陕西省课改中考试题及答案

陕西省基础教育课程改革实验区 2006 年 初 中 毕 业 学 业 考 试数 学 试 卷第Ⅰ卷（选择题 共30分）一、选择题（共10小题，每小题3分，计30分．每小题只有一个选项是符合题意的） 1．2-的相反数为（ ） A ．2B ．2-C ．12D ．12-2．下面四个图形中，经过折叠能围成如图只有三个面上印有图案的正方体纸盒的是（ ）3．不等式组2030x x +>⎧⎨-⎩，≥的解集是（ ）A ．23x -≤≤B ．2x <-，或3x ≥C ．23x -<<D ．23x -<≤4．将我省某日11个市、区的最高气温统计如下： 最高气温 10℃ 14℃ 21℃ 22℃ 23℃ 24℃ 25℃ 26℃市、区个数1 1 3 1 12 1 1 该天这11个市、区最高气温的平均数和众数分别是（ ）A ．2121℃，℃ B ．2021℃，℃ C ．2122℃，℃ D ．2022℃，℃ 5．中国人民银行宣布，从2007年6月5日起，上调人民币存款利率，一年定期存款利率上调到3.06%．某人于2007年6月5日存入定期为1年的人民币5000元（到期后银行将扣除20%的利息锐）．设到期后银行应向储户支付现金x 元，则所列方程正确的是（ ） A ．50005000 3.06%x -=⨯B ．500020%5000(1 3.06%)x +⨯=⨯+C ．5000 3.06%20%5000(1 3.06%)x +⨯⨯=⨯+D ．5000 3.06%20%5000 3.06%x +⨯⨯=⨯ A ． B ． D ． （第2题图）7．如图，一次函数图象经过点A ，且与正比例函数y x =-的 图象交于点B ，则该一次函数的表达式为（ ） A ．2y x =-+ B ．2y x =+C ．2y x =-D ．2y x =--8．抛物线247y x x =--的顶点坐标是（ ）A ．(211)-，B ．(27)-，C ．(211)，D ．(23)-，9．如图，在矩形ABCD 中，E 为CD 的中点，连接AE 并 延长交BC 的延长线于点F ，则图中全等的直角三角形共有（ ）A ．3对B ．4对C ．5对D ．6对10．如图，在等边ABC △中，9AC =，点O 在AC 上， 且3AO =，点P 是AB 上一动点，连结OP ，将线段OP 绕点O 逆时针旋转60得到线段OD ．要使点D 恰好落在BC 上，则AP 的长是（ ）A ．4B ．5C ．6D ．8第Ⅱ卷（非选择题 共90分）二、填空题（共6小题，每小题3分，计18分） 11．计算：221(3)3x y xy ⎛⎫-=⎪⎝⎭． 12．在ABC △的三个顶点(23)(45)(32)A B C ----，，，，，中，可能在反比例函数(0)ky k x=>的图象上的点是 ． 13．如图，50ABC AD ∠=，垂直平分线段BC 于点D ABC ∠，的 平分线BE 交AD 于点E ，连结EC ，则AEC ∠的度数是 ．（第7题图）（第9题图）（第10题图）Ｂ14．选作题．．．（要求在（1）、（2）中任选一题作答） （1）用计算器计算：3sin382-≈ （结果保留三个有效数字）．（2）小明在楼顶点A 处测得对面大楼楼顶点C 处的 仰角为52，楼底点D 处的俯角为13．若两座楼AB 与CD 相距60米，则楼CD 的高度约为 米．（结果保留三个有效数字）．（sin130.2250cos130.9744tan130.2309sin520.7880cos520.6157≈≈≈≈≈，，，， tan 52 1.2799≈）15．小说《达芬奇密码》中的一个故事里出现了一串神密排列的数，将这串令人费解的数按从小到大的顺序排列为：112358，，，，，，…，则这列数的第8个数是 ． 16．如图，要使输出值y 大于100，则输入的最小正整数x 是 ．三、解答题（共9小题，计72分．解答应写出过程） 17．（本题满分5分） 设23111x A B x x ==+--，，当x 为何值时，A 与B 的值相等？ 18．（本题满分6分）如图，横、纵相邻格点间的距离均为1个单位．（1）在格点中画出图形ABCD 先向右平移6个单位，再向上平移2个单位后的图形； （2）请写出平移前后两图形应对点之间的距离．（第16题图）13（第14题图）19．（本题满分7分）如图，在梯形ABCD 中，45AB DC DAAB B ∠=∥，⊥，，延长CD 到点E ，使DE DA =，连接AE ． （1）求证：AE BC∥；（2）若31AB CD ==，，求四边形ABCE 的面积．20．（本题满分8分）2006年，全国30个省区市在我省有投资项目，投资金额如下表：省区市 广东 福建 北京 浙江 其它金额（亿元）124 67 66 47 119 根据表格中的信息解答下列问题：（1）求2006年外省区市在陕投资总额； （2）补全图①中的条形统计图；（3）2006年，外省区投资中有81亿元用于西安高新技术产业开发区，54亿元用于西安经济技术开发区，剩余资金用于我省其它地区．请在图②中画出外省区市在我省投资金额使用情况的扇形统计图（扇形统计图中的圆心角精确到1，百分比精确到1%）．21．（本题满分8分） 为了迎接暑期旅游，某旅行社推出了一种价格优惠方案：从现在开始，各条旅游线路的价格每人y （元）是原来价格每人x （元）的一次函数．现知道其中两条旅游线路原来旅游价格分别为每人2100元和2800元，而现在旅游的价格分别为每人1800元和2300元． （第19题图） 图① 图②2006年外省区市 在陕投资金额使用情况统计图 （第20题图）东建京江它2006年外省区市在陕投资金额统计图（1）请写出在ABCD 内．（不包括边界）横、纵坐标均为 整数的点，且和为零的点的坐标；（2）在ABCD 内．（不包括边界）任取一个横、纵坐标均为 整数的点，求该点的横、纵坐标之和为零的概率． 23．（本题满分8分）如图，AB 是半圆O 的直径，过点O 作弦AD 的垂线交切线AC 于点C OC ，与半圆O 交于点E ，连结BE DE ，．（1）求证：BED C ∠=∠；（2）若58OA AD ==，，求AC 的长．24．（本题满分10分） 如图，在直角梯形OBCD 中，8110OB BC CD ===，，．（1）求C D ，两点的坐标；（2）若线段OB 上存在点P ，使PD PC ⊥，求过D P，，三点的抛物线的表达式．25．（本题满分12分）如图，O 的半径均为R ．（1）请在图①中画出弦AB CD ，，使图①为轴对称图形而不是．．中心对称图形；请在图②中画出弦AB CD ，，使图②仍为中心对称图形；（2）如图③，在O 中，(02)AB CD m m R ==<<，且AB 与CD 交于点E ，夹角为锐角α．求四边形ACBD 面积（用含m α，的式子表示）； （3）若线段AB CD ，是O 的两条弦，且AB CD ==，你认为在以点A B C D，，，为顶点的四边形中，是否存在面积最大的四边形？请利用图④说明理由．C A O B E D（第24题图） （第25题图①） （第25题图②） （第25题图③） （第25题图④）陕西省基础教育课程改革实验区2007年初中毕业学业考试数 学 答案及评分标准一、选择题1．A 2．