广州市113中学生2011届高二会考最后冲刺之四

12011年广州市普通高中毕业班综合测试（二）数学（理科）一．选择题：本大题共8小题，每小题5分，满分40分．在每小题给出的四个选项中，只有一项是符合题目要求的． 1．复数i z a b =+(),a b ∈R 的虚部记作()Im z b =，则1Im 2i ⎛⎫=⎪+⎝⎭A ．13B ．25C ．13-D ．15-2．已知全集{}1,2,3,4,5,6,7U A B == ，(){}2,4,6U A B = ð，则集合B =A ．{}2,4,6B ．{}1,3,5C ．{}1,3,5,7D ．{}1,2,3,4,5,6,7 3．设随机变量ξ服从正态分布()3,4N ，若()()232P a P a ξξ<-=>+，则a 的值为A ．73B ．53C ．5D ．34．已知函数()()32120f x x ax xa a =++>，则()2f 的最小值为A． B ．16 C ．288a a++D ．1128a a++5．已知()1sin cos f x x x =+，()1n f x +是()n f x 的导函数，即()()21f x f x '=，()()32f x f x '=，…，()()1n n f x f x +'=，n ∈*N ，则()2011f x =A ．sin cos x x --B ．sin cos x x -C ．sin cos x x -+D ．sin cos x x + 6．一条光线沿直线220x y -+=入射到直线50x y +-=后反射，则反射光线所在的直线方程为A ．260x y +-=B ．270x y -+=C ．30x y -+=D ．290x y +-= 7．三个共面向量a 、b 、c 两两所成的角相等，且1=a ，2=b ，3=c ，则a +b +c 等于AB ．6 C． 6D ．3或68．正方形A B C D 的边长为2，点E 、F 分别在边A B 、B C 上，且1A E =，12B F =，将此正方形沿D E 、D F 折起，使点A 、C 重合于点P ，则三棱锥P D EF -的体积是A ．13B．6C．9D ．3二、填空题：本大题共7小题，考生作答6小题，每小题5分，满分30分． （一）必做题（9～13题）29．已知函数()sin 6f x x πω⎛⎫=+⎪⎝⎭()0ω>，若函数()f x 图象上的一个对称中心到对称轴的距离的最小值为3π，则ω的值为 ．10．已知函数()f x 是定义在R 上的偶函数，当0x ≤时，()32f x x x =-，则当0x >时，()f x 的解析式为 ．11．若1223211C 3C 3C 3C385n n n nn n n---+++++= ，则 n 的值为 ． 12．如图1为某质点在4秒钟内作直线运动时，速度函数()v v t =的图象，则该质点运动的总路程s = 厘米．13．将正整数12分解成两个正整数的乘积有112⨯，26⨯，34⨯三种，其中34⨯是这三种分解中，两数差的绝对值最小的，我们称34⨯为12的最佳分解．当()*,p q p q p q ⨯≤∈N且是正整数n 的最佳分解时，我们规定函数()p f n q=，例如()3124f =.关于函数()f n 有下列叙述：①()177f =，②()3248f =，③()4287f =，④()914416f =.其中正确的序号为 （填入所有正确的序号）．（二）选做题（14～15题，考生只能从中选做一题）14．（几何证明选讲选做题）在梯形A B C D 中，AD BC ，2AD =，5B C =，点E 、F 分别在A B 、C D 上，且EF AD ，若34A E E B=，则E F 的长为 ．15．（坐标系与参数方程选做题）设点A 的极坐标为2,6π⎛⎫⎪⎝⎭，直线l 过点A 且与极轴所成的角为3π，则直线l 的极坐标．．．方程为 ．三、解答题：本大题共6小题，满分80分．解答须写出文字说明、证明过程和演算步骤． 16．（本小题满分12分）如图2，渔船甲位于岛屿A 的南偏西60方向的B 处，且与岛屿A 相距12海里，渔船乙以10海里/小时的速度从岛屿A 出发沿正北方向航行，若渔船甲同时从B 处出发沿北偏东α的方向追赶渔船乙，刚好用2小时追上． （1）求渔船甲的速度； （2）求sin α的值．17．（本小题满分12分）某地区对12岁儿童瞬时记忆能力进行调查．瞬时记忆能力包括听觉记忆图160ABC东南西北 α3能力与视觉记忆能力.某班学生共有40人，下表为该班学生瞬时记忆能力的调查结果．例如表中听觉记忆能力为中等，且视觉记忆能力偏高的学生为3人．力为中等或中等以上的概率为25．（1）试确定a 、b 的值；（2）从40人中任意抽取3人，求其中至少有一位具有听觉记忆能力或视觉记忆能力超常的学生的概率；（3）从40人中任意抽取3人，设具有听觉记忆能力或视觉记忆能力偏高或超常的学生人数为ξ，求随机变量ξ的数学期望E ξ． 18．（本小题满分14分）一个几何体是由圆柱11AD D A 和三棱锥E A B C -组合而成，点A 、B 、C 在圆O 的圆周上，其正（主）视图、侧（左）视图的面积分别为10和12，如图3所示，其中EA ABC ⊥平面， AB AC ⊥，A B A C =，2A E =．（1）求证：A C B D ⊥；（2）求二面角A B D C --的平面角的大小．19．（本小题满分14分）已知数列{}n a 的前n 项和()12nn n a S +=，且11a =．（1）求数列｛a n ｝的通项公式；（2）令ln n n b a =，是否存在k （2,k k *≥∈N ），使得k b 、1k b +、2k b +成等比数列．若存在，求出所有符合条件的k 值；若不存在，请说明理由．AODE正（主）视图 E A侧（左）视图A 1 D 1A D 1A 1E BCOD 图3420．（本小题满分14分） 已知双曲线C ：()222210x y a b ab-=>>和圆O ：222x y b +=（其中原点O 为圆心），过双曲线C 上一点()00,P x y 引圆O 的两条切线，切点分别为A 、B ．（1）若双曲线C 上存在点P ，使得90APB ∠= ，求双曲线离心率e 的取值范围； （2）求直线A B 的方程；（3）求三角形O A B 面积的最大值．21．（本小题满分14分）已知函数()ln f x ax x x =+的图象在点e x =（e 为自然对数的底数）处的切线斜率为3． （1）求实数a 的值； （2）若k ∈Z ，且()1f x k x <-对任意1x >恒成立，求k 的最大值；（3）当4n m >≥时，证明()()mnn m m n nm >．参考答案一、选择题：二、填空题： 9．3210．()32f x x x =-- 11．4 12．11 13．①③ 14．23715．sin 13πρθ⎛⎫-=⎪⎝⎭或cos 16πρθ⎛⎫+= ⎪⎝⎭或4sin 13πρθ⎛⎫-= ⎪⎝⎭cos sin 20θρθ--=三、解答题：16．解：（1）依题意，120BAC ∠=，12AB =，10220A C =⨯=，B C A α∠=．在△ABC 中，由余弦定理，得2222cos BC AB AC AB AC BAC =+-⨯⨯∠60ABC东南西北 α522122021220cos120784=+-⨯⨯⨯=． 解得28B C =．所以渔船甲的速度为142B C =海里/小时．答：渔船甲的速度为14海里/小时．（2）方法1：在△ABC 中，因为12AB =，120BAC ∠= ，28B C =，B C A α∠=，由正弦定理，得sin sin 120A B B C α=．即12sin 1202sin 2814AB BCα⨯===． 答：sin α14方法2：在△ABC 中，因为12AB =，20AC =，28B C =，B C A α∠=，由余弦定理，得222cos 2AC BC ABAC BCα+-=⨯．即22220281213c o s 2202814α+-==⨯⨯．因为α为锐角，所以sin α===14． 答：sin α1417．解：（1）由表格数据可知，视觉记忆能力恰为中等，且听觉记忆能力为中等或中等以上的学生共有()10a +人．记“视觉记忆能力恰为中等，且听觉记忆能力为中等或中等以上”为事件A ，则102()405a P A +==，解得6a =．所以40(32)40382b a =-+=-=．答：a 的值为6，b 的值为2．（2）由表格数据可知，具有听觉记忆能力或视觉记忆能力超常的学生共有8人．方法1：记“至少有一位具有听觉记忆能力或视觉记忆能力超常的学生”为事件B ，则“没有一位具有听觉记忆能力或视觉记忆能力超常的学生”为事件B ，所以332340C 124123()1()11C247247P B P B =-=-=-=．答：从这40人中任意抽取3人，其中至少有一位具有听觉记忆能力或视觉记忆能力超常的学生的概率为123247．方法2：记“至少有一位具有听觉记忆能力或视觉记忆能力超常的学生”为事件B ， 所以()122138328328340C C C C C 123C247P B ++==．答：从这40人中任意抽取3人，其中至少有一位具有听觉记忆能力或视觉记忆能力超常的学生的概率为123247．6（3）由于从40位学生中任意抽取3位的结果数为340C ，其中具有听觉记忆能力或视觉记忆能力偏高或超常的学生共24人，从40位学生中任意抽取3位，其中恰有k 位具有听觉记忆能力或视觉记忆能力偏高或超常的结果数为32416C C k k -，所以从40位学生中任意抽取3位，其中恰有k 位具有听觉记忆能力或视觉记忆能力偏高或超常的概率为32416340C C ()Ck kP k ξ-==，()0,1,2,3k =．ξ的可能取值为0，1，2，3，因为032416340C C 14(0)C247P ξ===， 122416340C C 72(1)C247P ξ===，212416340C C 552(2)C 1235P ξ===，32416340C C 253(3)C 1235P ξ===． 所以ξ的分布列为0E ξ=⨯142471+⨯722472+⨯55212353+⨯25312352223912355==．答：随机变量ξ的数学期望为95．18．方法1：（1）证明：因为EA ABC ⊥平面，C A ABC ⊂平面，所以E A A C ⊥，即E D A C ⊥． 又因为AC AB ⊥，AB ED A = ，所以A C ⊥平面EBD ．因为BD EBD ⊂平面，所以A C B D ⊥． （2）解：因为点A 、B 、C 在圆O 的圆周上，且AB AC ⊥，所以B C 为圆O 的直径．设圆O 的半径为r ，圆柱高为h ，根据正（主）视图、侧（左）视图的面积可得12210,2122212.2rh r rh r ⎧+⨯=⎪⎪⎨⎪+⨯⨯=⎪⎩解得2,2.r h =⎧⎨=⎩所以4B C =，AB AC ==过点C 作C H BD ⊥于点H ，连接A H ，由（1）知，A C B D ⊥，AC CH C = ，所以B D ⊥平面A C H ．因为AH ⊂平面A C H ，所以BD AH ⊥．所以A H C ∠为二面角A B D C --的平面角．由（1）知，A C ⊥平面ABD ，AH ⊂平面ABD ，所以A C A H ⊥，即△C A H 为直角三角形．在R t △BAD 中，AB =2AD =，则BD ==AB AD BD AH ⨯=⨯，解得3AH =．因为tan A C AH C A H∠==A H C ∠60=．所以二面角A B D C --的平面角大小AD 1A 1EBCO D7为60 ．方法2：（1）证明：因为点A 、B 、C 在圆O 的圆周上，且AB AC ⊥，所以B C 为圆O 的直径． 设圆O 的半径为r ，圆柱高为h ，根据正（主）视图、侧（左）视图的面积可得， 12210,2122212.2rh r rh r ⎧+⨯=⎪⎪⎨⎪+⨯⨯=⎪⎩解得2,2.r h =⎧⎨=⎩所以4B C =，AB AC ==以点D 为原点，1DD 、D E 所在的射线分别为x 轴、z 轴建立如图的空间直角坐标系D xyz -，则()0,0,0D ，()14,0,0D ，()0,0,2A ，()2,2,2B ，()2,2,2C -，()2,2,0AC =- ，()2,2,2DB =．因为()()2,2,02,2,20A CD B =-=，所以A C D B⊥．所以A C B D ⊥．