上海市新世纪高二上期中考试
2023-2024学年上海市黄浦区高二数学上学期期中考试卷附答案解析
2023-2024学年上海市黄浦区高二数学上学期期中考试卷2023-11（试卷总分为100分，考试时间为120分钟.）一、填空题（本大题共12题，每题3分，满分36分）1．已知一个球的半径为3，则这个球的表面积为.2．若平面l αβ= ，直线a α⊂，直线,b a b M β⊂⋂=，则点M 与l 的位置关系为．3．若向量()4,2,1a =-与向量()2,,b x y =共线，则x y -=.4．在正方体1111ABCD A B C D -中，异面直线11B D 与CD 所成角的大小是.5．如图所示，'''A B O V 是利用斜二测画法画出的ABO 的直观图，已知'''A B y P 轴，''4O B =，且ABO 的面积为16，过A '作'''AC x ⊥轴，则''A C 的长为．6．已知一个圆柱的轴截面为正方形，且它的侧面积为16π，则该圆柱的体积为.7．如图，在四棱锥P ﹣ABCD 中，PA ⊥底面ABCD ，底面ABCD 是边长为1的正方形，PA ＝1，则侧面PCD 与底面ABCD 所成的二面角的大小是．8．如图，在正三棱柱111ABC A B C -中，122AB AA ==，N 为11A C 的中点，M 为线段1AA 上的点.则MN MB+的最小值为9．已知向量(2,3,0)a = ，(1,0,3)b = ,则向量a 在向量b方向上投影向量的坐标为．10．正方体1111ABCD A B C D -的棱长为2,E 是棱1DD 的中点，则平面1AC E 截该正方体所得的截面面积为.11．如图所示，在平行六面体1111ABCD A B C D -中，底面ABCD 为菱形，且11π3A AB A AD BAD ∠=∠=∠=，则侧棱1AA 与底面ABCD 所成的角为．12．如图,在正方体1111ABCD A B C D -中,点P 在线段1BC 上运动,有下列判断:①平面1PB D ⊥平面1ACD ;②1//A P 平面1ACD ;③异面直线1A P 与1AD 所成角的取值范围是0,3π⎛⎤ ⎥⎝⎦;④三棱锥1D APC -的体积不变.其中,正确的是(把所有正确判断的序号都填上).二、单选题（本大题共4题，每题4分，满分16分）13．若球的表面积扩大到原来的4倍，那么该球的体积扩大到原来的（）A ．64B ．32C ．16D ．814．如图，在直三棱柱111ABC A B C -中，AB BC ⊥，AB BC =，1AA AC ==E 为棱11A B 的中点，点F 是棱BC 上的一点，且3BF FC =，则直线AE 与1C F所成角的余弦值为（）A ．1699B ．3299C ．99D ．9915．阅读材料：空间直角坐标系O xyz -中，过点()000,,P x y z 且一个法向量为(),,n a b c =的平面a 的方程为()()()0000a x xb y yc z z -+-+-=；过点()000,,P x y z 且一个方向向量为(,,)(0)d u v w uvw =≠的直线l 的方程为000x x y y z z u v w ---==.利用上面的材料，解决下面的问题：已知平面a 的方程为3570x y z -+-=，直线l 的方向向量为()3,1,2m =- ，则直线l 与平面a 所成角的正弦值为（）A ．B ．C ．D ．16．已知正方体1111ABCD A B C D -的棱长为1，,P Q 分别为棱11C D ，1B C 上的动点，则四面体PQAD 的体积最大值为（）A ．16B ．14C ．13D ．12三、解答题（本大题共5题，满分48分）17．已知空间中的三点(2,0,2),(1,1,2),(3,0,4)P M N ---，a PM = ，b PN = .(1)求PMN 的面积；(2)当ka b + 与2ka b -的夹角为钝角时，求k 的范围．18．已知1111ABCD A B C D -是底面边长为1的正四棱柱，高12AA =．求：⑴异面直线BD 与1AB 所成的角的大小（结果用反三角函数表示）；⑵四面体11AB D C的体积．19．如图，将边长为2的正方形ABCD 沿对角线BD 折叠，使得平面ABD 与平面CBD 所成二面角为直角，⊥AE 平面ABD ，且AE =(1)求证：直线EC 与平面ABD 平行；(2)求点C 到平面BED 的距离.20．如图，在四棱锥P ABCD -中，已知PA ⊥平面ABCD ，且四边形ABCD 为直角梯形，2ABC BAD π∠=∠=，2PA AD ==，1AB BC ==．(1)证明：AB PD ⊥；(2)线段PC 上是否存在一点M ，使得直线AM 垂直平面PCD ，若存在，求出线段AM 的长，若不存在，说明理由．21．如图，AB 是圆O 的直径，点C 是圆O 上异于,A B 的点，PO 垂直于圆O 所在的平面，且1PO =OB =．（Ⅰ）若D 为线段AC 的中点，求证C A ⊥平面D P O ；（Ⅱ）求三棱锥-P ABC 体积的最大值；（Ⅲ）若2BC =E 在线段PB 上，求CE OE +的最小值．1．36π【分析】根据球的表面积公式求出答案.【详解】设3r =，则24π6π3r =，故这个球的表面积为36π.故答案为：36π2．M l∈【分析】根据基本事实3（公理2）求解即可.【详解】因为a b M = ，所以M ∈直线a ，M ∈直线b ，因为直线a α⊂，直线b β⊂，所以M ∈平面α，M ∈平面β，又平面l αβ= ，所以M l ∈.故答案为：M l ∈.3．12-##0.5-【分析】根据向量共线基本定理，可设,R a b λλ=∈，列出方程组，即可求得x 和y 的值，进而求出x y-的值.【详解】由向量()4,2,1a =-与向量()2,,b x y =平行，可设,R a b λλ=∈ ，则4221x y λλλ-=⎧⎪=⎨⎪=⎩，解得112x y =-⎧⎪⎨=-⎪⎩，所以11122x y -=-+=-.故答案为：12-.4．π4【分析】根据异面直线所成角定义，平移直线CD 到11C D 使其与11B D 相交，解三角形即可．【详解】如图，在正方体1111ABCD A B C D -中，因为11CD C D ，所以111C D B ∠为异面直线为11B D 与CD 所成角，又因为111C D B 是以1C ∠为直角的等腰直角三角形，所以111π=4C D B ∠，即异面直线11B D 与CD 所成角为π4．故答案为：π4．5．22【分析】结合已知条件利用直观图与原图之间的面积关系得到A B O '''的面积，进而得到A C ''.【详解】因为16ABO S =V ，'''24A B O ABOS S =V V ，''4O B =所以'''''''1422A B O S O B AC ==⨯⨯V ，即''2AC =故答案为：2.6．16π【分析】设圆柱底面的半径为R ，高为h ，根据题意，由2216h RRh ππ=⎧⎨=⎩求解.【详解】解：设圆柱底面的半径为R ，高为h ，则2216h R Rh ππ=⎧⎨=⎩，解得24R h =⎧⎨=⎩，所以圆柱的体积2π16πV R h ==.故选：16π7．45°【分析】由题意可证得CD ⊥平面PAD ，从而∠PDA 为侧面PCD 与底面ABCD 所成的二面角的平面角，求解即可.【详解】因为底面ABCD 是边长为1的正方形，所以AD ⊥CD ，又因为PA ⊥底面ABCD ，CD ⊂底面ABCD ，所以PA ⊥CD ，因为PA∩AD ＝A ，PA 、AD 在面PAD 内，所以CD ⊥平面PAD ，又因为PD ⊂平面PAD ，所以CD ⊥PD ，于是∠PDA 为侧面PCD 与底面ABCD 所成的二面角的平面角，因为PA ⊥底面ABCD ，AD ⊂底面ABCD ，PA ⊥AD ，又因为PA ＝1，AD ＝1，所以∠PDA ＝45°，于是侧面PCD 与底面ABCD 所成的二面角的大小为45°．故答案为：45°．8【分析】将侧面11ABB A 沿1A A展开，使得侧面11ABB A 与侧面11ACC A 在同一平面内，根据平面上两点间线段最短可求得答案.【详解】解：将侧面11ABB A 沿1A A展开，使得侧面11ABB A 与侧面11ACC A 在同一平面内，如图，连接BN 交1AA 于M ，则MN MB+的最小值为此时的BN，BN =，∴MN MB +.9．13(,0,)55【分析】根据投影向量的定义即可求解.【详解】向量a 在向量b 方向上投影向量为2113cos ,,0,10555b a b b a a b b b b b b ⋅⎛⎫==== ⎪⎝⎭，故答案为：13,0,55⎛⎫⎪⎝⎭10．【分析】利用平面的性质作出截面1AFC E，然后求解面积即可.【详解】如图所示，设F 为1BB 的中点，连接1,AF FC ，设G 为1CC 的中点，连接,EG GB ，由EG AB ∥且EG AB =，得ABGE 是平行四边形，则AE BG ∥且AE BG =，又1BG C F∥且1BG C F=，得1AE C F且1AE C F=，则1,,,A E C F共面，故平面1AC E截该正方体所得的截面为1AFC E.又正方体1111ABCD A B C D -的棱长为2，所以1111,23,22AF FC EC EA AC EF EF AC ==⊥===，故1AFC E 的面积为12223262S =⨯=故答案为：2611．3【分析】运用平行六面体的性质和三余弦定理即可求得.【详解】如图，连接,AC 作1A O ⊥平面ABCD 于点O ,1A E AB⊥于点E ，1A F AD⊥于点F ,连接,,OE OF 则易得：AB ⊥平面1,AOE AD ⊥平面1AOF ，故有，,AB OE AD OF ⊥⊥,又由11A AB A AD∠=∠可得：11,A AE A AF ≅ 从而有,AE AF =因底面ABCD 为菱形，故,OAE OAF ≅ 可得：=,OE OF 故点O 必在直线AC 上,且侧棱1AA 与底面ABCD 所成的角为1.A AO ∠在1Rt A OA 中，11cos ,AO A AO AA ∠=在Rt AOE △中，cos ,AE OAE AO ∠=在1Rt A AE △中，11cos ,AE A AE AA ∠=故可得：11cos cos cos ,A AO OAE A AE ∠⋅∠=∠而πcos cos6OAE ∠==解得112cos ,332A AO ∠==故得：1A AO ∠=故答案为：12．①②④【解析】根据线面关系,逐项判断,即可求得答案.【详解】对于①, 在正方体中,1B D ⊥平面1ACD ,1B D ⊂平面1PB D,∴平面1PB D ⊥平面1ACD ,故①正确;对于②,连接11,A B A C,如图:容易证明平面11A BC //平面1ACD ,又 1A P ⊂平面11A BC,∴AP ∥平面1ACD ,故②正确;对于③,1BC ∥1AD ,∴异面直线1A P 与1AD 所成的角就是直线AP 与1BC 所成的角,在11A BC V 中,易知所求角的范围是,32ππ⎡⎤⎢⎥⎣⎦,故③错误;对于④,11D APC C AD PV V --= 点C 到平面1AD P 的距离不变,且1AD P △的面积不变,∴三棱锥1D APC -的体积不变,故④正确.综上所述,正确的是①②④.故答案为:①②④.【点睛】本题主要考查线线、线面、面面的平行与垂直关系,异面直线所成的角,三棱锥的体积等知识,解题关键是掌握正方体的特征和数形结合,考查了分析能力和空间想象能力,属于中档题.13．D【分析】由球的表面积和体积公式可知，球的表面积之比为半径比的平方，体积比为半径比的立方.【详解】设扩大前后球半径分别为12,r r ，由表面积之比为22211122222444r r r r r r ππ⎛⎫=== ⎪⎝⎭，得122r r =，则体积之比为333131133222432843r r r r r r ππ⎛⎫==== ⎪⎝⎭.故选：D.14．D【分析】以B 为坐标原点，BC ，BA ，1BB 所在直线分别为x 轴，y 轴，z 轴，建立空间直角坐标系，利用向量法能求出直线AE 与1C F所成角的余弦值.【详解】由AB BC ⊥，AB BC =，AC =2AB BC ==，以B 为坐标原点，BC ，BA ，1BB所在直线分别为x 轴，y 轴，z 轴，建立空间直角坐标系，如图所示.所以3,0,02F ⎛⎫⎪⎝⎭，(12,0,C ，()0,2,0A，(0,1,E ，所以112FC ⎛= ⎝，(0,1,AE =- ，所以1cos ,FC AE ==，即直线AE 与1C F 所成角的余弦值为163399.故选：D.15．A【分析】利用给定信息，求出平面的法向量，再利用线面角的向量求法求解即得.【详解】因为平面a 的方程为3570x y z -+-=，则平面a 的法向量可取()3,5,1n =-r，而直线l 的方向向量为()3,1,2m =-，所以直线l 与平面a所成角的正弦值为||sin |cos ,|35||||m n m n m n θ⋅=〈〉==.故选：A 16．A【分析】作平行辅助线，借助线面平行关系，将所求几何体体积Q PADV -转化为G PADV -，再利用等体积法转化为A PGDV -即可运算求解.【详解】过点Q 作11QG B C ∥交1CC 于G ，连接,,PG GD DP ，又11B C BC AD ∥∥QG AD ∴∥，又QG ⊄平面PAD ，且AD ⊂平面PAD ，//QG ∴平面PAD ，则Q PAD G PAD A PGDV V V ---==，设CG t =，1PD s =，则[],0,1t s ∈，11111(1)(1)(1)2222PGD S s t s t st =-----=-△，故四面体PQAD 的体积()1111(1)13326Q PAD A PGD PGD V V S AD st st --==⋅=⨯-=- ，当0st =时，其最大值为16.故选：A.17．(1)32；(2)5,22k ⎛⎫∈- ⎪⎝⎭.【分析】（1）应用向量坐标表示有(1,1,0)a = ，(1,0,2)b =- ，由向量夹角的坐标运算可得10cos ,10a b <>= ，再求其正弦值，应用三角形面积公式求面积；（2）向量坐标表示得(1,,2)ka b k k +=- ，()22,,4ka b k k -=+- ，它们的夹角θ为钝角，即cos 0θ<，即可求参数范围，注意排除向量反向共线的情况.【详解】（1）由题设(1,1,0)a = ，(1,0,2)b =-，则cos ,||||a b a b a b ⋅<>===，所以cos 10MPN ∠=，故在PMN中sin 10MPN ∠=，故PMN的面积为131032102.（2）由（1）知：(1,,2)ka b k k +=- ，()22,,4ka b k k -=+- ，且它们夹角θ为钝角，所以2cos 0θ=<，即()()21280k k k -++-<，所以()()22102520k k k k +-=+-<，可得522k -<<，当它们反向共线，即(2)ka b ka b λ+=-且0λ<时，有1(2)24k k k k λλλ-=+⎧⎪=⎨⎪=-⎩，无解，综上，5(,2)2k ∈-.18．（1）（2）23【详解】解：⑴连1111,,,BD AB B D AD ，∵1111//,BD B D AB AD =，∴异面直线BD 与1AB 所成角为11AB D ∠，记11AB D θ∠=，2221111111cos 2AB B D AD AB B Dθ+-==⨯∴异面直线BD 与1AB 所成角为．⑵连11,,AC CB CD ，则所求四面体的体积11111111242433ABCD A B C D C B C D V V V --=-⨯=-⨯=．19．(1)证明见解析(2)1d =【分析】（1）取BD 的中点F ，连接CF 、AF ，证明//EC 平面ABD 即可得解；（2）在三棱锥C BED -中，利用等体积法即可求出点C 到平面BED 的距离.【详解】（1）证明：取BD 的中点F ，连接CF 、AF，如图，依题意，在BCD △中，,BC CD BC CD =⊥，则CF BD ⊥，而平面ABD 与平面CBD 所成二面角为直角，即平面ABD ⊥平面CBD ，又平面ABD ⋂平面CBD BD =，CF ⊂平面CBD ，于是得CF ⊥平面ABD，且CF =因⊥AE 平面ABD，且AE =//AE CF ，且AE CF =，从而得四边形AFCE 为平行四边形，//EC AF ，又AF ⊂平面ABD ，EC ⊂/平面ABD ，则//EC 平面ABD ；（2）解：由（1）可得AF ⊥平面,//,CBD EC AF EC AF ==于是得EC ⊥平面CBD，EB ED ===则等腰BED 底边BD上的高2h =，12BED S BD h =⋅= 而2BCD S =，设点C 到平面BED 的距离为d ，由C BEDE BCD V V --=得1133BED BCD S d S EC⋅=⋅ ，即2=1d =，所以点C 到平面BED 的距离为1.20．(1)证明见解析(2)存在，AM =【分析】（1）由线面垂直得线线垂直PA AB ⊥，再由底面上的AB AD ⊥，可得AB ⊥平面ADP ，从而证得线线垂直；（2）建立如图所示的空间直角坐标系，用空间向量法表示线面垂直，求得AM ，得其长度．【详解】（1）证明：∵在四棱锥P ABCD -中，PA ⊥面ABCD ，AB ⊂面ABCD ，AD ⊂面ABCD ，∴PA AB ⊥，PA AD ⊥.在直角梯形ABCD 中，AB AD ⊥，2ABC BAD π∠=∠=.又AD ⊂面,ADP AP ⊂面ADP ，AD AP A = ,∴AB ⊥平面ADP ，又PD ⊂面ADP ，∴AB PD ⊥；（2）由题意及（1）得，存在一点M ，使得直线AM 垂直平面PCD .在四棱锥P ABCD -中，2PA AD ==，1AB BC ==，以,,AB AD AP 为,,x y z 轴建立空间直角坐标系如图所示：根据题意可得：()()()()()0,0,0,1,0,0,1,1,0,0,2,0,0,0,2A B C D P ，∴()()()1,1,2,1,1,0,0,2,2PC CD PD =-=-=-.根据点M 在线段CP 上，∴PM PC∥.设(,,2)PM t PC t t t ==- ，则(,,22)AM AP PM t t t =+=-+ ，由面AM PCD ⊥得()00022220AM CD t t AM PD t t ⎧⋅=-++=⎪⎨⋅=+--+=⎪⎩，得23t =，∴222,,333AM ⎛⎫= ⎪⎝⎭ ，∴AM ==.21．（Ⅰ）详见解析；（Ⅱ）13；（Ⅲ）2+．【详解】（Ⅰ）在C ∆AO 中，因为C OA =O ，D 为C A 的中点，所以C D A ⊥O ．又PO 垂直于圆O 所在的平面，所以C PO ⊥A ．因为D O PO =O ，所以C A ⊥平面D P O ．（Ⅱ）因为点C 在圆O 上，所以当C O ⊥AB 时，C 到AB 的距离最大，且最大值为1．又2AB =，所以C ∆AB 面积的最大值为12112⨯⨯=．又因为三棱锥C P -AB 的高1PO =，故三棱锥C P -AB 体积的最大值为111133⨯⨯=．（Ⅲ）在∆POB 中，1PO =OB =，90∠POB = ，所以PB ==同理C P =C C PB =P =B ．在三棱锥C P -AB 中，将侧面C B P 绕PB 旋转至平面C B 'P ，使之与平面ABP 共面，如图所示．当O ，E ，C '共线时，C E +OE 取得最小值．又因为OP =OB ，C C 'P ='B ，所以C O '垂直平分PB ，即E 为PB 中点．从而222C C O '=OE +E '=，亦即C E +OE 的最小值为262．考点：1、直线和平面垂直的判定；2、三棱锥体积．。
2023-2024学年上海市浦东新区高二数学上学期期中试卷附答案解析
2023-2024学年上海市浦东新区高二数学上学期期中试卷2023.11（考试时间90分钟，满分100分）一、填空题（每小题3分，共36分）1．公理2：不在同一直线上的点确定一个平面.2．直线与平面垂直的判定定理：如果一条直线与一个平面上的直线都垂直，那么此直线与该平面垂直.3．三垂线定理：平面上的一条直线和这个平面的一条斜线垂直的充要条件是它和这条斜线在垂直.4．已知球的半径为3，则该球的体积为.5．一个圆柱的底面半径为3cm ，高为4cm ，则它的侧面积为2cm .6．已知斜线段的长度是斜线段在这个平面内射影的长的两倍，则这条斜线和这个平面所成的角的大小为.7．一个正四棱柱底面边长为1，高为2，则它的表面积是.8．如图是一个正方体的平面展开图，在这个正方体中，下列说法中，正确的序号是.（1）直线AF 与直线DE 相交；（2）直线CH 与直线DE 平行；（3）直线BG 与直线DE 是异面直线；（4）直线CH 与直线BG 成60︒角.9．若空间三条直线a c ⊥，b c ⊥，则a ，b 的位置关系是.10．在正方体1111ABCD A B C D -中，平面11A BCD 与平面ABCD 所成的锐二面角的大小是.11．如图，正方体1111ABCD A B C D -的所有棱中，其所在的直线与直线1BA 成异面直线的共有条.12．在《九章算术》中，将四个面都为直角三角形的三棱锥称之为鳖臑()bienao ．已知在鳖臑M ABC -中，MA ⊥平面,2ABC MA AB BC ===，则该鳖臑的外接球的表面积为．二、选择题（每小题3分，共12分）13．“两条直线没有公共点”是“两条直线平行”的（）A ．充分非必要条件B ．必要非充分条件C ．充要条件D ．非充分非必要条件14．设m ，n 是两条不同的直线，α是平面，则下列命题正确的是（）A ．若//,//m n αα，则//m nB ．若//,m n αα⊂，则//m nC ．若//,//m n n α，则//m αD ．若//,,m n m n αα⊄⊂，则//m α15．如图，A 、B 、C 、D 是某长方体四条棱的中点，则直线AB 和直线CD 的位置关系是（）A ．相交B ．平行C ．异面D ．垂直16．已知棱长为1的正四面体的四个顶点都在一个球面上，则这个球的体积为（）A ．B ．C D 三、解答题17．正四棱柱1111ABCD A B C D -，的底面边长2AB =，若异面直线1A A 与1B C 所成角的大小为1arctan2，求正四棱柱1111ABCD A B C D -的侧面积和体积.18．如图，正三棱锥-P ABC 的底面边长为2，侧棱长为3.（1）求正三棱锥-P ABC 的表面积；（2）求正三棱锥-P ABC 的体积.19．如图所示，已知四棱锥P ABCD -中，底面ABCD 是直角梯形//AD BC ，AB BC ⊥，1AB AD ==，2BC =，PB ⊥平面ABCD ，1PB =.（Ⅰ）求证：CD PD ⊥；（Ⅱ）求四棱锥P ABCD -的表面积.20．如图，已知圆锥的顶点为P ，底面圆心为O ，高为2．(1)求该圆锥的侧面积；(2)设OA OB 、为该圆锥的底面半径，且90AOB ∠=︒，M 为AB 的中点，求二面角P AB O --的大小（用反三角表示）21．如图，PA ⊥平面ABCD ，四边形ABCD 是矩形，PA AD =，M ，N 分别是AB ，PC 的中点．（1）求证：//MN 平面PAD ；（2）求证：平面MND ⊥平面PCD ．1．三##3【分析】根据公理2判断可得；【详解】解：公理2：不在同一直线上的三点确定一个平面故答案为：三2．两条相交【分析】根据直线与平面垂直的判定定理得解；【详解】解：直线与平面垂直的判定定理：如果一条直线与一个平面上的两条相交直线都垂直，那么此直线与该平面垂直.故答案为：两条相交3．平面上的射影【分析】由三垂直线定理及其逆定理可得答案.【详解】解：由三垂线定理得：平面上的一条直线与平面的一条斜线在该平面内的射影垂直，则它也与这条斜线垂直；由三垂线定理的逆定理得：平面上的一条直线和这个平面的一条斜线垂直，则它与这条斜线在平面上的射影垂直；所以平面上的一条直线和这个平面的一条斜线垂直的充要条件是它和这条斜线在平面上的射影垂直，故答案为：平面上的射影.4．36π【分析】根据球的体积公式计算可得；【详解】解：因为球的半径3R =，所以球的体积334433633V R πππ==⨯=；故答案为：36π5．24π【分析】由圆柱的侧面积公式计算可得答案.【详解】解：圆柱的底面半径为3cm ，高为4cm ，则它的侧面积为23424ππ⨯⨯=2cm ，故答案为：24π.6．3π##60︒【分析】根据线面角的定义计算可得；【详解】解：因为斜线段的长度是斜线段在这个平面内射影的长的两倍，记这条斜线和这个平面所成的角为θ，则1cos 2θ=，因为0,2πθ⎛⎤∈ ⎥⎝⎦，所以3πθ=故答案为：3π7．10【分析】利用正四棱柱的性质进行计算即可【详解】因为正四棱柱底面边长为1，高为2，所以它的表面积为21141210⨯⨯+⨯⨯=，故答案为：108．（3）（4）##(4)(3)【分析】还原正方体ABCD EFGH -，结合图形即可判断（1）（2）（3），再连接AH ，AC ，则AHC ∠为异面直线CH 与直线BG 所成的角，根据三角形的性质即可求出异面直线所成角；【详解】解：由正方体的平面展开图可得正方体ABCD EFGH -，可得AF 与ED 为异面直线，故（1）错误；CH 与DE 为异面直线，故（2）错误；直线BG 与直线DE 是异面直线，故（3）正确；连接AH ，AC ，由正方体的性质可得//AH BG ，所以AHC ∠为异面直线CH 与直线BG 所成的角，因为AHC为等边三角形，所以60AHC ∠=︒，即直线CH 与直线BG 所成角为60︒，故（4）正确；故答案为：（3）（4）．9．平行，相交或异面【分析】根据空间直线的位置关系判断可得；【详解】解：因为空间三条直线a c ⊥，b c ⊥，所以a 与b 的位置关系是平行，相交或异面；故答案为：平行，相交或异面10．4π##45︒【分析】利用正方体的几何性质以及二面角的定义找到对应的平面角，在三角形中求解即可．【详解】正方体1111ABCD A B C D -中，BC ⊥平面11ABB A ，又1A B ⊂平面11ABB A ，所以1A B BC ⊥，又AB BC ⊥，所以1A BA ∠是平面11A BCD 与平面ABCD 所成的锐二面角的平面角，在直角1ABA △中，14ABA π∠=，所以平面11A BCD 与平面ABCD 所成的锐二面角的大小是4π．故答案为：4π．11．6【解析】根据几何体依次写出与直线1BA 成异面的直线即可得解.【详解】正方体1111ABCD A B C D -的所有棱中，其所在的直线与直线1BA 成异面直线如下：111111,,,,,AD DC DD B C C D C C ，一共6条.故答案为：6【点睛】此题考查异面直线的辨析，关键在于根据几何体特征准确找出与直线1BA 成异面的直线.12．12π．【分析】证明BC BM ⊥，可得MC 是外接球的直径，求得长度后可球表面积．【详解】因为MA ⊥平面ABC ，BC ⊂平面ABC ，所以MA BC ⊥，同理MA AC ⊥，又AB BC ⊥，AB MA A = ，,AB MA ⊂平面MAB ，所以BC ⊥平面MAB ，又MB ⊂平面MAB ，所以BC MB ⊥，所以MC 的中点O 到,,,M A B C 四点距离相等，为四面体M ABC -外接球球心，又由已知得AC ==MC =所以外接球表面积为2412S ππ=⨯=．故答案为：12π．【点睛】关键点点睛：本题考查求三棱锥外接球表面积，解题关键是打到外接球球心，求出球半径．三棱锥的外接球球心在过各面外心与该面垂直的直线上．13．B【分析】找出“两条直线没有公共点”的等价条件，结合充分条件、必要条件的定义判断可得出结论.【详解】“两条直线没有公共点”⇔“两条直线平行或异面”，所以，“两条直线没有公共点”是“两条直线平行”的必要非充分条件.故选：B.14．D【分析】根据线面位置关系的判定定理和性质定理，逐项判定，即可求解.【详解】对于A 中，若//,//m n αα，则m 与n 相交，平行或异面，故A 错误；对于B 中，若//,m n αα⊂，则m 与n 平行或异面，故B 错误；对于C 中，若//,//m n n α，则m 有可能在平面α内，故C 错误；对于D 中，若//,,m n m n αα⊄⊂，由直线与平面平行的判定定理，可得//m α，所以D 是正确的．故选：D 15．A【分析】如图，延长GM 到N,使12MN GM=,连接AN,DN.由AB 和DC 分别平行于正方体的两条相交的对角线，从而得AB 与DC 相交．【详解】如图，延长GM 到N,使12MN GM=,连接AN,DN.//AB FM ，AN ∥FM,∴A,B,N 三点共线，同理D,C,N 三点共线，AB ∴与DC 相交，故选：A ．【点睛】本题考查两直线的位置关系的判断，考查空间中线线、线面、面面间的位置关系等基础知识，考查运算求解能力，是中档题．16．A【分析】将正四面体放入正方体中，可得正方体的棱长为，求出正方体外接球的体积即为正四面体外接球的体积.【详解】如图将棱长为1的正四面体11B ACD -放入正方体1111ABCD A B C D -中，且正方体的棱长为1cos 452⨯=，所以正方体的体对角线162AC =，所以正方体外接球的直径12R AC =，所以正方体外接球的体积为3344πππ3348R ⎛=⨯= ⎝⎭，因为正四面体的外接球即为正方体的外接球，所以正四面体的外接球的体积为π8，故选：A.17．3216S V ==，.【分析】首先根据异面直线所成的角，求1BB ，再求正四棱柱的侧面积和体积.【详解】11//AA BB ，∴面直线1A A 与1B C 所成角是1CB B ∠，111tan 2BC CB B BB ∴∠==，2BC AB == ，14BB ∴=，∴正四棱柱1111ABCD A B C D -的侧面积42432S =⨯⨯=，体积22416V =创=.18．（1）；（2）3.【解析】（1）取BC 的中点D ，连接PD ，利用勾股定理求得PD ，可得三角形PBC 的面积，进一步可得正三棱锥-P ABC 的侧面积，再求出底面积，则正三棱锥-P ABC 的表面积可求；（2）连接AD ，设O 为正三角形ABC 的中心，则PO ⊥底面ABC .求解PO ，再由棱锥体积公式求解.【详解】（1）取BC 的中点D ，连接PD ，在Rt PBD △中，可得PD ==∴12PBC S BC PD =⋅=△.∵正三棱锥的三个侧面是全等的等腰三角形，∴正三棱锥-P ABC 的侧面积是3PBC S =△∵正三棱锥的底面是边长为2的正三角形，∴122sin 602ABC S =⨯⨯⨯︒=△.则正三棱锥-P ABC 的表面积为（2）连接AD ，设O 为正三角形ABC 的中心，则PO ⊥底面ABC .且1333OD AD ==.在Rt POD 中，PO .∴正三棱锥-P ABC 的体积为133ABC S PO ⋅=△.【点睛】本小题主要考查锥体的表面积和体积的求法，属于中档题.19．（Ⅰ）见解析；（Ⅱ）62+.【分析】（Ⅰ）在梯形ABCD 中，易求得CD BD ⊥，又由PB ⊥平面ABCD ，得PB CD ⊥，利用线面垂直的判定定理，即可得到CD ⊥平面PBD ，即可得到CD PD ⊥．（Ⅱ）由（Ⅰ）求得PCD S ∆=，进而根据PB ⊥平面ABCD ，得到,,PAD PBA PCD ∆∆∆，PCD ∆都为直角三角形，分别求得,,,PAD PAB PBC ABCD S S S S ∆∆∆梯形的面积，即可求解．【详解】（Ⅰ）在梯形ABCD 中，易求CD BD PD PA ====2,BC CD BD =∴⊥ .PB ⊥ 平面，,ABCD PB CD ∴⊥，又,PB BD B CD ⋂=∴⊥平面PBD ，又PD ⊂平面,PBD CD PD ∴⊥，.（Ⅱ）由（Ⅰ）知1622PCD S ∆=.又//,,DA BC BC AB PB ⊥⊥ 平面ABCD ，,,PAD PBA PCD ∴∆∆∆都为直角三角形.1,,122PAD PAB PBC S S S ∆∆∆∴===，所以32ABCD S ∴=梯形.∴四棱锥P ABCD -的表面积为6213626122222++++=.【点睛】本题主要考查了空间中位置关系的判定与证明，及几何体的表面积的计算，其中解答中熟记线面位置关系的判定定理与性质定理，以及准确计算几何体中每个面的面积是解答的关键，着重考查了空间想象能力，以及推理与运算能力，属于基础题．20．(1)8π(2)arc 【分析】（1）根据题意，由勾股定理求出圆锥的母线，结合圆锥的侧面积公式计算即可求解；（2）如图，由题意可得PM AB ⊥、OM AB ⊥，则PMO ∠为二面角P AB O --所成角.在Rt POM 中，解三角形即可求解.【详解】（1）由题意知，OP ⊥平面OAB ，OB ⊂平面OAB ，所以OP ⊥OB ，所以圆锥的母线4l =，所以圆锥的侧面积π8πS lr ==；（2）如图，连接PM ，M 为AB 的中点，PA PB =，则PM AB ⊥，又OAB 为等腰三角形，OA OB =，所以OM AB ⊥，所以PMO ∠为二面角P AB O --所成角.在等腰直角OAB 中，2OA OB ==，所以OM =在Rt POM 中，OP OM =tan OP PMO OM ∠=所以arctan PMO ∠==.21．（1）证明见解析（2）证明见解析【分析】（1）取CD 的中点E ，连接NE ，ME ，可证//NE PD ，//EM DA ，从而面//NEM 面PDA ，即可证明//MN 平面PAD ；（2）证明MN CD ⊥，由PM MC =，M 、N 分别是AB 、PC 的中点，可证MN PC ⊥，CD PC C = ，可知MN ⊥平面PCD ，从而得证．【详解】证明：（1）取CD 的中点E ，连接NE ，ME ，M 、N 分别是AB 、PC 的中点，//NE PD ∴，//EM DA又NE ⊄面PDA ，PD ⊂面PDA ，所以//NE 面PDA又ME ⊄面PDA ，AD ⊂面PDA ，所以//ME 面PDA因为NE ME E = ，,NE ME ⊂面NEM∴平面//NEM 平面PDA ，因为MN ⊂面NEM//MN ∴平面PAD ；（2） 底面ABCD 是矩形，PA ⊥平面ABCD ，CD PA ∴⊥，CD AD ⊥，PA AD A ⋂=，AD ⊂平面PAD ，PA ⊂平面PADCD \^平面PAD ，PD ⊂ 平面PADCD PD ∴⊥，//EN PDEN CD∴⊥又CD EM ⊥ ，EM EN E = CD \^平面ENMMN CD∴⊥,,PA AD BC AD AM MB===PM MC ∴=，M 、N 分别是AB 、PC 的中点，MN PC ∴⊥，CD PC C = ，,CD PC ⊂面PCD MN ∴⊥平面PCD ,又MN ⊂平面MND ，∴平面MND ⊥平面PCD .。
新世纪版高二第一学期Mid-termExamination期中考试
Mid-term ExaminationI. Listening Comprehension (略)H. Grammar and VocabularySection ADii ections: After readmg the passages below, fill m die blanks to make the passages coherent and granmiatically correct. For the blanks with a given word, fill in each blank with the proper form of die given word; for die other blanks, use one word that best fits each blank.(A)Don't Take the Fun Out of Youth SportsWhen I jomed a piivate football league a few years ago, the sport meant everything to me. My coach said I had lots of potential, and I became captain of my team. That was before all the fun (25) (take) out of playing.At first, everyone on the team got equal playing time. Then the team moved up to the top division after winning all its games, and the pressure started. Some parents, who (26) ________________ (pay) die coach extra (27) ______ retiini for tlieir daughters* private one-on-one training, got angry whenshe didn't giv已them more playing time. Tlie coach was replaced. Tlie new coach, however; took all the fim out of the game: All we did during practice was nm. We rail so much that, afterwards, we had trouble (28) ____________________ (breathe).Yoiuiger people shouldn't be domg exercises (29) __________ (design) for 18-year-olds.I was tlim before I started fbotbalL but as a member of tliis team I wouldn't eat much, because I thouglit to (30) ________ t liat I was afraid of being too fiill to run. Finally, I ended up leaving the football team. Four other girls did tlie same, two of (31) _______________ stopped playing football completely.That's sad because they had so much potential. They were just burned-out with all the pressure they felt from the coach or tlieir parents.I continued playing football at school and rediscovered my love for it. I jomed a private team and the coach told me I needed to relax because I looked nervous. After I calmed down. I played better. When you enjoy something, it is a lot (32) __________________ (easy) to do it well.How to Be a WinnerSteven Redgrave…Winner of 5 Olympic Gold MedaLs"Last year I was found to have developed lung disease. (33) _________ (believe) my career was over, I felt extremely low. Then one of the specialists said there was no reason why I should stop training and competing. Tliat was it——the encouragement I needed. I could still be a winner (34) I believed in myself. I am not saying that it isnt difficult sometimes. But I wanted to prove to myself that hard (35) t he condition was, I wasn't finished yet. Nothingis to stand in my way.HKaren Pickering-Swimming World Championn I swim 4 hours a day, 6 days a week. I manage that sort of workload by putting it on top of my diary. Tliis is the key to success 一you (36) ________________not bear following a career in any field without bemg well・orgamzed. List (37) ______ y ou believe you can achieve. Trust yourself write down your goals for the day, however small they are, and yoifll be a step closer to achieving them."Kii sten Best-Poet &Writer"When things are getting hard, a voice inside my head tells me I cant achieve something. Them there are other influencing factors, such as family or hobbies. Tlie key is to concentrate, (38) helps a lot to repeat words such as ' calin\ 'peace' or *focus\ either out loud or silently ill my mind when I feel tense. It makes me (39) ___________________________________( feel) more ill control and mcreases my confidence. Tliis is a habit that can become second nature quite easily and is (40) _________ owerfUl psychological tool.HSection BDiiections: Complete the following passage by using die words in the box. Each word can only be used once. Note tliat there is one word more than you need.anxiety can 41 itself in physical symptoms like nausea or sweaty palms, or in emotional symptoms like panic and irritability. Often, individuals will become nervous long before die exam date because theyfeel 42 by the amount of material they must master. The first step in 43 this anxiety is realizing tliat test anxiety is a natural reaction, and that it can be 44 . Iii order to regain control ovei anxiety, one should break the exam 45 into smallersections. It will seem much easier to master a series of small subjects than to handle a huge subject all at once.Test-takers should make a study schedule weeks or months ahead of the exam, and give themselves a small amount to learn each period. When the exam day 46 arrives, anxiety will be reduced by the feeluig of preparedness. Apart from that, anxiety can be lessened iii a few easy ways. First, avoid having any excessively stimulating foods, such as those 47 cafieiiie or sugar, on the day of the exam. Getting a good mglit s sleep and a little bit of exercise will 48 one's bram and allow for easy 49 of die learned material. Durmg the exam, breathe deeply and sit in a comfortable position. On exams where tliis is possible, many people find comfort in gomg tlirougli and answering all of die easy questions before 50 the more difficult ones.III. Reading ComprehensionSection ADiiections: Directions: For each blank in the following passage there are four words or phrases marked A,B f C and D ・ Fill in each blank with the word or phrase that best fits the context.When you say that someone has a good memory, what exactly do you mean? Are you saying tliat die person has fast recall or that he or she _________ 51 __ information quickly? Or maybe you just mean that the person remembers a lot about her or his childhood. The truth is that it is ________ 52 __ to say exactly what memory is. Even scientists who have been studying memory for decades say tliey are still trying to ___________ 53 __ exactly what it is. We do know that a particular memory is notjust one thing stored somewhere in the bram. ______ 54 __ , a memory is made up of bits and piecesof mfbrmation stored all over die bram. Perhaps the best way to _____ 55 ___ m emory is to say that itis a process—a process of recording, stormg. and getting back iiiformation. Practice and repetition can help to ______ 56 __ the pieces that make up our memory of tliat information.Memory can be 57 a fleeted by a number of things. 