B 3．D 4．A 5．C 6．C 7．B 8．A 9．B 10．C 二、填空题11．33x y - 12．B 13．115°（填115不扣分） 14．（1）0.433 （2）90.6 15．21 16．21 三、解答题17．解：当A B =时，23111x x x =+--． 311(1)(1)x x x x =+-+-． ····································································································· 1分 方程两边同时乘以(1)(1)x x +-，得(1)3(1)(1)x x x x +=++-． ································································································ 2分 2231x x x +=+-．2x =． ··································································································································· 3分检验：当2x =时，(1)(1)30x x +-=≠．2x =∴是分式方程的根． ···································································································· 4分 因此，当2x =时，A B =． ································································································· 5分18．解：（1）画图正确得4分．（2）个单位． ············································································································· 6分 19．解：（1）证明：45AB DC DA AB B ⊥∠=∵∥，，°， 135C DA DE ∠=⊥∴°，． ································································································· 1分 又DE DA =∵， 45E ∠=∴°． ······················································································································· 2分 180C E ∠+∠=∴°． ··········································································································· 3分 AE BC ∴∥． ······················································································································· 4分 （2）解：AE BC CE AB ∵∥，∥， ∴四边形ABCE 是平行四边形． ·························································································· 5分 3CE AB ==∴．2DA DE CE CD ==-=∴． ····························································································· 6分326ABCESCE AD ==⨯=∴·． ························································································ 7分 20．解：（1）2006年外省区市在陕投资总额为： 124676647119423++++=（亿元）． ··········································································· 2分 （2）如图①所示． ················································································································· 5分 2006年外省区市在陕投资金额计图 2006年外省区市在陕投资金额使用情况统计图（3）如图②所示． ················································································································· 8分 21．解：（1）设y 与x 的函数关系式为y kx b =+， ·························································· 1分由题意，得2100180028002300k b k b +=⎧⎨+=⎩，， ··························································································· 3分（第18题答案图）（第20题答案图①） （第20题答案图②） 东建京江它省区 市13%西安高新技术 19%解之，得57300k b ⎧=⎪⎨⎪=⎩，．················································································································ 5分 y ∴与x 的函数关系式为53007y x =+． ··········································································· 6分 （2）当5600x =时，5560030043007y =⨯+=元． ····················································· 7分 ∴王老师旅游这条线路的价格是4300元． ·········································································· 8分22．