2022年上海普陀区初三数学一模试卷及答案.docx

名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -普陀区 2022 学年度第一学期初三质量调研数 学 试 卷（时间： 100 分钟,满分： 150 分）考生留意：1本试卷含三个大题,共 25 题答题时,考生务必按答题要求在答题纸规定的位置上作答,在草稿纸、本试卷上答题一律无效2除第一、二大题外,其余各题如无特殊说明,都必需在答题纸的相应位置上写出证明或计 算的主要步骤 一、挑选题： （本大题共6 题,每题4 分,满分24 分） 以下各题的四个选项中,有且只有一个选项是正确的,挑选正确项的代号并填涂在答题纸的相应位置上 1已知x3,那么以下等式中,不肯定正确选项（ ）3 5y5（A ） 5 =3y ；（B）x y8；（C）x y8；（D）xxy5yy2以下二次函数中,假如函数图像的对称轴是y 轴,那么这个函数是（ ）2 1（A ）y2 x2x ； （B）yx22x1； （C）yx22；（D）yx3已知在 Rt ABC 中,C90,sinA1,那么以下说法中正确选项（ ）3（A ）cosB1；（B）cotA1；（C）tanA2 2；（D）cotB2 233334以下说法中,正确选项（ ）（A ）假如k0r, a是非零向量,那么r ka0；r（B）假如 e是单位向量,那么r e1；r（C）假如 br ar,那么 br ar 或 br a；（D）已知非零向量r a, 假如向量r b5r ar,那么 ar b1 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 1 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -5假如二次函数yxm2n 的图像如图1 所示 ,y那么一次函数 y mx n 的图像经过（ ）（A ）第一、二、三象限；（B）第一、三、四象限；O x（C）第一、二、四象限；（D）其次、三、四象限图 16如图 2,在 RtABC 中,ACB 90, CD AB ,垂足为点 D ,假如 CADC 3,CCDB 2AD 9,那么 BC 的长是（ ）C（A ）4；（B）6；（C） 2 13 ； （D） 3 10 二、填空题： （本大题共 12 题,每题 4 分,满分 48 分）A图 2 D B17化简 ：2 a b a b 228抛物线 y a 2 x 在对称轴左侧的部分是上升的,那么 a 的取值范畴是29已知函数 f x 3 x 2 x 1,假如 x 2,那么 f x 210假如抛物线 y ax 2 ax c 与 x 轴的一个交点的坐标是 1,0 ,那么与 x 轴的另一个交点的坐标是211将二次函数 y x 2 x 2 的图像向下平移 m m 0 个单位后,它的顶点恰好落在 x轴上,那么 m 的值等于12已知在 RtABC 中,C 90,cot B 1,BC 2,那么 AC313如图 3, ABC 的中线 AD 、 CE 交于点 G ,点 F 在边 AC 上, GF / BC ,那么GF 的BC值是14如图 4,在 ABC 与 AED中,AB BC,要使 ABC 与 AED相像,仍需添加AE ED一个条件,这个条件可以是（只需填一个条件）AA ADEFGEB D C B C C D B图 3 图 4 图 52 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 2 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -15如图 5,在 RtABC中,C90,AD 是三角形的角平分线, 假如AB3 5,AC2 5,那么点 D 到直线 AB 的距离等于16如图 6,斜坡 AB长为 100 米,坡角 ABC 30,现因“ 改小坡度” 工程的需要,将斜坡 AB改造成坡度 i 1:5 的斜坡 BD （A、D、 C 三点在地面的同一条垂线上）,那么由点 A 到点 D 下降了米（结果保留根号）A A ADDOC,B图 6CC图 7BB图 817如图 7,在四边形ABCD 中,ABC90,对角线 AC 、 BD 交于点 O , AOCO ,CDBD ,假如CD3,BCPC35,那么 AB90,AC5,sinB5,点 P 为边 BC 上一点,18如图 8,在 Rt ABC 中,C13将 ABC 绕点 P 旋转得到 A B C （点 A 、B 、C 分别与点 A 、B 、C 对应）,使 B C/ AB ,边 A C 与边 AB 交于点 G ,那么 A G 的长等于三、解答题： （本大题共7 题,满分78 分）19（此题满分10 分）运算：2 2sin 60cos602 tan 604cos4520.（此题满分10 分）如图 9,在 ABC 中,点 D 、 E 、F 分别在边 AB、 AC 、 BC 上, DE / BC , EF / AB ,AD AB1:35时,求 FC 的长；uuur, EAr（用向量 ar、b表示）（1）当DEuuur（2）设 ADr auuur, CFr buuur,那么 FEABFDEC图 93 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 3 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -21（此题满分 10 分）如图 10,在 ABC 中,点 P 、D 分别在边 BC 、AC 上,PAAB ,垂足为点 A,DPBC ,垂足为点 P ,AP PDBPCD（1）求证：APDC ；（2）假如AB3,DC2,求 AP 的长ADBPC图 1022（此题满分10 分）A3, k2,函数ym与函数yx（ m 、 k 为不等于零的常数）的图像有一个公共点xk其中正比例函数y 的值随 x 的值增大而减小,求这两个函数的解析式23（此题满分12 分）已知：如图11,四边形 ABCD 的对角线 AC 、 BD 相交于点 O ,SAODSBOC（1）求证：DOCO；OBOASC（2）设 OAB 的面积为 S ,CDk,求证：S 四边形ABCDk2 1ABDOA B图 114 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 4 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -24（此题满分 12 分）在平面直角坐标系 xOy 中（如图 12）,已知抛物线 y ax 2 a 8 x c a 0 经3过点 A 3, 2,与 y 轴交于点 B 0, 2,抛物线的顶点为点 C ,对称轴与 x 轴交于点 D （1）求抛物线的表达式及点 C 的坐标；（2）点 E 是 x轴正半轴上的一点,假如 AED BCD ,求点 E 的坐标；（3）在（ 2）的条件下,点 P是位于 y 轴左侧抛物线上的一点,假如 PAE是以 AE 为直角边的直角三角形,求点 P的坐标1O 1图 125 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 5 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -25（此题满分 14 分）如图 13,在梯形 ABCD 中, AD / BC ,C90,AD2,BC5,DC3,点 E 在边 BC 上, tanAEC3点 M 是射线 DC 上一个动点（不与点D 、 C 重合）,联结 BM 交射线 AE 于点 N ,设 DMx , ANy （1）求 BE 的长；（2）当动点 M 在线段 DC 上时,试求y与 x之间的函数解析式,并写出函数的定义域；（3）当动点 M 运动时, 直线 BM 与直线 AE 的夹角等于 45 ,请直接写出这时线段 DM 的长A DMNBEC图 13A DBEC备用图6 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 6 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -普陀区 2022 学年度第一学期初三质量调研数学试卷参考答案及评分说明一、挑选题： （本大题共6 题,每题 4 分,满分 24 分）5B；6 C1B ；2C；3A ；4D；二、填空题：（ 本大题共12 题,每题 4 分,满分 48 分）7a2b ；8a2；AC9 7；103 0；11 1；E（AB AE12 6；131 