58 nutrition can affect a person s ability to store information. Excessive alcohol use can also weaken memory and cause permanent 59 to the brain ovei the long term. A vision or hearing problem may affect aperson s ability to notice certam things, thus makuig it________ 60 __ to register mfbrmation in the brain.When people talk about memory, they often _______ 61 _ short-term memory and long-term memory. If you want to call a store or an office that you don't call often, you look in die telephone book for the number. You dial the munber, and then you forget it? You use your short-termmemory to remember the niunber. Your short-temi memory lasts about 30 seconds, or half a niuiute. 62 __________ . you don't need to look ill die telephone book for your best friend's numberbecause you already know it. This number is ill your long-term memory, which _____________ 63 __ mforination about things you have learned and experienced through the years.Why do you forget things sometimes? Tlie major reason for forgetting somethuig is that you didnot learn it well enougli _______ 64 __ . For example, if you meet some new people and riglit away forget their names, it is because you did not ______ 65 __ the names at the first few seconds whenyou heard them.51. A. collects B. processes C. publishes D. absorbs52. A. necessary B. miportant C. difficult D. convenient53. A. figure out B. take out C. put out D. give out54. A. After all B. Instead C. By contrast D Besides55. A. recall B. refresh C. describe D. decrease56. A. lose B. organize C. identify D. strengthen57. A. positively B. negatively C. actively D. directly58. A. Poor B. Adequate C. Special D. Various59. A. benefit B. ofience C. effect D. damage60. A. easier B. more impressive C. harder D. more convenient61. A. refer to B. apply for C. come across D. break down62. A. Furthermore B. However C. Consequently D. Othenvise63. A. leaks B. transmits C. checks D. stores64. A. in the middle B. at the end C. in the beginning D. ahead of time65. A. restore B. record C.replace D. respondSection BDiiections: Read the followmg tliree passages. Each passage is followed by several questions or unfinished statements. For each of them there are four choices marked A, B, C and D. Choose the one that fits best according to the information given in the passage you hate just read.(A)Car HireHiring a sel&dnve car really adds to the enjoyment of your holiday. There are so many places of interest to tisit, and if you enjoy seeing more than just the city center, there s no better way to explore thanby car.Hire Charges! I! What are included !i ia)Unlimited mileage (英里数). ;I ib)Expenses o n oil,maintenance and repairs, which will be repaid on production of ;:invoices (发栗).! c) Full insurance except personal accident ( see below ) and contents.What arc not included! a) Personal accident insurance・■ib) Garaging, petrol, parking and traffic fines.I ____________________________________________________________________________ JConditions of Hii e★The shortest rental period at these special low prices is three days. For prices for periods of one or two days you only see our representative at the hotel.★Car hire must be booked six weeks or more before arrival in London to guarantee a car. But if you have been unable to make a booking ill advance, please see our representative at the hotel who may still be able to help you.★The car types on the sheet are examples of the types of cars in each price range, but a particular car camiot be guaranteed.★Upon delivery die diiver(s) will be asked to sign the car hire company's Conditions of Hire.★If you decide to hire a car, just fill in the Bookmg Form and retiini it to us. A booking fee of £ 12 as part of the car hire cost is required.★Should you be forced to cancel your car hire booking after payment in fiill (two weeks before date of hire) , a cancellation charge of £12 will be made.66.From the advertisement we can see a car hirer will pay _______ .A.msuraiice against damage to the carB. insurance against injury to the driverC. die cost of maintenance of the carD. the cost of repairs to die car67.Which of the following is NOT true according to die advertisement?A.You hav已to pay parking fine yourselfB.Tlie company can't provide all kinds of cars you need.C.When you hire a car from the company, a driver will go together with you.D.You havi to pay a booking fee when you hire a car from the company.68.If car liirers change their minds after paying die whole cost of hiring, the £ 12 booking fee isA.partly returnedB. double paidC. not returned at allD. retimied witliiii six weeks69.The prices for car lure are especially low when _______ .A. it is booked for at least three daysB. it is booked two weeks in advanceC. it is booked for two daysD. the bookuig is made in London(B)If you are a recent social science graduate who lias had to listen to jokes about unemployment &om your computer major classmates, you may have had the last laugh・ There are many advantages for tlie social science major because this lugli-tech ^Information Age'' demands people who are flexible and who have good conmiumcation skills.There are many social science majors m large companies who fill important positions. For example, a number of research studies found that social science majors had achie\・ed greater managerial success than those who had technical traiimig or pre-professional courses. Studies show that social science majors are most suited for change, which is the leadmg feature of the kmd of high speed, high-pressure, high-tech world we now live in.Social science majors are not only experiencing success m their long-term con^aiiy jobs, but tliey are also finding jobs more easily. A study showed that many companies had filled a large percentage of their entry・level positions with social science graduates. The study also showed that die most souglit-after quality in a person who was looking for a job was communication skills, noted as "very important' by 92 percent of the companies. Social science majors have these skills, often without knowing how important they are. It is probably due to these skills that they have been ofiered a wide variety of positions.Fmally, althougli some social science majors may still find it more difficult than their teclmically trained classmates to land the first job, recent graduates report that they doit regret tlieir choice of study.pared with graduates of other subjects, social science graduates _______ .A.are ready to change when situations changeB.are better able to deal with difficultiesC.are equally good at computer skillsD.are likely to give others pressure71.According to the text, what has made it easy for social science graduates to find jobs?A. Willingness to take low-paid jobs.B. Readmess to gain high-tech knowledge.C. Skills in expressing themselves.D. Part-time work experience.72.The underlined word fc land?，in the last paragraph probably means _________ .A. keep for some timeB. successfully getC. iimnediately startD. lose regretfiillySection CDirections: Complete the following passage using the sentences giveri below. Each sentence can be used O7ilv once・ Note that there are two more sentences than you need.Weekend Sleep Imbalance Bad for Blood SugarFew tilings are more satisfying than sleepmg late on weekends. But tliougli the extra zs may impiove your mood, they do not appear to improve your health. Because a new study shows that so-called 'recovery sleep' cannot reset the bodys* metabolic（新陈代谢的）clock., and may actually lead to some serious health issues.73 ________ It can increase our risk for cardiovascular disease, it can cause weight gam, itcan decrease insulin sensitivity, so it can increase our risk of diabetes/ says Christopher Depner, an assistant professor in die department of integrative （综合的）physiology at the University of Colorado,B oulder.A lot of us shut off the alann on Saturday and Sunday 74 _______________ 'So we were really mterested in how the sort of cycle of benveen insufficient sleep, weekend recovery sleep, insufficient sleep, can intact your risk of obesity and metabolic disease, says Depner. He and his colleagues invited volunteers to a nme-day snooze-a-tlion.One group was allowed to get a fiill niglit s sleep. The next was kept to just five hours each night. And the third group went back and fortli. restricted to five hours of shut-eye during die workweek, allowedas much sleep as they wanted o*ver the weekend, and then back to five hours for the last couple days.The study shows that when we maintain insufficient short sleep schedules durmg a typical work or school week, we find that this leads people to eat more tliaii they need and this leads to weight gain. 75 _________________________ And tliis altogether also leads to a reduced ability to regulate ourblood sugar levels, says Ken Wright, professor of integrative physiology at University of Colorado Boulder. But even more surprising, sleeping in on the weekend doesnt help-and even makes tlimgs worse.*We found that after the weekend, when they went back to getting insufficient sleep durmg die work or school week, we found that their liveT and their muscle insulin sensitivity or blood sugar regulation was reduced. And this is not sometliing we had found m people who mamtained chronic insufficient sleep schedules/says WriglitSo it's possible that this is a worsening of the body's ability to regulate blood sugar for those specific tissues after the weekend. 76 ____________________ And trade the sweets for sweet dreams.r\： Summary WritingDirections: Read the following passage・ Su/mnanze the main idea and the main points of the passage no more than 60 words ・ Use your ow?i words as far as possible ・Witli the average number of children in a British family falling under an verage of 2.0, compared witli the traditional 2.4 children, the population of the UK has been follmg for quite a few years.Why aren't British people hating as many children as they used to? Well, there are many reasons. One of them is tliat British people are now having their children at a much older age tliaii previously, meaning they have fewer years in which they can have children. Most young people today want a well-paid job, so tliey often go to university after high school after years of study at university, they then need a few years of work experience before they can get the job tliey want. So it's not imtil they are about 30 years old that they can start to think about settling down and having children.Another reason is that it is relatively expensive to bring up a child in the UK. Usually both parents need to work to pay for their home and living expenses. Tliey need to pay someone to look after tlieir child durmg the day, which is expensive.In addition, problems in the global economy often affect British families. For example, when times are tough and there is an economic downturn, it becomes difficult to find a job or keep the existing one. This kind of uncertainty discourages people &om having larger families.So what is Britaui domg to try and save British families? First of all. the government is trying to make it cheaper to ha\・e cliildreii. For example, there ha\・e been mcreases ill money families can gam from the state each montli. Also, there is an mcreasmg amoirnt of money for nursery schools, so that parents don't need to pay so much for childcare. In addition, there are now laws allowing parents to take more time off work so that they can look after their children themselves rather than having to pay others to do it.V TranslationDii ections: Translate die following sentences mto English, using the words given in die brackets.1.前天，许多物理学家出席了颁奖典礼。
上海市高二第一学期物理期中试题
上海市高二第一学期物理期中试题（90分钟内完成，满分100分）有用的信息：1、本卷重力加速度g取10 m/s2，2、静电力恒量k＝9.0⨯109 N·m2/C2，3、电子和质子的电量均为1.6⨯10-19 C。
一、单选题（每小题3分，共24分）1．以下说法正确的是（）（A）密立根用摩擦起电的实验发现了电子（B）密立根用摩擦起电的实验测定了元电荷的电荷量（C）密立根用油滴实验发现了电子（D）密立根用油滴实验测定了元电荷的电荷量2．以下说法正确的是（）（A）只要带电体的体积足够小，就可以把它看作点电荷（B）只要带电体的带电量足够小，就可以把它看作点电荷（C）可以把两块带等量异号电荷的平行板看作点电荷（D）研究两个相距非常近的带电小球相互作用时就不能把它们看作点电荷3．A、B是如图所示电场线上的两点，据此可以判断（）（A）A点电场强度一定大于B点电场强度（B）A点的电势一定高于B点的电势（C）电荷在A点的电势能一定大于在B点的电势能（D）该电场一定是匀强电场4．关于电场强度的定义式E＝Fq，以下叙述正确的是（）（A）E与F成正比（B）E与q成反比（C）E由F和q的比值决定（D）E表示电场本身性质，与q无关，其大小可由F和q的比值来测定5．正确的防静电措施是（）（A）印染厂的车间要保持干燥通风（B）油罐车后要用一根铁链条拖在地上（C）飞机起落架上的轮胎要用绝缘橡胶制成（D）购买电脑CPU配件时要用手握住其金属部分6．设A、B两个点电荷间的距离保持恒定，把电荷C移近A、B，则A、B两点电荷之间的静电作用力将（）（A）变大（B）变小（C）不变（D）都有可能A B7．在如图装置中，电键一闭合，保险丝就熔断，断开电键，重新换上保险丝，再闭合电键时，保险丝又被熔断，设电路中只有一处短路，则电路中短路处位于（ ） （A ）插头处 （B ）电键处 （C ）灯泡接线处 （D ）插头到电键处8．下列由基本门组成的电路中，能使蜂鸣器发出声音的是（ ）（A ）只有（1） （B ）只有（2） （C ）只有（3） （D ）（1）、（2）、（3） 二、填空题（每小题4分，共24分）9．在氢原子中，电子和质子的平均距离是0.53×10-10 m ，质子在这个距离处产生的电场强度大小为_______N/C ，电子受到的电场力的大小为_______N 。
上海市高二上学期期中数学试卷含答案(共3套)
上海市高二第一学期数学期中考试试卷（满分：100分 考试时间：90分钟）一、填空题（本大题满分36分）本大题共有12题，只要求直接填写结果，每小 题填对得3分，否则一律得零分.1. 已知()1,3a =-，则a =___________.2. 方程组21320x y x y -=⎧⎨+=⎩的增广矩阵为_______________________.3. 行列式101213131--- 中3-的代数余子式的值为___________.4. 已知R a ∈，若11321lim22=+--+∞→n n n an n ,则=a ___________． 5. 1134lim 34n nn n n ++→∞-=+____________． 6. 若首项为2的无穷等比数列{}n a 的各项的和为10，则公比q =___________.7. 已知3a =，4b =，5a b +=，则a 与b 的夹角为 . 8. 已知()1,2a =，(),4b m =，()||2a a b +，则实数m 的值为_____________. 9. 设向量()3,0a =-，()2,6b =-，则b 在a 上的投影为______________. 10. 已知数列}{n a 是首项为1，公差为2的等差数列，n S 是其前n 项和，则=∞→2limnnn a S __________．11. 已知向量a ，b 是同一平面内的两个向量，其中()1,2a =，()1,1b =，a 与a b λ+的夹角为锐角，则实数λ的取值范围是____________________.12. 如图所示：矩形n n n n A B P Q 的一边n n A B 在x 轴上，另两个顶点,n n P Q 在函数22()(0)1xf x x x =>+的图像上（其中点n B 的坐标为()*,0(2,)n n n N ≥∈），矩形n n n n A B P Q 的面积记为n S ，则lim n n S →∞= .二、选择题（本大题满分12分）本大题共有4题，每题都给出代号为A 、B 、C 、D 的四个结论，其中有且只有一个结论是正确的.13. 下列命题中，真命题为………………………………………………………（ ）（A ）若0 =a ，则0=a； （B ）若b a =，则b a =或b a -=；（C ）若a 与b 是平行的向量，则a 与b是相等的向量；（D ）若a b -=，则0=+b a ． 14. 数列{}n a 的通项公式是1(1)2nn a +-=，则此数列…………………………（ ）（A ）有极限，其值是整数； （B ）有极限，其值是分数； （C ）有两个极限； （D ）lim n n a →∞不存在．15. 在数列{}n a 中，111111234212n a n n=-+-++--，则1k a +=…………（ ） (A) 121k a k ++ (B) 112224k a k k +-++ (C) 122k a k ++ (D)112122k a k k +-++ 16. 有下列四个命题：①若22lim A a n n =∞→，则A a n n =∞→lim ； ②若0>n a ，A a n n =∞→lim ，则0>A ；③若()0lim =-∞→n n n b a ，则n n n n b a ∞→∞→=lim lim ；④若A a n n =∞→lim ，则22lim A a n n =∞→.其中正确命题的个数是……………………………………………………………（ ） （A ）1个 （B ） 2个 （C ） 3个 （D ）4个三、解答题（本大题满分52分）本大题共有5题，解答下列各题必须写出必要的步骤.17.（本题满分10分）已知)10,5(),4,3(---B A ，O 为坐标原点， (1) 求向量AB 的坐标及AB ；(2) 若OB OA OC +=，求与OC 同向的单位向量的坐标．18.（本题满分10分）用行列式的方法解关于x 、y 的二元一次方程组1323mx y mx my m +=-⎧⎨-=+⎩，并对解的情况进行讨论.19. （本题满分10分）已知O 为坐标原点，()3,4OA =-，()6,3OB =-，()5,3OC m m =---. （1）若A ，B ，C 三点共线，求m 的值；（2）若△ABC 是以角A 为直角顶点的直角三角形，求m 的值以及此时三角形的面积.20．（本题满分10分）已知等比数列{}n a ，首项为1a ，公比为q ，11lim()12n n a q q →∞-=+，求首项1a 的取值范围.21.（本题满分12分）已知点的序列(),0,*,n n A x n N ∈，其中()120,0,x x a a ==>，3A 是线段12A A 的中点，4A 是线段23A A 的中点，n A 是线段21n n A A --的中点，（1）写出n x 与12,n n x x --之间的关系式()3n ≥；（2）设1n n n a x x +=-，计算123,,,a a a 由此推测数列{}n a 的通项公式，并加以证明.第一学期高二数学期中考试试卷答案及评分细则注：填空题结果只要等价就得分；解答题其他解法相应给分。
2023-2024学年高二上学期期中语文试题含参考答案
2023-2024学年高二上学期期中语文试题含参考答案一、选择题组阅读下面的文字。
完成各题。
新世纪以来，我国的文艺事业进入一个空前的繁荣时期，尤其是文学创作走上了“快车道”，大量作品竞相问世，百卉千葩让人（）。
新世纪文学带来海量阅读的同时，也引发了我们对文学创作的思考？纵观历史，有司马迁历时十余载成就“史家之绝唱，无韵之离骚”的《史记》；有曹雪芹（），“披阅十载，增删五次”方得“字字看来皆是血，十年辛苦不寻常”的《红楼梦》；回眸当代，有鲁迅痛定思痛、以笔为枪完成“揭出病苦，引起疗救注意”的《呐喊》；有沈从文朴素自然、细腻入微展现“优美，健康而又不悖乎人性”的《边城》……（），。
同时这些著者更是以一丝不苟与精益求精的精神，完美地诠释了“如切如磋，如琢如磨”的创作品质。
而如今文学的“大众化”“生活化”“社会化”虽催生了创作的繁荣，但在繁荣背后也存在（）的现象。
面对“走得快”还是“走得远”，文学创作该何去何从？我认为只有在追求创作多元化的同时，坚守“载道与言志”，才可迎来文学创作的“春风十里”。
1．依次填入文中括号内的词语，最恰当的一组是（）A．应接不暇/呕心沥血/毋庸置疑/鱼目混珠B．应接不暇/呕心沥血/毋庸置疑/泥沙俱下C．美不胜收/殚精竭虑/不言而喻/鱼目混珠D．美不胜收/殚精竭虑/不言而喻/泥沙俱下2．在选文横线处填入句子，表述最恰当的一项是（）A．这些堪称经典的文学作品，无一不是著者以执着的信念和坚忍的意志精雕细琢而成的。
B．这些堪称经典的文学作品，无一不是著者经过执着的信念和坚忍的意志精雕细琢而成的。
C．这些无一不是著者以执着的信念和坚忍的意志精雕细琢而成的堪称经典的文学作品。
D．这些无一不是著者经过执着的信念和坚忍的意志精雕细琢而成的堪称经典的文学作品。
3．对选文中涉及的文学文化常识解说恰当的一项是（）A．《红楼梦》，古代章回体长篇小说，又名《石头记》《情僧录》《风月宝鉴》《金陵十二钗》等，被称为“中国封建社会的百科全书”，传统文化的集大成者。
上海市某中学高二上学期期中检测数学试题含答案
高二第一学期期中考试数学试卷 2012.11一、填空题：（本大题共12小题，每小题4分，满分48分）1.数列{}n a 满足：()*11,0N n n a a a n n ∈+==+，则数列{}n a 的通项公式=n a2.如图1所示算法流程图输出的结果是3.已知数列{}n a 的前n 项和12++=n s nn ，则=+31a a4.如图2给出一个数阵，其中每行每列均为等差数列，且数阵从左至右以及从上到下都有无限个数．①第三列前n 项和为 ；②数阵中数100共出现 次5.数列{}n a 中，1,273==a a ，数列⎭⎬⎫⎩⎨⎧+11n a 是等差数列，则=11a6.设{}n a 是等比数列，公比2=q ,n s 为{}n a 的前n 项和.记n T =1217+-n n n a S S ，*N n ∈.设0n T 为数列{}=0n T n 的最大项，则7.甲、乙两人于同一天分别携款1万元到银行储蓄，甲存五年期定期储蓄，年利率为2.88%，乙存一年期定期储蓄，年利率为2.25%，并在每年到期时将本息续存一年期定期储蓄，按规定每次计息时，储户须交纳利息的20%作为利息税，若存满五年后两人同时从银行取出存款，则甲与乙所得本息之和的差为 元．（假定利率五年内保持不变，结果精确到1分） 8.给出数列{}n a 的条件如下：①设n n a b 2=，{}n b 是等差数列；②设)2(11≥+=--n a a b n n n ，{}n b 是等差数列；③前n 项的和12+=n S n ；④设12-=n n a b ，数列{}n b 前n 项和为2n ．其中使数列{}n a 是等差数列的条件的正确序号是9.在1，2之间插入n 个正数,21,......,,n a a a ，使这n+2个数成等比数列，则=n a a a a (321)图1图210.正项无穷等比数列{}n a 的前n 项和为n S ，若1lim1=+∞→n nn S S ，则其公比q 的取值范围是11.数列()()⎭⎬⎫⎩⎨⎧+-12121n n 的前n 项和为n S ，使T S n <恒成立的最小正数T 是12.2n 个正数排成n 行n 列，如图3，其中每行 数都成等比数列，每列数都成等差数列，且所 有公比都相等，已知,18,6,5565424===a a a 则=+1422a a二、选择题：（本大题共4小题，每小题4分，每题有且只有一个正确答案，满分16分） 13.用数学归纳法证明不等式6412721......412111>++++-n ,*N n ∈成立，起始值至少应取为（ ）A.7B.8C.9D.1014.设命题甲：△ABC 的一个内角为60°，命题乙：△ABC 的三内角的度数成等差数列．那么（ ） A ．甲是乙的充分条件，但不是必要条件 B ．甲是乙的必要条件，但不是充分条件C ．甲是乙的充要条件D ．甲不是乙的充分条件，也不是乙的必要条件15.已知{}n a 为等比数列，下面结论中正确的是（ ）A.2312a a a ≥+B.2223212a a a ≥+ C.若31a a =，则21a a = D.若13a a >，则24a a >16.若矩阵726967656259817468645952857976726964228219211204195183A ⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭是表示我校学生高二上学期的期中成绩矩阵，A 中元素(1,2,3,4;1,2,3,4,5,6)ij a i j ==的含义如下：1i =表示语文成绩，2i =表示数学成绩，3i =表示英语成绩，4i =表示语数外三门总分成绩*,j k k N =∈表示第50k 名分数。
2023-2024学年上海市高二上册期中数学质量检测模拟试题(含解析)
