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    2008年宁波市初中毕业生学业考试中考数学试卷及解析

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    宁波市2008年初中毕业生学业考试
    数学试题
    一、选择题(每小题3分,共36分,在每小题给出的四个选项中,只有一项符合题目要求1．比3大的实数是(A．5
    B．0

    C．3

    D．2
    2．下列运算正确的是(A．xxx
    3
    3
    6
    B．2x3x6x
    23
    C．(2x6x
    33
    (2xxx2xD．
    2
    3．下列事件是不确定事件的是(
    A．宁波今年国庆节当天的最高气温是35℃B．在一个装着白球和黑球的袋中摸球,摸出红球C．抛掷一石头,石头终将落地
    D．有一名运动员奔跑的速度是20米/秒
    4．如图,已知∠1∠2∠355,则∠4的度数是(A．110
    B．115
    C．120
    D．125
    4
    1
    3
    2
    （第4题）
    5．2008年5月12日,四川汶川发生了特大地震．震后,国内外纷纷向灾区捐物捐款,截至5月26日12时,捐款达308.76亿元．把它用科学记数法表示为(A．30.87610元
    119

    B．3.087610元D．3.087610元
    11
    10
    C．0.3087610元
    y
    6．如图,正方形ABOC的边长为2,反比例函数y则k的值是(A．2B．2
    k
    过点A,x
    AB
    CO
    x
    C．4

    D．4
    （第6题）
    7．在平面直角坐标系中,点(3，2关于原点对称的点是(A．(2，3
    B．(3，2
    C．(3，2

    D．(3，2
    8．已知圆锥的母线长为5,底面半径为3,则圆锥的表面积为(．．．
    A．15B．24C．30D．39
    9．已知半径分别为5cm和8cm的两圆相交,则它们的圆心距可能是(A．1cmB．3cmC．10cmD．15cm
    10．由一些完全相同的小立方块搭成的几何体的三视图如图所示,
    主视图那么搭成这个几何体所用的小立方块的个数是(
    A．8B．7C．6D．5
    11．甲、乙、丙三个同学排成一排拍照,则甲排在中间的概率是(俯视图

    左视图
    （第10题）


    A．
    16
    B．
    14
    C．
    13
    D．
    12
    12．如图,某电信公司提供了A，B两种方案的移动通讯费用y(元与通话时间x(元之间的关系,则以下说法错误的是(．．
    70A．若通话时间少于120分,则A方案比B方案便宜20元
    50B．若通话时间超过200分,则B方案比A方案便宜12元
    30
    C．若通讯费用为60元,则B方案比A方案的通话时间多
    D．若两种方案通讯费用相差10元,则通话时间是145分或185分
    y（元）
    A方案
    B方案
    120170200250
    x（分）
    （第12题）
    试题卷Ⅱ
    二、填空题(每小题3分,共18分13．计算3(3．
    2
    14．若实数x，y满足x2(y30,则xy的值是．
    20
    15．分解因式2x12x18．
    16．课外活动小组测量学校旗杆的高度．如图,当太阳光线与地面成
    A
    2
    35时,测得旗杆AB在地面上的投影BC长为23.5米,则旗杆AB
    的高度约是米(精确到0.1米
    17．宁波市2008年初中毕业生学业考试各科的满分值如下:
    科目满分值
    语文120
    数学120
    英语110
    科学150
    社政80
    B
    35°
    （第16题）体育30
    C
    若把表中各科满分值按比例绘成扇形统计图,则表示数学科学的扇形的圆心角应是度(结果保留3个有效数字．
    18．如图,菱形OABC中,∠A120,OA1,将菱形OABC绕点O按顺时针方向旋转90,则图中由BB,BA,AC,CB
    BA
    C
    A
    BC
    O
    （第18题）
    围成的阴影部分的面积是．
    三、解答题(第19~21题各6分,22题9分,23题8分,24题9分,25题10分,26题12分,共66分
    a1a2a
    19．化简．a1a21

    3(x2≥x4，
    20．解不等式组x1
    1.2





    21．(1如图1,△ABC中,∠C90,请用直尺和圆规作一条直线,把△ABC分割成两个等腰三角形(不写作法,但须保留作图痕迹．
    (2已知内角度数的两个三角形如图2、图3所示．请你判断,能否分别画一条直线把它们分割成两个等腰三角形？若能,请写出分割成的两个等腰三角形顶角的度数．
    CCC