3；14 B等）； 15 2 ；AD1715 41820 1316 5010 3 ；三、解答题（本大题共 7 题,其中第 19-22 题每题 10 分,第 23、24 题每题 12 分,第 25 题 14 分,满分78 分）3 2 12 19解：原式 2 2················· ····· ············· ········· ····· ····· ····· ········ （4 分） 3 24 223 12 2······ ·················· ····· ··· ····· ····· ·············· ····· ··· ····· ·················· （3 分）3 2 23 2 2 ···· ··················· ····· ········ ····· ·············· ············· ·················· （3 分）20解：（1）DE / BC , EF / AB , DE BF · ·············· ····· ····· ········ ····· ·············· ············· ·················· （1 分）DE 5,BF 5···· ····· ·················· ············· ·············· ····· ····· ··· ····· （1 分） DE / BC ,AD ABDE BC · ·············· ····· ····· ········ ····· ·············· ············· ·················· （1 分）AD AB1,5 BC1·· ····· ·················· ············· ·············· ····· ····· ··· ····· （1 分）337 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 7 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -解得BC15, ············· ····· ····· ············· ·············· ······························· （1 分）FC 10······ ·················· ····· ··· ····· ····· ·············· ····· ··· ····· ·················· （1 分）（2） FE uuur2a r, EA uuura r 1b r ············· ····· ·············· ········ ····· ····· ······ （2 分+2 分）221解：（1）PA AB , DP PC ,BAP CPD 90······················· ············· ········· ····· ····· ····· ········ （1 分）在 Rt ABP与 RtPCD 中,AP BP,PD CD RtABP RtPCD ······················ ············· ········· ····· ····· ····· ········ （1 分）APB PDC ··· ····· ····· ·················· ·············· ·················· ············· （1 分）DPB APB APD ,DPB PDC C ,得 APD C ··········· ····· ····· ············· ·············· ······························· （2 分）（ 2）Rt ABPRt PCD B C AB AC · ·············· ····· ····· ········ ····· ·············· ············· ·················· （1 分）AB 3,DC 2,AD 1 ········· ····· ·············· ········ ····· ····· ············· （1 分）APD C ,PAD CAP ,APDACP · ····· ····· ·················· ············· ··················· ········ ····· （1 分）AD AP············· ····· ····· ············· ·············· ······························· （1 分）AP AC得 AP 3·· ·············· ····· ····· ········ ····· ·············· ············· ·················· （1 分）22解：由点 A 3, k 2 在函数 y x的图像上,可得kk 2 3 ············ ····· ····· ············· ·············· ······························· （1 分）k2整理,得 k 2 k 3 0 ················· ····· ············· ········· ····· ····· ····· ········ （1 分）解得 k 1 3,k 2 1························ ········ ····· ·············· ····· ····· ··· ····· （2 分）正比例函数 y 的值随 x 的值增大而减小,k 1······ ·················· ····· ··· ····· ····· ·············· ····· ··· ····· ·················· （2 分）8 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 8 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -得yx ,点 A 3, 3 ·············· ····· ············· ········· ····· ····· ············· （2 分）由点 A 3, 3 在函数 y m 的图像上,可得xm 9 · ·············· ····· ····· ········ ····· ·············· ············· ·················· （1 分）y 9 ···· ··················· ····· ········ ····· ·············· ············· ·················· （1 分）x两个函数的解析式分别为 y x ,y 9x23证明：（1）过点 A 作 AH BD ,垂足为点 H .