2023-2024学年上海市高二上册期中数学质量检测模拟试题一、填空题1．空间内，两异面直线所成角的取值范围是______.（用区间表示）【正确答案】π(0,2【分析】利用异面直线所成角的定义直接写出范围作答.【详解】由异面直线所成角的定义知，两异面直线所成角的取值范围是π(0,]2.故π(0,]22．已知底面边长为4的正三棱柱侧面积为9，则其体积为______.【正确答案】【分析】根据侧面积可求正三棱柱的高，进而可求体积.【详解】设正三棱柱的高为h ，则349h ⨯=，解得34h =，所以体积21π4sin 23V Sh h ==⨯⨯⨯=故答案为.3．圆柱的底面半径为1，高为2，则其表面积为______.【正确答案】6π【分析】直接利用表面积公式计算得到答案.【详解】表面积22π2π2π4π6πS r rh =+=+=.故6π4．在两平面平行的判定定理中，假设,αβ为两不同平面，,l m 为两不同直线，若要得到//αβ，则需要在条件“,,//,//l m l m αββ⊂”之外补充条件______.【正确答案】l m ≠∅【分析】确定,l m 为平面α内的两条相交直线，//,//l m ββ，故//αβ，得到答案.【详解】因为一个平面内两条相交直线平行于另一个面，则这两个面平行，所以要证//αβ，需要,l m α⊂，//,//l m ββ，以及l m ≠∅ ，共五个条件，所以需要在条件“,,//,//l m l m αββ⊂”之外补充条件是l m ≠∅ .故答案为.l m ≠∅5．{}n a 为等差数列，n S 为其前n 项和，若344S a =，则10a =______.【正确答案】0【分析】根据等差数列的性质及通项公式计算即可得解.【详解】因为334S a =，所以2434a a =，即2234(2)a a d =+，所以280a d +=，所以10280a a d =+=.故06．记等比数列{}n a 的前n 项和为n S ，若58327S S S =-，则该等比数列的公比q =______.【正确答案】13【分析】排除1q ≠，由等比数列求和公式代入方程求得公比.【详解】因为58327S S S =-，易得1q ≠，所以()()()51831111271111a q qa q a q q q--=-----，解得q =13.故答案为.137．在正方体1111ABCD A B C D -中，,E F 分别是线段1,BC C D 的中点，则直线1A B 与直线EF 的位置关系是______.（从相交，平行，异面中选填）【正确答案】相交【分析】连接111,,BD CD CD 与1C D 交于点F ，易得11A BCD 是平行四边形，根据平面的基本性质即可判断直线1A B 与直线EF 的位置关系.【详解】如图所示：连接111,,BD CD CD 与1C D 交于点F ，由题意，易得四边形11A BCD 是平行四边形，在平行四边形11A BCD 中，,E F 分别是线段1,BC CD 的中点，∴1//EF BD ，又11A B BD B ⋂=且1,,,A B E F 共面，则直线1A B 与直线EF 相交.故相交.8．如图的四面体OABC 中，所以棱长均相等，每个面都是全等的正三角形，,M N 分别是棱,OA BC 的中点，则直线OA 与平面CMN 所成角的大小为______.【正确答案】2π【分析】由题意得，四面体OABC 为正四面体，进而可以证明OA ⊥平面CMN ，求出线面角.【详解】如图，连接,CM BM ，由题意得，四面体OABC 为正四面体，所以CM OA ⊥，BM OA ⊥，因为CM BM 与点M ，CM ⊂平面CMN ，BM ⊂平面CMN ，所以OA ⊥平面CMN ，所以直线OA 与平面CMN 所成角的大小为π2.故答案为.π29．已知等差数列{}n a 满足，253,9a a ==，等比数列{}n b 的公比111,55q b ==，令*,N ,n n n c a b n =∈{}n c 的前n 项和为n S ，若“0n n ≥”是“1010412021n n S c +->”的充分条件，则正整数0n 的最小值为______.【正确答案】6【分析】计算21n a n =-，15n n b ⎛⎫= ⎪⎝⎭，得到()1215nn n n c a b n ⎛⎫==- ⎪⎝⎭，利用错位相减得到()111141121255nn n S n -⎡⎤⎛⎫=+---⨯ ⎪⎢⎥⎣⎦⎝⎭，代入不等式解得答案.【详解】213a a d =+=，5149a a d =+=，11a =，2d =，故21n a n =-；15n n b ⎛⎫= ⎪⎝⎭，故()1215nn n n c a b n ⎛⎫==- ⎪⎝⎭，()()211111323215555n nn S n n -⎛⎫⎛⎫⎛⎫=+⨯++-⨯+-⨯ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭()()231111113232155555nn n S n n +⎛⎫⎛⎫⎛⎫⎛⎫=+⨯++-⨯+-⨯ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭两式相减得()231411111221555555n n n S n +⎡⎤⎛⎫⎛⎫⎛⎫⎛⎫=++++--⨯⎢⎥ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭⎢⎥⎣⎦，所以()111141121255nn n S n -⎛⎫⎛⎫=+---⨯ ⎪ ⎪⎝⎭⎝⎭，因为1010412021n n S c +->，所以()()111111010112121125552021nnn n n -⎛⎫⎛⎫⎛⎫+---⨯+-⨯->⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭，整理得到510105n >，553125=，6515625=，N n *∈，所以6n ≥.正整数0n 的最小值为6.故610．如图，曲线1C 是一个圆心位于()1,0,半径为1得四分之一圆弧，2C 是直线:L y x =上的线段，两者交于()1,1，1C ，2C 与x 轴共同构造一个封闭区域G ，将G 绕y 轴旋转一周得到几何体Ω，现已知：过点()0,y 作Ω的水平截面，所得的截面积S 与y 之间的函数关系式为())22π2π01S y y y =-+≤≤，利用()S y 的表达式与祖暅原理，考虑一个长方体，一个四棱锥和一个平放的半圆柱，计算几何体Ω的体积为______.【正确答案】24ππ32+【分析】利用祖暅原理，通过一个长方体减去一个四棱锥加上一个半圆柱的组合体，使其与Ω的水平截面表达式相等，算出构造的组合体体积即可.【详解】如图，取一个宽2AB =，长πAD =，高1AM =的长方体ABCD MLON -挖空一个四棱锥D MLON -，再加12个半径1XH =，高πEF =的圆柱，当高()01AQ XK y y ==≤≤时，水平截面（阴影部分）面积QRSP VUTP IHGJ S S S S =-+矩形矩形矩形，由1PV DP PT yNM DN NO ===，可得22πVUTP S y =矩形，由2IH KH ==，可得IHGJ S =矩形，则此组合体水平截面面积S 与y 之间的函数关系式为())22π2π01S y y y =-+≤≤，所以此组合体体积与几何体Ω的体积相等，1=2ABCD MLON BEFC D MLON V V V V Ω---+长方体圆柱棱锥22114ππ=2π12π1π1π3232⨯⨯-⨯⨯⨯+⨯⨯⨯=+.故答案为.24ππ32+二、单选题11．如图所示，用符号语言可表述为（）A ．m αβ= ，n ⊂α，m n A =B ．m n m n Aαβα⋂=∈⋂=，，C ．m n A mA n αβα⋂=⊂⊂⊂，，，D ．m n A m A n αβα⋂=∈∈∈，，，【正确答案】A【分析】由题可知两平交于直线m ，直线n 在平面α内，两直线交于点A ，从而可得答案.【详解】由题可知平,αβ交于直线m ，直线n 在平面α内，两直线,m n 交于点A ，所以用符号语言可表示为m αβ= ，n ⊂α，m n A = ，故选：A.12．已知一个圆锥的底面半径1r =，若其体积V 与侧面积S 侧之间满足9S V =侧，则该圆锥的母线长度为（）A BCD ．2【正确答案】C【分析】设圆锥的高为h ，根据体积和表面积公式得到21π9π3r h =⨯，解得218h =，再计算母线长得到答案.【详解】设圆锥的高为h ，则12ππ2S r =⨯=侧21π3V r h =，9S V =侧，即21π9π3r h =⨯，解得218h =，4==.故选：C.13．如图是一棱长为1的正方体，则异面直线1A B 与11B D 之间的距离为（）A 3B ．33C ．12D ．22【正确答案】B【分析】建立空间直角坐标系，求出与11D B 和1A B uuu r垂直的向量坐标，求出异面直线间的距离.【详解】以D 为原点，DA ，DC ，1DD 分别为x ，y ，z轴，建立如图空间直角坐标系，则11(1,1,0)D B = ，1(0,1,1)A B =- ，设(,,)n x y z =与11D B 和1A B uuu r 都垂直，则11100D B n A B n ⎧⋅=⎪⎨⋅=⎪⎩ ，即00x y y z +=⎧⎨-=⎩，取(1,1,1)n =--r ，又因为11(1,0,0)D A = ，所以异面直线11D B 和1A B 间的距离为11333D A n n⋅= 故选：B.14．空间内水平放置的两个封闭图形分别为（i ）长为3、宽为2的矩形；（ii ）边长为3的正三角形，记（i ）中原图形面积为1S ，斜二测画法得到的直观图面积为1S '，（ii ）中原图形面积为2S ，斜二测画法得到的直观图面积为2S '，对以下两个命题：①1124S S '=；②2224S S '=，以下判断正确的是（）A ．①为真命题，②为假命题B ．①为假命题，②为真命题C ．①为真命题，②为真命题D ．①为假命题，②为假命题【正确答案】C【分析】根据斜二测画法的知识求得正确答案.【详解】①，长为3、宽为2的矩形：原图：直观图：所以原图面积为1326S =⨯=，直观图的面积为11π13sin 2242S ⎛⎫'=⨯⨯⨯⨯= ⎪⎝⎭，所以1124S S '=，①为真命题.②，边长为3的正三角形：原图：直观图：所以原图面积为21π9333sin 234S =⨯⨯⨯=，直观图的面积为21333π6sin 2224416S ⎛⎫'=⨯⨯⨯⨯= ⎪ ⎪⎝⎭，所以2224S S '=，②为真命题.故选：C.三、解答题15．如图，1AC 为长方体1111ABCD A B C D -的体对角线，(1)写出所在直线与直线1AA 异面的所有棱；(2)若2,3DA DC ==，且长方体的表面积为22，求异面直线1AC 与BC 所成的角大小.【正确答案】(1)1111,,,BC B C DC D C ；(2)147.【分析】(1)根据异面直线的概念即得；(2)连接1AB ，则11AC B ∠即可判断出为所求的角，解三角形即可.【详解】（1）由异面直线的性质得1111,,,BC B C DC D C 与直线1AA 异面；（2）连接1AB ，因为11//BC B C，所以11AC B ∠即为所求角，因为2,3DA DC ==，且长方体的表面积为22，所以()11222AD DC AD DD DC DD =⨯+⨯+⨯，所以11DD =，易得1AC =，由题可知11B C ⊥平面11ABB A ，1AB ⊂平面11ABB A ，所以11B C ⊥1AB ，即11AC B 为直角三角形，所以11111cos B C AC B AC ∠==所以异面直线1AC 与BC所成的角大小为.16．数列{}n a 的前n 项和为n S ，已知()2112,322n n n a S a n +*+==-+∈N .(1)n *∈N 时，写出1n a +与n a 之间的递推关系；(2)求{}n a 的通项公式.【正确答案】(1)1142n n n a a ++=+(2)42n nn a =-【分析】（1）根据n a 与n S 之间的关系分析运算，注意分2n ≥和1n =两种情况讨论；（2）根据题意利用构造法结合等比数列求通项公式.【详解】（1）因为21322n n n S a ++=-+①，所以当2n ≥时，11322n n n S a +-=-+②，-①②得：()11322n n n n a a a n ++=--≥，即1142(2)n n n a a n ++=+≥，在①中：令1n =得322113221242a a a =+-==+，也符合上式，所以1142n n n a a ++=+.（2）因为1142n n n a a ++=+，则()11242n n n n a a +++=+，且1240a +=≠所以数列{}2n n a +是以4为首项，4为公比的等比数列，所以24n n n a +=，故42n n n a =-.17．如图，四棱锥P ABCD -的底面为矩形，PD ⊥平面ABCD ，M 为BC 的中点，且PB AM ⊥.(1)证明：平面PAM ⊥平面PDB ；(2)若6PD DC ==，求该四棱锥的体积.【正确答案】(1)证明见解析(2)【分析】（1）利用线面垂直的性质定理得到PD AM ⊥，然后利用线面垂直的判定定理即可得到AM ⊥平面PDB ，即可得证；（2）以D 为原点，分别以,,DA DC DP 为,,x y z 轴建立空间直角坐标系，得到,6,02x AM ⎛⎫=- ⎪⎝⎭，(),6,0DB x = ，利用AM BD ⊥可求出x =【详解】（1）因为PD ⊥平面ABCD ，AM ⊂平面ABCD ，所以PD AM ⊥，又因为PB AM ⊥，PD PB P = ，PD ⊂平面PDB ，PB ⊂平面PDB ，所以AM ⊥平面PDB ，因为AM ⊂平面PAM ，所以平面PAM ⊥平面PDB ；（2）以D 为原点，分别以,,DA DC DP 为,,x y z 轴建立空间直角坐标系，设DA x =，则()(),0,0,,,6,02x A x M B x ⎛⎫ ⎪⎝⎭，所以,6,02x AM ⎛⎫=- ⎪⎝⎭，(),6,0DB x = ，由AM ⊥平面PDB ，BD ⊂平面PDB 可得AM BD ⊥，所以AM BD ⊥ ，即3602x x -⨯+=，解得x =所以四棱锥的体积为1133ABCD V S PD DC DA PD =⨯=⨯⨯=18．如图，正方形ABCD 、ABEF 的边长都是1，而且平面ABCD 、ABEF 互相垂直．点M在AC 上移动，点N 在BF 上移动，若(0CM BN a a ==<．(1)求MN 的长；(2)a 为何值时，MN 的长最小；(3)当MN 的长最小时，求面MNA 与面MNB 所成二面角α的大小．【正确答案】(1)MN a =<(2)当a =MN (3)1πarccos 3-【分析】（1）作MP ∥AB 交BC 于点P ，NQ ∥AB 交BE 于点Q ，则MNQP 是平行四边形，根据MN ＝PQ ，即可求出MN 的长；（2）根据（1）的结果，结合二次函数的性质，即可求出MN 的最小值；（3）取MN 的中点G ，由题意知AG ⊥MN ，BG ⊥MN ，根据二面角的平面角的定义可知∠AGB 即为二面角的平面角，在三角形AGB 中利用余弦定理求解即可．【详解】（1）作MP ∥AB 交BC 于点P ，NQ ∥AB 交BE 于点Q ，连接PQ ，依题意可得MP ∥NQ ，且MP ＝NQ ，即MNQP 是平行四边形，∴MN ＝PQ ．由已知，CM ＝BN ＝a ，CB ＝AB ＝BE ＝1，∴AC ＝BF ，11CP BQ ==CP BQ =MN PQ ∴==a =<<．（2）由（1）得，MN =0a <<，所以，当2a =时，MN 的长取最小值2.即当M 、N 分别为AC 、BF 的中点时，MN .（3）取MN 的中点G ，连接AG 、BG ，由（2）可得2AM AN BM BN ====∵G 为MN 的中点，∴AG ⊥MN ，BG ⊥MN ，则∠AGB 即为所求二面角的平面角α，又AG ＝BG所以由余弦定理有2211cos 3α+-=-，故所求二面角1πarccos 3α=-．19．已知数列{}n a 有递推关系1191069,,,5655n n n n a a n a a a *+-⎛⎫=∈≠= ⎪-⎝⎭N (1)记,n n a b k =+若数列{}n b 的递推式形如(1,,n n n rb b p q r pb q +=∈+R 且),0p r ≠，也即分子中不再含有常数项，求实数k 的值；(2)求{}n a 的通项公式.【正确答案】(1)1或2(2)()4141nn n n a =+--【分析】（1）根据题意整理可得()219551510556n n n k b k k b b k +--+-=+-，即2515100k k -+-=，运算求解即可；（2）取1k =，可得1451n n n b b b +=-，利用构造法结合等比数列求通项公式.【详解】（1）因为n n a b k =+，且191056n n n a a a +-=-，所以()()()211910955151056556n n n n n n b k k b k k b a k k b k b k +++---+-=-==+-+-，则2515100k k -+-=，解得1k =或2；（2）由（1）可得：当1k =时，则1n n a b =+，且1451n n n b b b +=-，可得1511115444n n n n b b b b +-==-⨯+，则1111114n n b b +⎛⎫-=-- ⎪⎝⎭，且111104b -=≠，故数列11n b ⎧⎫-⎨⎬⎩⎭是以14为首项，14-为公比的等比数列，∴()111111444n n n n b --⎛⎫-=⨯-=- ⎪⎝⎭，则()441nn n n b =--，故()4141nn n n a =+--.20．无穷数列{}n a 和{}n b 满足：①{}0,1,2,;n a n N *∈∈②()111,24n n n n n b a a n N b *++=--∈，记{}n b 的前n 项积为n T ，(1)是否存在1234,,,,a a a a 使得{}n b 的前四项依次成等差数列？若存在则写出一组这样的1234,,,,a a a a 若不存在，则说明理由；(2)若11b =，求2021T 的最大值.【正确答案】(1)不存在，理由见解析(2)102010012⎛⎫ ⎪⎝⎭【分析】（1）假设存在，对1b 正负进行讨论，找出矛盾得出结论；（2）根据（1）分析出41424340,0,0,0n n n n b b b b ---<<>>，进而对112,,,n n n n a a a a +++的取值进行讨论，最终得到结果.【详解】（1）假设存在，设n b 的前四项公差为d ，当10b >时，易得2340,0,0b b b <<>，所以21430,0d b b d b b =-=-，矛盾；同理10b ＜时，也矛盾，故不存在；（2）因为110b =>，由（1）得41424340,0,0,0n n n n b b b b ---<<>>，设10n n n b q b +=>，则1113,,,,124424n n n n a a q q +⎧⎫=-∈⎨⎬⎩⎭，所以21n n n n b q q b ++=⋅⋅，又因为1n n q q +⋅的值从大到小依次为3911,,,4162①若11n n q q +⋅=，则111n n q q +=⎧⎨=⎩，则112(,)(2,0)(,)(2,0)n n n n a a a a +++=⎧⎨=⎩，1n a +不能同时存在，故不成立；②若134n n q q +⋅=，则1134n n q q +=⎧⎪⎨=⎪⎩或1341n n q q +⎧=⎪⎨⎪=⎩，此时112(,)(2,0)(,)(2,1)n n n n a a a a +++=⎧⎨=⎩或112(,)(2,1)(,)(2,0)n n n n a a a a +++=⎧⎨=⎩，1n a +不能同时存在，故不成立；③若1916n n q q +⋅=，则13434n n q q +⎧=⎪⎪⎨⎪=⎪⎩，则112(,)(2,1)(,)(2,1)n n n n a a a a +++=⎧⎨=⎩，1n a +不能同时存在，故不成立；所以112n n q q +⋅≤，此时112(,)(2,0)(,)(0,2)n n n n a a a a +++=⎧⎨=⎩或112(,)(0,2)(,)(2,0)n n n n a a a a +++=⎧⎨=⎩可取到，所以2112n n n n n b q q b b ++=⋅⋅≤，所以112111122n n n b b ---⎛⎫⎛⎫≤= ⎪ ⎝⎭⎝⎭，111221111222n n n n b b b ---⎛⎫⎛⎫⎛⎫≤≤≤ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭，所以20211232021132021242020T b b b b b b b b b b =⋅⋅=⋅⋅ 210102100910201001111111112222222⎛⎫⎛⎫⎛⎫⎛⎫⎛⎫≤⨯⨯⨯⨯⨯⨯⨯⨯⨯= ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭⎝⎭ ，此时{}n a 为2,0,2,0,因为20210T >，所以1020100202112T ⎛⎫≤ ⎪⎝⎭，即2021T 的最大值为102010012⎛⎫ ⎪⎝⎭.分类讨论思想是高中数学一项重要的考查内容.分类讨论思想要求在不能用统一的方法解决问题的时候，将问题划分成不同的模块，通过分块来实现问题的求解，体现了对数学问题的分析处理能力和解决能力.。
上海市高二上学期期中数学试卷含答案
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2023-2024学年上海市上海中学高二上学期期中考试数学试卷含详解
上海中学2023学年第一学期高二年级数学期中2023.11一､填空题（本大题共有12题，满分42分，第16题每题3分，第7-12题每题4分）1.向量()()1,0,1,,1,2a b x == 且3⋅= a b ，则x =__________.2.已知两条相交直线a ，b ，且a //平面α，则b 与α的位置关系是____________.3.将一个圆心角为2π3，面积为3π的扇形卷成一个圆锥，那么该圆锥的体积为__________.4.如图，我们将一本书打开放置在桌面上（每页书都有一边恰好落在桌面上）.根据我们所学的__________定理，我们可以证明书脊所在的直线AB 垂直于桌面.5.已知四棱锥P ABCD -的高为2，其底面ABCD 水平放置的直观图（斜二测画法）是边长为1的正方形，则该四棱锥的体积为__________.6.已知正方体ABCD-A 1B 1C 1D 1的棱长为1,M 为棱CC 1的中点,则点M 到平面A 1BD 的距离是___.7.正三棱柱ABC A B C '''-中，1,2AB AA '==，则直线BC '与平面ABB A ''所成角的正弦值为______.8.下列说法正确的是__________.①一条直线和平面平行的充要条件是直线的方向向量垂直于平面的法向量.②如果直线AB 与CD 是异面直线，那么向量AB 与CD不共面③两条异面直线的公垂线段，是连接两条异面直线所有线段中的最短线段.④直三棱柱任意两个侧面的面积之和大于第三个侧面的面积.9.设AB 和CD 都是平面α的垂线，其垂足分别为,B D .已知5,9,3AB CD BD ===，那么线段AC =__________.10.设12,O O 分别是圆柱P 的上､下底面12,ππ的中心，i Q 是以i O 为顶点，3i π-为底面的圆锥体()1,2i =，若圆柱P 的体积为1，那么圆锥12,Q Q 的公共部分的体积为__________.11.正四棱柱1111ABCD A B C D -中，已知12,1AB AA ==，那么以A 为球心，半径为2的球面与该四棱柱表面交线的总长度为__________.12.已知空间四个单位向量1234,,,e e e e 满足：1234123421+=+=+++= e e e e e e e e ，则13⋅ e e 的最大值为__________.二､选择题（本大题共有4题，满分16分，每小题4分）13.南水北调工程缓解了北方一些地区水资源短缺问题，其中一部分水蓄入某水库.已知该水库水位为海拔1485m ．时，相应水面的面积为21400km ．；水位为海拔1575m ．时，相应水面的面积为21800km ．，将该水库在这两个水位间的形状看作一个棱台，则该水库水位从海拔1485m ．上升到1575m ． 2.65≈）（）A.931.010m ⨯B.931.210m ⨯C.931.410m ⨯D.931.610m ⨯14.已知平面αβγ、、两两垂直，直线a b c 、、满足：,,a b c αβγ⊆⊆⊆，则直线a b c 、、不可能满足以下哪种关系A .两两垂直 B.两两平行 C.两两相交 D.两两异面15.已知二面角l αβ--为060，点P 、Q 分别在、内且PQ l ⊥，P 到，Q 到的距离为2,则PQ 两点之间的距离为A.B.1C.2D.16.如图，已知正三棱柱1111,ABC A B C AC AA -=，E ，F 分别是棱11,BC A C 上的点．记EF 与1AA 所成的角为α，EF 与平面ABC 所成的角为β，二面角F BC A --的平面角为γ，则（）A.αβγ≤≤B.βαγ≤≤C.βγα≤≤D.αγβ≤≤三､解答题（本大题共有4题，满分42分）17.如图，在正四棱锥P ABCD -中，,PA AB a E ==是棱PC 的中点；（1）求证：PA 平面EDB ；（2）求三棱锥-E BDC 的体积.18.如图，在四面体ABCD 中，3AB =，2AC AD ==，23BAD CAD π∠=∠=，2BAC π∠=，点M ，N 分别在棱AB ，BC 上，且AM BM =，2CN BN =.（1）用AB ，AC ，AD 表示AN ，DM ；（2）求异面直线AN ，DM 所成角的余弦值.19.在底面为正三角形的三棱柱111ABC A B C -中，平面ABC ⊥平面11BCC B ，160CBB ∠=︒，124A A AB ==.（1）证明：111B C AC ⊥；（2）求二面角1C AB A --的余弦值.20.在长方体1111ABCD A B C D -中，（1）已知P ､Q 分别为棱AB ､1CC 的中点（如图1），做出过点1D ，P ，Q 的平面与长方体的截面.保留作图痕迹，不必说明理由；（2）如图2，已知13AB =，5AD =，112AA =，过点A 且与直线CD 平行的平面α将长方体分成两部分.现同时将两个球分别放入这两部分几何体内，则在平面α变化的过程中，求这两个球的半径之和的最大值.上海中学2023学年第一学期高二年级数学期中2023.11一､填空题（本大题共有12题，满分42分，第16题每题3分，第7-12题每题4分）1.向量()()1,0,1,,1,2a b x == 且3⋅= a b ，则x =__________.【答案】1【分析】利用向量数量积的坐标公式求解即可.【详解】·1·0·11·223a b x x =++=+= ，所以1x =.故答案为：12.已知两条相交直线a ，b ，且a //平面α，则b 与α的位置关系是____________.【答案】b //平面α或b 与平面α相交【分析】画出图形不难看出直线b 与平面α的位置关系，平行或相交.【详解】由题意画出图形，当,a b 所在平面与平面α平行时，b 与平面α平行，当,a b 所在平面与平面α相交时，b 与平面α相交.故答案为:b //平面a 或b 与平面α相交.【点睛】本题考查空间中直线与平面之间的位置关系，考查空间想象能力，是基础题.3.将一个圆心角为2π3，面积为3π的扇形卷成一个圆锥，那么该圆锥的体积为__________.【答案】22π3【分析】求出扇形的半径，进而求出圆锥的底面半径和高，利用体积公式求出答案.【详解】设扇形的半径为l ，则22π3π312l ⨯=，解得3l =，即卷成的圆锥母线长为3，设圆锥的底面半径为r ，则π23π32π2π23r l ⨯===，解得1r =，则圆锥的高2222h l r =-=故圆锥的体积为2π12π233r h =故答案为：22π34.如图，我们将一本书打开放置在桌面上（每页书都有一边恰好落在桌面上）.根据我们所学的__________定理，我们可以证明书脊所在的直线AB 垂直于桌面.【答案】线面垂直的判定【分析】略【详解】略5.已知四棱锥P ABCD -的高为2，其底面ABCD 水平放置的直观图（斜二测画法）是边长为1的正方形，则该四棱锥的体积为__________.【答案】423423【分析】由直观图得到原图形，进而得到底面ABCD 的面积，利用锥体体积公式求出答案.【详解】直观图如图1，则原图形如图2，则原图形为平行四边形，面积为12222⨯=故底面ABCD 的面积为22，故该四棱锥的体积为14222233⨯=.故答案为：4236.已知正方体ABCD-A 1B 1C 1D 1的棱长为1,M 为棱CC 1的中点,则点M 到平面A 1BD 的距离是___.【答案】32【分析】建立空间直角坐标系，得到平面的法向量，根据空间向量的运算求得点到平面的距离．【详解】以D 为原点，DA 为x 轴，DC 为y 轴，DD 1为z 轴，建立如下图所示的空间直角坐标系则D （0，0，0），A 1（1，0，1），B （1，1，0），M （0，1，12）所以()()111,1,0,0,1,,1,0,12DB DM DA ⎛⎫=== ⎪⎝⎭ 设平面A 1BD 的法向量为(),,m x y z=，根据向量垂直可得100m DB x y m DA x z ⎧⋅=+=⎪⎨⋅=+=⎪⎩ ，令x=1，解得()1,1,1m =--所以M 到平面A 1BD 的距离为3322m DM d m ⋅=== 【点睛】本题考查了空间向量在求点到平面距离中的基本应用，属于中档题．7.正三棱柱ABC A B C '''-中，1,2AB AA '==，则直线BC '与平面ABB A ''所成角的正弦值为______.【答案】1510【分析】建立空间直角坐标系，利用空间向量的夹角坐标公式即可求出结果.【详解】以B 为原点，以过B 作BC 的垂线为x 轴，以,BC BB '为,y z 轴，建立空间直角坐标系，如图，则31(0,0,0),(0,0,2),(0,1,2),,,0)22B B C A ''，所以31(0,0,2),,,0)22BB BA ==' ，平面ABB A ''的一个法向量设为(,,)n x y z = ，则2031022n BB z n BA x y ⎧⋅==⎪⎨⋅='+=⎪⎩，令1x =得(1,n = ，又(0,1,2)BC ='，设直线BC '与平面ABB A ''所成的角为θ，π[0,2θ∈，则sin cos ,10n BC n BC n BCθ⋅===='⋅'' .故答案为：15108.下列说法正确的是__________.①一条直线和平面平行的充要条件是直线的方向向量垂直于平面的法向量.②如果直线AB 与CD 是异面直线，那么向量AB 与CD 不共面③两条异面直线的公垂线段，是连接两条异面直线所有线段中的最短线段.④直三棱柱任意两个侧面的面积之和大于第三个侧面的面积.【答案】③④【分析】利用直线方向向量和平面法向量的位置关系判断①，利用向量可以平移的性质判断②，根据异面直线公垂线段的定义判断③，根据直三棱柱的结构特征判断④.【详解】由“一条直线和平面平行”可得“直线的方向向量垂直于平面的法向量”，所以充分性成立，由“直线的方向向量垂直于平面的法向量”可得“直线平行于平面或直线在平面内”，所以必要性不成立，综上一条直线和平面平行的充分不必要条件是直线的方向向量垂直于平面的法向量，①说法错误；直线AB 与CD 是异面直线，因为向量是可以自由平移的，所以向量AB 与CD 可以平移到同一平面，②说法错误；两条异面直线的距离是两条异面直线的公垂线段的长度，两条异面直线的公垂线段，是连接两条异面直线所有线段中的最短线段，③说法正确；因为直三棱柱的侧棱垂直于底面，三角形的任意两边之和大于第三边，所以直三棱柱任意两个侧面的面积之和大于第三个侧面的面积，④说法正确；故答案为：③④9.设AB 和CD 都是平面α的垂线，其垂足分别为,B D .已知5,9,3AB CD BD ===，那么线段AC =__________.【答案】5【分析】根据线面垂直的性质，证得//AB CD ，且,AB BD CD BD ⊥⊥，分,A C 在平面的同侧和,A C 在平面的两侧，两种情况讨论，分别作AE CD ⊥和AF CF ⊥，在直角ACE △和直角ACF △中，结合勾股定理，即可求解.【详解】如图所示，因为AB 和CD 都是平面α的垂线，其垂足分别为,B D ，可得//AB CD ，且,AB BD CD BD ⊥⊥，如图（1）所示，当点,A C 在平面的同侧时，过点A 作AE CD ⊥，垂足为E ，则//AE BD ，又因为5,9,3AB CD BD ===，可得4CE CD DE CD AB =-=-=，在直角ACE △中，可得5AC ==.如图（2）所示，当点,A C 在平面的两侧时，过点A 作CD 的延长线的垂线，设AF CF ⊥，垂足为F ，则//AF BD ，又因为5,9,3AB CD BD ===，可得14CF CD DF CD AB =+=+=，在直角ACF △中，可得AC ===.故答案为：510.设12,O O 分别是圆柱P 的上､下底面12,ππ的中心，i Q 是以i O 为顶点，3i π-为底面的圆锥体()1,2i =，若圆柱P 的体积为1，那么圆锥12,Q Q 的公共部分的体积为__________.【答案】112【分析】根据题意，得到圆锥12,Q Q 的公共部分为同底的圆锥1OO 和2OO ，结合圆柱和圆锥的体积公式，即可求解.【详解】如图所示，根据题意，以i Q 是以i O 为顶点，3i π-为底面的圆锥体()1,2i =，则圆锥12,Q Q 的公共部分为同底的圆锥1OO 和2OO 设圆锥P 底面圆的半径为R ，高为h ，可得2π1R h =，由11O AB O CD ∽，且根据几何体的对称性，可得AB R =，即公共部分的圆锥的底面圆的半径为12R ，且每个小圆锥的高为12h ,所以公共部分的体积为2211111π()2π3221212V R h R h =⨯⋅⨯==.故答案为：112.11.正四棱柱1111ABCD A B C D -中，已知12,1AB AA ==，那么以A 为球心，半径为2的球面与该四棱柱表面交线的总长度为__________.【答案】5π+32.【分析】利用球与正四棱柱的特征求轨迹长度即可.【详解】如图所示，以A 为球心，半径为2的球面与该四棱柱的表面交线为四段弧 FB BD FG DG、、、，分别在平面11111111ABB A ABCD ADD A A B C D 、、、上，易知 1πππ22663AF AA FAB FB DG =⇒∠=⇒=⨯==， π=2=π2BD ⨯， π3π=22GF ，所以交线长为5π3π+32.故答案为：5π3π32+12.已知空间四个单位向量1234,,,e e e e 满足：1234123421+=+=+++= e e e e e e e e ，则13⋅ e e 的最大值为__________.【答案】731516+【分析】将该四个单位向量平移至共起点置于球中，利用空间向量的数量积计算1234,e e e e ++ ，借助圆锥图形确定13,e e 即可.【详解】如图所示，令1234,,,e e e e 共起点O ，由题意易得2212112212121121cos ,2e e e e e e e e e e +=⇒+⋅+=⇒⋅=-= ，同理34123412πcos ,,,23e e e e e e =-⇒== ，设1234,e e a e e b +=+= ，则13π,,3e a e b == ，根据条件有221172cos ,248a b a a b b a b a b +=⇒+⋅+=⇒⋅=-= ，所以13,e e 分别在以,a b 所在直线为轴，O 为顶点，以夹角π3旋转一周得到的圆锥12OO OO 、的侧面上，观察图形可知当13,,e e ,a b 在平面12OO O 内时，此时13,e e 夹角最小，易知15,πsin ,8a b a b <⇒= ，则13max 2π2π2π7315cos ,cos ,cos ,cos sin ,sin cos 33316e e a b a b a b +=-=+= ，所以13137cos ,16e e e e +⋅=≤ .故答案为：731516+.二､选择题（本大题共有4题，满分16分，每小题4分）13.南水北调工程缓解了北方一些地区水资源短缺问题，其中一部分水蓄入某水库.已知该水库水位为海拔1485m ．时，相应水面的面积为21400km ．；水位为海拔1575m ．时，相应水面的面积为21800km ．，将该水库在这两个水位间的形状看作一个棱台，则该水库水位从海拔1485m ．上升到1575m ． 2.65≈）（）A.931.010m ⨯ B.931.210m ⨯ C.931.410m ⨯ D.931.610m ⨯【答案】C【分析】根据题意只要求出棱台的高，即可利用棱台的体积公式求出．【详解】依题意可知棱台的高为157.5148.59MN =-=(m)，所以增加的水量即为棱台的体积V ．棱台上底面积262140.014010S ==⨯km m ，下底面积262180.018010S '==⨯km m ，∴((66119140101801033V h S S =++=⨯⨯⨯+⨯'(()679933320109618 2.6510 1.43710 1.410(m )=⨯+⨯≈+⨯⨯=⨯≈⨯．故选：C ．14.已知平面αβγ、、两两垂直，直线a b c 、、满足：,,a b c αβγ⊆⊆⊆，则直线a b c 、、不可能满足以下哪种关系A.两两垂直B.两两平行C.两两相交D.两两异面【答案】B【分析】通过假设//a b ，可得,a b 平行于,αβ的交线，由此可得c 与交线相交或异面，由此不可能存在////a b c ，可得正确结果.【详解】设l αβ= ，且l 与,a b 均不重合假设：////a b c ，由//a b 可得：//a β，//b α又l αβ= ，可知//a l ，//b l又////a b c ，可得：//c l因为,,αβγ两两互相垂直，可知l 与γ相交，即l 与c 相交或异面若l 与a 或b 重合，同理可得l 与c 相交或异面可知假设错误，由此可知三条直线不能两两平行本题正确选项：B【点睛】本题考查空间中的直线、平面之间的位置关系，关键在于能够通过线面关系得到第三条直线与前两条线之间的位置关系，从而得到正确结果.15.已知二面角l αβ--为060，点P 、Q 分别在、内且PQ l ⊥，P 到的距离为3，Q 到的距离为32,则PQ 两点之间的距离为A.3 B.1 C.2 D.2【答案】A【分析】由题意分别作,PC QD βα⊥⊥，过C 作CM l ⊥，连接,PM QM ，在,Rt PMC Rt QMD ∆∆中，分别求出,QM PM ，再在PMQ ∆中，利用余弦定理即可求解.【详解】如图，作,PC QD βα⊥⊥，过C 作CM l ⊥，连接,PM QM ，由l αβ= ，所以,PC l QD l ⊥⊥，又PQ l ⊥，l ⊥平面QCDP ，即l ⊥平面QMP由二面角l αβ--为060，P 到的距离为3Q 到的距离为32,在Rt QMD ∆中，32QD =,60QMD ∠= ，21sin 60QM ==在Rt PCM ∆中，PC =,60QMD ∠= ，32sin 60PM == ，在PMQ ∆中，22212cos 60142232QP QM PM QM PM =+-⋅=+-⨯⨯= ，所以PQ =故选：A 【点睛】本题考查了由面面角求距离、余弦定理解三角形，考查了空间想象能力，属于基础题.16.如图，已知正三棱柱1111,ABC A B C AC AA -=，E ，F 分别是棱11,BC A C 上的点．记EF 与1AA 所成的角为α，EF 与平面ABC 所成的角为β，二面角F BC A --的平面角为γ，则（）A.αβγ≤≤B.βαγ≤≤C.βγα≤≤D.αγβ≤≤【答案】A【分析】先用几何法表示出αβγ，，，再根据边长关系即可比较大小．【详解】如图所示，过点F 作FP AC ⊥于P ，过P 作PM BC ⊥于M ，连接PE ，则EFP α=∠，FEP β=∠，FMP γ=∠，tan 1PE PE FP AB α==≤，tan 1FP AB PE PE β==≥，tan tan FP FP PM PEγβ=≥=，所以αβγ≤≤，故选：A ．三､解答题（本大题共有4题，满分42分）17.如图，在正四棱锥P ABCD -中，,PA AB a E ==是棱PC 的中点；（1）求证：PA 平面EDB ；（2）求三棱锥-E BDC 的体积.【答案】（1）证明见解析；（2）3224.【分析】（1）连接AC 交BD 于O ，由中位线的性质可得出//PA OE ，再利用线面平行的判定定理可证得结论成立；（2）计算出点E 到底面的距离以及BCD △的面积，再利用锥体的体积公式可求得三棱锥的体积.【小问1详解】证明：在正四棱锥P ABCD -中，四边形ABCD 为正方形，连接AC 交BD 于O ，则O 为AC 的中点，又因为E 为PC 的中点，所以//PA OE ，.因为OE ⊂平面EDB ，PA ⊄平面EDB ，所以PA 平面EDB.【小问2详解】在正四棱锥P ABCD -中，O 为底面的中心，则PO ⊥底面ABCD ，所以POA 为直角三角形，22PO ==因为E 为PC 的中点，则点到E 平面ABCD 的距离1224h PO ==，因此，三棱锥-E BDC 的体积3211122··332424BCD V S h a ==⋅⋅= .18.如图，在四面体ABCD 中，3AB =，2AC AD ==，23BAD CAD π∠=∠=，2BAC π∠=，点M ，N 分别在棱AB ，BC 上，且AM BM =，2CN BN =.（1）用AB ，AC ，AD 表示AN ，DM ；（2）求异面直线AN ，DM 所成角的余弦值.【答案】（1）2133AN AB AC =+ ，12DM AB AD =- （2）17370370【分析】（1）根据向量的线性运算直接表示各向量；（2）利用转化法求向量数量积及夹角.【小问1详解】因为点M ，N 分别在棱AB ，BC 上，且AM BM =，2CN BN =，所以12AM AB = ，13BN BC = ，所以()11213333AN AB BN AB BC AB AC AB AB AC =+=+=+-=+ ，12DM DA AM AB AD =+=- ；【小问2详解】因为3AB =，2AC AD ==，23BAD CAD π∠=∠=，2BAC π∠=，所以32cos 02AB AC π⋅=⨯⨯= ，232cos 33AB AD π⋅=⨯⨯=- ，222cos 23AC AD π⋅=⨯⨯=- 所以2222144110339993AN AB AC AB AB AC AC ⎛⎫=+=+⋅+ ⎪⎝⎭，372DM==，211332AN DM AB AC AB AD⎛⎫⎛⎫⋅=+⋅-⎪ ⎪⎝⎭⎝⎭212111733633AB AB AD AB AC AC AD=-⋅+⋅-⋅=，所以17173703cos,37032AN DMAN DMAN DM⋅==⋅，即异面直线AN，DM所成角的余弦值为370.19.在底面为正三角形的三棱柱111ABC A B C-中，平面ABC⊥平面11BCC B，160CBB∠=︒，124A A A B==.（1）证明：111B C AC⊥；（2）求二面角1C AB A--的余弦值.【答案】（1）证明见解析（2）5【分析】（1）求出1B C，利用勾股定理证明111B C B C⊥，再根据面面垂直的性质可得1B C⊥平面111A B C，再根据线面垂直的性质即可得证；（2）以1B为原点，1B C，11B C所在直线分别为x轴，y轴建立空间直角坐标系，利用向量法即可求出答案.【小问1详解】证明：因为160CBB∠=︒，124A A A B==，所以22211112cos12B C BC BB BC BB CBB=+-⋅⋅∠=，则1B C=，所以2221111B C B C CC+=，即111B C B C⊥，因为平面ABC∥平面111A B C，平面ABC⊥平面11BCC B，所以平面111A B C⊥平面11BCC B，因为平面111A B C Ç平面1111BCC B B C =，所以1B C ⊥平面111A B C ，又11AC ⊂平面111AB C ，所以111B C AC ⊥；【小问2详解】解：如图，以1B 为原点，1B C ，11B C 所在直线分别为x 轴，y 轴建立空间直角坐标系，则()10,0,0B，()C，()2,0B -，(1A ，所以(11B A =，()12,0B B =- ，设平面1ABA 的法向量为()1,,n x y z = ，则1111100n B A n B B ⎧⋅=⎪⎨⋅=⎪⎩，即020y y ⎧+=⎪⎨-=⎪⎩，取x =1，则()11n =- ，又因为x 轴⊥平面ABC ，所以取平面ABC 的法向量()21,0,0n =u u r，所以112122cos ,55n n n n n n ⋅== ，由图可知，二面角为锐角，所以二面角1C AB A --的余弦值为55.20.在长方体1111ABCD A B C D -中，（1）已知P ､Q 分别为棱AB ､1CC 的中点（如图1），做出过点1D ，P ，Q 的平面与长方体的截面.保留作图痕迹，不必说明理由；（2）如图2，已知13AB =，5AD =，112AA =，过点A 且与直线CD 平行的平面α将长方体分成两部分.现同时将两个球分别放入这两部分几何体内，则在平面α变化的过程中，求这两个球的半径之和的最大值.【答案】（1）答案见解析（2）最大值为16538【分析】（1）运用基本事实3：两面有一个公共点，则必有一条过该点的交线，基本事实3是做截面问题的基础；（2）用CBP ∠的三角函数将两圆的半径分别表示出来，构造新函数，通过函数单调性求得问题的最值.【小问1详解】①延长1D Q 交DC 延长线于点E ；②连接PE 与BC 交于点F ，并延长EP 交DA 延长线于点G ；③连接1D G 交1AA 于点H ；④分别连接线段1D H ，HP ，PF ，FQ ，1QD ，则五边形1D HPFQ 及其内部（图中阴影部分）即为所求截面.【小问2详解】如图所示，平面ABMN 将长方体分成两部分，MN 有可能在平面11CDD C 上或平面1111A D C B 上，但是若MN 在平面1111A D C B 上运动，两部分几何体都是细长形状，放入的两个小球由于棱长AD 限制，易知要使两球半径和的最大，需在平面11CDD C 上运动.延长11B C 与BM 交于点P ，作1O Q BC ⊥于Q 点，设1CBP BPB α∠=∠=，圆1O 对应的半径为1r ，根据三角形内切圆的性质，在1Rt O QB △中，12QBO α∠=，15BQ BC CQ r =-=-，111tan 25O Q r BQ r α==-，则15tan5251tan 1tan 22r ααα==-++，又当BP 与1BC 重合时，1r 取得最大值，由内切圆等面积法求得1512251213r ⨯≤=++，则2tan 23α≤设圆2O 对应的半径为2r ，同理可得266tan2r α=-，又252r ≤，解得7tan 212α≥.故1255566tan 1761tan 221tan 1tan 22r r αααα⎛⎫+=-+-=--+ ⎪⎝⎭++，72tan 1223α≤≤，设1tan 2x α=+，则195,123x ⎡⎤∈⎢⎣⎦，()5176f x x x =--，由对勾函数性质易知195,123x ⎡⎤∈⎢⎣⎦，函数()f x 单减，则()195191651761912123812f x f ⎛⎫≤=--⨯= ⎪⎝⎭，即最大值为16538.故两个球的半径之和的最大值为16538.【点睛】本题考查截面问题，考查球的综合问题，考查构造函数思想以及数形结合思想，借助三角函数表示边长，从而把问题转化为函数问题，再通过单调性解决最值问题，属于较难题目.。
上海交大附中高二英语上学期期中考试 上海新世纪版 新课标 【会员独享】