    84°104°24°24°52°BAABBA

    图1图2图3

    （第21题）

    22．2008年8月8日,第29届奥运会将在北京举行．现在,奥运会门票已在世界各地开始销售,下图是奥运会部分项目的门票价格:
    北京2008年奥运会部分项目门票价格统计图
    价格（元）
    1200
    1000
    1000
    800800800800
    800600500
    400200500
    田径
    50篮球
    60跳水
    40足球
    30
    50
    项目

    (1从以上统计图可知,同一项目门票价格相差很大,分别求出篮球项目门票价格的极差和跳水项目门票价格的极差．
    (2求出这6个奥运会项目门票最高价的平均数、中位数和众数．
    (3田径比赛将在国家体育场“鸟巢”进行,“鸟巢”内共有观众座位9.1万个．从安全角度考虑,正式比赛时将留出0.6万个座位．某场田径赛,组委会决定向奥运赞助商和相关部门赠送还1.5万张门票,其余门票全部售出．若售出的门票中最高价门票占10%至15%,其他门票的平均价格是300元,你估计这场比赛售出的门票收入约是多少万元？请说明理由．
    23．如图,
    最低价最高价
    游泳乒乓球
    8,以点C为顶点的抛物线ABCD中,AB4,点D的坐标是(0，
    yD
    C
    yax2bxc经过x轴上的点A，B．
    (1求点A，B，C的坐标．
    (2若抛物线向上平移后恰好经过点D,求平移后抛物线的解析式．

    OAB
    x
    （第23题）



    24．如图,点C是半圆O的半径OB上的动点,作PCAB于C．点D是半圆上位于PC左侧的点,连结BD交线段PC于E,且PDPE．(1求证:PD是O的切线．
    P
    2
    (2若O的半径为43,PC83,设OCx，PDy．①求y关于x的函数关系式．②当x
    D
    E
    A
    B
    OC（第24题）
    3时,求tanB的值．

    25．2008年5月1日,目前世界上最长的跨海大桥——杭州湾跨海大桥通车了．通车后,苏南A地到宁波港的路程比原来缩短了120千米．已知运输车速度不变时,行驶时间将从原来的3时20分缩短到2时．
    (1求A地经杭州湾跨海大桥到宁波港的路程．
    (2若货物运输费用包括运输成本和时间成本,已知某车货物从A地到宁波港的运输成本是每千米1.8元,时间成本是每时28元,那么该车货物从A地经杭州湾跨海大桥到宁波港的运输费用是多少元？
    (3A地准备开辟宁波方向的外运路线,即货物从A地经杭州湾跨海大桥到宁波港,再从宁波港运到B地．若有一批货物(不超过10车从A地按外运路线运到B地的运费需8320元,其中从A地经杭州湾跨海大桥到宁波港的每车运输费用与(2中相同,从宁波港到B地的海上运费对一批不超过10车的货物计费方式是:一车800元,当货物每增加1车时,每车的海上运费就减少20元,问这批货物有几车？
    26．如图1,把一张标准纸一次又一次对开,得到“2开”纸、“4开”纸、“8开”纸、“16开”纸…．已知标准纸的短边长为a．．．．(1如图2,把这张标准纸对开得到的“16开”张纸按如①标准纸“2开”纸、“4
    开”纸、“8开”纸、“16下步骤折叠:
    开”纸……都是矩形．
    第一步将矩形的短边AB与长边AD对齐折叠,点B②本题中所求边长或面积
    都用含a的代数式表示．落在AD上的点B处,铺平后得折痕AE；
    第二步将长边AD与折痕AE对齐折叠,点D正好与点E重合,铺平后得折痕AF．
    则AD:AB的值是,AD，AB的长分别是,．(2“2开”纸、“4开”纸、“8开”纸的长与宽之比是否都相等？若相等,直接写出这个比值；若不相等,请分别计算它们的比值．
    (3如图3,由8个大小相等的小正方形构成“L”型图案,它的四个顶点E，F，G，H分别



    在“16开”纸的边AB，BC，CD，DA上,求DG的长．
    (4已知梯形MNPQ中,MN∥PQ,∠M90,MNMQ2PQ,且四个顶点
    M，N，P，Q都在“4开”纸的边上,请直接写出2个符合条件且大小不同的直角梯形的面
    积．
    4开
    a
    2开
    8开
    16开图1
    A
    B
    DF
    AE
    H
    DG
    B
    E图2
    C
    B
    F图3
    C
    （第26题）
    宁波市二2008年初中毕业生学业考试
    数学试题参考答案及评分标准

