2009年湖南省张家界市中考数学真题及答案.doc

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2009 年湖南省张家界市中考数学真题及答案考生注意：本学科试卷共三道大题 25 小题，满分 120 分，考试时量 120 分钟．一、选择题（本大题共 8 小题，每小题 3 分，满分 24 分） 1．在实数 0， 2 ，  ，0.74， π 中，无理数有（ 1 3 ） A．1 个 B．2 个 C．3 个 D．4 个 2．用计算器求 32 值时，需相继按“2”，“∧”，“3”，“  ”键，若小红相继按“ ”，“2”，“∧”，“4”， “  ”键，则输出结果是（ A．4 3．下图所示的几何体的主视图是（ D．16 B．5 C．6 ）） 4．不等式组 A． 1) 7 5( x    24 6 3 x    4 2 x x    B． C． D．的解集在数轴上表示为（） 0 6 A． 0 6 B． 0 6 C． 0 6 D． 5．下列运算正确的是（） A． 2 ab 2 2 ab  1 B． tan 45 sin 45 ° 1 ° C． 2 x x x 3 D． 2 3 )a ( 5 a ） 6．下列不是必然事件的是（ A．两直线相交，对顶角相等 B．三角形的外心到三个顶点的距离相等 C．三角形任意两边之和大于第三边 D．两相似多边形面积的比等于周长的比 7．如图， AB CD∥ ，且 1 115   °，则 E 的度数是（ A．30° C． 40° 8．为了预防“HINI”流感，某校对教室进行药熏消毒，药品燃烧时，室内每立方米的含药量与时间成正比；燃烧后，室内每立方米含药量与时间成反比，则消毒过程中室内每立方米含药量 y 与时间t 的函数关系图象大致为（ B．50° D．60° A  °， 75 ）） E C 1 D A B y O y t O t y O y t O t A． B． C． D．

二、填空题（本大题共 8 小题，每小题 3 分，满分 24 分） A ． 9． 3 的绝对值为 10．如图， O 是 ABC△ 的切点分别为 D E F，，，若 11．张家界国际乡村音乐周活动中，来自中、日、美的三名音乐家准备在同一节目中依次演奏本国的民族音乐，若他们出场先后的机会是均等的，则按“美—日—中”顺序演奏的概率是的内切圆，与边 BC CA AB A  °，则 EDF ，，  70 O D ． B ． E F C 12．将函数 y   3 x  的图象向上平移 2 个单位，得到函数 3 的图象． 13．分解因式 3 a 2 ab  ． 14．我市甲、乙两景点今年 5 月上旬每天接待游客的人数如图所示，甲、乙两景点日接待游客人数的方差大小关系为： 2S甲 2S乙．人数 2800 2600 2400 2200 2000 1800 甲乙 1 2 3 4 5 6 7 8 9 10 日 15．对于正实数 a b，作新定义： a b b a a b   ，在此定义下，若9   x  ，则 x 的值为 55 ． 16．如图，等腰梯形 ABCD 中， AD BC∥ ，且 AD  1 2 BC ， E 为 AD 上一点， AC 与 BE 交于点 F ，若 : AE DE  2 :1 ，则 △ △ AEF CBF 的面积的面积  ．三、解答题（本题共 9 小题，满分 72 分） 17．（本小题 6 分）计算  11    2     (5  3)  ° 2sin 45  ° 1 2 1  A E D F B C 18．（本小题 6 分）小明将一幅三角板如图所示摆放在一起，发现只要知道其中一边的长就可以求出其它各边的长，若已知 CD  ，求 AC 的长． 2 D B A C

19．先化简，后求值（本小题 6 分） 4  4 2 a  2  2 a  1  2 a 其中 a  2 2  20．（本小题 6 分）在建立平面直角坐标系的方格纸中，每个小方格都是边长为 1 的小正方形， ABC△ P 的坐标为 ( 1 0)  ，，请按要求画图与作答的顶点均在格点上，点（1）把 ABC△ （2）把 ABC△   （3） A B C △ 绕点 P 旋转180°得 A B C    向右平移 7 个单位得 A B C   与 A B C △ △ △ ．．  是否成中心对称，若是，找出对称中心 P ，并写出其坐标． A B y C P O x 21．列方程解应用题（本小题 9 分） “阳黄公路”开通后，从长沙到武陵源增加了一条新线路，新线路里程在原线路长 360Km 的基础上缩短了 50Km，今有一旅游客车和小车同时从长沙出发前往武陵源，旅游客车走新线路，小车因故走原线路，中途停留 6 分钟．若小车速度是旅游客车速度的 1.2 倍，且两车同时到达武陵源，求两车的速度各是多少？ 22．（本小题 9 分）如图，有两个动点 E F，分别从正方形 ABCD 的两个顶点 B C，同时出发，以相同速度分别沿边 BC 和CD 移动，问：（1）在 E F，移动过程中， AE 与 BF 的位置和大小有何关系？并给予证明．（2）若 AE 和 BF 相交点O ，图中有多少对相似三角形？请把它们写出来．