上海交通大学附属中学09-10学年高二上学期期中考试英语试卷（满分100分，100分钟完成，答案一律写在答题卡、答题纸上）I. Listening Comprehension (24%)Section A Short Conversation (10%)Direction: In Section A, you will hear ten short conversations between two speakers. At the end of each conversation, a question will be asked about what was said. Each conversation and question will be spoken only once. After you hear a conversation and the question about it, read the four possible answers and decide which one is the best answer to the question you have heard.1. a. find a larger room b. sell the old tablec. buy two bookshelvesd. rearrange some furniture2. a. guest and receptionist b. passenger and air hostessc. customer and shop assistantd. patient and nurse3. a. Tom survived the accident b. Tom was put into prisonc. The wall was knocked downd. It did little damage to Tom’s car4. a. The man didn’t want the woman to heave her hair cut.b. The woman followed the man’s advice.c. The woman is wearing long hair now.d. The man didn’t care if the woman had her hair cut or not.5. a. in a restaurant b. in a hospitalc. in a toy stored. in a zoo6. a. She seldom reads books from cover to cover.b. She has no interest in reading novels.c. She read only parts of the book.d. She was eager to know what the book was about.7. a. 2:50 p.m. b. 2:15p.m. c. 1:50 p.m. d. 1:15 p.m.8. a. Jack always arrives on time.b. Jack used to be late for work.c. Jack knows nothing about his work.d. Jack is often late for work.9. a. by going on a diet b. by doing physical exercisec. by having fewer mealsd. by eating fruit and vegetables10. a. Most people killed in the accidents are heavy drinkers.b. Passengers should pay more attention when crossing the road.c. She doesn’t agree with what the man said.d. Drunk driving causes many accidents on the road.Section B Passage (6 %)Direction: In Section B, you will hear two short passages, and you will be asked three questions on each of the passages. The passages will be read twice, but the questions will be spoken onlyonce. When you hear a question, read the four possible answers in your paper and decide which one could be the best answer to the question you have heard.Questions 11 through 13 are based on the following passage.11. a. to warn people of the danger of smoking during pregnancy.b. to advise people with breathing problems to stop smoking.c. to show people the risks of having a low birth weight baby.d. to offer some tips on how to give up smoking.12. a. It may cause the difficult birth of the baby.b. The babies are more likely to suffer heart disease.c. The babies may weigh less than other children.d. The danger of lung cancer may increase for pregnant mothers.13. a. a boy whose mother smokesb. a girl whose mother smokesc. a boy whose father smokesd. a girl whose father smokesQuestions 14 through 16 are based on the following news.14. a. There were too many people living in the building.b. Electric heaters were not carefully used.c. The old house was in bad condition and needed repairing.d. Children in the nearby school played with fire.15. a. customers and staffb. non-governmental organizationsc. charitiesd. environmental protection groups16. a. about 20 million b. 15.3 million c. 130 million d. 77 millionSection C Longer Conversations (8%)Direction: In Section c, you will hear two longer conversations, you are required to fulfill the task by filling in the numbered blanks with the information you have heard.Complete the form. Write NO MORE THAN THREE WORDS for each answer.Ⅱ. Grammar and V ocabulary 26%Section ADirections: Beneath each of the following sentences there are four choices marked a, b, c and d. Choose the one answer that best completes the sentence.25.A bullet hit the soldier and he was wounded in___________leg.a. ab. onec. thed. his26. The cost of hiring a car should be no more than $800 for a full day, and in some places. ______.a. much as halfb. as half muchc. half as muchd. half much as27. Whom would you rather _______ us solve the traffic problem in Shanghai, Mr. Brown or Mr. Smith?a. had helpedb. have to helpc. have helpd. have helped28. The explosion on the sun cannot be heard, there _______ no air between the earth’s atmosphere and the sun.a. isb. wasc. havingd. being29. ___________competitive in this international economy, a company must open itself to infor-mation and ideas.a. Stayingb. Having stayedc. To stay d To be staying30. The emperor ordered that the weavers ____ some gold in order that they ____ their work at once.a. should give; could beginb. be given; might beginc. would be give; could startd. should be given; may start31. It was well known that Napoleon always requested that his soldiers ____ the same three questions, and usually in the same order.a. answeredb. could have answeredc. answerd. should have answered32. I would like to do it right now, but I ______ time.a. have nob. hadn’tc. didn’t haved. will have no33.___ the books not properly marked and catalogued, they ____ be so easy to find.a. If; couldb. Were; couldn’tc. Had; couldn’td. Were; could34. Why? It's unbelievable that the maid who had taken so good care of the baby _____ like that in the past two years.a. could have been treatedb. should have been treatedc. may have been treatedd. must have been treated35. Grown people should never say or do anything wrong before children lest they ______ thema bad example.a. will setb. are to setc. should setd. would set36. ________, he retuned to his hometown from abroad.a. Every few yearsb. Every a few yearsc. Each a few yearsd. Each few years37. Mr. Baker bought a pair of leather shoes to ______ his new suit, which was sent to him as the birthday present.a. suitb. matchc. fitd. agree38. These days he was _______ doing experiments and he always had _____ noodles for meals.a. engaged in…fastb. engaged in… instantc. engaged with…convenientd. engaged with…instant39. As for Latin America generally, there is no reason why, __________available resources, it cannot become a highly industrialized area.a. at the mercy ofb. in terms ofc. at the turn ofd. at the cost of40. A lot of people from all parts of the world _______ to have seen UFOs each year, but not a case has been proved so far.a. indicateb. informc. claimd. declare41. They were disappointed when their son ______ bottom in the examination.a. came upb. came outc. came tod. came offSection BDirections: Complete the following passage by using the words in the box. Each word can only be used once. Note that there is one word more than you need.A project called the World Community Grid has found a way for computers connected to the Internet to help solve humanitarian problems. The World Community Grid is making technology ___42_____ to public and non-profit organizations to use in humanitarian research.Scientists at the University of Washington, for example, are using the technology to study ways to ___43__ the nutritional value of rice. Another research project ___44__ by the World Community Grid is studying mathematical ways to design drugs to treat the disease AIDS. Other projects are studying cancer. And still others are studying climate change in Africa.The success of the World Community Grid depends upon ___45____ collectively donating their extra computer power. This is based on the idea that most computers are ____46___ most of the time. During the time they are not used, they can help solve complex scientific or engineering problems.The IBM Corporation started the World Community Grid more than two years ago. The company continues to provide ___47____ and support for the project. Stanley Litow heads community relations for IBM. He says anyone in the world with a computer connected to the Internet can join the project.V olunteers download a program from the World Community Grid website. Every so often the program uploads or downloads more information to be ___48____. Individuals can also findout how much work their computer power has done on the website.____49____, about one million personal computers in 100 countries are involved in the World Community Grid. Mr. Litow hopes that another million computers will join the project. Then, he says, the World Community Grid will become the world’s largest super computer able to do so many ____50___ a year.III. Reading Comprehension 35%Section ADirections: For each blank in the following passages there are four words or phrases marked a, b, c and d. Fill in each blank with the word or phrase that best fits the context.It seems to be a simple thing: teaching your student to say “I am sorry”. But far between simply repeat a conditioned response, truly understanding the nature of an apology, and being able to deliver one ___51__, requires a level of social ability that many __52____ find difficult, let alone the teenage with autism(我向思考). Breaking that “simple” __53____ down into understandable steps of progress goes a long way towards helping a student with autism comprehend and __54_____ this critical interpersonal skill.The need for an apology may come at a time when emotions are running high. If this is the case, an apology need not come ___55___ to be effective. There is a huge gap between merely reciting words of apology and feeling true __56____. In the heat of anger, the student may need a cooling-off period before he or she can _57______real feelings of being sorry and apologize sincerely. The cooling-off period will __58___from person to person. This requires striking a balance—you want the ___59___ handled in a timely and relatively immediate way, but an insincere apology is not an ____60____ apology. Remember, attempting to teach anything when the student is still ___61____ overstressed is a wasted effort.In order to issue a sincere apology, a student needs first to understand why the apology is necessary. Le arning to recite the words “I’m sorry.” i sn’t enough—it doesn’t help the student understand what he or she did and the __62_____ it had on another person. All students, not just those with autism, have __63____ of selfishness, defensiveness in their ____64___. Encourage them to think about other people’s feelings, and explain why we make apologies is important and helps them to develop ___65____. This will come increasingly over a long period.51. a. meaningfully b. sincerely c. effectively d. intelligently52. a. philosophers b. psychologists c. infants d. adults53. a. method b. apology c. situation d. theory54. a. test b. acquire c. apply d. possess55. a. immediately b. mentally c. naturally d. normally56. a. regret b. sympathy c. innocence d. satisfaction57. a. appreciate b. resist c. emphasize d. recognize58. a. change b. vary c. range d. arrange59. a. conflict b. incident c. condition d. occasion60. a. false b. adequate c. negative d. worthless61. a. alternatively b. physically c. emotionally d. subconsciously62. a. effect b. comment c. impression d. intension63. a. definitions b. materials c. elements d. functions64. a. qualifications b. attitudes c. manners d. personalities65. a. sharing b. reasoning c. undertaking d. understandingSection BDirections：Read the following four passages. Each passage is followed by several questions or unfinished statements. For each of them there are four choices marked a, b, c and d. Choose the one that fits best according to the information given in the passage you have just read.*Embassy Suites Hotel San Luis is currently accepting applications for Front Desk Services Agent. *Full time day and part time night position open.*Availability must include weekends and holidays.*This position has a combination of duties mainly related, but not limited to checking-in hotel guests.We are seeking candidates who have the ability to:*Understanding guest inquires and provides responses in a helpful, courteous(礼貌的) manner. *Promote positive relations with all individuals who approach the Front Desk and enter the hotel. *Focus on the guest needs remaining calm and courteous.*Work well under pressure.*Input and access data in the computer.*Ensure security and confidentiality of guest and hotel information.*Work cooperatively with other departments and co-workers as part of a team. REQUIREMENTSCustomer service experience preferred.Please apply online by submitting a resume, including a cover letter.OR:Apply in person, by submitting an application for employment, from the executive offices located within the hotel.66. Which of the following best describes the job advertised?a. Technical Engineerb. Door Manc. Reservation Officerd. Front Desk Services Agent67. Those interested in this job may contact Rebecca Hyer _________.a. by e-mailb. by faxc. by sending short messagesd. by post68. Which of the following is needed for the job?a. College diploma or aboveb. Necessary PC skillsc. Speaking at least three languagesd. Five-year working experienceBOn June 17, 1744, the officials from Maryland and Virginia held a talk with the Indians of the Six Nations. The Indians were invited to send boys to William and Mary College. In a letter the next day he refused the offer as follows:We know that you have a high opinion of the kind of learning taught in your colleges, and that the costs of living of our young men, while with you, would be very expensive to you. We are convinced that you mean to do us good by your proposal; and we thank you heartily. But you must know that different nations have different ways of looking at things, and you will therefore not be offended if your ideas of this kind of education happen not t be the same as yours. We have had some experience of it. Several of our young people were formerly brought up at the colleges of the northern provinces: they were taught all your sciences; but, when they came back to us, they were bad runners, ignorant of every means of living in the woods… they were totally good for nothing.We are, however, not the less obliged by your kind offer, though we refuse to accept it; and, to show our grateful sense of it, if the gentlemen of Virginia will send up a dozen of their sons, we will take care of their education, teach them in all we know, and make men of them.69. The passage is about ________.a. the talk between the Indians and the officialsb. the colleges of northern provincesc. the educational values of the Indiansd. the problems of the Americans in the mid-eighteenth century70. The Indians’ chief purpose in writing the letter seems to ________.a. politely refuse a friendly offerb. express their opinion on equal treatmentc. show their prided. describe Indian customs71. According to the letter, the Indians believed that ________.a. it would be better for their boys to receive some schoolingb. they were being insulted by the offerc. they knew more about science than the officialsd. they had a better way of educating young men72. Different from the officials’ view of education, the Indians though t ________.a. young women should also be educatedb. they had different goals of educationc. they taught different branches of scienced. they should teach the sons of the officials firstCA Canadian architect has designed a completely different kind of city house—it’s independent of the electricity grid (net), the water system, and sewage lines. “We wanted a house that co uld remain independent of the city utilities, “ explains Toronto architect, Martin Liefhebber, “ So we will heat it with an efficient wood stove, provide electricity from solar panels, and recycle all waste with the help of a self-composting toilet and gre enhouse.”Liefhebber estimates that the two-story, 992-square-foot home could be built for less than$100,000, nothing will be wasted. The roof catches snow and rainwater will be treated for drinking water; the toilet, which will use only a cup of water per flush, will produce fertilizer for the green house garden situated on the home’s roof. The wall will be built of blocked made from concrete mixed with cellulose fibers-light and expensive, yet a very good insulator(隔热材料). Liefhebbers has named his house the Codicile, “The name reflects the philosophy that if we use less resources, we’ll have more to pass on to future generations.” he says.Toronto’s laws have so far prevented the house from going up, but Liefhebber i s hopeful that his new idea will have wide appeal and plans to build one later this year.73. The house designed by a Canadian architect, _______ city utilities.a. usesb. is free ofc. is equipped withd. is connected with74. Drinking water comes from _________.a. roofb. snow and rainc. piped waterd. both b and c75. The blocks of which the walls will be built are ________.a. semi-conductorsb. light but expensivec. not only lightweight but cheapd. both a and c76. The architect designed the house in order to ________.a. save resourcesb. recycle all the wastec. want more from natured. both a and bDTwo men on a touring holiday of Britain were injured by an explosion in their motor van in the center of Norwich yesterday.Shoppers, traders and businessmen in Red Lion Street were shocked by a loud bang. And seconds later the two men jumped over from the vehicle, which had stopped outside Barclays Bank. Several people rushed to give assistance and help to put out the fire inside the vehicle, a light American truck converted to provide living accommodation, before Norwich firemen arrived.The men-American Mr. Gary House, age 25, of Ohio, who was driving , and his passenger Mr. Charles Lynn, 23, of Vancouver—were taken to Norfolk and Norwich Hospital with minor burns. They were allowed to leave after treatment.“I heard this explosion. It was pretty loud. I thought it could have been a bomb,” said Mr. Leslie Webster, manager of the market, who was working in his office in Re d Lion Street. “I looked out of the window and saw this lad leap from the van and roll on the ground. Then another lad came out of the van. He seemed to be in a worse state—parts of his trousers were hanging below his knee.”“I came downstairs to get a fir e extinguisher, but by the time I got outside someone from the bank was in the van with an extinguisher.”Mr. Webster, who lives at 71, Trinity Street, Norwich, said both victims were shocked. One was taken into the market’s office to await an ambulance. “The second man insisted on going back into the van to see if everything was all right, and five minutes later he came out with a drawer that was blazing, ”he added.The explosion was also heard inside the bank. Staff provided a fire extinguisher and telephoned for an ambulance.Although a plastic window was blown out, damage inside the vehicle was mainly superficial.The two men have spent the last six months touring the continent and had traveled to Norwich from Snettleton. At the time of the accident their wives were shopping in the city.77. The two men in the van _________.a. were slightly hurtb. were badly injuredc. were shocked but unhurtd. needed careful treatment78. The word “leap” most probably means _________.a. fall offb. get offc. jump offd. step down79. Mr. Webster thought that the explosion _________.a. was a bombb. wasn’t a bombc. was like a bombd. was almost a bomb80. Which of the following is NOT TRUE according to the passage?a.Mr. Webster was among those who happened to see the accident.b.Mr. Webster didn’t take a fire extinguisher to the van.c.Mr. Webster helped to put out the fire.d.Mr. Webster was willing to be a helper in the accident.Section C: Read the following text and choose the most suitable heading from A-F for each paragraph. There is one extra heading which you do not need. F=A+EBefore you argue with your boss, check with the boss’s secreta ry to determine his mood. If he eats nails for breakfast, it is not a good idea to ask him for something. Even without the boss’s secretary, ther e are keys to timing; don’t approach the boss when he’s on deadline; don’t go in right before lunch, when he is apt to be distracted and rushed; don’t go in just before or after he has taken a vacation.82. _____________________If you are mad, that will only make your boss mad. Calm down first. And don’t let a particular concern open the floodgates for all your accumulated frustration. The boss will feel that you think negatively about the company and it is hopeless trying to change your mind. Then, maybe he will dismiss you.83. ____________________Terrible disputes can result when neither the employer nor the employee knows what is the problem the other wants to discuss. Sometimes the fight will go away when the issues are made clear. The employee has to get his point across clearly in order to make the boss understand it. 84. ____________________Your boss ha s enough on his mind without your adding more. If you can’t put forward an immediate solution, at least suggest how to approach the problem. People who frequently present problem without solutions to their bosses soon find they ca n’t get past the secretary.85. _____________________To deal effectively with a boss, it’s important to consider his goals and pressures. If you can put yourself in the position of being a partner to the boss, then he will be naturally more inclined to work with you to achieve your goals.IV. Translation 15%Directions: Translate the following sentences into English, using the word or phrase given in the brackets.86. 重要的是你能从错误中吸取教训。
上海市东昌中学高二英语上学期期中考试 上海新世纪版 新课标【会员独享】