（2）解：设(),,x y z =n 是平面BC D 的法向量，因为()0,4,0BC =-，所以0,0.B C D B ⎧=⎪⎨=⎪⎩n n 即40,2220.y x y z -=⎧⎨++=⎩ 取1z =-，则()1,0,1=-n 是平面B C D 的一个法向量．由（1）知，A C B D ⊥，又AC AB ⊥，AB BD B = ，所以A C ⊥平面ABD ．所以()2,2,0AC =-是平面ABD 的一个法向量．因为1cos ,2ACAC AC⋅===⋅ n n n ，所以,60AC =n ．而,ACn 等于二面角A B D C --的平面角，所以二面角A B D C --的平面角大小为60 ．方法3：（1）证明：因为EA ABC ⊥平面，C A ABC ⊂平面，所以E A A C ⊥，即ED AC ⊥．又因为AC AB ⊥，AB ED A = ，所以A C ⊥平面EBD ．因为BD EBD ⊂平面，所以A C B D ⊥． （2）解：因为点A 、B 、C 在圆O 的圆周上，且AB AC ⊥，所以B C 为圆O 的直径． 设圆O 的半径为r ，圆柱高为h ，根据正（主）视图、侧（左）视图的面积可得， 12210,2122212.2rh r rh r ⎧+⨯=⎪⎪⎨⎪+⨯⨯=⎪⎩解得2,2.r h =⎧⎨=⎩ 所以4B C =，AB AC ==AD 1A 1E BCODAD 1A 1EBC O D8以点D 为原点，1DD 、D E 所在的射线分别为x 轴、z 轴建立如图的空间直角坐标系D xyz -，则()0,0,0D ，()14,0,0D ，()0,0,2A ，()2,2,2B ，()2,2,2C -，()0,4,0BC =- ，()2,2,2DB =设(),,x y z =n 是平面BC D 的法向量，则0,0.B C D B ⎧=⎪⎨=⎪⎩ n n即40,2220.y x y z -=⎧⎨++=⎩ 取1z =-，则()1,0,1=-n 是平面BC D 的一个法向量．由（1）知，A C B D ⊥，又AC AB ⊥，AB BD B = ，所以A C ⊥平面ABD ．所以()2,2,0AC =-是平面ABD 的一个法向量．因为1cos ,2ACAC AC⋅===⋅n n n ，所以,60AC =n ．而,ACn 等于二面角A B D C --的平面角，所以二面角A B D C --的平面角大小为60 ．19．解：（1）解法1：当2n ≥时，()11122nn n n n n a na a S S --+=-=-，即11n n a a nn -=-()2n ≥． 所以数列n a n ⎧⎫⎨⎬⎩⎭是首项为111a =的常数列．所以1n a n =，即n a n =()n ∈*N ．所以数列{}na 的通项公式为n a n =()n ∈*N ．解法2：当2n ≥时，()11122nn n n n n a na a S S --+=-=-， 即11n n a n a n -=-()2n ≥．所以1321122113211221n n n n n a a a a nn a a n a a a a n n ----=⨯⨯⨯⨯⨯=⨯⨯⨯⨯⨯=-- ．因为11a =，符合n a 的表达式．所以数列{}n a 的通项公式为n a n =()n ∈*N ．（2）假设存在k ()2,,k m k ≥∈*N ，使得k b 、1k b +、2k b +成等比数列，则2k k b b +=21k b +．因为ln ln n n b a n ==（n≥2），所以()()2222ln 2ln ln 2ln ln(2)22k k k k k k b b k k +⎡⎤+++⎡⎤⎢⎥=⋅+<=⎢⎥⎢⎥⎣⎦⎣⎦9()()22221ln 1ln 12k k k b +⎡⎤+<=+=⎡⎤⎢⎥⎣⎦⎢⎥⎣⎦．这与2k k b b +=21k b +矛盾． 故不存在k （2,k k *≥∈N ），使得k b 、1k b +、2k b +成等比数列．20．解：（1）因为0a b >>，所以1b a<，所以c e aa===<．由90APB ∠= 及圆的性质，可知四边形P A O B是正方形，所以OP =．因为OP a =≥，所以2b a≥，所以c e aa===2≥．故双曲线离心率e的取值范围为2⎡⎢⎣⎭． （2）方法1：因为22222200PA O P O A x y b =-=+-，所以以点P 为圆心，PA 为半径的圆P 的方程为()()222220000x x y y x y b -+-=+-．因为圆O 与圆P 两圆的公共弦所在的直线即为直线A B ， 所以联立方程组()()222222220000,.x y b x x y y x y b ⎧+=⎪⎨-+-=+-⎪⎩消去2x ，2y ，即得直线A B 的方程为200x x y y b +=． 方法2：设()11,A x y ()22,B x y ，已知点()00,P x y ，则P A k =0101y y x x --，11O A y k x =()101,0x x x ≠≠其中．因为P A O A ⊥，所以1PA O A k k =-，即0110111y y y x x x -⨯=--． 整理得22010111x x y y x y +=+．因为22211x y b +=，所以20101x x y y b +=．因为O A O B =，PA PB =，根据平面几何知识可知，A B O P ⊥．因为00O P y k x =，所以00AB x k y =-．所以直线A B 方程为()0110x y y x x y -=--．即0010x x y y x xy y +=+．所以直线A B 的方程为200x x y y b +=．方法3：设()()1122,,,A x y B x y ，已知点()00,P x y ， 则P A k =0101y y x x --，11O A y k x =()101,0x x x ≠≠其中．10因为P A O A ⊥，所以1PA O A k k =-，即0110111y y y x x x -⨯=--．整理得22010111x x y y x y +=+．因为22211x y b +=，所以20101x x y y b +=．这说明点A 在直线200x x y y b +=上． 同理点B 也在直线200x x y y b +=上．所以200x x y y b +=就是直线A B 的方程．（3）由（2）知，直线A B 的方程为200x x y y b +=，所以点O 到直线A B的距离为2bd =．因为A B ===，所以三角形O A B 的面积20012S AB d x y =⨯⨯=+以下给出求三角形O A B 的面积S 的三种方法： 方法1：因为点()00,P x y 在双曲线22221x y ab-=上，所以2200221x y ab-=，即22222002b x a by a-=()22xa≥．设t ==≥322b t S t b=+．因为()()()3222bt b t b S tb-+-'=+，所以当0t b <<时，0S '>，当t b >时，0S '<．所以322b t S t b=+在()0,b 上单调递增，在(),b +∞上单调递减．当b ≤，即b a <≤时，322212b b S b b b⨯==+最大值，当b >，即a >时，2S ab==+最大值综上可知，当b a <≤时，212S b =最大值；当a >时，S a=最大值．11方法2：设t =33222b t b S b t bt t==++．因为点()00,P x y 在双曲线22221x y ab-=上，即2200221x y ab-=，即22222002b x a by a-=()22xa≥．所以t ==≥令()2b g t t t=+，则()()()2221t b t b b g t tt+-'=-=．所以当0t b <<时，()0g t '<，当t b >时，()0g t '>．所以()2b g t t t=+在()0,b 上单调递减，在(),b +∞上单调递增．当b ≤，即b a <≤时，32212b S b bb b==+最大值，当b >，即a >时，322bbS a==最大值．综上可知，当b a <≤时，212S b =最大值；当a >时，S a=最大值．方法3：设2200t x y =+，则S bt==．因为点()00,P x y 在双曲线22221x y ab-=上，即2200221x y ab-=，即22222002b x a by a-=()22xa≥．所以22222200021b t x y x b a a ⎛⎫=+=+-≥ ⎪⎝⎭．令()2222221124g u b u u b u b b ⎛⎫=-+=--+ ⎪⎝⎭，所以()g u 在21,2b ⎛⎫-∞ ⎪⎝⎭上单调递增，在21,2b ⎛⎫+∞ ⎪⎝⎭上单调递减．因为t a ≥，所以2110,u t a ⎛⎤=∈ ⎥⎝⎦， 当22112b a≤，即b a <≤时，()22max1124g u g b b ⎛⎫==⎡⎤ ⎪⎣⎦⎝⎭，此时321122S b b b =⨯=最大值．当22112ba>，即a >时，()2224m ax1a bg u g a a -⎛⎫==⎡⎤ ⎪⎣⎦⎝⎭，此时S a=最大值．12综上可知，当b a <≤时，212S b =最大值；当a >时，2bS a=最大值．21．（1）解：因为()ln f x ax x x =+，所以()ln 1f x a x '=++．因为函数()ln f x ax x x =+的图像在点e x =处的切线斜率为3，所以()e 3f '=，即l n e 13a ++=．所以1a =．（2）解：由（1）知，()ln f x x x x =+，所以()1f x k x <-对任意1x >恒成立，即ln 1x x x k x +<-对任意1x >恒成立．令()ln 1x x x g x x +=-，则()()2ln 21x x g x x --'=-，令()ln 2h x x x =--()1x >，则()1110x h x xx-'=-=>，所以函数()h x 在()1,+∞上单调递增．因为()()31ln 30,422ln 20h h =-<=->，所以方程()0h x =在()1,+∞上存在唯一实根0x ，且满足()03,4x ∈．当01()0x x h x <<<时，，即()0g x '<，当0()0x x h x >>时，，即()0g x '>，所以函数()ln 1x x x g x x +=-在()01,x 上单调递减，在()0,x +∞上单调递增．所以()()()()()000000m in001ln 123,411x x x x g x g x x x x ++-====∈⎡⎤⎣⎦--．所以()()0min 3,4k g x x <=∈⎡⎤⎣⎦．故整数k 的最大值是3． （3）证明1：由（2）知，()ln 1x x x g x x +=-是[)4,+∞上的增函数，所以当4n m >≥时，l n l n 11n n n m m m n m ++>--．即()()()()11ln 11ln n m n m n m -+>-+．整理得 ()l n l n l n l n m n n m mm n mn nn m+>++-． 因为n m >， 所以ln ln ln ln m n n m m m n m n n +>+． 即ln ln ln ln mn m mn nn m m n +>+．即()()ln ln mnmmnnn mmn>． 所以()()mnn mm n nm>．证明2：构造函数()ln ln ln ln f x mx x m m mx m x x =+--，则()()1l n 1l n f x m x mm m '=-+--．因为4x m >≥，所以()()1ln 1ln 1ln 0f x m m m m m m m '>-+--=-->．所以函数()f x 在[),m +∞13上单调递增．因为n m >， 所以()()f n f m >．所以ln ln ln ln m n n m m m n m n n +-->22ln ln ln ln 0m m m m m m m m +--=． 即ln ln ln ln m n n m m m n m n n +>+． 即ln ln ln ln mn m mn n n m m n +>+． 即()()ln ln mn m mn n n m m n >．所以()()mnn m m n nm >．。