FD O A C E B 23．（本小题 9 分）我市今年初三体育考试结束后，从某县 3000 名参考学生中抽取了 100 名考生成绩进行统计分析（满分 100 分，记分均为整数），得到如图所示的频数分布直方图，请你根据图形完成下列问题：（1）本次抽样的样本容量是（2）请补全频数分布直方图．（3）若 80 分以上（含 80 分）为优秀，请你据此．估算该县本次考试的优秀人数．．人数 40 30 20 10 30 20 2 3 5 39.5 49.5 59.5 69.5 79.5 89.5 100 分数 24．（本小题 9 分）有若干个数，第 1 个数记为 1a ，第 2 个数记为 2a ，第 3 个数记为 3a ， 第 n 个数记为 na ，若 1 a   ， 1 3 从第二个数起，每个数都等于．．．．．．．．．．．．．1．与前面那个数的差的倒数．．．．．．．．．．．． a （1）分别求出 2 4 a，，的值． a 3 a （2）计算 1  a 2  a 3   a 36 的值． 25．（本小题 12 分）在平面直角坐标系中，已知 ( 4 0) A  ，， (1 0) B ，，且以 AB 为直径的圆交 y 轴的正半轴于点 (0 2) C ，，过点C 作圆的切线交 x 轴于点 D ．（1）求过 A B C，，三点的抛物线的解析式（2）求点 D 的坐标（3）设平行于 x 轴的直线交抛物线于 E F，两点，问：是否存在以线段 EF 为直径的圆，恰好与 x 轴相切？ y C 2 A B D

若存在，求出该圆的半径，若不存在，请说明理由？张家界市 2009 年初中毕业学业考试数学试卷答案一、选择题 1．B 二、填空题 2．A 3．B 4．A 5．C 6．D 7．C 8．A 9． 3 10．55° 11． 1 6 12． y   3 x  5

13． ( a a b a b  )(  ) 14． 2 S 甲 2 S 乙 15．16 16． 1 9 三、解答题 17．原式 2 1 2     2 2  1 2 1   2 1  2 1  ··························································3 分 2 1    2 ( 2 1)   ····················································································· 4 分  2 2 1 2 1  ······················································································· 5 分    2 ·············································································································· 6 分 18．解： BD CD  2  ·················································································· 2 分 2 2 BC     ，则 AB x 2   x (2 2) 2 2 2 2 2  2 x AC 2 (2 ) x  ····················································································· 4 分 x  2 6 3 ···································································································· 5 分 AC  2 AB  19．解：原式 4 3  6 ··························································································6 分 4 2)( ( a  a  2)  2  2 a  1  2 a    4 2)( ( a  a  2) 2(  2) a  2)( a   2)  ( a a  2)( 2 a  2) ( a  ·················································· 2 分 4 2(  ( a 2) ( a   2)( a   a  2) 2) ····················································································3 分 1  2a ········································································································· 4 分当 a  2 2  时 1  1 a  1 2 2 1    1 2 1   2 1  ··························································································· 6 分 20．注：每问 2 分 y A C B P O C A A C x B P B

P ，（3） (2.5 0)     x  x  1 10 是方程的根，且合题意1.2 100 120 21．解：设旅游客车速度为 x Km/h，则小车为1.2x Km/h·········································· 1 分 310 360 ·····························································································3 分 1.2 x x 解方程得 100 ···························································································· 7 分经检验 120 Km/时·····································8 分答：小车的平均速度为 120Km/时·········································································9 分 22．解：（1）在正方形 ABCD 中， AB BC BE CF ABE △ EAB  CBF   EAB  在 ABO△ ··································································································· 1 分（SAS）··············································································2 分 ·························································································· 3 分 90  °··················································································4 分 90  ° AOB  BCF FBC ABO ABO  ≌△        ° ) 90    ° BCD ABO ABC EAB 180     中， 90 ° ，  ( AE BF   ···································································································6 分（2）有 5 对相似三角形···················································································· 7 分 △ △ ···························································9 分 ∽△ ∽△ ∽△ ∽△ BEO BFO △ △ ∽△ ABO ABO BEO BCF AEB BFC ABO ABE △ 23．（1）100··································································································· 2 分（2）············································································································ 5 分（3）3000 0.6 1800 该县优秀人数约为 1800 人················································································· 9 分   人数 40 30 20 10 1     1 3    1 24．解：（1） 2 a  40 30 20 5  39.5 49.5 59.5 69.5 79.5 89.5 100 3 3 ································································· 2 分 4 分数  2 1 4 3  4 ·························································································· 4 分 a  3 1  1 3 4 a  4 1 1 4   1 1 4 1   3 ······························································································ 6 分

a （2） 1  a 2  a 3   a 36 34     4 1 3     12 53  ··············································· 9 分    25．解：（1）令二次函数 y  2 ax  bx  ，则 c a  16 4 b c        0 a b c    c 2 0 ····························································································1 分 1 2 3 2    a      b   2 c   ···································································································· 2 分 过 A B C，，三点的抛物线的解析式为 y   （2）以 AB 为直径的圆圆心坐标为 O    3 0  ，  2  21 x 2  3 2 x  ···································4 分 2 O O  ····················································································· 5 分  CD ····································································· 6 分 3 2  OC 90  ° 90  ° 5 2  O C CD 为圆O 切线  DCO O CD     O CO CO O  △ O CO ∽△ 3 / 2 2 /  2 OD   CDO OD DCO   /  / CO O    O O OC OC OD 8  3 ······················································ 8 分 D 坐标为    80 ， ···························································································9 分 3    （3）存在·····································································································10 分 3(   ，或 2 r r ) F 3( r   ， 2 r ) 抛物线对称轴为 X   3 2 设满足条件的圆的半径为 r ，则 E 的坐标为而 E 点在抛物线 y      r 1 2 1 r    1 (   r 2 )  3 2 29 2  3 2 (   21 x 2 3 2 3 2 x  上 2 r ) 2  1 r    2 29 2

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2009年湖南省张家界市中考数学真题及答案.doc

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