上海市东昌中学09-10学年高二上学期期中测试英语试卷时间：90分钟满分：125分（2009-11）I. Listening Comprehension 30%Section A Short ConversationDirections: In Section A, you will hear ten short conversations between two speakers. At the end of each conversation, a question will be asked about what was said. The conversations and the questions will be spoken only once. After you hear a conversation and the question about it, read the four possible answers on your paper, and decide which one is the best answer to the question you have heard.1. A. On a highway. B. At the manager’s office.C. In a restaurant.D. At the post office.2. A. Husband and wife. B. Doctor and patient.C. Professor and student.D. Boss and assistant.3. A. For a coupe of days B. For a month.C. For a weekD. For two weeks.4. A. 514, Fourteenth Street. B. 540, Fourth Street.C. 514, Fourth Street.D. 513, Fourth Street.5. A. Because he was too tired. B. Because he was ill.C. Because he was too old.D. Because he was cold.6. A. Pop music. B. Light music . C. Classical music. D. Rock and roll.7. A. He is borrowing some money. B. He is helping the woman.C. He is asking the way.D. He is waiting for the woman.8. A. It is quite romantic. B. It doesn’t look very delicious.C. It tastes inviting(诱人的).D. It’s a pity he can’t see what he’s eating.9. A. He had to work overtime. B. He was held up in traffic.C. His car ran out of gas.D. He had a traffic accident.10. A. The doctor only has time on Tuesday. B. The doctor is busy on Tuesday morning.C. The man must come more than one time.D. The man must arrive on time.Section B PassagesDirections: In Section B, you will hear two short passages, and you will be asked three questions on each of the passages. The passages will be read twice, but the questions will be spoken only once. When you hear a question, read the four possible answers on your paper and decide which one would be the best answer to the question you have heard.QUESTIONS 11 THROUGH 13 ARE BASED ON THE FOLLOWING PASSAGE.11. A. An expensive blue umbrella and a large handbag.B. A large expensive handbag and an umbrella.C. An expensive handbag and an umbrella.D. A large funny hat.12. A. Because she was not sitting in her own seat.B. Because she was shouting loudly.C. Because she was saying something not very nice about a passenger’s hat.D. Because she was staring at the woman’s hat.13. A. Children tend to say what they think.B. Sometimes children tend to shout loudly while their mothers keep quiet.C. Children are always active on a bus.D. Children always like a funny hat.QUESTIONS 14 THROUGH 16 ARE BASED ON THE FOLLOWING PASSAGE.14. A. red B. yellow C. green D. white15. A. the cheapest B. the smallest C. the best D. the most useful16. A. blue B. green C. red D. yellowSection C Longer conversationDirections: In Section C, you will hear two longer conversations. The conversations will be read twice. After you hear each conversation, you are required to fill in the numbered blanks with the information you have heard. Write your answers on your answer sheet.Blanks 17 through 20 are based on the following conversation.Complete the form. Write ONE WORD for each answer.Blanks 21 through 24 are based on the following conversation.II. Choices 20%25. In ______________, fashions changed and many factories closed down.A. 1830’sB. 1830sC. the 1830sD. the 1830s’26. Have you any comment(s) to make _____ the recent developments?A. toB. out ofC. onD. at27. The famous pop singer came to Shanghai last week to ____ her newly- released album.A. illustrateB. promoteC. appreciateD. program28. ______ the introduction of school uniforms, the number of fights increased by five percent in schools with youth gangs.A. AlthoughB. As ifC. In spite ofD. Despite of29. You should bring an umbrella ____.A. in case it will rainB. in case of rainingC. just in caseD. in the case of rain30. The teacher asked the students ____ stay after class.A. referred to toB. referred toC. referring toD. referring to to31. According to the new CEO, these problems _______ years of bad management.A. as a result ofB. resulted fromC. are resulted fromD. resulted in32. The washing machine is environmentally friendly because it uses ___ water and electricity than ___ models.A. less…olderB. less…elderC. fewer…olderD. fewer…elder33. Are you kidding? You have bought ______ at such a low price.A. such many clothingB. so many clothingC. so many clothesD. such many clothes34. In the hot weather, the food______.A. easy went badB. goes bad easilyC. goes badly easyD. easily goes badly35. I can’t make a decision now. I need ____ more time to think it over.A. ratherB. fairlyC. quiteD. too36. The more certificates (证书) you get before graduation, _______ for this desired job.A. the more you are qualifiedB. the more qualified you areC. the more qualified for youD. you are more qualified37. At the sight of the snake, the little girl _____ move.A. dares notB. doesn’t dareC. dare not toD. dares not to38. Gary won a contest and went to see the sights of London, so he ____ Laura’s party last night.A. mustn’t have attendedB. couldn’t attendC. mustn’t attendD. couldn’t have attended39. Since you have to catch an early train, you _______ now.A. might as well leaveB. ought to have leftC. must as well leaveD. had better to leave40. You say you won’t do it, but I say you _____ do it.A. willB. wouldC. needD. shall41. Some of aspects of a pilot’s job ___ be boring, and pilots often ___ work at inconvenient hours.A. can…have toB. may…canC. have to …mayD. ought to …must42. I was ill that day, otherwise I _____ the sports meeting.A. would have taken part inB. took part inC. had taken part inD. would take part in43. The judge made a recommendation to the court _____ the man ______ be set free.A. where… couldB. that… shouldC. which… needD. that …must44. Without electricity, human life____ quite different today.A. isB. will beC. would have beenD. would beIII. Blank-filling 9%Teachers always have found that teaching and learning become interesting and enjoyable when students are actively involved in the lesson. Their active engagement, however, does not come 45 . In most 46 , teachers have to find ways to get them involved.Hai K. P. Huynh, an English teacher at American-Vietnamese International English Centre in Da Nang, Vietnam, demonstrates how teachers can get students actively engaged in the learning process and take charge of their learning by giving them the 47 to find and correct mistakes themselves.The effective way is to prepare an activity where students take over the role of correcting mistakes which is 48 done by the teacher. The teacher 49 becomes the ‘mistake maker ’.This technique can bring forth several benefits. For instance, by switching the role of the ‘mistake corrector’, the teacher can often observe that students get excited. The 50 of excitement is even increased when the class is divided into two teams to compete with each other in finding and correcting the mistakes.Another benefit is that they have the opportunity to 51 the possible mistakes themselves instead of the teacher telling them what the mistakes are. This technique can help the teacher 52 his or her students’ understanding of a grammar point or co mprehension of a reading task. It can be used to reinforce and improve their production skills such as writing and pronunciation.The activity is a simple technique but it is very effective in getting students' attention and participation. It can easily be developed by teachers to 53 their needs and their students’ as well.IV. Cloze 15%Eleven-year-old Angela had something wrong with her nervous system(神经系统）.She was unable to walk. In fact, she could hardly make any__54__. Although she believed that she had a good chance of recovering, the doctors said that__55__, if any, could come back to normal after getting this disease. Having heard this, the little girl was not __56__. There, lying in her hospital bed, she __57__ that no matter what the doctors said, her going back to school was__58__.She was moved to a specialized health center, and whatever method could be tried was used. Still she would not __59__. It seemed that she was undefeatable. The doctors were all fond of her and taught her about __60__that she could make it. Every day Angela would lie there, faithfully doing her mental exercise.One day, __61__she was imagining her legs moving again, it seemed as though a miracle(奇迹）happened: The bed began to__62__!“Look, what I'm doing! Look! I can do it! I moved! I moved!" she__63__.However, at this very moment everyone else in the hospital was __64__. More importantly, they were running for safety. People were crying, and equipment was __65__. You see, it was an earthquake. But don't __66__that to Angela. She has __67__that she did it, just as she had never doubted that she would recover. And now only a few years later, she's back in school. You see, to such a person who can __68__the earth, such a disease is a small problem, isn't it?54. A. progress B. difference C. movement D. achievement55. A. few B. all C. some D. most56. A. satisfied B. delighted C. surprised D. discouraged57. A. insisted B. sighed C. feared D. promised58. A. true B. doubtful C. certain D. impossible59. A. get up B. give up C. turn up D. stand up60. A. thinking B. expecting C. pretending D. imagining61. A. as B. since C. after D. before62. A. fly B. move C. roll D. speak63. A. jumped B. wondered C. screamed D. recovered64. A. frightened B. pleased C. touched D. encouraged65. A. rising B. falling C. missing D. gathering66. A. tell B. do C. give D. show67. A. noticed B. supposed C. believed D. discovered68. A. push B. shock C. shake D. saveV. Reading Comprehension 27%AWhat time is it? Most people are pretty accurate(精确的) in their answer. And if you don’t know for sure, it’s very likely that you can find out. There may be a watch on your wrist, there may be a clock on the wall, desk, or computer screen; or maybe you’re riding in a car that has a clock in the dashboard (仪表板).Even if you don’t have a timepiece of some sort nearby, your body keeps its own beat. Humans have an internal（内部的）clock that regulates（管理）the beating of our heart, the pace of our breathing, the discharge (排出) of chemicals within our bloodstream, and many other bodily functions.Time is something from which we can’t escape. Even if we ignore it, it’s still going by, ticking away, second by second, minute by minute, hour by hour. So the main issue in using your time well is, “Who’s in charge?” We c an allow time to slip(溜，滑) by and let it be our enemy. Or we can take control of it and make it our ally.By taking control of how you spend your time, you’ll increase your chances of becoming a more successful student. Perhaps more importantly, the better you are at managing the time you devote to your studies, the more time you’ll have to spend on your outside interests.The aim of time management is not to schedule every moment so we become slaves of a timetable that governs every waking moment of the day. Instead, the aim is to make informed choices as to how we use our time. Rather than letting the day go by, largely without our awareness （意识）, what we are going to discuss next can make us better able to control time for our own purposes.69. The underlined word “ally” in Para. 3 more likely means somebody or something that is_________.A. your slave and serves youB. your supporter and helps youC. under your control and obeys youD. under your influence and follows you70. The author intends to tell us that time _____________.A. could be regulated by a timepiece such as a clock or a watchB. could be managed by the internal clock of human bodiesC. should be well managed for our own interestD. should be saved for outside interests71. In the next part, the author would most probably discuss with you ________.A. how to keep up with the timesB. how to make up for lost timeC. how to have a good timeD. how to make good use of timeBMany of the world’s pollution problems have been caused by the crowding of large groups of people into the cities. Supply for the needs of the people leads to further pollution by industry. If the rapid increase in human population continues at the present rate, there may be much greater harm. Some scientists speak of the increase in numbers of people as “population pollution”.About 2,000 years ago, the world population was probably about 250 million. It reached a billion in 1850. By 1930 the population was two billion. It is now three and a half billion. It is expected to double by the year 2000. If the population continues to grow at the same rate, there would be 25 billion people in the world a hundred years from now!Man has been using the Earth’s resources more and more rapidly over the years. Some of them are almost gone. Now many people believe that man’s greatest problem is how to control the growth of his own population. The materials in the world will not support the human population in time to come, if the present rate of increase continues. Already, there is overcrowding in the cities and hunger in some countries. Can man’s rate of increase continue? Many people belie ve that human survival in the future depends on the answer to this question.72. The world’s pollution problems would be less serious if _____.A. fewer people moved from countryside to the citiesB. more people moved from countryside to the citiesC. industry developed less fastD. many people moved to the cities without the fast development of industry73. The increasing move of the people to the cities is thought to be _____.A. city pollutionB. population pollutionC. industry pollutionD. countryside pollution74. In the year 2000, it is expected that world population will be _____.A. 3.5 billionB. 25 billionC. 7 billionD. 4 billion75. The most serious problem that man is facing is thought to be ____.A. the us ing of the Earth’s resourcesB. the control over the waste of the natural materialsC. how to slow down the increase of populationD. how to stop the birth of new babiesCAttitude(态度) is an internal state（状态）that influences the choices of personal action made by the individual. Some researchers consider that attitudes come from differences between beliefs and ideas: others believe that attitudes come from emotional states. Here, we focus on the effects of attitudes upon behavior, that is, upon the choices of action made by the individual.The kinds of actions taken by human beings are obviously influenced greatly by attitudes. Whether one listens to classical music or rock, whether one obeys the speed limit while driving, whether one enco urages one’s husband or wife to express his or her own ideas－all are influenced by attitudes. These internal states are acquired(获得) throughout life from situations one is faced with in the home, in the streets, and in the school.Of course, the course of action chosen by an individual in any situation will be largely determined by the particulars(细节,详细) of that situation. An individual who has a strong attitude of obeying laws may drive too fast when he is in a hurry and no police cars in sight. A child who has a strong attitude of honesty may steal a penny when she thinks no one will notice. But the internal state which remains unchanged over a period of time, and which makes the individual behave regularly in a variety of situations, is what is meant by an attitude.Attitudes are learned in a variety of ways. They can result from single incidents, as when an attitude toward snakes is acquired by an experience in childhood at the sudden movement of a snake. They can result from the individual’s ex periences of success and pleasure, as when someone acquires（获得）a positive（积极的）attitude toward doing crossword puzzles by being able to complete some of them, and frequently, they are learned by copying other people’s behavior, as when a child learns how to behave toward foreigners by observing the actions of his parents. Regardless of these differences, there is something in common in the learning and modification(修正) of attitudes.76. According to the passage, attitudes __________.A. come from differe nt situations in one’s lifeB. are largely affected by one’s behaviorC. remain unchanged in one’s daily lifeD. could be chosen according to one’s will77. The author uses the examples in Paragraph 3 to show ______.A. people often make mistakes when they are not noticedB. people with good attitudes may sometimes do bad deeds（行为）C. particulars of a situation may influence an individual’s actionD. an individual may change his or her attitude fairly easily78. Which of the following is TURE about the learning of attitudes mentioned in the last paragraph?A. Attitudes are o nly learned through one’s success.B. Attitudes learned in danger will last longer.C. Copying others’ behavior is not a good idea.D. Attitudes can be learned from one’s parents.79. What would be the best title for the passage?A. Differences of Attitudes.B. Nature (本质) of Attitude.C. Choices of Attitudes.D. Modification of Attitude.D80. __________In recent decades, scientists have discovered that the ordinary matter that makes up stars, planets, even human beings, accounts for（占）only 5 percent of everything in the universe. The rest belongs to dark matter and dark energy, phenomena that scientists are just now learning about. What is dark matter made of and where does it reside？What is dark energy？Researchers hope to find answers.81. __________In the 17th century, French philosopher and mathematician Rene Descartes declared that mindand body are entirely separate, leaving the debate over the nature of consciousness to other philosophers. Today scientists are challenging that notion with a view that consciousness arises from the properties and the organization of neurons in the brain. Experimental work to figure out those properties and processes has only just begun.82. __________There are hundreds of billions of stars in our own galaxy（银河系）, the Milky Way, and hundreds of billions of galaxies in the universe. Close to home, scientists have already discovered 150 planets moving round nearby stars. In short, scientists say the universe is likely full of places where the conditions are ripe for intelligent life to evolve. The really big question is when, if ever, we'll have the technological means to reach out and touch such intelligence.83. __________Scientists know that the world is warming and that humans are the cause behind most of this global climate change. Researchers are much less certain about how warm the Earth will become in response to the doubling of the greenhouse gas carbon dioxide that is expected in this century. Modeling studies suggest Earth will warm by at least 2.7 degrees Fahrenheit（1.5 degrees Celsius）and maybe by as much as 19.8 degrees Fahrenheit（11 degrees Celsius）. A better understanding of the climate system’ and more robust models will be needed to better forecast the temperature increase.84. __________The price of oil and the demand for energy are increasing: Oil supplies are decreasing and the polar ice caps are melting. So the time is ripe for humans to transition from oil to a new source of energy. Alternative energy sources are available but need to be scaled up and made more cost-efficient to replace oil. Advances in new technology may be the answer. But will they come soon enough to avoid an energy crisis?VI. Verb-filling 10%85. I only buy books which are worth _____ (read) a second time.86. My brother wasted his time _____ (look) for a pair of trainers in London, which I think looks ugly.87. Independent learning allows those who can’t afford the time during workdays ___ (continue)their education.88. I would appreciate you ____ (call) back tomorrow morning.89. Our school is the first one ___ (come) up with the slogan that honesty is the best policy.90. I _____(write) a composition all morning but still haven’t finished it.91. _____(enrol) by a good university, you have to work much harder.92. School uniforms are not likely ____ (adopt) all over the country.93. When first ____ (introduce) into the schools, uniforms met strong resistance from many parents.94. It is better to focus in class than ___ (spend) extra time in a tutorial centre.VII. Translation: 14%95. 这双最酷的运动鞋花了我所有的积蓄。
上海交大附中09-10学年高二英语上学期期中考试 上海新世纪版 新课标 【会员独享】