数学（文科）答案A 第 1 页 共 9 页2011年广州市普通高中毕业班综合测试（二）数学（文科）试题参考答案及评分标准说明：1．参考答案与评分标准指出了每道题要考查的主要知识和能力，并给出了一种或几种解法供参考，如果考生的解法与参考答案不同，可根据试题主要考查的知识点和能力比照评分标准给以相应的分数．2．对解答题中的计算题，当考生的解答在某一步出现错误时，如果后继部分的解答未改变该题的内容和难度，可视影响的程度决定后继部分的得分，但所给分数不得超过该部分正确解答应得分数的一半；如果后继部分的解答有较严重的错误，就不再给分．3．解答右端所注分数，表示考生正确做到这一步应得的累加分数．4．只给整数分数，选择题和填空题不给中间分． 一、选择题：本大题主要考查基本知识和基本运算．共10小题，每小题5分，满分50分．二、填空题：本大题主要考查基本知识和基本运算．本大题共5小题，考生作答4小题，每小题5分，满分20分．其中14～15题是选做题，考生只能选做一题． 11．3 12．2 13．①③ 14．23715．sin 13πρθ⎛⎫-=⎪⎝⎭或cos 16πρθ⎛⎫+= ⎪⎝⎭或4sin 13πρθ⎛⎫-= ⎪⎝⎭cos sin 20θρθ--= 三、解答题：本大题共6小题，满分80分.解答须写出文字说明、证明过程和演算步骤. 16．（本小题满分12分）（本小题主要考查概率与统计的概念，考查运算求解能力等．）解：（1）由表格数据可知视觉记忆能力恰为中等，且听觉记忆能力为中等或中等以上的学生有()10a +人． 记“视觉记忆能力恰为中等，且听觉记忆能力为中等或中等以上”为事件A ，则102()405a P A +==， …………………………………………………………………………………4分 解得6a =． ………………………………………………………………………………………………5分因为3240a b ++=，所以2b =．答：a 的值为6，b 的值为2．……………………………………………………………………………7分 （2）由表格数据可知，听觉记忆能力恰为中等，且视觉记忆能力为中等或中等以上的学生有()11b +人，由（1）知，2b =，即听觉记忆能力恰为中等，且视觉记忆能力为中等或中等以上的学生共有13人．………………………9分记“听觉记忆能力恰为中等，且视觉记忆能力为中等或中等以上”为事件B ， 则()11134040b P B +==． 答：听觉记忆能力恰为中等，且视觉记忆能力为中等或中等以上的概率为1340．…………………12分数学（文科）答案A 第 2 页 共 9 页（本小题主要考查方位角、正弦定理、余弦定理等基础知识，考查运算求解能力等．）解：（1）依题意，120BAC ∠= ，12AB =，10220AC =⨯=，BCA α∠=．………………………2分在△ABC 中，由余弦定理，得2222cos BC AB AC AB AC BAC =+-⨯⨯∠ ……………………4分22122021220cos120784=+-⨯⨯⨯=．解得28BC =．………………………………………………………6分所以渔船甲的速度为142BC=海里/小时． 答：渔船甲的速度为14海里/小时．…………………………………7分（2）方法1：在△ABC 中，因为12AB =，120BAC ∠=，28BC =，BCA α∠=，由正弦定理，得sin sin120AB BCα=．……………………………………………………………………9分即12sin1202sin 2814AB BCα===答：sin α12分 方法2：在△ABC 中，因为12AB =，20AC =，28BC =，BCA α∠=，由余弦定理，得222cos 2AC BC AB AC BC α+-=⨯．…………………………………………………………9分即22220281213cos 2202814α+-==⨯⨯． 因为α为锐角，所以sin α===．答：sin α的值为14．………………………………………………………………………………12分60ABC东南西北 α数学（文科）答案A 第 3 页 共 9 页（本小题主要考查等差数列、等比数列、不等式等基础知识，考查方程思想以及运算求解能力．） 解：（1）设等差数列{}n a 的公差为d ，则()112n n n S na d -=+．………………………………………1分 由已知，得111091055,2201920210.2a d a d ⨯⎧+=⎪⎪⎨⨯⎪+=⎪⎩………………………………………………………………………3分 即112911,21921.a d a d +=⎧⎨+=⎩解得11,1.a d =⎧⎨=⎩…………………………………………………………………………5分所以1(1)n a a n d n =+-=（n *∈N ）．………………………………………………………………6分 （2）假设存在m 、k ()2,,k m m k >≥∈N ，使得1b 、m b 、k b 成等比数列，则21m k b bb =．……………………………………………………………………………………………7分 因为11n n n a nb a n +==+，…………………………………………………………………………………8分 所以11,,211m k m k b b b m k ===++． 所以21121m k m k ⎛⎫=⨯ ⎪++⎝⎭．……………………………………………………………………………9分 整理，得22221m k m m =-++．…………………………………………………………………………10分 以下给出求m ，k 的三种方法：方法1：因为0k >，所以2210m m -++>．………………………………………………………11分解得11m <<12分 因为2,m m ≥∈*N ， 所以2m =，此时8k =．故存在2m =、8k =，使得1b 、m b 、k b 成等比数列．……………………………………………14分方法2：因为k m >，所以22221m k m m m =>-++．…………………………………………………11分数学（文科）答案A 第 4 页 共 9 页即221021m m m +<--，即221021m m m -<--．解得11m -<<11m <<………………………………………………………………12分 因为2,m m ≥∈*N ， 所以2m =，此时8k =．故存在2m =、8k =，使得1b 、m b 、k b 成等比数列．……………………………………………14分方法3：因为2k m >≥，所以222221m k m m =>-++．……………………………………………11分 即221021m m m +<--，即22221021m m m m --<--．解得1m <<1m <<12分 因为2,m m ≥∈*N ， 所以2m =，此时8k =．故存在2m =、8k =，使得1b 、m b 、k b 成等比数列．……………………………………………14分19．（本小题满分14分）（本小题主要考查锥体体积，空间线线、线面关系，三视图等知识，考查化归与转化的数学思想方法，以及空间想象能力、推理论证能力和运算求解能力．）（1）证明：因为EA ABC ⊥平面，C A ABC ⊂平面，所以EA AC ⊥，即ED AC ⊥．又因为AC AB ⊥，AB ED A = ，所以AC ⊥平面EBD ．因为BD EBD ⊂平面，所以AC BD ⊥．………………………………………………………………4分 （2）解：因为点A 、B 、C 在圆O 的圆周上，且AB AC ⊥，所以BC 为圆O 的直径．设圆O 的半径为r ，圆柱高为h ，根据正（主）视图、侧（左）视图的面积可得，12210,2122212.2rh r rh r ⎧+⨯=⎪⎪⎨⎪+⨯⨯=⎪⎩…………………………………………6分 解得2,2.r h =⎧⎨=⎩所以4BC =，AB AC ==………………………………………………………………………8分AD 1A 1EBCOD数学（文科）答案A 第 5 页 共 9 页以下给出求三棱锥E BCD -体积的两种方法： 方法1：由（1）知，AC ⊥平面EBD ，所以13E BCD C EBD EBD V V S CA --∆==⨯．………………………………………………………………10分 因为EA ABC ⊥平面，AB ABC ⊂平面， 所以EA AB ⊥，即ED AB ⊥．其中224ED EA DA =+=+=，因为AB AC ⊥，AB AC ==，所以11422EBD S ED AB ∆=⨯⨯=⨯⨯=．…………………………………………………13分所以11633E BCD V -=⨯=．…………………………………………………………………14分方法2：因为EA ABC ⊥平面，所以111333E BCD E ABC D ABC ABC ABC ABC V V V S EA S DA S ED ---∆∆∆=+=⨯+⨯=⨯．