· ····· ·············· ············· ·················· （1 分） S AOD= 1 DO AH , S AOB= 1 OB AH , 2 2S AOD 12 DO AH DO . ··········· ····· ·············· ········ ····· ····· ············· （2 分）S AOB 1 OB AH OB2同理,S BOC CO · ····· ····· ·················· ············· ··················· ········ ····· （1 分）S AOB OASAOD SBOC,DO CO············ ····· ····· ············· ·············· ······························· （1 分）OB OA（2）DO CO,COD AOB,OB OA OCD OAB ··· ····· ·················· ············· ·············· ····· ····· ··· ····· （1 分）CD DO CO k ················· ····· ············· ········· ····· ····· ····· ········ （1 分）AB BO AO2S OCD CD k 2················ ····· ············· ········· ····· ····· ············· （1 分）S OAB AB OAB 的面积为 S,S OCD k 2 S . ·············· ·················· ············· （1 分）又S AOD DO k,S AOD k S·············· ·················· ············· （1 分）S OAB OB同理,S BOC k S . ·· ····· ·················· ············· ·············· ····· ····· ··· ····· （1 分）9 细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 9 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -S 四边形ABCDSAOBSBOCSCODSDOAS k S k 2 S k S k 2 2 k 1 S k 1 2 S··············· ····· ············· ········· ····· ····· ············· （1 分）2 824解：（1）由抛物线 y ax a x c 经过点 A 3, 2 和点 B 0, 2,3c 2,a 4 ,得 8 解得 3·· ·············· ······························· （2 分）9 a 3 a3 c 2. c 2.抛物线的表达式是 y 4 x 24 x 2· ·············· ······························· （1 分）3点 C 的坐标是 3 , 5················· ····· ············· ········· ····· ····· ····· ········ （1 分）2（2）联结 AB交 CD 于点 F ,过点 A 作 AH OD , H 为垂足 A 3, 2, B 0, 2,AB 3由对称性可得 BF 3························ ········ ····· ·············· ····· ····· ··· ····· （1 分）2CD 5,CF 3在 Rt BCF 中,tan BCF BF 1 ···· ·············· ······························· （1 分）CF 2在 Rt AEH 中, tan AEH AH,EHAED BCD ,AH 1EH 4························ ········ ····· ·············· ····· ····· ··· ····· （1 分）EH 2OH 3,OE 1点 E 的坐标是 1,0 ··· ····· ·················· ············· ·············· ····· ····· ··· ····· （1 分）（3） PAE 是以 AE 为直角边的直角三角形,PAE 90 或 PEA 90设点 P 点的坐标为 m , 4 m 24 m 23当 PAE 90 时,点 P 只能在 AE 的下方过点 P 作 PG AH , G 为垂足10细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 10 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -PGm3,AG42 m4m 3GAE AHE AEH ,GAE PAE PAG ,PAG AEH tan PAG tan AEH PG AHm 3 1·· ····· ·············· ············· ·················· （1 分）AG EH 4m 24 m 233解得 m 3,m2m 3 不合题意舍去,m 323点 P 的坐标是 , 5 ·············· ····· ············· ········· ····· ····· ············· （1 分）2当 PEA 90 时同理可得点 P 的坐标是 9 129 13 , 129 ··· ········· ····· ····· ····· ········ （2 分）4 225解：（1）过点 A 作 AH BC , H 为垂足 AH BC ,AHE 90C 90,AHE C AH / DC AD / BC ,DC 3AH DC 3 ·······························································（1 分）同理可得 HC AD 2···························································································（1 分）在 Rt AEH 中,AHE 90, tan AEH 3,AH 3HEEH 1················································································································（1 分）BC 5,BE 2·····························································································（1 分）（ 2）延长 BM 、 AD 交于点 G ·············································································（1 分） DG / BC ,DG DMBC MC由 DM x,DC 3,BC 5,得 DG x,解得 DG 5 x·········································································（1 分）5 3 x 3 xAG 6 3 x ·········································································································（1 分）3 x AG / BC ,AN AGBN BE在 Rt AEH 中,AHE 90,EH 1,AH 3,11细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 11 页,共 12 页 - - - - - - - - - 名师归纳总结 精品学习资料 - - - - - - - - - - - - - - -可得AE10········································································································（1 分）63x3）····························································（2 分）由yy32x,10化简,得y3 10x6 10（ 0x12x（ 3）当点 M 在线段 DC 上时,DM1 2 ·························································（2 分）当点 M 在线段 DC 的延长线上时,点N 在线段 AE 的延长线上,DM13（ 2 分）12细心整理归纳 精选学习资料 - - - - - - - - - - - - - - - 第 12 页,共 12 页 - - - - - - - - -