某某交通大学附属中学09-10学年高二上学期期中考试英语试卷〔总分为100分，100分钟完成，答案一律写在答题卡、答题纸上〕I. Listening Comprehension (24%)Section A Short Conversation (10%)Direction: In Section A, you will hear ten short conversations between two speakers. At the end of each conversation, a question will be asked about what was said. Each conversation and question will be spoken only once. After you hear a conversation and the question about it, read the four possible answers and decide which one is the best answer to the question you have heard.1. a. find a larger room b. sell the old tablec. buy two bookshelvesd. rearrange some furniture2. a. guest and receptionist b. passenger and air hostessc. customer and shop assistantd. patient and nurse3. a. Tom survived the accident b. Tom was put into prisonc. The wall was knocked downd. It did little damage to Tom’s car4. a. The man didn’t want the woman to heave her hair cut.b. The woman followed the man’s advice.c. The woman is wearing long hair now.d. The ma n didn’t care if the woman had her hair cut or not.5. a. in a restaurant b. in a hospitalc. in a toy stored. in a zoo6. a. She seldom reads books from cover to cover.b. She has no interest in reading novels.c. She read only parts of the book.d. She was eager to know what the book was about.7. a. 2:50 p.m. b. 2:15p.m. c. 1:50 p.m. d. 1:15 p.m.8. a. Jack always arrives on time.b. Jack used to be late for work.c. Jack knows nothing about his work.d. Jack is often late for work.9. a. by going on a diet b. by doing physical exercisec. by having fewer mealsd. by eating fruit and vegetables10. a. Most people killed in the accidents are heavy drinkers.b. Passengers should pay more attention when crossing the road.c. She doesn’t agree with what the man said.d. Drunk driving causes many accidents on the road.Section B Passage (6 %)Direction: In Section B, you will hear two short passages, and you will be asked three questions on each of the passages. The passages will be read twice, but the questions will be spoken onlyonce. When you hear a question, read the four possible answers in your paper and decide which one could be the best answer to the question you have heard.Questions 11 through 13 are based on the following passage.11. a. to warn people of the danger of smoking during pregnancy.b. to advise people with breathing problems to stop smoking.c. to show people the risks of having a low birth weight baby.d. to offer some tips on how to give up smoking.12. a. It may cause the difficult birth of the baby.b. The babies are more likely to suffer heart disease.c. The babies may weigh less than other children.d. The danger of lung cancer may increase for pregnant mothers.13. a. a boy whose mother smokesb. a girl whose mother smokesc. a boy whose father smokesd. a girl whose father smokesQuestions 14 through 16 are based on the following news.14. a. There were too many people living in the building.b. Electric heaters were not carefully used.c. The old house was in bad condition and needed repairing.d. Children in the nearby school played with fire.15. a. customers and staffb. non-governmental organizationsc. charitiesd. environmental protection groups16. a. about 20 million b. 15.3 million c. 130 million d. 77 millionSection C Longer Conversations (8%)Direction: In Section c, you will hear two longer conversations, you are required to fulfill the task by filling in the numbered blanks with the information you have heard.Complete the form. Write NO MORE THAN THREE WORDS for each answer.Ⅱ. Grammar and V ocabulary 26%Section ADirections: Beneath each of the following sentences there are four choices marked a, b, c and d. Choose the one answer that best completes the sentence.25.A bullet hit the soldier and he was wounded in___________leg.a. ab. onec. thed. his26. The cost of hiring a car should be no more than $800 for a full day, and in some places. ______.a. much as halfb. as half muchc. half as muchd. half much as27. Whom would you rather _______ us solve the traffic problem in Shanghai, Mr. Brown or Mr. Smith?a. had helpedb. have to helpc. have helpd. have helped28. The explosion on the sun cannot be heard, there _______ no air between the earth’s atmosphere and the sun.a. isb. wasc. havingd. being29. ___________competitive in this international economy, a company must open itself to infor-mation and ideas.a. Stayingb. Having stayedc. To stay d To be staying30. The emperor ordered that the weavers ____ some gold in order that they ____ their work at once.a. should give; could beginb. be given; might beginc. would be give; could startd. should be given; may start31. It was well known that Napoleon always requested that his soldiers ____ the same three questions, and usually in the same order.a. answeredb. could have answeredc. answerd. should have answered32. I would like to do it right now, but I ______ time.a. have nob. hadn’tc. didn’t haved. will have no33.___ the books not properly marked and catalogued, they ____ be so easy to find.a. If; couldb. Were; couldn’tc. Had; couldn’td. Were; could34. Why? It's unbelievable that the maid who had taken so good care of the baby _____ like that in the past two years.a. could have been treatedb. should have been treatedc. may have been treatedd. must have been treated35. Grown people should never say or do anything wrong before children lest they ______ thema bad example.a. will setb. are to setc. should setd. would set36. ________, he retuned to his hometown from abroad.a. Every few yearsb. Every a few yearsc. Each a few yearsd. Each few years37. Mr. Baker bought a pair of leather shoes to ______ his new suit, which was sent to him as the birthday present.a. suitb. matchc. fitd. agree38. These days he was _______ doing experiments and he always had _____ noodles for meals.a. engaged in…fastb. engaged in… instantc. engaged with…convenientd. engaged with…instant39. As for Latin America generally, there is no reason why, __________available resources, it cannot become a highly industrialized area.a. at the mercy ofb. in terms ofc. at the turn ofd. at the cost of40. A lot of people from all parts of the world _______ to have seen UFOs each year, but not a case has been proved so far.a. indicateb. informc. claimd. declare41. They were disappointed when their son ______ bottom in the examination.a. came upb. came outc. came tod. came offSection BDirections: Complete the following passage by using the words in the box. Each word can only be used once. Note that there is one word more than you need.A project called the World Community Grid has found a way for computers connected to the Internet to help solve humanitarian problems. The World Community Grid is making technology ___42_____ to public and non-profit organizations to use in humanitarian research.Scientists at the University of Washington, for example, are using the technology to study ways to ___43__ the nutritional value of rice. Another research project ___44__ by the World Community Grid is studying mathematical ways to design drugs to treat the disease AIDS. Other projects are studying cancer. And still others are studying climate change in Africa.The success of the World Community Grid depends upon ___45____ collectively donating their extra computer power. This is based on the idea that most computers are ____46___ most of the time. During the time they are not used, they can help solve complex scientific or engineering problems.The IBM Corporation started the World Community Grid more than two years ago. The company continues to provide ___47____ and support for the project. Stanley Litow heads community relations for IBM. He says anyone in the world with a computer connected to the Internet can join theproject.V olunteers download a program from the World Community Grid website. Every so often the program uploads or downloads more information to be ___48____. Individuals can also findout how much work their computer power has done on the website.____49____, about one million personal computers in 100 countries are involved in the World Community Grid. Mr. Litow hopes that another million computers will join the project. Then, he says, the World Community Grid will become the world’s largest super computer able to do so many ____50___ a year.III. Reading Comprehension 35%Section ADirections: For each blank in the following passages there are four words or phrases marked a, b, c and d. Fill in each blank with the word or phrase that best fits the context.It seems to be a simple thing: teaching your student to say“ I am sorry〞. But far between simply repeat a conditioned response, truly understanding the nature of an apology, and being able to deliver one ___51__, requires a level of social ability that many __52____ find difficult, let alone the teenage with autism(我向思考). Breaking that“simple〞__53____ down into understandable steps of progress goes a long way towards helping a student with autism comprehend and __54_____ this critical interpersonal skill.The need for an apology may come at a time when emotions are running high. If this is the case, an apology need not come ___55___ to be effective. There is a huge gap between merely reciting words of apology and feeling true __56____. In the heat of anger, the student may need a cooling-off period before he or she can _57______real feelings of being sorry and apologize sincerely. The cooling-off period will __58___from person to person. This requires striking a balance—you want the ___59___ handled in a timely and relatively immediate way, but an insincere apology is not an ____60____ apology. Remember, attempting to teach anything when the student is still ___61____ overstressed is a wasted effort.In order to issue a sincere apology, a student needs first to understand why the apology is necessary. Learning to recite the words “I’m sorry.〞i sn’t enough—it doesn’t help the student understand what he or she did and the __62_____ it had on another person. All students, not just those with autism, have __63____ of selfishness, defensiveness in their ____64___. Encourage them to think about other people’s feelings, and explain why we make apologies is im portant and helps them to develop ___65____. This will come increasingly over a long period.51. a. meaningfully b. sincerely c. effectively d. intelligently52. a. philosophers b. psychologists c. infants d. adults53. a. method b. apology c. situation d. theory54. a. test b. acquire c. apply d. possess55. a. immediately b. mentally c. naturally d. normally56. a. regret b. sympathy c. innocence d. satisfaction57. a. appreciate b. resist c. emphasize d. recognize58. a. change b. vary c. range d. arrange59. a. conflict b. incident c. condition d. occasion60. a. false b. adequate c. negative d. worthless61. a. alternatively b. physically c. emotionally d. subconsciously62. a. effect b. comment c. impression d. intension63. a. definitions b. materials c. elements d. functions64. a. qualifications b. attitudes c. manners d. personalities65. a. sharing b. reasoning c. undertaking d. understandingSection BDirections：Read the following four passages. Each passage is followed by several questions or unfinished statements. For each of them there are four choices marked a, b, c and d. Choose the one that fits best according to the information given in the passage you have just read.*Embassy Suites Hotel San Luis is currently accepting applications for Front Desk Services Agent. *Full time day and part time night position open.*Availability must include weekends and holidays.*This position has a combination of duties mainly related, but not limited to checking-in hotel guests.We are seeking candidates who have the ability to:*Understanding guest inquires and provides responses in a helpful, courteous(礼貌的) manner. *Promote positive relations with all individuals who approach the Front Desk and enter the hotel. *Focus on the guest needs remaining calm and courteous.*Work well under pressure.*Input and access data in the computer.*Ensure security and confidentiality of guest and hotel information.*Work cooperatively with other departments and co-workers as part of a team. REQUIREMENTSCustomer service experience preferred.Please apply online by submitting a resume, including a cover letter.OR:Apply in person, by submitting an application for employment, from the executive offices located within the hotel.66. Which of the following best describes the job advertised?a. Technical Engineerb. Door Manc. Reservation Officerd. Front Desk Services Agent67. Those interested in this job may contact Rebecca Hyer _________.a. by e-mailb. by faxc. by sending short messagesd. by post68. Which of the following is needed for the job?a. College diploma or aboveb. Necessary PC skillsc. Speaking at least three languagesd. Five-year working experienceBOn June 17, 1744, the officials from Maryland and Virginia held a talk with the Indians of the Six Nations. The Indians were invited to send boys to William and MaryCollege. In a letter the next day he refused the offer as follows:We know that you have a high opinion of the kind of learning taught in your colleges, and that the costs of living of our young men, while with you, would be very expensive to you. We are convinced that you mean to do us good by your proposal; and we thank you heartily. But you must know that different nations have different ways of looking at things, and you will therefore not be offended if your ideas of this kind of education happen not t be the same as yours. We have had some experience of it. Several of our young people were formerly brought up at the colleges of the northern provinces: they were taught all your sciences; but, when they came back to us, they were bad runners, ignorant of every means of living in the woods… they were totally good for nothing.We are, however, not the less obliged by your kind offer, though we refuse to accept it; and, to show our grateful sense of it, if the gentlemen of Virginia will send up a dozen of their sons, we will take care of their education, teach them in all we know, and make men of them.69. The passage is about ________.a. the talk between the Indians and the officialsb. the colleges of northern provincesc. the educational values of the Indiansd. the problems of the Americans in the mid-eighteenth century70. The Indians’ chief purpose in writing the letter seems to ________.a. politely refuse a friendly offerb. express their opinion on equal treatmentc. show their prided. describe Indian customs71. According to the letter, the Indians believed that ________.a. it would be better for their boys to receive some schoolingb. they were being insulted by the offerc. they knew more about science than the officialsd. they had a better way of educating young men72. Different from the officials’ view of education, the Indians though t ________.a. young women should also be educatedb. they had different goals of educationc. they taught different branches of scienced. they should teach the sons of the officials firstCA Canadian architect has designed a completely different kind of city house—it’s independent of the electricity grid (net), the water system, and sewage lines. “We wanted a house that could remain independent of the city utilities, “ explains Toronto architect, Martin Liefhebber, “ So we will heat it with an efficient wood stove, pro vide electricity from solar panels, and recycle all waste with the help of a self-composting toilet and greenhouse.〞Liefhebber estimates that the two-story, 992-square-foot home could be built for less than$100,000, nothing will be wasted. The roof catches snow and rainwater will be treated for drinking water; the toilet, which will use only a cup of water per flush, will produce fertilizer for the green house garden situated on the home’s roof. The wall will be built of blocked made from concrete mixed with cellulose fibers-light and expensive, yet a very good insulator(隔热材料). Liefhebbers has named his house the Codicile, “The name reflects the philosophy that if we use less resources, we’ll have more to pass on to future generations.〞he says.Toronto’s laws have so far prevented the house from going up, but Liefhebber is hopeful that his new idea will have wide appeal and plans to build one later this year.73. The house designed by a Canadian architect, _______ city utilities.a. usesb. is free ofc. is equipped withd. is connected with74. Drinking water comes from _________.a. roofb. snow and rainc. piped waterd. both b and c75. The blocks of which the walls will be built are ________.a. semi-conductorsb. light but expensivec. not only lightweight but cheapd. both a and c76. The architect designed the house in order to ________.a. save resourcesb. recycle all the wastec. want more from natured. both a and bDTwo men on a touring holiday of Britain were injured by an explosion in their motor van in the center of Norwich yesterday.Shoppers, traders and businessmen in Red Lion Street were shocked by a loud bang. And seconds later the two men jumped over from the vehicle, which had stopped outside Barclays Bank. Several people rushed to give assistance and help to put out the fire inside the vehicle, a light American truck converted to provide living accommodation, before Norwich firemen arrived.The men-American Mr. Gary House, age 25, of Ohio, who was driving , and his passenger Mr. Charles Lynn, 23, of Vancouver—were taken to Norfolk and Norwich Hospital with minor burns. They were allowed to leave after treatment.“I heard this explosion. It was pretty loud. I thought it could have been a bomb,〞said Mr. Leslie Web ster, manager of the market, who was working in his office in Red Lion Street. “I looked out of the window and saw this lad leap from the van and roll on the ground. Then another lad came out of the van. He seemed to be in a worse state—parts of his trousers were hanging below his knee.〞“I came downstairs to get a fire extinguisher, but by the time I got outside someone from the bank was in the van with an extinguisher.〞Mr. Webster, who lives at 71, Trinity Street, Norwich, said both victims were shocked. One was taken into the market’s office to await an ambulance. “The second man insisted on going back into the van to see if everything was all right, and five minutes later he came out with a drawer that was blazing, 〞he added.The explosion was also heard inside the bank. Staff provided a fire extinguisher and telephoned for an ambulance.Although a plastic window was blown out, damage inside the vehicle was mainly superficial.The two men have spent the last six months touring the continent and had traveled to Norwich from Snettleton. At the time of the accident their wives were shopping in the city.77. The two men in the van _________.a. were slightly hurtb. were badly injuredc. were shocked but unhurtd. needed careful treatment78. The word“leap〞most probably means _________.a. fall offb. get offc. jump offd. step down79. Mr. Webster thought that the explosion _________.a. was a bombb. wasn’t a bombc. was like a bombd. was almost a bomb80. Which of the following is NOT TRUE according to the passage?a.Mr. Webster was among those who happened to see the accident.b.Mr. Webster didn’t take a fire extinguisher to the van.c.Mr. Webster helped to put out the fire.d.Mr. Webster was willing to be a helper in the accident.Section C: Read the following text and choose the most suitable heading from A-F for each paragraph. There is one extra heading which you do not need. F=A+EBefore you argue with your boss, check with the boss’s secreta ry to determine his mood. If heeats nails for breakfast, it is not a good idea t o ask him for something. Even without the boss’s secretary, there are keys to timing; don’t approach the boss when he’s on deadline; don’t go in right before lunch, when he is apt to be distracted and rushed; don’t go in just before or after he has taken a vacation.82. _____________________If you are mad, that will only make your boss mad. Calm down first. And don’t let a particular concern open the floodgates for all your accumulated frustration. The boss will feel that you think negatively about the company and it is hopeless trying to change your mind. Then, maybe he will dismiss you.83. ____________________Terrible disputes can result when neither the employer nor the employee knows what is the problem the other wants to discuss. Sometimes the fight will go away when the issues are made clear. The employee has to get his point across clearly in order to make the boss understand it. 84. ____________________Your boss ha s enough on his mind without your adding more. If you can’t put forward an immediate solution, at least suggest how to approach the problem. People who frequently present problem without solutions to their bosses soon find they can’t get past the secretary.85. _____________________To deal effectively with a boss, it’s important to consider his goals and pressures. If you can put yourself in the position of being a partner to the boss, then he will be naturally more inclined to work with you to achieve your goals.IV.Translation 15%Directions: Translate the following sentences into English, using the word or phrase given in the brackets.86. 重要的是你能从错误中吸取教训。
2024-2025学年高二上学期期中模拟考试数学试题(上海专用,测试范围：沪教版)含解析
2024-2025学年高二数学上学期期中模拟卷（考试时间：120分钟试卷满分：150分）注意事项：1．答卷前，考生务必将自己的姓名、准考证号填写在答题卡上。
2．回答选择题时，选出每小题答案后，用铅笔把答题卡上对应题目的答案标号涂黑。
如需改动，用橡皮擦干净后，再选涂其他答案标号。
回答非选择题时，将答案写在答题卡上。
写在本试卷上无效。
3．考试结束后，将本试卷和答题卡一并交回。
4．测试范围：沪教版2020必修第三册第十~十一章。
5．难度系数：0.65。