…………………10分其中224ED EA DA =+=+=，因为AB AC ⊥，AB AC ==，所以11422ABC S AC AB ∆=⨯⨯=⨯=．…………………………………………………13分 所以1164433E BCDV -=⨯⨯=．…………………………………………………………………………14分20．（本小题满分14分）（本小题主要考查分段函数、导数、函数的单调性和最值等基础知识，考查分类讨论思想，以及运算求解能力和推理论证能力等．） 解：（1）因为函数()2f x x =的定义域(),F =-∞+∞，函数()lng x a x =的定义域()0,G =+∞，所以()22ln ,0,,0.x a x x h x x x ⎧+>⎪=⎨⎪⎩≤……………………………………………………………………4分（2）当0x ≤时，函数()2h x x =单调递减，所以函数()h x 在(],0-∞上的最小值为()00h =．……………………………………………………5分当0x >时，()2ln h x x a x =+．若0a =，函数()2h x x =在()0,+∞上单调递增．此时，函数()h x 不存在最小值．……………6分若0a >，因为()2220a x ah x x x x+'=+=>，………………………………………………………7分 所以函数()2ln h x x a x =+在()0,+∞上单调递增．此时，函数()h x 不存在最小值．……………8分数学（文科）答案A 第 6 页 共 9 页若0a <，因为()222x x x a h x x x⎛+ +⎝⎭⎝⎭'==，……………………………………9分 所以函数()2ln h x x a x =+在⎛ ⎝上单调递减，在⎫+∞⎪⎪⎭上单调递增．此时，函数()h x的最小值为h ．…………………………………………………………………………………10分因为ln 1ln 222222a a a a a a h a ⎡⎤⎛⎫⎛⎫=-+=-+-=--- ⎪ ⎪⎢⎥⎝⎭⎝⎭⎣⎦，………………………11分 所以当2e 0a -<≤时，0h ≥，当2e a <-时，0h <．…………………………13分综上可知，当0a >时，函数()h x 没有最小值；当2e 0a -≤≤时，函数()h x 的最小值为()00h =；当2e a <-时，函数()h x的最小值为1ln 22a a h ⎡⎤⎛⎫=--- ⎪⎢⎥⎝⎭⎣⎦．……………………………14分21．（本小题满分14分）（本小题主要考查圆、双曲线、直线方程和不等式等基础知识，考查运算求解能力和推理论证能力，以及分类讨论思想与创新意识等．）解：（1）因为0a b >>，所以1b a <，所以c e a===<1分 由90APB ∠=及圆的性质，可知四边形PAOB是正方形，所以OP =．因为OP a =≥，所以2b a ≥，所以c e a ===2≥．……………3分故双曲线离心率e的取值范围为⎣．…………………………………………………………4分（2）方法1：因为22222200PA OP OA x y b =-=+-，所以以点P 为圆心，PA 为半径的圆P 的方程为()()222220000x x y y x y b -+-=+-．………5分因为圆O 与圆P 两圆的公共弦所在的直线即为直线AB ，……………………………………………6分所以联立方程组()()222222220000,.x y b x x y y x y b ⎧+=⎪⎨-+-=+-⎪⎩………………………………………………7分数学（文科）答案A 第 7 页 共 9 页消去2x ，2y ，即得直线AB 的方程为200x x y y b +=．………………………………………………8分方法2：设()11,A x y ()22,B x y ，已知点()00,P x y ， 则PA k =0101y y x x --，11OA yk x =()101,0x x x ≠≠其中．因为PA OA ⊥，所以1PA OA k k =-，即0110111y y y x x x -⨯=--．…………………………………………5分整理得22010111x x y y x y +=+．因为22211x y b +=，所以20101x x y y b +=．……………………………………………………………6分 因为OA OB =，PA PB =，根据平面几何知识可知，AB OP ⊥． 因为00OP y k x =，所以00AB xk y =-．………………………………………………………………………7分 所以直线AB 方程为()0110x y y x x y -=--． 即000101x x y y x x y y +=+．所以直线AB 的方程为200x x y y b +=．………………………………………………………………8分 方法3：设()()1122,,,A x y B x y ，已知点()00,P x y ， 则PA k =0101y y x x --，11OA yk x =()101,0x x x ≠≠其中．因为PA OA ⊥，所以1PA OA k k =-，即0110111y y y x x x -⨯=--．…………………………………………5分整理得22010111x x y y x y +=+．因为22211x y b +=，所以20101x x y y b +=．……6分这说明点A 在直线200x x y y b +=上． …………7分同理点B 也在直线200x x y y b +=上．所以200x x y y b +=就是直线AB 的方程． ……8分 （3）由（2）知，直线AB 的方程为200x x y y b +=，所以点O 到直线AB 的距离为2d =．因为AB===，所以三角形OAB的面积0012S AB d=⨯⨯=……………………………………10分以下给出求三角形OAB的面积S的三种方法：方法1：因为点()00,P x y在双曲线22221x ya b-=上，所以2200221x ya b-=，即22222002b x a bya-=()22x a≥．设t==≥所以322b tSt b=+．………………………………………………………………………………………11分因为()()()3222b t b t bSt b-+-'=+，所以当0t b<<时，0S'>，当t b>时，0S'<．所以322b tSt b=+在()0,b上单调递增，在(),b+∞上单调递减．……………………………………12分b≤，即b a<≤时，322212b bS bb b⨯==+最大值，…………………………………13分b>，即a>时，()3222b bSab==+最大值综上可知，当b a<≤时，212S b=最大值；当a>时，S=最大值．………14分方法2：设t=33222b t bSbt btt==++．…………………………………………11分因为点()00,P x y在双曲线22221x ya b-=上，即2200221x ya b-=，即22222002b x a bya-=()22x a≥．所以t==数学（文科）答案A 第 8 页共 9 页数学（文科）答案A 第 9 页 共 9 页令()2b g t t t =+，则()()()2221t b t b b g t t t +-'=-=． 所以当0t b <<时，()0g t '<，当t b >时，()0g t '>．所以()2b g t t t=+在()0,b 上单调递减，在(),b +∞上单调递增．…………………………………12分b ≤，即b a <≤时，32212b S b b b b==+最大值，……………………………………13分b >，即a >时，32b S ==最大值．综上可知，当b a <≤时，212S b =最大值；当a >时，2b S a=最大值．………14分 方法3：设2200t x y =+，则S b ==11分 因为点()00,P x y 在双曲线22221x y a b -=上，即2200221x y a b-=，即22222002b x a b y a -=()220x a ≥． 所以22222200021b t x y x b a a ⎛⎫=+=+-≥ ⎪⎝⎭．令()2222221124g u b u u b u b b ⎛⎫=-+=--+ ⎪⎝⎭，所以()g u 在21,2b ⎛⎫-∞ ⎪⎝⎭上单调递增，在21,2b ⎛⎫+∞ ⎪⎝⎭上单调递减．………………………………12分 因为t a ≥，所以2110,u t a ⎛⎤=∈ ⎥⎝⎦， 当22112b a ≤，即b a <≤时，()22max 1124g u g b b⎛⎫==⎡⎤ ⎪⎣⎦⎝⎭，此时321122S b b b =⨯=最大值． ………………………………13分当22112b a >，即a >时，()2224max 1a b g u g a a -⎛⎫==⎡⎤ ⎪⎣⎦⎝⎭，此时2b S a =最大值．综上可知，当b a <≤时，212S b =最大值；当a >时，S =最大值．………14分。