    一、选择题(每小题3分,共36分题号1234567
    二、填空题(每小题3分,共18分题号
    13
    14
    15
    16
    17
    18
    89101112
    C
    D
    答案CBADBDDBCA
    2
    答案232(x316.570.8
    89
    23π32

    三、解答题(共66分
    注:1．阅卷时应按步计分,每步只设整分；
    2．如有其它解法,只要正确,都可参照评分标准,各步相应给分．19．解:原式
    a1a(a1
    ····································································2分a1(a1(a1

    a1a
    ·············································································4分a1a11······················································································6分a1
    20．解:解不等式(1,得x≥1．·······································································2分解不等式(2,得x3．·····················································································4分
    原不等式组的解是1≤x3．····································································6分



    21．解:(1如图,直线CM即为所求
    CC
    或
    AABMBM
    ···············································································3分(作图正确,不写结论不扣分(2图2能画一条直线分割成两个等腰三角形,························································4分分割成的两个等腰三角形的顶角分别是132和84．··············································5分图3不能分割成两个等腰三角形．·····································································6分22．解:(1篮球项目门票价格的极差是100050950(元······································1分跳水项目门票价格的极差是50060440(元····················································2分(2这6个奥运会项目门票最高价的平均数是
    11
    (10005008004783(元63
    ····································································4分(写成783.33,783.3或783都不扣分
    中位数800元,众数800元．··············································································6分(3(答案不唯一,合理即正确,如2520万元,理由如下:···············································7分售出的门票共9.10.61.57(万张································································8分这场比赛售出的门票最低收入为:710%800(7710%3002450(万元
    这场比赛售出的门票最高收入为:715%800(7715%3002625(万元········9分23．解:(1在
    ABCD中,CD∥AB且CDAB4,
    点C的坐标为(4，····················································································1分8·
    设抛物线的对称轴与x轴相交于点H,则AHBH2,···························································································2分
    点A，B的坐标为A(2，，······························································4分0B(6，0．·
    (2由抛物线yaxbxc的顶点为C(4，8,
    可设抛物线的解析式为ya(x48,····························································5分把A(2，0代入上式,
    解得a2．································································································6分设平移后抛物线的解析式为y2(x48k
    把(0，··············································································7分8代入上式得k32·
    2
    2
    2
    平移后抛物线的解析式为y2(x4240．················································8分
    即y2x16x8．

    2


    24．解:(1连结OD,OBOD,
    ····················································································1分OBDODB．·
    PDPE,
    ····················································································2分PDEPED．·
    PDOPDEODEPEDOBDBECOBD
    90,
    ·····························································································3分PDOD．·
    ···················································································4分PD是圆O的切线．·
    (2①连结OP,在Rt△POC中,
    OP2OC2PC2
    x192．···························································································5分在Rt△PDO中,
    ························································································6分PD2OP2OD2·
    2
    x2144．
    yx2144(0≤x≤43．······································7分(x取值范围不写不扣分
    ②当x
    3时,y147,
    PD73,································································································8分
    EC3,
    而CB33,在Rt△ECB中,
    CE1
    ·························································································9分．·
    CB3
    25．解:(1设A地经杭州湾跨海大桥到宁波港的路程为x千米,
    x120x
    由题意得····················································································2分,·
    1023
    解得x180．
    ·········································4分A地经杭州湾跨海大桥到宁波港的路程为180千米．·
    (21.8180282380(元,
    该车货物从A地经杭州湾跨海大桥到宁波港的运输费用为380元．························6分tanB



    (3设这批货物有y车,
    由题意得y[80020(y1]380y8320,······················································8分整理得y60y4160,
    解得y18,y252(不合题意,舍去,··································································9分
    2
    这批货物有8车．·····················································································10分
    26．解:(12，
    21
    a，a．···············································································3分44
    (2相等,比值为2．···························5分(无“相等”不扣分有“相等”,比值错给1分(3设DGx,
    在矩形ABCD中,BCD90,
    HGF90,
    DHGCGF90DGH,
    △HDG∽△GCF,
    DGHG1,CFGF2
    ·····················································································6分CF2DG2x．·
    同理BEFCFG．EFFG,△FBE≌△GCF,
    1
    ·················································································7分BFCGax．·
    4
    CFBFBC,
    122xaxa,···················································································8分
    44
    解得x
    21
    a．4
    21
    a．························································································9分4
    即DG(4
    32
    ···································································································10分a,·
    16
    271822
    a．12分8






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