解：（1）(11)(00)(11)--，，，，，． ······················································································ 3分 （2）∵在ABCD 内横、纵坐标均为整数的点有15个， 其中横、纵坐标和为零的点有3个， ···················································································· 6分31155P ==∴． ····················································································································· 8分 23．解：（1）证明：AC ∵是O 的切线，AB 是O 直径， AB AC ⊥∴． 则1290∠+∠=°． ················································································································ 1分 又OC AD ⊥∵， 190C ∠+∠=∴°． ·············································································································· 2分 2C ∠=∠∴． ······················································································································· 3分 而2BED ∠=∠， BED C ∠=∠∴． ················································································································· 4分 （2）解：连接BD ． AB ∵是O 直径， 90ADB ∠=∴°．6BD ===∴． ··········································································· 5分 OAC BDA ∴△∽△． ········································································································· 6分::OA BD AC DA =∴． 即5:6:8AC =．··················································································································· 7分203AC =∴． ························································································································ 8分24．解：（1）过点C 作CE OD ⊥于点E ，则四边形OBCE 为矩形．C AOBED（第23题答案图）1 27OD DE OE =+=∴．C D ∴，两点的坐标分别为(81)(07)C D ，，，．···································································· 4分 （2）PC PD ⊥∵， 1290∠+∠=∴°． 又1390∠+∠=°， 23∠=∠∴．Rt Rt POD CBP ∴△∽△．::PO CB OD BP =∴．即:17:(8)PO PO =-．2870PO PO -+=∴． 1PO =∴，或7PO =．∴点P 的坐标为(10)，，或(70)，． ······················································································· 6分 ①当点P 的坐标为(10)，时，设经过D P C ，，三点的抛物线表达式为2y ax bx c =++，则706481c a b c a b c =⎧⎪++=⎨⎪++=⎩，，．∴2528221287a b c ⎧=⎪⎪⎪=-⎨⎪=⎪⎪⎩，．∴所求抛物线的表达式为：22522172828y x x =-+． ·························································· 9分 ②当点P 为(70)，时， 设经过D P C ，，三点的抛物线表达式为2y ax bx c =++，则749706481c a b c a b c =⎧⎪++=⎨⎪++=⎩，，． ∴141147a b c ⎧=⎪⎪⎪=-⎨⎪=⎪⎪⎩．∴所求抛物线的表达式为：2111744y x x =-+． ···························································· 10分 （说明：求出一条抛物线表达式给3分，求出两条抛物线表达式给4分）25．解：（1）答案不唯一，如图①、②（只要满足题意，画对一个图形给2分，画对两个给3分）················································································································································· 3分 （2）过点A B ，分别作CD 的垂线，垂足分别为M N ，．11sin 22ACD S CD AM CD AE α==△∵···，11sin 22BCD S CD BN CD BE α==△···． ············································································· 5分ACD BCD ACBD S S S =+△△四边形∴11sin sin 22CD AE CD BE αα=+····1()sin 2CD AE BE α=+·· 1sin 2CD AB α=·· 21sin 2m α=． ···········································7分（3）存在．分两种情况说明如下： ···················································································· 8分 ①当AB 与CD 相交时，（第25题答案图①） （第25题答案图②）AD （第25题答案图③）。