2011年高考语文最后冲锋抢分高效模拟卷5本试卷共8页，满分为150分。

考试时间150分钟。

注意事项：1．答卷前，考生务必将答题卷座位号及密封线的项目填写清楚。

2．选择题每小题选出答案后，用2B铅笔把答题卷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；答案不能答在试题上。

3．非选择题必须用黑色字迹钢笔或签字笔作答，答案必须写在答题卷各题目指定区域的相应位置上；如需改动，先划掉原来的答案，然后再写上新的答案；不准使用铅笔或涂改液。

不按以上要求作答的答案无效。

4．考生必须保持答题卷的整洁，考试结束后，将本试题和答题卷一并交回。

一、本大题4小题，每小题3分，共12分。

1．下列词语中加点的字，每对读音都不相同的一组是（）A．休憩．/休戚．沤．肥/讴．歌倒．戈/颠倒．B．掸．子/胆．子沉湎．/缅．怀吐．露/吐．血C．擎．天/晴．天湖滨．/摈．弃炮．仗/炮．弹D．噎．气/咽．气蜿．蜒/手腕．散．居/分散．2．下列各句中，加点的熟语使用恰当的一项是（）A．为了建设和谐社会，城市学校应主动和乡镇学校结对子，鼎力相助．．．．薄弱学校。

