQ 所在直线的函数表达式为3611x ．············································ 10分 当5013t时，5013PA ，10013BQ ，2813OP ，28013P ∴，． 设Q 点的坐标为()x y ，，则有x BQOA BA ，10013610x ∴ 6013x ∴． 当6013x时，46024831313y ， Q ∴的坐标为60241313，． ······································································· 11分 设PQ 的表达式为ykxb ，则281360241313k bk b，342113kb∴，PQ∴的表达式为321413y x．···········12分。