一、填空题（本大题共有12题，满分54分，第1-6题每题4分，第7-12题每题5分）1．“点A 在直线l 上”用符号语言可以表示为.【答案】A l∈【解析】A 在直线l 上，即A l∈故答案为：A l∈2．在正方体1111ABCD A B C D -中，12AA =，则直线AB 到平面11CDD C 的距离为．【答案】2【解析】根据正方体的性质可知，//AB CD .又AB ⊄平面11CDD C ，CD ⊂平面11CDD C ，所以，//AB 平面11CDD C .所以，点A 到平面11CDD C 的距离，即等于直线AB 到平面11CDD C 的距离.又AD ⊥平面11CDD C ，所以点A 到平面11CDD C 的距离即为12AD AA ==.所以，直线AB 到平面11CDD C 的距离为2.故答案为：2.3．已知圆柱的底面半径为2，高为2，则该圆柱的侧面积是．【答案】8π【解析】圆柱的侧面展开为矩形，其中矩形的一条边长为圆柱底面周长，即2π24π⨯=，另一边长为2，故圆柱的侧面面积为24π8π⨯=.故答案为：8π4．如图，在正方体1111ABCD A B C D -中，异面直线1A B 与AC 所成的角为．正方体111ABCD A B C D -因此1ACD ∠是异面直线A 而112AD CD AC AB ===所以异面直线1A B 与AC 故答案为：60o5．圆锥的母线长为2，母线所在直线与圆锥的轴所成角为30︒，则该圆锥的高为．【答案】3【解析】由已知得该圆锥的高为故答案为：3.6．一个梯形的直观图是一个如图所示的等腰梯形，且132A B O C O A '''=='''=，，，则原梯形的面积为.易得24OA O A ''==，1AB A B ''==，OC O =故原梯形的面积为：113482S =⨯+⨯=（），故答案为：8.7．已知斜线段的长度是斜线段在这个平面内射影的长的为.【答案】4π/45 【解析】设斜线和平面所成角为02παα⎛≤≤ ⎝8．已知球的两个平行截面的面积分别为49π，400π且两个截面之间的距离是9，则球的表面积为．9．如图，在四棱锥P﹣ABCD中，PA⊥底面ABCD，底面ABCD是边长为1的正方形，PA＝1，则侧面PCD 与底面ABCD所成的二面角的大小是．【答案】45°【解析】因为底面ABCD是边长为1的正方形，所以AD⊥CD，又因为PA⊥底面ABCD，CD⊂底面ABCD，所以PA⊥CD，因为PA∩AD＝A，PA、AD在面PAD内，所以CD⊥平面PAD，又因为PD⊂平面PAD，所以CD⊥PD，于是∠PDA为侧面PCD与底面ABCD所成的二面角的平面角，因为PA⊥底面ABCD，AD⊂底面ABCD，PA⊥AD，又因为PA＝1，AD＝1，所以∠PDA＝45°，于是侧面PCD与底面ABCD所成的二面角的大小为45°．故答案为：45°．10．如图是一个正方体的平面展开图，在这个正方体中，下列说法中，正确的序号是.（1）直线AF 与直线DE 相交；（2）直线CH 与直线DE 平行；（3）直线BG 与直线DE 是异面直线；（4）直线CH 与直线BG 成60︒角.【答案】（3）（4）/(4)(3)【解析】解：由正方体的平面展开图可得正方体ABCD EFGH -，可得AF 与ED 为异面直线，故（1）错误；CH 与DE 为异面直线，故（2）错误；直线BG 与直线DE 是异面直线，故（3）正确；连接AH ，AC ，由正方体的性质可得//AH BG ，所以AHC ∠为异面直线CH 与直线BG 所成的角，因为AHC 为等边三角形，所以60AHC ∠=︒，即直线CH 与直线BG 所成角为60︒，故（4）正确；故答案为：（3）（4）．11．设AB 和CD 都是平面α的垂线，其垂足分别为,B D .已知5,9,3AB CD BD ===，那么线段AC =.12．如图，平面OAB ⊥平面α，OA α⊂，OA AB =，120OAB ∠=︒．平面α内一点P 满足PA PB ⊥，记直线OP 与平面OAB 所成角为θ，则tan θ的最大值是．【答案】612【解析】如图，过点B 作BH OA ⊥，交OA 的延长线于点H ，连接取AH 的中点为E ，连接PE ，过点P 作PF OA ⊥，垂足为当且仅当PE OP ⊥，即OP 是圆E 的切线时，角tan θ取得最大值为：22PE PE OP OE PE ==-故答案为：612.二、选择题(本题共有4题，满分18分，第13-14题每题4分，第15-16题每题5分；每题有且只有一个正确选项)13．下列说法错误的是（）A ．一个棱柱至少有5个面B ．斜棱柱的侧面中没有矩形C ．圆柱的母线平行于轴D ．正棱锥的侧面是全等的等腰三角形【答案】B【解析】由棱柱的性质可知A 正确，B 错误；由圆柱的性质可知C 正确；由正棱锥的性质可知D 正确.故选：B14．已知l 是直线，,αβ是两个不同平面，下列命题中的真命题是（）A ．若//,//l l αβ，则//αβB ．若,//αβα⊥l ，则l β⊥C ．若,//l l αβ⊥，则αβ⊥D ．若//,//l ααβ，则//l β【答案】C 【解析】若,//,,m l m l l αβαβ⋂=⊄⊄，则有//,//l l αβ，故可判断A 错误.若,//,m l m l αβα⋂=⊄，则//l β或l β⊂，故B 错误.若,//l l αβ⊥，则β存在直线与l 平行，所以αβ⊥，故C 正确.若//,//l ααβ，则//l β或l β⊂，故D 错误.故选：C.15．《九章算术》中所述“羡除”，是指如图所示五面体ABCDEF ，其中////AB DC EF ，“羡除”形似“楔体”．“广”是指“羡除”的三条平行侧棱之长a 、b 、c ，“深”是指一条侧棱到另两条侧棱所在平面的距离m 、“袤”是指这两条侧棱所在平行直线之间的距离n （如图）．羡除的体积公式为()6a b c mn V ++=，过线段AD ，BC 的中点G ，H 及直线EF 作该羡除的一个截面α，已知α刚好将羡除分成体积比为5:4的两部分．若4AB =、2DC =，则EF 的长为（）A ．2B ．3C ．4D ．6故选：B16．如图，在正方体1111ABCD A B C D -中，点P 在线段1B C 上运动，则以下命题正确的序号为（）①直线1BD ⊥平面11AC D②平面1B CD 与平面BCD 的夹角大小为π2③三棱锥11P AC D -的体积为定值④异面直线AP 与1A D 所成角的取值范围是ππ,42⎡⎤⎢⎥⎣⎦A ．①②B ．①③C ．①③④D ．①④三、解答题(本大题共有5题，满分78分，第17-19题每题14分，第20、21题每题18分.)17．如图，已知,,,E F G H 分别是正方体1111ABCD A B C D -的棱111,,,AB BC CC C D 的中点，且EF 与HG 相交于点Q ．(1)求证：点Q 在直线DC 上；(2)求异面直线EF 与11A B 所成角的大小．【解析】（1）平面ABCD 平面11CDD C DC =，由于Q EF ∈⊂平面ABCD ，Q HG ∈⊂平面11CDD C ，所以Q DC ∈，也即点Q 在直线DC 上.（6分）（2）根据正方体的性质可知11//A B DC ，所以异面直线EF 与11A B 所成角为DQE ∠，（8分）由于//,,AB DC E F 分别是,AB BC 的中点，所以45DQE FEB ∠=∠=︒，所以异面直线EF 与11A B 所成角的大小为45︒.（14分）18．如图，在四棱锥P ABCD -中，底面ABCD 为平行四边形，O 是AC 与BD 的交点，=45ADC ∠ ，2AD AC ==，⊥PO 平面ABCD ，2PO =，M 是PD 的中点．(1)证明：//PB 平面ACM(2)求直线AM 与平面ABCD 所成角的大小．【解析】（1）连接MO ，在平行四边形ABCD 中，因为O 为AC 与BD 的交点，所以O 为BD 的中点，（2分）又M 为PD 的中点,所以//PB MO .因为PB ⊄平面,ACM MO ⊂平面ACM ，所以//PB 平面ACM .（6分）（2）取DO 中点N ,连接MN ,AN，因为M 为PD 的中点,所以//MN PO ,且112MN PO ==,由⊥PO 平面ABCD ，得MN ⊥平面ABCD ，所以MAN ∠是直线AM 与平面ABCD 所成的角.（8分）因为底面ABCD 为平行四边形，且45ADC ∠=o ，2AD AC ==，所以45ACD ∠= ，则90DAC ∠= ，在Rt DAO 中,2,1AD AO ==,所以DO =从而122AN DO ==，因为MN ⊥平面ABCD ,AN ⊂平面ABCD ,MN AN ∴⊥，所以在Rt ANM中，tan 5MN MAN AN ∠==，0,2MAN π⎡⎤∠∈⎢⎥⎣⎦，19．某种“笼具”由上、下两层组成，上层和下层分别是一个圆锥和一个圆柱，其中圆柱与圆锥的底面半径相等，如图所示：圆锥无底面，圆柱无上底面有下底面，内部镂空，已知圆锥的母线长为20cm ，圆柱高为30cm ，底面的周长为24πcm .(1)求这种“笼具”的体积（结果精确到30.1cm ）；(2)现要使用一种纱网材料制作这样“笼具”的保护罩（包括底面）50个，该保护罩紧贴包裹“笼具”，纱网材料（按实测面积计算）的造价为每平方米．．．．8元．，共需多少元？（结果精确到0.1元）【解析】（1）设圆锥的底面半径为r ，母线长为l ，高为1h ，圆柱高为2h ，则由题意有2π24πr =，得12cm r =，圆锥高116cm h ==，所以“笼具”的体积2232111πππ14430144165088π15984.4cm 33V r h r h ⎛⎫=+=⨯+⨯⨯=≈ ⎪⎝⎭.（6分）（2）圆柱的侧面积2122π720πcm S rh ==，圆柱的底面积22π144πS r ==，圆锥的侧面积3π240πS rl ==，所以“笼具”的侧面积21231104πcm S S S S =++=侧.（12分）故造50个“笼具”的最低总造价为41104π5081104π138.71025⨯⨯=≈元.（14分）答：这种“笼具”的体积约为315984.4cm ；生产50个笼具需要138.7元.20．如图，已知四棱锥P ABCD -的底面为直角梯形，//AD BC ，90BCD ∠=︒，PA PB =，PC PD =．(1)证明：CD 与平面PAD 不垂直；(2)证明：平面PAB ⊥平面ABCD ；(3)如果CD AD BC =+，二面角P BC A --等于60︒，求二面角P CD A --的大小．【解析】（1）若CD ⊥平面PAD ，则CD PD ⊥，由已知PC PD =，得90PCD PDC ∠=∠<︒，（2分）这与CD PD ⊥矛盾，所以CD 与平面PAD 不垂直．（4分）（2）取AB 、CD 的中点E 、F ，连接PE 、PF 、EF ，由PA PB =，PC PD =，得PE AB ⊥，PF CD ⊥，EF ∴为直角梯形的中位线，（6分）EF CD ∴⊥，又PF EF F = ，∴C ⊥平面P ，（8分）由PE ⊂平面PEF ，得CD PE ⊥，又AB PE ⊥且梯形两腰AB 、CD 必交，PE ∴⊥平面ABCD ,又PE ⊂平面PAB ，∴平面PAB ⊥平面ABCD ,（10分）（3）由（2）及二面角的定义知PFE ∠为二面角P CD A --的平面角,作EG BC ⊥于G ，连PG ，由于PE ⊥平面ABCD ,⊂BC 平面ABCD ,故PE BC ⊥,EG BC ⊥,,,EG PE E EG PE ⋂=⊂平面PEG ,故⊥BC 平面PEG PG ⊂平面PEG ,所以PG BC⊥故PGE ∠为二面角P BC A --的平面角,（12分）即60PGE ∠=︒，由已知，得11()22EF AD BC CD =+=，又12EG CF CD ==．EF EG ∴=，Rt F Rt PE PEG ∴≅ ．60PEF PGE ∴∠=∠=︒，故二面角P CD A --的大小为60︒．（18分）21．如图，斜三棱柱111ABC A B C -中，AC BC =，D 为AB 的中点，1D 为11A B 的中点，平面ABC ⊥平面11ABB A ．(1)求证：直线1//A D 平面11BC D ；(2)设直线1AB 与直线1BD 的交点为点E ，若三角形ABC 是等边三角形且边长为2，侧棱1AA =直线1BC 与1AB 互相垂直，求异面直线1A D 与1BC 所成角；(3)若12,2AB AC BC A AB ===∠=，在三棱柱111ABC A B C -内放置两个半径相等的球，使这两个球相切，且每个球都与三棱柱的三个侧面及一个底面相切．求三棱柱111ABC A B C -的高．【解析】（1）斜三棱柱111ABC A B C -中，1D 为11A B 的中点，D 为AB 的中点，所以11111122A D AB AB BD ===，且11A D BD //，所以四边形11A D BD 为平行四边形，所以11//A D BD ，（2分）因为1BD ⊂平面11BC D ，1A D ⊄平面11BC D ，所以1//A D 平面11BC D ；（4分）（2）因为AC =BC ，D 为AB 的中点，所以CD ⊥AB ，因为平面ABC ⊥平面11ABB A ，交线为AB ，CD ⊂平面ABC ，所以CD ⊥平面11ABB A ，故11C D ⊥平面11ABB A ，所以111C D AB ⊥,又1BC 与1AB 互相垂直，1111BC C D C ⋂=,111,BC C D ⊂面11BC D 故1AB ⊥面11BC D ,得11⊥AB BD .即11B D E 为直角三角形，（6分）在11ABB A 中，1,D D 为中点，11//A D BD ，所以E 为1AB 的三等分点，设1B E t =，由余弦定理可得：()22222211111111111132cos 21232t B E AB A B AA t A B A B D AB A B t +-+-⎝⎭∠====⋅⨯⨯解之：t =，所以11π,6A B A ∠=故112D E =11111113//,,.22D E B D A B AB BD EB AB ∴==∴=11C D ⊥平面11ABB A ，111,C D BD ∴⊥在11BD C △中，11tan 3D BC ∠=.1A D 与1BC所成的角为arctan（10分）（3）过B 作1BP AA ⊥于P ，过P 作1FP CC ⊥于F ，连BFBPF ∴ 为直截面，小球半径为BPF △的内切圆半径因为2,2AB AC BC ===，所以222AC BC AB +=，故AC ⊥BC ，则112CD AB ==（12分）设2,BP t =所以2AP t =，由222AB BP AP =+解得63t =，232633BP AP =由最小角定理112cos cos cos 263A AC A AB BAC ∠=∠∠=⨯=12sin 3PF AC A AC =∠=（14分）由CD ⊥面11ABB A ,易知1BP CC ⊥，23BF PF BP ∴===内切圆半径为：13r =则12362sin .9h r r r A AB =++∠=（18分）。
上海民办新世纪中学2022年高二语文测试题含解析
上海民办新世纪中学2022年高二语文测试题含解析一、现代文阅读（35分，共3题）14.阅读下面的文字，完成（1）～（4）题。
（21分）我寻求挫败——张晓风①我一直在寻求挫败，寻求被征服被震慑被并吞的喜悦。
②有人出发去“征山”，我却从来不是，而且刚好相反。
我爬山，是为了被山征服。
有人飞舟，是为了“凌驾”水，而我不是。
如果我去亲近水，我需要的是涓水归川的感受，是自身的消失，是自我复归于零的一次冒险。
③记得故事中那个叫“独孤求败”的第一剑侠吗？终其一生，他遇不到一个对手，人间再没有可以挫阻自己的高人，天地间再没有可匹敌的力量，真要令人忽忽如狂啊！④生来有一块通灵宝玉的贾宝玉是幸福的，但更大的幸福却发生在他掷玉的刹那。
那时，他初遇黛玉，一见面，彼此惊为旧识，仿佛已相契了万年。
他在惊愕中竟把那块玉胡乱砸在地上，那种自我的降服和破碎是动人的，是真爱情最醇美的倾注。
⑤文学史上也不乏这样的例子。
陈师曾一遇见黄山谷，就烧掉自己的书稿，虚心去向黄山谷学习。
一个人能碰见令自己心折首俯的人，并能一把火烧尽自己的旧作，应该算是一种极幸福的际遇。
⑥《新约》中的先知约翰，一见耶稣，便屈身降志。
他对人们说：“我仅仅以水为你们施洗礼，他却以灵为你们施洗礼，我之于他，只能算是一声开道的吆喝声。
”《红拂传》里的虬髯客一见李靖，便知天下大势已定，于是飘然远隐。
那使无数人为之倾倒的大唐盛世的李靖，我多么想见他一眼啊！清朝末年的孙中山也有如此风仪，使四方豪杰甘于俯首受命。
生的悲剧原不在头断血流，而在于没有大英雄可以为之赴命，没有大理想供其追求。
⑦我一直在寻求挫败，人生天地间，还有什么比失败更快乐的事？站在千丘万壑的大峡谷前，感到自己渺小得像蝼蚁一样，还有什么时候能如此心甘情愿地卑微下来，享受大化的赫赫天威？曾记得一个夏夜，卧在海滩上，看到满天繁星如雨阵如箭簇，一时几乎惊得昏了过去，感到一种投身在伟大之下的绝望，知道人类永远不能去逼近那百万光年之外的光体。
2022-2023学年上海民办新世纪中学高二物理测试题含解析
2022-2023学年上海民办新世纪中学高二物理测试题含解析一、选择题：本题共5小题，每小题3分，共计15分．每小题只有一个选项符合题意1. 如图所示，是一定质量的理想气体的三种升温过程，那么，以下四种解释中，哪些是正确的()A．a→d的过程气体体积增加B．b→d的过程气体体积不变C．c→d的过程气体体积增加D．a→d的过程气体体积减小参考答案：AB2. 关于天然放射性，下列说法正确的是A. 所有元素都可能发生衰变B. 放射性元素的半衰期与外界的温度有关C. 放射性元素与别的元素形成化合物时仍具有放射性D. α、β和γ三种射线中，γ射线穿透能力最弱参考答案：C【详解】A．原子核可以自发地衰变，但不是所有元素都可以发生衰变，故A错误。
B．放射性元素的半衰期由原子核本身决定的，与外界的温度无关，故B错误。
C．放射性元素的放射性与核外电子无关，故放射性元素与别的元素形成化合物时仍具有放射性，故C正确。
D．α、β和γ三种射线，γ射线的穿透力最强，电离能力最弱，故D错误。
3. （多选）在“验证力的平行四边形”实验中，橡皮条的一端固定在木板上，用两个弹簧秤把橡皮条的另一端拉到某一位置O点，以下操作中错误的是（）A．同一次实验过程中，O点位置允许变动B．在实验中，弹簧秤必须保持与木板平行，读数时视线要正对弹簧秤刻度C．实验中，先将其中一个弹簧秤沿某一方向拉到最大量程，然后只需调节另一弹簧秤拉力的大小和方向，把橡皮条的结点拉到O点D．实验中，把橡皮条的结点拉到O点时，两弹簧之间的夹角应取90°不变，以便于算出合力的大小参考答案：ACD4. 使带电的金属球靠近不带电的验电器，验电器的箔片张开。
下列各图表示验电器上感应电荷的分布情况，正确的是（）参考答案：B5. （单选）下列关于物体的动量和动能的说法，正确的是( )A、物体的动量发生变化，其动能一定发生变化B、物体的动能发生变化，其动量一定发生变化C．若两个物体的动量相同，它们的动能也一定相同D．两物体中动能大的物体，其动量也一定大参考答案：A二、填空题：本题共8小题，每小题2分，共计16分6. 如图所示卢瑟福做原子核人工转变实验的装置．容器中充有________气，放射性物质A 射出的α粒子________穿过铝箔F(填“能”或“不能”)。
2021-2022学年上海中学高二(上)期中数学试卷-附答案详解
2021-2022学年上海中学高二（上）期中数学试卷一、单选题（本大题共4小题，共12.0分）1.投掷一枚骰子，下列事件中是对立事件的是()A. 向上的点数是1与向上的点数是5B. 向上的点数小于3与向上的点数大于3C. 向上的点数是奇数与向上的点数是偶数D. 向上的点数大于3与向上的点数小于52.设四面体的六条棱的长分别为1，1，1，1，√2和a，且长为a的棱与长为√2的棱异面，则a的取值范围是()A. (0,√2)B. (0,√3)C. (1,√2)D. (1,√3)3.已知a、b、c是空间中的三条不重合的直线，α、β、γ是三个不同的平面，下列命题中真命题是()A. 若a//α，α//β，则a//βB. 若α⊥β，β⊥γ，则α//γC. 若a⊥b，b⊥c，则a//cD. 若b⊥β，b⊥γ，则β//γ4.已知正方体ABCD−A′B′C′D′的棱长为1，点M，N分别为线段AB′，AC上的动点，点T在平面BCC′B′内，则|MT|+|NT|的最小值是()A. √2B. 2√33C. √62D. 1二、单空题（本大题共12小题，共36.0分）5.三个平面两两垂直，它们的交线交于一点O，空间有一点P到三个面的距离分别为3、4、5，则OP的长为．6.已知球的表面积为16π，则该球的体积为________．7.随机投掷一枚均匀的硬币两次，则两次都正面朝上的概率为______．8.半径为2的半圆卷成一个圆锥，则它的体积为______ ．9.已知正四棱台的侧棱长为3，两底面边长分别为2和4，则该四棱台的体积为______．10.已知圆台的上、下底面半径分别为2和4，圆台的高为2，则该圆台的侧面积为______．11.正方体ABCD−A′B′C′D′中，异面直线AB′与BD所成的角为______ ．12.正方体ABCD−A1B1C1D1中，直线C1D与平面ACC1A1所成角大小为______．13. △ABC 的三边AB =10，BC =12，CA =14，D 、E 、F 分别是AB 、BC 、CA 的中点，沿DF 、EF 、ED 将△ADF ，△CEF ，△BED 折起，使得A 、B 、C 重合于P ，则四面体P −DEF 的体积为______．14. 棱长为6的正方体内有一个棱长为x 的正四面体，且该四面体可以在正方体内任意转动，则x 的最大值为______ ．15. 已知e 1⃗⃗⃗ 、e 2⃗⃗⃗ 是空间单位向量，e 1⃗⃗⃗ ⋅e 2⃗⃗⃗ =12，若空间向量b ⃗ 满足b ⃗ ⋅e 1⃗⃗⃗ =2，b ⃗ ⋅e 2⃗⃗⃗ =52，且对任意x 、y ∈R ，|b ⃗ −(x e 1⃗⃗⃗ +y e 2⃗⃗⃗ )|≥|b ⃗ −(x 0e 1⃗⃗⃗ +y 0e 2⃗⃗⃗ )|=1(x 0、y 0∈R)，则x 0+y 0+|b ⃗ |=______．16. 如图，已知三个两两互相垂直的半平面α，β，γ交于点O ，矩形ABCD 的边BC 在半平面γ内，顶点A ，D 分别在半平面α，β内，AD =2，AB =3，AD 与平面α所成角为π4，二面角A −BC −O 的余弦值为13，则同时与半平面α，β，γ和平面ABCD 都相切的球的半径为______．三、解答题（本大题共5小题，共60.0分）17. 如图，在圆柱OO 1中，它的轴截面ABB 1A 1是一个边长为2的正方形，点C 为棱BB 1的中点，点C 1为弧A 1B 1的中点．(1)求异面直线OC 与A 1C 1所成角的大小；(2)求直线CC 1与圆柱OO 1底面所成角的正弦值．18.如图，已知正方形ABCD和矩形ACEF所在的平面互相垂直，AB=√2，AF=1，M是线段EF的中点．(1)求证：AM//平面BDE；(2)求二面角A−DF−B的大小．19.如图，三棱锥P−ABC中，底面ABC是正三角形，PA⊥底面ABC，AG⊥平面PBC，垂足为G．(1)G是否可能是△PBC的垂心？请说明理由；(2)若G恰是△PBC的重心，且△ABC的边长为2，求点C到平面ABG的距离．20. 已知平面向量中有如下两个结论：结论1：若OA ⃗⃗⃗⃗⃗ 、OB ⃗⃗⃗⃗⃗⃗ 是不共线的两个平面向量，OC ⃗⃗⃗⃗⃗ =λOA ⃗⃗⃗⃗⃗ +μOB⃗⃗⃗⃗⃗⃗ ，则A 、B 、C 三点共线的充要条件是λ+μ=1；结论2：若OA ⃗⃗⃗⃗⃗ 、OB ⃗⃗⃗⃗⃗⃗ 是不共线的两个平面向量，OP ⃗⃗⃗⃗⃗ =λOA ⃗⃗⃗⃗⃗ +μOB⃗⃗⃗⃗⃗⃗ ，若点P 在与AB 平行的直线上，则λ+μ=k(k 为定值)．将上述两个结论推广至空间向量(无需写出推广结论)解决以下问题：已知OA ⃗⃗⃗⃗⃗ 、OB ⃗⃗⃗⃗⃗⃗ 、OC ⃗⃗⃗⃗⃗ 是两两垂直的单位向量，P 是空间中一点．(1)若OP ⃗⃗⃗⃗⃗ =x OA ⃗⃗⃗⃗⃗ +y OB ⃗⃗⃗⃗⃗⃗ +z OC ⃗⃗⃗⃗⃗ 且x +2y +4z =1，求|OP ⃗⃗⃗⃗⃗ −OA ⃗⃗⃗⃗⃗ −OB⃗⃗⃗⃗⃗⃗ |的最小值； (2)若OP ⃗⃗⃗⃗⃗ =x OA ⃗⃗⃗⃗⃗ +y OB ⃗⃗⃗⃗⃗⃗ +z OC ⃗⃗⃗⃗⃗ 且满足{0≤x,y,z ≤11≤x +y +z ≤2，求动点P 的轨迹所围成的区域的体积．21. 如图，在矩形ABCD 中，M 、N 分别是线段AB 、CD 的中点，AD =2，AB =4，将△ADM 沿DM 翻折，在翻折过程中A 点记为P 点．(1)从△ADM 翻折至△NDM 的过程中，求点P 运动的轨迹长度；(2)翻折过程中，二面角P −BC −D 的平面角为θ，求tanθ的最大值．答案和解析1.【答案】C【解析】解：对于A，向上的点数是1与向上的点数是5不能同时发生，但能同时不发生，是互斥不对立事件，故A错误；对于B，向上的点数小于3与向上的点数大于3不能同时发生，但能同时不发生，是互斥不对立事件，故B错误；对于C，向上的点数是奇数与向上的点数是偶数既不能同时发生，也不能同时不发生，是对立事件，故C正确；对于D，向上的点数大于3与向上的点数小于5能同时发生，不是互斥事件，更不是对立事件，故D错误．故选：C．利用对立事件、互斥事件的定义直接求解．本题考查对立事件的判断，考查对立事件、互斥事件的定义等基础知识，是基础题．2.【答案】A【解析】解：设四面体的底面是BCD，BC=a，BD=CD=1，顶点为A，AD=√2，在三角形BCD中，因为两边之和大于第三边可得：0<a<2(1)，取BC中点E，∵E是中点，直角三角形ACE全等于直角DCE，)2，所以在三角形AED中，AE=ED=√1−(a2∵两边之和大于第三边，∴√2<2√1−(a) 2得0<a<√2(负值0值舍)(2)，2由(1)(2)得0<a<√2．故选：A．先在三角形BCD中求出a的范围，再在三角形AED中求出a的范围，二者相结合即可得到答案．本题主要考查三角形三边关系以及异面直线的位置．解决本题的关键在于利用三角形两边之和大于第三边这一结论．3.【答案】D【解析】解：若a//α，α//β，则a//β或a⊂β，故A错误；若α⊥β，β⊥γ，则α//γ或α与γ相交，故B错误；若a⊥b，b⊥c，则a//c或a与c相交或a与c异面，故C错误；若b⊥β，b⊥γ，由直线与平面垂直的性质可得β//γ，故D正确．故选：D．由平面与平面平行得线面关系判定A；由垂直于同一平面的两平面的位置关系判定B；由垂直于同一直线的两直线的位置关系判定C；由直线与平面垂直的性质判断D．本题考查空间中直线与直线、直线与平面、平面与平面位置关系的判定及应用，考查空间想象能力与思维能力，是基础题．4.【答案】B【解析】解：A点关于BC的对称点为E，N关于BC的对称点为N′，设d为异面直线AB′与CE之间的距离，则|MT|+|NT|=|MT|+|N′T|≥|MN′|≥d，因为CE//DB，DB//D′B′，所以CE//B′D′，又因为△AB′D′为正三角形，所以∠AB′D′=60°，所以直线AB′与CE所成角为60°，四面体AB′CE的体积V AB′CE=13⋅12|AB′|⋅|CE|sin60°d=√36d，又因为V AB′CE=13S ACE|BB′|=13⋅12|AE|⋅|BC|⋅|BB′|=13，所以√36d=13，解得d=2√33，所以|MT|+|NT|的最小值为2√33，故选：B．先根据对称性和三角不等式把问题转化为AB′与CE上两点连线长最小值问题，再利用异面直线距离求解．本题考查了正方体的基本性质，考查了异面直线成角和距离问题，考查了四面体体积计算问题，属于中档题．5.【答案】5√2【解析】【分析】本题主要考查棱柱的结构特征，属于基础题．根据题意，以OP为对角线，P到三个平面的垂线段分别为长、宽、高，建立长方体，利用长方体的提对角长公式加以计算，即可得到OP的长．【解答】解：根据题意，过点P分别作三个平面的垂线，垂足分别为B、D、F，∵三个平面两两垂直，∴分别以PB、PD、PF为长、宽、高，作长方体OABC−EDPF，如图所示．∵点P到三个面的距离分别为3、4、5，∴长方体的对角线长为√3 2+4 2+5 2=5√2，即OP的长为5√2．故答案为：5√2π6.【答案】323【解析】【分析】本题考查球的表面积、体积的计算，考查计算能力，公式的应用，属于基础题．通过球的表面积求出球的半径，然后求出球的体积【解答】解：一个球的表面积是16π，所以球的半径为：2，所以这个球的体积为：4π3×23=323π．故答案为：323π．7.【答案】14【解析】解：分别用0和1表示硬币正面朝上和反面朝上，投掷两次的所有可能情况为(0,0)，(0,1)，(1,0)，(1,1)，共4种，而两次正面朝上的情况为(0,0)，只有1种，所以所求事件的概率为14．故答案为：14．用列举法写出投掷两次的所有可能情况和两次正面朝上的情况，再由古典概型，即可得解．本题考查古典概型，事件的概率，考查逻辑推理能力和运算能力，属于基础题．8.【答案】√33π【解析】解：半径为R的半圆卷成一个圆锥，则圆锥的母线长为R，设圆锥的底面半径为r，则2πr=πR，即r=1，∴圆锥的高ℎ=√R2−r2=√3，∴圆锥的体积V=13π⋅12√3=√33π，故答案为：√33π．半径为R的半圆卷成一个圆锥，则圆锥的母线长为R，底面半径r=1，求出圆锥的高后，代入圆锥体积公式可得答案．本题考查旋转体，即圆锥的体积，意大利考查了旋转体的侧面展开和锥体体积公式等知识．9.【答案】28√73【解析】解：如图所示，正四棱台ABCD−A1B1C1D1中，设下底面中心为O，上底面中心为O1，则OO1为正棱台的高，过点B1作B1E⊥BD，则B1E=OO1，在Rt△BEB1中，BB1=3，BE=2√2−√2=√2，所以B1E=√32−(√2)2=√7，又S ABCD=42=16,S A1B1C1D1=22=4，所以该四棱台的体积为V=13(16+√16×4+4)×√7=28√73．故答案为：28√73．根据正四棱台的性质求出高，然后由棱台的体积公式求解即可．本题考查了正四棱台的几何性质，正棱台体积公式的应用，考查了逻辑推理能力与化简运算能力，属于中档题．10.【答案】12√2π【解析】解：由题意可得，AO1=2，BO2=4，O1O2=AC=2，所以BC=BO2=CO2=2，故圆台的母线长为AB=√AC2+BC2=2√2，所以圆台的侧面积为S=π(R+r)l=π(4+2)×2√2=12√2π．故答案为：12√2π．作出圆台的轴截面，利用勾股定理求出圆台的母线，再根据圆台的侧面积公式计算即可．本题考查了圆台的几何性质的理解与应用，圆台的轴截面的理解与应用，圆台侧面积公式的运用，考查了逻辑推理能力与化简运算能力，属于基础题．11.【答案】60°【解析】解：如图，连接B′D′，则BD//B′D′，∴∠AB′D′即为异面直线AB′与BD所成角，连接AD′，可得△AB′D′为等边三角形，则∠AB′D′=60°．故答案为：60°．由题意画出图形，连接B′D′，则BD//B′D′，则∠AB′D′即为异面直线AB′与BD所成角，连接AD′，可得△AB′D′为等边三角形，从而可得∠AB′D′=60°．本题考查异面直线所成角，考查了数形结合的解题思想方法，是中档题．12.【答案】30°【解析】解：因为ABCD−A1B1C1D1是正方体，所以BD⊥AC，BD⊥C1C，又因为C1C∩AC=C，所以BD⊥平面ACC1A1，所以C1E是C1D在平面ACC1A1内投影，所以C1D与平面ACC1A1所成角为∠DC1E，因为DE =12⋅BD =12⋅C 1D ，所以∠DC 1E =30°， 故答案为：30°．先找直线C 1D 与平面ACC 1A 1成角是∠DC 1E ，再解直角三角形．本题考查了正方体的结构特征，考查了直线与平面成角问题，属于中档题．13.【答案】2√95【解析】解：由题意知，PE =DF =6，DE =PF =7，EF =PD =5，将四面体P −DEF 嵌入如图所示的长方体中，设长方体的三条棱长分别为a ，b ，c ， 则{a 2+c 2=DE 2=49b 2+c 2=EF 2=25a 2+b 2=PE 2=36，解得{a =√30b =√6c =√19， 四面体P −DEF 的体积为长方体体积减去四个三棱锥，且所有三棱锥的体积均相同均为长方体的16， ∴V =(1−4×16)×√30×√6×√19=2√95．故答案为：2√95．由已知得PE =DF =6，DE =PF =7，EF =PD =5，将其嵌入某指定长方体中，再由长方体的体积减去四个三棱锥的体积即可．本题考查多面体体积的求法，训练了分割补形法，考查转化思想及运算求解能力，属于中档题．14.【答案】2√6【解析】解：由题意得：该四面体在棱长为6的正方体的内切球内， ∴该四面体内接于球时棱长最大， ∵棱长为6的正方体的内切球半径R =3，∴(23√x 2−(12x)2)2+(√a 2−(23√x 2−(12x)2)2−3)2=32，解得x =2√6． 故答案为：2√6．该四面体在棱长为6的正方体的内切球内，从而该四面体内接于球时棱长最大。
2021-2022学年上海中学高二(上)期中数学试卷
2021-2022学年上海中学高二（上）期中数学试卷试题数：21，总分：01.（填空题，3分）三个平面两两垂直，它们的交线交于一点O，空间有一点P到三个面的距离分别为3、4、5，则OP的长为___ ．2.（填空题，3分）已知球的表面积为16π，则该球的体积为___ ．3.（填空题，3分）随机投掷一枚均匀的硬币两次，则两次都正面朝上的概率为 ___ ．4.（填空题，3分）半径为2的半圆卷成一个圆锥，则它的体积为___ ．5.（填空题，3分）已知正四棱台的侧棱长为3，两底面边长分别为2和4，则该四棱台的体积为 ___ ．6.（填空题，3分）已知圆台的上、下底面半径分别为2和4，圆台的高为2，则该圆台的侧面积为 ___ ．7.（填空题，3分）正方体ABCD-A′B′C′D′中，异面直线AB′与BD所成的角为___ ．8.（填空题，3分）正方体ABCD-A1B1C1D1中，直线C1D与平面ACC1A1所成角大小为 ___ ．9.（填空题，3分）△ABC的三边AB=10，BC=12，CA=14，D、E、F分别是AB、BC、CA 的中点，沿DF、EF、ED将△ADF，△CEF，△BED折起，使得A、B、C重合于P，则四面体P-DEF的体积为 ___ ．10.（填空题，3分）棱长为6的正方体内有一个棱长为x的正四面体，且该四面体可以在正方体内任意转动，则x的最大值为___ ．11.（填空题，3分）已知e1⃗⃗⃗ 、e2⃗⃗⃗ 是空间单位向量，e1⃗⃗⃗ •e2⃗⃗⃗ =12，若空间向量b⃗满足b⃗•e1⃗⃗⃗ =2，b⃗•e2⃗⃗⃗ =52，且对任意x、y∈R，| b⃗ -（x e1⃗⃗⃗ +y e2⃗⃗⃗ ）|≥| b⃗ -（x0e1⃗⃗⃗ +y0e2⃗⃗⃗ ）|=1（x0、y0∈R），则x0+y0+|b⃗| =___ ．12.（填空题，3分）如图，已知三个两两互相垂直的半平面α，β，γ交于点O，矩形ABCD 的边BC在半平面γ内，顶点A，D分别在半平面α，β内，AD=2，AB=3，AD与平面α所成角为π4，二面角A-BC-O的余弦值为13，则同时与半平面α，β，γ和平面ABCD都相切的球的半径为 ___ ．13.（单选题，3分）投掷一枚骰子，下列事件中是对立事件的是（）A.向上的点数是1与向上的点数是5B.向上的点数小于3与向上的点数大于3C.向上的点数是奇数与向上的点数是偶数D.向上的点数大于3与向上的点数小于514.（单选题，3分）设四面体的六条棱的长分别为1，1，1，1，√2和a，且长为a的棱与长为√2的棱异面，则a的取值范围是（）A.（0，√2）B.（0，√3）C.（1，√2）D.（1，√3）15.（单选题，3分）已知a、b、c是空间中的三条不重合的直线，α、β、γ是三个不同的平面，下列命题中真命题是（）A.若a || α，α || β，则a || βB.若α⊥β，β⊥γ，则α || γC.若a⊥b，b⊥c，则a || cD.若b⊥β，b⊥γ，则β || γ16.（单选题，3分）已知正方体ABCD-A'B'C'D'的棱长为1，点M，N分别为线段AB'，AC上的动点，点T在平面BCC'B'内，则|MT|+|NT|的最小值是（）A. √2B. 2√33C. √62D.117.（问答题，0分）如图，在圆柱OO1中，它的轴截面ABB1A1是一个边长为2的正方形，点C为棱BB1的中点，点C1为弧A1B1的中点．（1）求异面直线OC 与A 1C 1所成角的大小；（2）求直线CC 1与圆柱OO 1底面所成角的正弦值．18.（问答题，0分）如图，已知正方形ABCD 和矩形ACEF 所在的平面互相垂直，AB= √2 ，AF=1，M 是线段EF 的中点．（Ⅰ）求证：AM || 平面BDE ；（Ⅱ）求二面角A-DF-B 的大小．19.（问答题，0分）如图，三棱锥P-ABC 中，底面ABC 是正三角形，PA⊥底面ABC ，AG⊥平面PBC ，垂足为G ．（1）G 是否可能是△PBC 的垂心？请说明理由；（2）若G 恰是△PBC 的重心，且△ABC 的边长为2，求点C 到平面ABG 的距离．20.（问答题，0分）已知平面向量中有如下两个结论：结论1：若 OA ⃗⃗⃗⃗⃗ 、 OB ⃗⃗⃗⃗⃗ 是不共线的两个平面向量， OC ⃗⃗⃗⃗⃗ =λOA⃗⃗⃗⃗⃗ +μOB ⃗⃗⃗⃗⃗ ，则A 、B 、C 三点共线的充要条件是λ+μ=1；结论2：若 OA ⃗⃗⃗⃗⃗ 、 OB ⃗⃗⃗⃗⃗ 是不共线的两个平面向量， OP ⃗⃗⃗⃗⃗ =λOA⃗⃗⃗⃗⃗ +μOB ⃗⃗⃗⃗⃗ ，若点P 在与AB 平行的直线上，则λ+μ=k （k 为定值）．将上述两个结论推广至空间向量（无需写出推广结论）解决以下问题：已知 OA ⃗⃗⃗⃗⃗ 、 OB ⃗⃗⃗⃗⃗ 、 OC ⃗⃗⃗⃗⃗ 是两两垂直的单位向量，P 是空间中一点．（1）若 OP ⃗⃗⃗⃗⃗ =xOA ⃗⃗⃗⃗⃗ +yOB ⃗⃗⃗⃗⃗ +zOC ⃗⃗⃗⃗⃗ 且x+2y+4z=1，求 |OP ⃗⃗⃗⃗⃗ −OA ⃗⃗⃗⃗⃗ −OB ⃗⃗⃗⃗⃗ | 的最小值；（2）若 OP ⃗⃗⃗⃗⃗ =xOA ⃗⃗⃗⃗⃗ +yOB ⃗⃗⃗⃗⃗ +zOC ⃗⃗⃗⃗⃗ 且满足 {0≤x ，y ，z ≤11≤x +y +z ≤2，求动点P 的轨迹所围成的区域的体积．21.（问答题，0分）如图，在矩形ABCD 中，M 、N 分别是线段AB 、CD 的中点，AD=2，AB=4，将△ADM 沿DM 翻折，在翻折过程中A 点记为P 点．（1）从△ADM 翻折至△NDM 的过程中，求点P 运动的轨迹长度；（2）翻折过程中，二面角P-BC-D 的平面角为θ，求tanθ的最大值．2021-2022学年上海中学高二（上）期中数学试卷参考答案与试题解析试题数：21，总分：01.（填空题，3分）三个平面两两垂直，它们的交线交于一点O，空间有一点P到三个面的距离分别为3、4、5，则OP的长为___ ．【正确答案】：[1]5 √2【解析】：根据题意，以OP为对角线，P到三个平面的垂线段分别为长、宽、高，建立如图所示长方体，利用长方体的对角长公式加以计算，即可得到OP的长．【解答】：解：根据题意，过点P分别作三个平面的垂线，垂足分别为B、D、F，∵三个平面两两垂直，∴分别以PB、PD、PF为长、宽、高，作长方体OABC-EDPF，如图所示．∵点P到三个面的距离分别为3、4、5，∴长方体的对角线长为√3 2+4 2+5 2 = 5√2，即OP的长为5√2．故答案为：5 √2【点评】：本题给出两两垂直的三个平面，求点P到它们的交点之间的距离．着重考查了面面垂直的性质、长方体的对角线长公式和空间距离的计算等知识，属于中档题．2.（填空题，3分）已知球的表面积为16π，则该球的体积为___ ．π【正确答案】：[1] 323【解析】：通过球的表面积求出球的半径，然后求出球的体积【解答】：解：一个球的表面积是16π，所以球的半径为：2，所以这个球的体积为：4π3×23 = 323π．故答案为：323π．【点评】：本题是基础题，考查球的表面积、体积的计算，考查计算能力，公式的应用．3.（填空题，3分）随机投掷一枚均匀的硬币两次，则两次都正面朝上的概率为 ___ ．【正确答案】：[1] 14【解析】：用列举法写出投掷两次的所有可能情况和两次正面朝上的情况，再由古典概型，即可得解．【解答】：解：分别用0和1表示硬币正面朝上和反面朝上，投掷两次的所有可能情况为（0，0），（0，1），（1，0），（1，1），共4种，而两次正面朝上的情况为（0，0），只有1种，所以所求事件的概率为14．故答案为：14．【点评】：本题考查古典概型，事件的概率，考查逻辑推理能力和运算能力，属于基础题．4.（填空题，3分）半径为2的半圆卷成一个圆锥，则它的体积为___ ．【正确答案】：[1] √33π【解析】：半径为R的半圆卷成一个圆锥，则圆锥的母线长为R，底面半径r=1，求出圆锥的高后，代入圆锥体积公式可得答案．【解答】：解：半径为R的半圆卷成一个圆锥，则圆锥的母线长为R，设圆锥的底面半径为r，则2πr=πR，即r=1，∴圆锥的高h= √R2−r2 = √3，∴圆锥的体积V= 13π•12√3 = √33π，故答案为：√33π．【点评】：本题考查旋转体，即圆锥的体积，意大利考查了旋转体的侧面展开和锥体体积公式等知识．5.（填空题，3分）已知正四棱台的侧棱长为3，两底面边长分别为2和4，则该四棱台的体积为 ___ ．【正确答案】：[1] 28√73【解析】：根据正四棱台的性质求出高，然后由棱台的体积公式求解即可．【解答】：解：如图所示，正四棱台ABCD-A1B1C1D1中，设下底面中心为O，上底面中心为O1，则OO1为正棱台的高，过点B1作B1E⊥BD，则B1E=OO1，在Rt△BEB1中，BB1=3，BE= 2√2−√2=√2，所以B1E= √32−(√2)2=√7，又S ABCD=42=16，S A1B1C1D1=22=4，所以该四棱台的体积为V=13(16+√16×4+4)×√7 = 28√73．故答案为：28√73．【点评】：本题考查了正四棱台的几何性质，正棱台体积公式的应用，考查了逻辑推理能力与化简运算能力，属于中档题．6.（填空题，3分）已知圆台的上、下底面半径分别为2和4，圆台的高为2，则该圆台的侧面积为 ___ ．【正确答案】：[1] 12√2π【解析】：作出圆台的轴截面，利用勾股定理求出圆台的母线，再根据圆台的侧面积公式计算即可．【解答】：解：由题意可得，AO1=2，BO2=4，O1O2=AC=2，所以BC=BO2=CO2=2，故圆台的母线长为AB=√AC2+BC2=2√2，所以圆台的侧面积为S=π(R+r)l=π(4+2)×2√2=12√2π．故答案为：12√2π．【点评】：本题考查了圆台的几何性质的理解与应用，圆台的轴截面的理解与应用，圆台侧面积公式的运用，考查了逻辑推理能力与化简运算能力，属于基础题．7.（填空题，3分）正方体ABCD-A′B′C′D′中，异面直线AB′与BD所成的角为___ ．【正确答案】：[1]60°【解析】：由题意画出图形，连接B′D′，则BD || B′D′，则∠AB′D′即为异面直线AB′与BD 所成角，连接AD′，可得△AB′D′为等边三角形，从而可得∠AB′D′=60°．【解答】：解：如图，连接B′D′，则BD || B′D′，∴∠AB′D′即为异面直线AB′与BD 所成角，连接AD′，可得△AB′D′为等边三角形，则∠AB′D′=60°．故答案为：60°．【点评】：本题考查异面直线所成角，考查了数形结合的解题思想方法，是中档题．8.（填空题，3分）正方体ABCD-A1B1C1D1中，直线C1D与平面ACC1A1所成角大小为 ___ ．【正确答案】：[1]30°【解析】：先找直线C 1D 与平面ACC 1A 1成角是∠DC 1E ，再解直角三角形．【解答】：解：因为ABCD-A 1B 1C 1D 1是正方体，所以BD⊥AC ，BD⊥C 1C ，又因为C 1C∩AC=C ，所以BD⊥平面ACC 1A 1，所以C 1E 是C 1D 在平面ACC 1A 1内投影，所以C 1D 与平面ACC 1A 1所成角为∠DC 1E ，因为DE= 12•BD = 12• C 1D ，所以∠DC 1E=30°，故答案为：30°．【点评】：本题考查了正方体的结构特征，考查了直线与平面成角问题，属于中档题．9.（填空题，3分）△ABC 的三边AB=10，BC=12，CA=14，D 、E 、F 分别是AB 、BC 、CA 的中点，沿DF 、EF 、ED 将△ADF ，△CEF ，△BED 折起，使得A 、B 、C 重合于P ，则四面体P-DEF 的体积为 ___ ．【正确答案】：[1]2 √95【解析】：由已知得PE=DF=6，DE=PF=7，EF=PD=5，将其嵌入某指定长方体中，再由长方体的体积减去四个三棱锥的体积即可．【解答】：解：由题意知，PE=DF=6，DE=PF=7，EF=PD=5，将四面体P-DEF 嵌入如图所示的长方体中，设长方体的三条棱长分别为a ，b ，c ，则 {a 2+c 2=DE 2=49b 2+c 2=EF 2=25a 2+b 2=PE 2=36 ，解得 {a =√30b =√6c =√19，四面体P-DEF 的体积为长方体体积减去四个三棱锥，且所有三棱锥的体积均相同均为长方体的 16 ，∴V=（1-4× 16 ）× √30 × √6 × √19 =2 √95 ．故答案为：2 √95 ．【点评】：本题考查多面体体积的求法，训练了分割补形法，考查转化思想及运算求解能力，属于中档题．10.（填空题，3分）棱长为6的正方体内有一个棱长为x的正四面体，且该四面体可以在正方体内任意转动，则x的最大值为___ ．【正确答案】：[1] 2√6【解析】：该四面体在棱长为6的正方体的内切球内，从而该四面体内接于球时棱长最大。
上海市2021-2022学年高二数学上学期期中试题(含解析)
上海市上海师范大学附属中学2021-2021学年高二数学上学期期中试题（含解析）一、填空题（本大题共有12小题，满分54分）考生应在答题纸相应编号的空格内直接填写结果，1-6题每个空格填对得4分，7-12题每个空格填对得5分，否则一律得0分． 1.已知直线1:10l x y ++=，2:2l x =，则1l 与2l 的夹角为_________． 【答案】45︒ 【解析】 【分析】先根据两直线的方程求出他们的斜率及倾斜角，在坐标系中画出图形，结合图形求出直线1l 与2l 的夹角θ的大小．【详解】解：直线1:10l x y ++=，2:2l x =，∴直线1l 的斜率为1-，倾斜角为135α=︒，2l 的斜率不存在，如图所示：故直线1l 与2l 的夹角为4πθ=，故答案为4π． 【点睛】本题考查直线的倾斜角和斜率的关系，两直线的夹角的定义，体现了数形结合的数学思想．2.向量(1,2),(3,4)a b =-=-，则a 在b 方向上的投影为_________． 【答案】115【解析】 【分析】根据平面向量的数量积运算与向量投影的定义，写出对应的运算即可． 【详解】解：(1,2),(3,4)a b =-=-，∴()()132411a b ⋅=⨯-+-⨯=-，2(54b =+；∴向量a 在向量b 方向上的投影为：11cos ,5a b a a b b ⋅-==．故答案为：115． 【点睛】本题考查了平面向量的数量积运算与向量投影的定义与应用问题，是基础题． 3.已知向量()()1,1,2,2m n λλ=+=+，若()()m n m n +⊥-，则=λ_________ . 【答案】3- 【解析】试题分析：因为()()m n m n +⊥-，所以()()220m n m n m n +⋅-=-=，22m n =，即222(1)1(2)2λλ++=++，解得3λ=-．考点：向量垂直的性质，考查学生的基本运算能力．4.若圆22(1)(4)5x y -+-=的圆心到直线0x y a -+=，则a 的值为_________． 【答案】5或1【解析】 【分析】直接利用点到直线的距离公式求出结果． 【详解】解：圆22(1)(4)5x y -+-=圆心坐标为：()1,4，则：圆心()1,4到直线0x y a -+=的距离d ==，解得：1a =或5． 故答案为：1或5【点睛】本题考查的知识要点：点到直线的距离公式的应用．5.过点()13A ，与点()25B -，且半径最小圆的标准方程为_________． 【答案】22113()(4)24x y ++-= 【解析】 【分析】过两点的半径最小的圆即是以这两点为直径的圆。