B．私立学校虽然缺乏教学管理经验，但可以向公办学校学习，可以在亦步亦趋．．．．的基础上，渐渐走出自己的路来。

C．现在是高考冲刺的黄金时段，请利用好每一分钟，等高考结束再做事后诸亮．．．．，悔之晚矣。

D．研究生导师配备互选，既加深了导师和学员之间的相互了解，避免了“乱点鸳鸯谱．．．．．”，又极激发了研究生导师的荣誉感和责任心。

3．下列各句中没有语病的一句是（）A．上网的诱惑无法令现代人拒绝，而昂贵的网上消费使网迷们心痛不已。

B．贪官黄利发在任财政局长的半年时间里，利用正当或不正当的手段，贪污了20万元。

C．在中央领导视察火车站的时候，承诺将在三天开通北上的列车，让老百姓尽早回家过年。

D．在教育问题上，网民的留言多集中在农村教育上，包括办学经费严重不足、教师待遇太低、人才流失严重、地方政府不执行中央编制政策等。

W-E-A-L-T-H 系列“Revision is the mother of success”, which was taught by my excellent English teacher when I was in Senior 2. I have practiced it and also taught my students like this for years. The experience shows that it really works. Catherine 广州市113中学生2011届高二会考最后冲刺之九完型填空专题Class___________No. _________ Name__________ Score ___________(1)Most parents, I suppose, have had the experience of reading a bedtime story to their children.And they must have 1 how difficult it is to write a 2 children's book.Either the author has aimed too 3 , so that the children can't follow what is in his (or more often, her) story, 4 the story seems to be talking to the readers.The best children's books are 5 very difficult nor very simple, and satisfy both the 6 who hears the story and the adult who 7 it.Unfortunately, there are in fact 8 books like this, 9 the problem of finding the right bedtime story is not 10 to solve.This may be why many of books regarded as 11 of children's literature were in fact written for 12 .“Alice's Adventure in Wonderland”is perhaps the most 13 of this.Children, left for themselves, often 14 the worst possible interest in literature.Just leave a child in bookshop or .library and he will 15 willingly choose the books written in an imaginative way, or have a look at most children's comics, full of the stories and jokes which are the .objections of teachers and right-thinking parents.Perhaps we parents should stop trying to brainwash childrensintos accepting our taste in literature.After all children and adults are so different that we parents should not expect that they will enjoy the same books.So I suppose we'll just have to compromise over that bedtime story.1.A.hoped B.realized C.told D.said2.A.short B.long C.bad D.good3.A.easy B.short C.high D.difficult4.A.and B.but C.or D.so5.A.both B.neither C.either D.very6.A.child B.father C.mother D.teacher.7.A.hears B.buys C.understands D.reads8.A.few B.many C.a great deal of D.a great number of9.A.but B.however C.so D.because10 A.hard B.easy C.enough D.fast11.A.articles B.work C.arts D.works12.A.grown ups B.girls C.boys D.children13.A.difficult B.hidden C.obvious D.easy14.A.are B.show C.find D.add15.A.more B.less C.able D.be(2)More and more students want to study in“hot”majors. 1 a result, many students want to 2 their interests and study in these 3 such as foreign languages, international business and law, etc.Fewer and fewer students choose scientific majors, 4 maths, physics and biology, and art majors, 5 history, Chinese and philosophy.6 students can study in these“hot”majors, because the number of these“hot”majors7 limited.If one 8 interest in his work or study, 9 can he do well? I 10 this from one of my classmates.He is 11 the countryside.His parents are farmers.Though he 12 biology, he chose“international business”.He 13 to live a life which is different 14 of his parents.In the end, he found he was not interested in doing business.He found all the subjects to be tiresome. 15 this wouldn't have happened if he had chosen his major according to his own interests.Choosing a major in university does not decide one's whole life.Majors which are not“hot”today may become the“hot”major of tomorrow.Choosing your major according to your own interests is the best way to succeed.1.A.Being B.For C.Having D.As2.A.give up B.appear C.give D.master3.A.place B.room C.areas D.space4.A.for example B.such as C.and so on D.as a result5.A.even B.like C.just D.or6.A.Only a few B.Quite a few C.Perhaps D.Many7.A.is B.are C.would be D.have been8.A.had no B.had C.has no D.has9.A.why B.and what C.how D.and how10.A.suggested B.guessed C.searched D.learned11.A.out of B.off C.in D.from12.A.studied B.likes C.learns D.succeeds to study13.A.wants B.doesn’t want C.enjoys D.doesn’t like14.A.from which B.from that C.for which D.for that15.A.So B.Then C.Just then D.Maybe( 3 )Shopping habits in the United States have changed greatly in the last quarter of the 20th century. 1 in the 1900s most American towns and cities had a Main Street. Main Street was always in the heart of a town. This street was 2 on both sides with many 3 businesses. Here, shoppers walked into stores to look at all sorts of merchandise: clothing, furniture, hardware, groceries. 4 ,some shops offered services.These shops included drugstores, restaurants, shoe-repair stores, and barber or hairdressing shops. 5 in the 1950s, a change began to 6 .Too many automobiles had crowded into Main Street 7 too few parking places were 8 shoppers. Because the streets were crowded, merchants began to look with interest at the open spaces outside the city limits. Open space is what their car-driving customers needed.And open space is what they got when the first shopping centre was built. Shopping centres, or rather malls, 9 as a collection of small new stores 10 crowded city centres. 11 by hundreds of free parking space, customers were drawn away from downtown areas to outlying malls. And the growing popularity of shopping centres led 12 to the building of bigger and better stocked stores. 13 the late 1970s,many shopping malls had almost developed into small cities themselves. In addition to providing the 14 of one stop shopping, malls were transformed into landscaped parks, 15 benches, fountains, and outdoor entertainment.1.A.As early as B.Early C.Early as D.Earlier2.A.built B.designed C.intented D.lined3.A.varied B.various C.sorted D.mixed up4.A.Apart from B.However C.In addition D.As well5.A.suddenly B.Abruptly C.Contrarily D.But6.A.be taking place B.take place C.be taken place D.havetaken place7.A.while B.yet C.though D.and then8.A.available for B.available to ed by D.ready for9.A.started B.founded C.set up anized10.A.out of B.away from C.next to D.near11.A.Attracted B.Surprised C.Delighted D.Enjoyed12.A.on B.in turn C.by turns D.further13.A.By B.During C.In D.Towards14.A.cheapness B.readiness C.convenience D.handiness15.A.because of B.and C.with D.provided( 4 )For many people today, reading is no longer relaxation. To keep up their work they must read letters, reports, trade publications, interoffice communications, not to mention newspapers and magazines: a never-ending flood of words. In 1 a job or advancing in one, the ability to read and comprehend 2 can mean the difference between success and failure. Yet the unfortunate fact is that most of us are 3 readers. Most of us develop poor reading 4 at an early age, and never get over them. The main deficiency 5 in the actual stuff of language itself-words. Taken individually, words have 6 meaning until they are strung together into phrased, sentences and paragraphs. 7 , however, the untrained reader does not read groups of words. He laboriously reads one word at a time, often regressing to 8 words or passages. Regression, the tendency to look back over 9 you have just read, is a common bad habit in reading. Another habit which 10 down the speed of reading is vocalization—sounding each word either orally or mentally as 11 reads.To overcome these bad habits, some reading clinics use a device called an accelerator, which moves a bar (or curtain) down the page at a predetermined speed. The bar is set at a slightly faster rate 12 the reader finds comfortable, in order to “stretch” him.The accelerator forces the reader to read fast, 13word-by-word reading, regression and subvocalization, practically impossible. At first comprehension is sacrificed for speed. But when you learn to read ideas and concepts, you will not only read faster, 14your comprehension will improve. Many people have found their reading skill drastically improved after some training.15Charlce Au, a business manager, for instance, his reading rate was a reasonably good 172 words a minute before the training, now it is an excellent 1,378 words a minute. He is delighted that how he can get through a lot more reading material in a short period of time.1.A.applying B.doing C.offering D.getting2.A.quickly B.easily C.roughly D.decidedly3.A.good B.curious C.poor D.urgent4.A.training B.habits C.situations D.custom5.A.lies bines C.touches D.involves6.A.some B.A lot C.little D.dull7.A.Fortunately B.In fact C.Logically D.Unfortunately8.A.reuse B.reread C.rewrite D.recite9.A.what B.which C.that D.if10.A.scales B.cuts C.slows D.measures11.A.some one B.one C.he D.reader12.A.then B.as C.beyond D.than13A.enabling B.leading C.making D.indicating14A.but B.nor C.or D.for15.A.Look at B.Take C.Make D.Consider( 5 )There are about fifteen hundred languages in the world.But ___1___ a few of them are very ___2___. English is one of these. Many, many people use it, not only in England and the U. S. A, but in other parts of the world. About 200, 000, 000 speak it as their own language. It is difficult to say how many people are learning it as a ___3___ language. Many millions are ___4___ to do so.Is it easy or difficult to learn English? Different people may have different ___5___. Have you ever ___6___ the ads of this kind in the newspapers or magazines?“Learn English in six month, or your ___7___ back ...”“Easy and funny? Our records and tapes ___8___ you master your English in a month. ___9___ the first day your ___10___ will be excellent. Just send ...”Of course, it never ___11___ quite like this.The only language that seems easy to learn is the mother tongue. We should ___12___ that we all learned our own language well when we were ___13___. If we could learn English in the same way, it would not seemso difficult. ___14___ what a small child does. He listens to what people say. He tries what he hears. When he is using the language, talking in it, and ___15___ in it all the time, just imagine how much practice that gets!So it is hard to say that learning English is easy, because a good command of English depends upon a lot of practice. And practice needs great effort and takes much time. Good teachers, records, tapes, books, and dictionaries will help. But they cannot do the student’s work for him.1. A. not B. quite C. only D. very2. A. difficult B. important C. necessary D. easy3. A. native B. foreign C. useful D. mother4. A. learning B. enjoying C. trying D. liking5. A. questions B. problems C. ideas D. answers6. A. found B. watched C. noticed D. known7. A. knowledge B. time C. money D. English8. A. make B. help C. let D. allow9. A. From B. On C. Since D. After10.A. spelling B. grammar C. English D. pronunciation11.A. happened B. know C. seemed D. felt12. A. know B. remember C. understand D. think13. A. students B. children C. babies D. grown-ups14. A. Imagine B. Mind C. Do D. Think of15. A. using B. thinking C. trying D. practicing（ 6 ）Jenna, a popular girl from Westwood Middle School, had graduated first in her class and was ready for new ___1___ in high school.___2___, high school was different. In the first week, Jenna went to tryouts (选拔赛) for cheerleaders (拉拉队队员). She was competing against very talented girls, and she knew it would be ___3___ for her to be selected. Two hours later, the judge read a list of the girls for a second tryout. Her heart ___4___ as the list ended without her name. Feeling awful, she walked home carrying her schoolbag full of homework.Arriving home, she started with math. She had always been a good math student, but now she was ___5___. She moved on to English and history, and was relieved to find that she didn’t have any trouble with those subjects. Feeling better, she decided not to ___6___ math for the time being.The next day, Jenna went to see Mrs. Biden about being on the school ___7___. Mrs. Biden wasn’t as ___8___ as Jenna. “I am sorry, but we have enough ___9___ for the newspaper already. Come back next year and we’ll talk then.” Jenna smiled weakly and left. “ Why is high school so ___10___?” She sighed.Later in ___11___ class, Jenna devoted herself to figuring out theproblems that had given her so much ___12___. By the end of class, she understood how to get them right. As she gathered her books, Jenna decidedshe’d continue to try to ___13___ at her new school. She wasn’t sureif she’d succeed, but she knew she had to ___14___. High school was justas her mother had said: “ You will feel like a small fish in a big pond___15___ a big fish in a small pond. The challenge is to become the bestfish you can be.”1. A. processes B. decisions C. challenges D. exercises2. A. Therefore B. However C. Otherwise D. Besides3. A. difficult B. easy C. boring D. interesting4. A. jumped B. sank C. stopped D. raced5. A. struggling B. improving C. working D. complaining6. A. put up B. prepare for C. worry about D. give up7. A. committee B. newspaper C. radio D. team8. A. enthusiastic B. artistic C. sympathetic D. realistic9. A. speakers B. readers C. cheerleaders D. writers10. A. similar B. ordinary C. different D. familiar11. A. physics B. history C. English D. math12. A. pleasure B. hope C. trouble D. sorrow13. A. fit in B. look out C. stay up D. get around14. A. swim B. try C. ask D. escape15. A. in return for B. in case of C. in terms of D. instead of( 7 )Leonardo da Vinci began painting the Mona Lisa in 1503．He wasworking 1 a special painting for a church at that time, 2 the churchpainting was not 3 well. An Italian businessman asked da Vinci to painta picture of his second 4 . This is the woman who 5 be seen in theMona Lisa.All in all, the Mona Lisa is a very good example of da Vinci’s 6 andit satisfied the husband．Da Vinci used 7 and light in a clever 8 inthe painting. Da Vinci loved science and 9 . Right away a person cansee that there is a lot of geometry(几何形状) in the Mona LisA．The faceof the Mona Lisa is made of many circles and 10 shapes like 11 . Evenher 12 can be seen as a small part of a large circle. The woman inthe 13 is sitting on a balcony(阳台), and 14 can be seen behind her.Da Vinci loved to study rocks, so these can be seen 15 in his otherpaintings. The woman is sitting with her knees to the side. Her headis turned to look out of the painting. Her hands are held together infront of her. This way of sitting is now used by many models when theyare being painted. The Mona Lisa is a remarkable master.1．A．up B．in C．on D．about2．A．but B．thus C．however D．so3．A．doing B．going C．making D．working4．A．servant B．daughter C．nurse D．wife5．A．must B．should C．might D．can6．A．works B．jobs C．novels D．photos 7．A．heaviness B．black C．darkness D．oils8．A．way B．picture C．hand D．eye 9．A．chemistry B．maths C．geography D．biology 10．A．square B．round C．long D．egg11．A．balls B．sticks C．vases D．boxes12．A．smile B．shout C．cry D．anger13．A．church B．painting C．sofa D．house14．A．trees B．buildings C．mountains D．flowers15．A．by and by B．here and there C．over and over D．up and down。

2011年广州市高二学业水平考试英语第一节完形填空（共15小题，每小题2分，满分30分）阅读下面短文，掌握其大意，然后从16~30各题所给的A、B、C、D中，选出最佳选项，并在答题卡上将该项涂黑。

St. Petersburg is a great city to visit all year round. But it is ___16___ beautiful in the summer when you can walk all night long, because, although the sun ___17___, it never gets dark. It‟s a city that couldn‟t be more ___18___ from Moscow.St. Petersburg was originally built as a ___19___ city, modeled on French and German cities, the opposite of Moscow which followed the ___20___ Russian design. Differences can be seen in the architecture, fashion and even___21___. In the 18th century, French was more ___22___ than Russian among rich people in St. Petersburg and many French words are still used by locals today. Just as was ___23___ by the city‟s original designers, St. Petersburg is Russia‟s window to Europe.Life in Moscow is ___24___ than life in the more relaxed St. Petersburg. In Petersburg you never need to ___25___. It‟s much smaller than Moscow so people like walking everywhere, even during t he ___26___ dark winter.People in St. Petersburg are more ___27___ than Moscow locals too. There is an old joke about this difference. One day, a young man ___28___ his seat to an old lady on a crowed Moscow train. The old lady said, “You must be from St.Petersburg.” “Yes. How did you know?” replied the man ___29___. The old lady said, “Moscow locals never give their seats.” The young man then said with a ___30___. “And you must be from Moscow, granny, because you didn‟t say …thanks‟ when I gave your my seat.”16. A finally B naturally C especially D immediately17. A sets B climbs C moves D appears18. A distant B absent C separate D different19. A typical B European C Asian D northern20. A traditional B logical C hopeful D worthy21. A music B art C language D history22. A popular B famous C interesting D beautiful23. A forgotten B supported C invented D planned24. A happier B faster C slower D colder25. A work B walk C think D hurry26. A long B warm C colorful D exciting27. A hardworking B devoted C brave D polite28. A sent B offered C led D allowed29. A in silence B in anger C in surprise D in hope30. A cry B shout C smile D look第二节语法填空（共10小题，每小题1.5分，满分15分）阅读下面短文，按照句子结构的语法性和上下文连贯的要求，在空格处填入一个适当的词或使用括号中间词语的正确形式填空，并将答案填在答题卡标号为31~40的相应位置上。

机密★启用前2011年广东省初中毕业生学业考试数学模拟试卷说明：1．全卷共6页，考试用时100分钟，满分为120分。

2．答卷前，考生务必用黑色字迹的签字笔或钢笔在答题卡填写自己的准考证号、姓名、试室号、座位号。

用2B铅笔把对应该号码的标号涂黑。

3．选择题每小题选出答案后，用2B铅笔把答题卡上对应题目选项的答案信息点涂黑，如需改动，用橡皮擦干净后，再选涂其他答案，答案不能答在试题上。

4．非选择题必须用黑色字迹钢笔或签字笔作答、答案必须写在答题卡各题目指定区域内相应位置上；如需改动，先划掉原来的答案，然后再写上新的答案；不准使用铅笔和涂改液。

不按以上要求作答的答案无效。

5．考生务必保持答题卡的整洁。

考试结束时，将试卷和答题卡一并交回。

一、选择题（本大题5小题，每小题3分，共15分）在每小题列出的四个选项中，只有一个是正确的，请把答题卡上对应题目所选的选项涂黑。

1．25的算术平方根是A． 5 B．±5 C．5 D．±52．从长度分别为3、5、7、9的4条线段中任取3条作边，能组成三角形的概率为（）（A）34（B）12（C）13（D）143．如左下图所示的几何体的正视图是（）A． B． C． D．4．下列运算正确的是（）A．3412a a a⋅=B．339()y y-=C．222264x x x-+=D．3252()m n m n=5. 如图1，把一块直角三角板的直角顶点放在直尺的一边上，如果∠1=32o，那么∠2的度数是（）(A) 32o； (B) 58o； (C) 68o； (D) 60o．二、填空题（本大题5小题，每小题4分，共20分）请将下列各题的正确答案填写在答题卡相应的位置上。

6. 分解因式：324a ab-＝ .7．请写出一个以直线3-=x为对称轴，且在对称轴右侧部分是下降的抛物线的表达式，这条抛物线的表达式可以是．8. 成功、精彩、难忘的中国2010年上海世博会，众多境外参观者纷至沓来。