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Mid-term Examination(满分150分，考试时间120分钟)第l卷(共103分)I．Listening Comprehension．Section ADirections：In Part A，you will hear ten short conversations between two speakers．At the end of each conversation, a question will be asked about what was said．The conversations and the questions will be spoken only once. After you hear a conversation and the question about it, read the four possible answers in your paper，and decide which one is the best answer to the question you have heard．(10分)( )1．A 27815389．B．27185398 C．27815398 D．27185389( )2．A．Riding B．By bus C．Driving．D．On foot．( ) 3. A 1l：00． B 12:00 C．1：15．D．11：45．( )4．A He prefers Chinese tea．B He prefers coke and coffee．C He prefers any other drink．D．He prefers water( )5．A．She’d like to move there．B．She knows someone who lived thereC．She used to have all apartment thereD．She’ll visit there soon．( )6．A．She has to meet someone at the station at l0:30B．Her train will arrive at 10:15.C．She doesn’t want to miss her train.D．She can’t drive as fast as the train．( )7．A．She can’t go this weekend．B．She dug a hole in the beach．C．They should stay at the beach longerD．They should n’t spend much money and time traveling( )8．A．At the Bank of China B．At the post officeC．At the next stop. D .At the bus station( ) 9. A. To do some business. B．To visit some friends.C .To go sightseeing. D. To buy clothes.( ) 10 A. She couldn’t afford the handbag B. She got the handbagfor$5.C. She did not like the handbag at allD. None of them．Section BDirections：In Part B，you will hear two passages，and you will be asked three questions on each of the passages．The passages will be read twice，but the questions will be spoken only once．When you hear a question，read the four possible answers in your paper and decide which one would be the best answer to the question you have heard．(12分) Questions 11 through 13 are based oil the following passage．( ) 11. A. Hel earns independently．B．He learns actively．C. He learns purposefully．D．Both A，B and C．( )12．A They depend on books． B They depend on teachers．C．They depend on themselves D．They depend on rules．( ) 13. A .They wait for a chance to use the languageB．They find the patterns and rules for themselves．C．They learn the language for exams．D．They won’t try anything to communicate．Questions 14 through 16 are based on the following passage．( ) 14 .A. Because spores help one earn money．B .Because sports help one keep healthy and happy as well as live longer．C．Because sports help one kill time．D．Because sports help one keep a riving．( )15．A．Horse—riding．B．Hunting．C．Fishing．D．All of the above．( )16．A．Football B．Skating．C．Swimming．D．SkiingSection CDirections：In Part C，you will hear a longer conversation．The conversation will be read twice. After you hear the conversation, you are required to fill in the blanks according to what you have heard．(8分)Blanks 17through 20 are based on the following conversation.Blanks 17 through 20 are based oil the following the conversation．17. _______ 18. _________ 19. _________ 20._________Blanks 21 through 24 are based on the following conversation.21．________ 22._________ 23. __________ 24. _____________ Complete the form. Write One Word for each blank.Ⅱ. Grammar and V ocabularySection ADirections：After rending the passages below, fill in the blanks to make the passages coherent and grammatically correct. For the blanks with a given word, fill each blank with the proper form of the given word; for the other blanks, use one word that best fitseach blank．(A)Birth, marriage and death: these are the (25) _______(great)events in human life. Many things, good and bad, can happen(26)_______ us in our lives. Yet there are three days(27) _______are usually marked by some kind of special ceremony：the day we are born, the day we get married and the day we die．These are the three main events in life. We only have a choice in(28)_______second of these. We(29)_______choose whether or not to marry．But we have no choice in birth and death. All human beings from the most primitive to the most sophisticated(高级的)(30)_______(affect)by these three events．The only thing that(31)_______(differ)in each society is the way these events are celebrated. Yet all societies share common characteristics. Birth is a time of joy. The proud parents receive congratulations and presents on behalf of the new-born. Marriage is also a time of joy．The young couple go through a special wedding ceremony and receive presents(32)_______ (help)them set up their home. Death is a time of sorrow and is marked by a special ceremony and mourning．The dates of all three events are usually remembered．(B)In North America，people are always in a hurry. Children have special lessons or sport activities after school. Parents often work late and don’t get home (33)_______ 7 or 8 o’clock a t night. More than 50％of women work at full-time jobs, and many people do part-time work. Most North American families don’t have time(34)_______(eat)meals together.When a family(35)_______ (take)time to eat a meal together，often there isn’t enough time to prepare the food. That is(36)_______ fast food is so popular in North America. People spend about 40％of their dollars on fast food.Fast food is food (37) _______pizza, sandwiches, or fried chicken. People usually buy the food from restaurant chains such as Pizza Hut, McDonald’s, or Kentucky Fried Chicken. Fast food saves work and time,(38)_______it is not very nutritious.Fast food is popular in many countries. American fast-food companies now have restaurants nearly all over the world．But not(39)______ is happy about the spread of North American fast food. A group of people in Italy want to fight against the spread of American fast food. They do not want any more fast food restaurants(40)________(open) in their country. They also want to fight against the spread of fast food everywhere in the world.Section BDirections．Complete the following passage by using the words in the box．Each wordsport. A lot watch sport on TV. But the number who takes part in sport is quite small. On the whole English people ___41___ to be fat rather than thin.The most popular sport in England is football．Football is played on Saturday afternoonin most towns and the___42___of a certain team will travel from one end of the country to the other to see their teams play. There are four divisions(级别)of the football league. Not___ 43___ the best teams are in the first division. But the best supporters are often in the fourth division. You have to be a good supporter to watch the fourth division football! Many other sports are also played in England, such as golf, in which you try to___44___a ball into a hole; basketball, in which you try to get a ball through a___45___; tennis, in which you try to___46___a ball so that your ___47___cannot hit it. As you see, if the ball had not been ___48___ there would have been no sport. Actually, that is not quite___49___．Athletics isn’t played with a ball; nor horseracing. Perhaps that is why they are not so___ 50____as football!Ⅲ．Reading comprehensionSection ADirections：For each blank in the following passage there are four words or phrases marked A, B．C and D．Fill in each blank with the word or phrase that best fits the context．There is no doubt that people and even highly educated adults vary greatly in the speed and efficiency of their reading．Some proceed very slowly throughout；others dash along too___51___and then have to go back. Poor readers in particular may lack the ability to ___52___ their tensions in reading. A great reader can move at a great speed through the text of a novel or___53___ tight reading matters He may be able to___54___ a page, picking up a word or two here and there, and gain a___55___idea what the text is about without really reading it. In reading more difficult material, with the intention of taking in the whole of it, he___56___ more slowly，but even then he will vary his pace, ___57___on the key words and passages and pass more quickly over the ___58___. A less efficient reader tends to maintain the same speed whatever material he reads.___59___ even light reading matters require special concentration at difficult___60___．A type of reading which requires careful attention to___61___is proofreading(校对), in which the reader，in order to___62___ misprints in a sample print, has to notice not so much the meaning of what he reads as die exact shape and order of letters and words in the text. This is extremely difficult for most people, since they are accustomed to___63___such details. In fact, extreme practice is required to practice this task___64___and it can be done only by reading very slowly, and by paying comparatively___65___attention to the general meaning on the text．( )51．A．carefully B．attentively C．smoothly D．quickly ( )52．A．estimate B．judge C．vary D．alter( )53．A．familiar B．similar C．complex D．professional ( )54．A．skim B．tear C．mark D．explore ( )55．A．vague B．general C．special D．accurate ( )56．A．continues B．persists C．proceed D．withdraws ( )57．A．depending B．counting C．passing D．focusing ( )58．A．remainder B．context C．article D．principle ( )59．A．Gradually B．Consequently C．Wholly D．Personally ( )60．A．occasions B．period C．points D．functions( )61．A．theme B．subject C．chapter D．detail( )62．A．decide B．detect C．produce D．improve ( )63．A．noticing B．examining C．overlooking D．explaining ( )64．A．modestly B．temporarily C．efficiently D．potentially ( )65．A．constant B．individual C．mote D．1ittleSection BDirection：Read the following passages. Each passage is followed by several questions or unfinished statements．For each of them there are four choices marked A．B．C. and D. Choose the one that fits best according to the information given in the passage you have just read(22分)(A)In the past，many disabled Chinese people led very restricted(受限制的)lives. They had little chance to find work or support themselves. But their situation has improved dramatically (引人注目地)over recent decades.In the past five years, the government has been stepping up efforts to help employ disabled people. Huge infestations (投资)have been made in training, and many disabled people have found jobs and become financially independent.When Wu Zihe was growing up, she was like a prisoner in her home. Having a job was something she could only dream about. But now she is famous for her art. Wu is working on a landscape. She said, “I can earn 30 or 40 yuan for paintings like this, if they are sold in groups. If I sell them separately, I can get about 100 yuan. It only takes me one day to finish one painting, so I can earn 1,000 to 2,000 yuan a month. ”Wu Zihe practices knife-drawing, which originated in her hometown Dunhua. If one has the eyes, the brain and hands, it takes about a year to learn the basic techniques.Dunhua government’s official Zhang Chunhua, said, “We think it suits disabled people very well. So the government spent 10 million yuan to set up this knife-painting training centre for them. ”Training in the center is free. Even the materials are provided by the government. Since last year, the center has trained over 200 disabled persons like Wu Zihe. They usually earn 500 to l,000 yuan per month. Sometimes they make as much as 5,000 yuan. That’s much higher than average salaries in the area．This year, the city has set up a center to sell these works. Art fans from more than ten countries and regions have bought paintings.( )66．How was life for the disabled people in the past?A．They had to stay at home and had little chance to find jobs．B．According to the law, they led very restricted lives．C．With the help of the government，they led a comfortable lifeD．They had to learn to paint to earn a living．( )67 From the passage we can learn that_______．A．the disabled had to earn a living by workingB．all the disabled have found jobs and become financially independent these years C．the disabled people’s living situation has improved greatly in recent yearsD．more and more disabled people will learn to paint( )68．From the last paragraph we can conclude thatA．Dunhua has set up a center to sell knife-drawing worksB．perhaps some of Wu Zihe’s paintings could be brought abroadC．only the disabled people’s paintings could be sold to foreignersD．people who like knife-drawing art have bought the disabled people’s paintings( )69．What would be the best title of the passage?A．A kind of folk art in China．B．Knife-drawing was first developed in DunhuaC．A knife—painting gaining center in Dunhua．D．China helps the disabled find jobs．(B)Food is life：it gives us the nourishment(营养)we need to stay alive and be healthy．Usually, we eat because we are hungry or need energy. Brian Wansink, a professor at the University of Illinois，says we also eat certain foods because they make us feel good, and remind us of happy memories. Wansink calls this kind of food comfort food. For some people, ice cream is a comfort food. For others, a bowl of noodles makes them feel good.How does a food become a comfort food? Professor Wansink believes that we connect food with important times, feelings and people in our lives. “When I was a child，my mother made a delicious soup: I loved it. Now, I often have this kind of food when 1 am tired or worried. And it helps me feel better,’’says one of Wansink’s coworkers.Do men and women choose different comfort foods? Wansink’s research at the University of Illinois says “yes”. In his study, the favorite comfort food for both men and women was ice cream. After this, men usually preferred hot, savory(香的)foods like soup or noodles. Women like sweet things such as chocolate and cookies. Men and women like to eat comfort foods when they are happy, but women eat these foods more when they are sad or worried．Not all comfort food is junk food．About 40 percent of the comfort foods in Wansink’s study were healthy main dishes or soups and vegetables. It shows, says Wansink, that a comfort food can taste good and be good for you.( )70．According to the passage，a kind of food that makes us feel good is called_____.A．fastfood B．healthy food C．Chinese food D comfort food ( )71．In Wansink’s s tudy, both men and women like to eat_________．A．soup B．ice cream C．noodles D．chocolates ( )72．The passage implies that__________.A．man probably eat more comfort foods than womenB．women usually eat comfort foods only when they are happyC．men usually eat comfort food when they are sadD．women probably eat comfort food more often than men(C)Suppose you work in a big company and find English very important for your job. Now you are looking for a place where you can improve your English，especially your spoken English.Here are some advertisements about English language training from newspapers. You may find the information you need．Global English Center◆General English in all four skills：listening, speaking, reading and writing◆3 months(700yuan), 6 months(1,200yuan)and one year(2,000yuan)◆Choice of morning or evening classes. 3 hours per day．Mon.—Fri．◆Experienced college English teachers◆Close to city center and bus stopsTel：67605272Add：105 Zhongshan Road, 100082Modern Language School◆Special coarse in English for business, travel, banking, hotel management◆Small classes(12一】6 students)on Sat．＆Sun．from 2：00—5：00 P．m．◆English teachers from Canada and USA◆Language lab and computers supplied◆3 months：I，050 yuan；6 months：1，850 ynan◆Write or phone：Modem Language School, 675 Park Road, 100056Tel：67353019The 21st Century English Training Center◆We offer morning or afternoon classes，both of which last three months and a half ata cost of 800 yuan．◆We also have a six-week TOEFL preparation class during winter and summer holidays．◆Entrance exams：June 1and Dec. 1◆Only 15-minute walk from city centerCall 67801642 for more informationThe International House of English◆Three／Six．month English courses for students at very low cost：60 yuan for 12 hours per week：convenient class hours：9：00—12：00 a．m．and 2：()0—5：00 p．m．◆A four-month evening program for developing speaking skills(same cost as day classes)◆Best foreign teachers experienced in teaching English as a second foreign language ◆Free sightseeing and social activities◆Very close to the Central ParkFor further information call 67432308．( )73．You work from 9：00 a．m，t04：30 P m．every day．You should choose______. A．Global English Center and the international HouseB．Global English Center and Modern Language SchoolC．Modern Language School and the 21st CenturyD．The 21st Century and the International House( )74．The 21st Century is different from the other three schools in thatA its teaching quality is better B．it is nearest to the city centerC．its courses are mole advanced D．it requires an entrance examination( )75．If you take the evening program at the International House, you will payabout_______.A 60 yuan B．240 yuan C．720 yuan D．1．000 yuan( )76. Yon will probably prefer to go to the International House because itA．offers free sightseeing and social activities B．has a special course in spoken English C．costs less than the other schools D．has foreign English teachers( )77．If your friend wants to take a course in English for hotel management．he may call at______ for class informationA．67605272 B 67353019 C．67801642 D．67432308Section CDirections：Read the passage carefully. Then answer the questions or complete the statements in the fewest possible words．Winter is a great time to experiment with new spots. The key is to find one that matches your interests and natural abilities. If you like to walk, keep walking—on snowshoes. If you want to try an endurance sport, go for cross-country skiing. Besides, snowboarding is just great fun．Not satisfied with these? Try downhill skiing, then. Downhill skiing is not as hard as it used to be—shorter, lightweight, curved skis make any beginner feel like an Olympic winner. These newer skis--along with another type of equipment called ski boards, which are even shorter than skis--help you control your speed and body movements．Consider testing the latest high-tech skis or snowboards? Check with your local sports shops or the rental places at the ski mountains about sample programs.You could also try sledding. Use a wood—framed sled with steel runners or a plastic sled to head down a snowy hill If you prefer ice to snow．think hockey or figure skating．Runners can also train during the winter in spite of wet or slippery roads. One of the easiest sports wound, snowshoeing can be excellent cold—weather cross—training for runners and cyclists—or anyone wanting to take a wintry walk in the woods. Snowshoes are smaller, lighter and better than ever. If you want to try them out, you may be able to rent a pair for a day at many of the larger outdoor or sporting goods stores.Whatever sport you choose，don’t rely on a friend for instruction. You wouldn’t let an inexperienced doctor perform a brain operation on you, but why let one teach you to ski or skate? That’s what instructors are for—help newcomers start out right．Instructors can give you advice about equipment, techniques, safety, and dealing with injuries if they do happen to you．Above all, if you want to progress, invest your time in learning the basic skills thoroughly. Everything else you do as a skier, boarder, or skater will be built on these first skills.(Note：Answer the questions or complete the statements in NO MORE THAN TEN WORDS．)78．How many different types of sports are mentioned in the first two paragraphs?______________________________________________________________________ 79．We learn from the passage that Downhill skiing used to be more difficult to learn because______________________________________________________________________ 80．The sentence “You wouldn’t let all inexperienced doctor perform a brain operation onyou but why let one teach you to ski or skate?”implies that it's_____________________________________________________________________ 81．Who are the most likely readers of the passage?______________________________________________________________________第Ⅱ卷(共47分)I．Translation．Directions：Translate the following sentences into English．Using the words given in the brackets.1．许多人并没意识到他们究竞消耗掉了多少食品和饮料。