W-E-A-L-T-H 系列I was about to search my mind what to share todaywhen I caught sight of one of the ten series I didon my own several years ago , I will share withyou all here. Hope they will do you good. ---Catherine广州市113中学2011届高二期末复习冲刺二语法填空Class_________ No. ___________ Name _____________ Score _________（1）People _____1___ (live) in different countries made different kinds ofwords. Today there are about fifteen hundred _____2_____ in the world.Each contains many thousands of words. A very large dictionary, forexample, contains four ___3___ five hundred thousand words. But we do notneed ____4____ these. To read short stories you need to know only abouttwo thousand words. ____5_____ you leave school, you will learn only onethousand or more.The words you know are called your vocabulary. You should try to make yourvocabulary ______6___ (big). Read as many books as we can. There are alot of books _____7____ (write) in easy English. You will enjoy them. Whenyou meet _____8____ new word, look it ____9_____ in your dictionary. Yourdictionary is your _____10_____ (much) useful book.（2）Finishing their shopping at the mall, a couple discovered that theirnew car ____1_____ (steal) .They filed a report____2____ the policestation and a detective drove them ____3____ to the parking lot to lookfor evidence.To their ____4____ (amaze), the car had been returned ___5____ there wasa note in it that said: “I apologize for taking your car. My wife washaving a baby and I had to rush her to the ____6_____. Please forget the inconvenience. There are two tickets ____7____ tonight's Mania Twainconcert.”Their faith in humanity restored. The couple attended ____8____ concert.But when they returned home, they immediately found ____9______theirhouse had been ransacked (洗劫). On the bathroom mirror was ____10_____note: “I have to put my kid through college somehow, don't I?”（3）We often think of future. We often wonder ____1_the world will be like a hundred years’ time.Think of ____2___ space. Perhaps a permanent station on the moon _____3___ (set up). Perhaps people will be able to visit the moon as _____4___. Cheap rockets for space travel will have been developed, _____5___ (permit) long journeys through the solar system. When that time comes, people will be taking holidays in space and visiting _____6____ planets. Great progress will have been made _____7___ medicine, too. Pollution will have been controlled in a hundred years’ time._____8_____ the world will have been developed—even Antarctica. We will have used up most of the earth’s land to build our cities, _____9____ floating cities will have been built. The Japanese have already plans of this kind. And there will be cities ____10____ the sea.（4）Why is setting goals so ______1_____? Because goals can help you do, be, and experience everything ____2_____ you want in life. Instead _____3____ just letting life happen to you, goals allow ______4_____ to make your life happen._____5_____ (success) and happy people have sets lots of goals to help them reach their aims. By setting goals you are taking control of your life. It’s _____6____ having a map to show you _____7____ you want to go.Winners in life set goals and follow through with them. Winners decide what they want in life and then get there by making plans and _____8___ (set) goals. Unsuccessful people just let life happen by accident. Goals aren’t difficult to set, and _____9_____aren’t difficult to reach. It is up to you to find out what your goals really are. You are ____10____one who must decide what to do and in what direction to aim your life.（5）We all know that computers have played and 1 play a very important part 2 various fields.In personal lives, computers can do lots of work. It can carry 3 people’s instructions.In agriculture, computers are used 4 (control) the growing conditions of vegetables and plants. In transportation, computers can work out the 5 ( good) distance 6 trains and operate the trains instead 7 human beings. In the fields of education, health and scientific research, computers are important 8 .People use them to store information 9 records.Because of the computers, life is easier and more 10 (convenience) for people.（6）Thirteen-year-old Andrew will never forget that Saturday. He was relaxing on his family’s sofa with his sister, Susan. Suddenly the heaterexploded. His sister, who was unhurt, managed to pull 1 away from the fire 2 his mum dialed for an ambulance. 3 ambulance soon arrived and rushed him to Children’s Hospital4 he was operated on immediately.Doctors told his mum there was little chance of Andrew being able to walk again. The damage to his body was so bad and the cuts in his legs so deep that the 5 (operate) took 11 hours. After a couple of weeks, he 6 (allow) to go home but had to spend the next six weeks in a wheelchair.7 the help of his doctor, this brave young boy returned to school after only two and a half months. 8 (look) back on the accident, Andrew says he believes children can do 9 they want 10 they put their mind to it.（7）We are what we eat. What we are 1 （accustom）to eating every day has a great effect on our health. Consequently, we should make 2 effort to change our bad eating habits. As we all know , baked or fried foods may be 3 （taste）, but eating too much of them will probably 4 some illnesses. In addition, some food which costs us a great deal of money is not healthy, 5 junk food.6 people like to eat is various from culture to culture. Compared with people in the West, Chinese people used to eat more grain, more vegetables and less meat,7 is a good eating habit.8 , things are quite different now . With the rapid economic9 （grow）of our country, we also eat a lot of food high in sugar and fat. If you take a casual look at people in the street, what impresses you will probably be the fact that there are a lot more fat people than ever before. We are at risk when eating out without 10 （consider）the balance of our food . To be healthy, we must keep it in our mind that the food we choose to eat should give us the nutrients we need.（8）Walt Disney, a great film-maker, was born in Chicago in 1901 1 diedin 1966. 2 a young man, he wanted very much to be an artist. He went to newspaper office in the hope 3 getting a job there, 4 he had no luck. He didn’t lose heart. He kept on 5 (draw) lots of pictures. Being poor, he had to sit in the family garage drawing pictures. Day 6 day, he worked hard. Several years 7 , he succeeded in 8 (make) his first cartoon character---Mickey Mouse. From then 9 , he became a successful cartoon maker. Many characters in his cartoons are very popular, such as Mickey Mouse and Donald Duck, 10 are loved by everyone, especially by children.（9）Few people would even think of beginning a new job at the age of 76, ___1___ one of America’s most famous artists did just that. Anna Mary Robertson, better known as “Grandma Moses”, turned to painting ___2___ she was too old to work on her farm.Grandma Moses was crazy about painting soon after she picked it up and worked hard at it. She painted ___3___(care) and her works were nice. She first painted only to please ___4___, and then began to sell her works ___5___a little money. In 1993, a collector, Louis Caldor happened to see several of Grandma Moses’ works ___6___(hang) in a shop. He liked them, ___7___(buy) them at once, and set out to look for ___8___. Caldor held ___9___ show to introduce the works of Grandma Moses to other artists. Grandma Moses, ___10___ was world-famous, died on December 13, 1961, at the age of 101.（10）Our home had little contact with Gentiles. But there were the Gentile washwomen 1 came to the house 2 (fetch) our laundry. My story is about one of these. She was a small woman, old and wrinkled. When she started washing for us, she was already 3 seventy. Most Jewish women of her age were 4 (sick), weak and broken in body. 5 this washwoman, small and thin 6 she was, possessed a strength that came from generations of peasant ancestors. Mother would count 7 to her a bag of laundry that 8 (accumulate) over several weeks. She would lift the heavy bag, load it on her shoulders, and carry 9 the long way home. It must have been 10 walk of an hour and a half.（ 11 ）It is said that long ago there 1 to be 10 suns in the sky. Each day, one of the suns would travel around the sky on a carriage 2 by Xihe, the mother of the suns. One day, 3 (expect), all 10 suns simultaneously appeared in the sky, 4 instantly dried the crops and caused disaster to the people on earth. Hou一i, a local archer, had great sympathy 5 people's sufferings from the blistering（酷热的）weather and decided to help them 6 . Houyi climbed up to the summit of Kunlun Mountains and shot down the suns leaving only 7 to benefit people. After he shot down the sun, Hou Yi became a hero 8 was revered by local people. 9 , Hou Yi married a beautiful girl—Chang'e. The young couple lived 10 happy and sweet life.（ 12 ）Long ago the Emperor decided to hold a swimming race between 13 different animals. The order in which they finished would decide the order of their 1 (appear) in the Chinese calendar. All the animals lined up on the river bank and 2 (give) the task of getting to the opposite shore. The cat wondered how he would get across 3 he was afraid ofwater. The ox worried how he would cross 4 his poor eyesight. The calculating rat suggested that he and the cat 5 (jump) onto the ox's back and guide him across. The ox, steady and hard-working, did not notice the two animals on his back. In the meanwhile, the rat ran behind the unsuspecting cat and pushed 6 into the water. As the ox came ashore, the rat jumped off and finished the race first. 7 the rat got the first year 8 (name) after him and the ox got the second year. 9 the lazy pig reached the finish line in 12th place the race was over and he was awarded the final place on the calendar. From that day the cat, `10 never finished and lost his place on the calendar, became the enemy of the rat.KEY JUST FOR REFERENCE广州市113中学2011届高二期末复习冲刺二语法填空(1) 1. living 2. languages 3. or 4. all 5. Before 6. bigger7. written 8. a 9 up 10 most(2) 1. had been stolen 2. at 3. back 4. amazement 5. and 6.hospital 7.for 8. the 9. that 10. another(3) 1. what 2. / 3. will have been set up 4.tourists/visitors/travelers 5. permitting 6. other 7. in 8 All 9.so/thus/therefore 10. under(4) 1. important 2. that 3. of 4. yourself 5. Successful 6. like 7.where 8. setting 9. they 10. the(5) 1. will 2. in 3. out 4. to control 5.best 6.between 7. of 8. too 9. and 10.convenient(6) 1. him 2.while 3.The 4.where 5.operation 6.was allowed 7.With8.Looking 9.whatever 10.if(7) 1. accustomed 2. an / every 3. tasty 4. result in/ lead to 5. but 6.What 7 .which 8 .However 9.growth10. considering(8)1．and 2. As 3. of 4. but 5. drawing 6. after 7. later 8. making9. on 10. which(9) 1. but 2. because 3. carefully 4. herself 5. for 6. hanging 7.bought 8. more 9. a 10. who(10)1: who 2: to fetch 3: over/past 4: sickly 5: But 6: as/though7: out 8: had accumulated 9: it 10: a(11)1. used 2. driven 3. unexpectedly 4. which. 5. for 6. out7. one 8. who 9. Later 10. a(12）1. appearance 2. were given 3. because / as 4. with 5.jump 6. him 7. So / Thus 8. named9. When / After 10. who。

2023-2024学年广东省广州市第一一三中学高考物理二模试卷注意事项1．考试结束后，请将本试卷和答题卡一并交回．2．答题前，请务必将自己的姓名、准考证号用0．5毫米黑色墨水的签字笔填写在试卷及答题卡的规定位置．3．请认真核对监考员在答题卡上所粘贴的条形码上的姓名、准考证号与本人是否相符．4．作答选择题，必须用2B铅笔将答题卡上对应选项的方框涂满、涂黑；如需改动，请用橡皮擦干净后，再选涂其他答案．作答非选择题，必须用05毫米黑色墨水的签字笔在答题卡上的指定位置作答，在其他位置作答一律无效．5．如需作图，须用2B铅笔绘、写清楚，线条、符号等须加黑、加粗．一、单项选择题：本题共6小题，每小题4分，共24分。

在每小题给出的四个选项中，只有一项是符合题目要求的。

1、如图所示的电路中，D1、D2是完全相同的灯泡，线圈L的自感系数较大，直流电阻不计。

先闭合开关S，电路稳定后再断开开关S，此时( )A．D1立刻熄灭B．D2立刻熄灭C．D1闪亮一下逐渐熄灭D．D2闪亮一下逐渐熄灭2、如图所示，由均匀导线绕成的直角扇形导线框OMN绕O点在竖直面内从匀强磁场边界逆时针匀速转动，周期为T，磁场的方向垂直于纸面向里，线框电阻为R，线框在进入磁场过程中回路的电流强度大小为I，则（）A．线框在进入磁场过程中回路产生的电流方向为顺时针B．线框在进入与离开磁场的过程中ON段两端电压大小相等C．线框转动一周过程中产生的热量为12I2RTD．线框转动一周过程中回路的等效电流大小为2I3、如图所示，在水平向右的匀强电场中，质量为m的带电小球以初速度 从M点竖直向上抛出，通过N点时，速度大小为3v，方向与电场方向相反，则小球从M点运动到N点的过程中（）A ．动能增加232mvB ．机械能增加24mvC ．重力势能增加212mvD ．电势能增加292mv 4、如图所示，理想变压器原、副线圈匝数比为1∶2，正弦交流电源电压为U ＝12 V ，电阻R 1＝1 Ω，R 2＝2 Ω，滑动变阻器R 3最大阻值为20 Ω，滑片P 处于中间位置，则A ．R 1与R 2消耗的电功率相等B ．通过R 1的电流为3 AC ．若向上移动P ，电压表读数将变大D ．若向上移动P ，电源输出功率将不变5、如图所示，定滑轮通过细绳OO ' 连接在天花板上，跨过定滑轮的细绳两端连接带电小球A 、B ，其质量分别为m 1 、m 2 （ m 1 ≠ m 2 ）。

数学学业水平测试 第 1 页 （共 13 页）2011学年度广州市高中二年级学生学业水平测试数 学本试卷共4页. 满分150分. 考试用时120分钟. 注意事项：1．答卷前，考生务必用黑色字迹钢笔或签字笔将自己的姓名和准考证号填写在答题卡指定的位置上.2．选择题每小题选出答案后，用2B 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，再选涂其它答案，答案不能答在试卷上.3. 非选择题必须用黑色字迹的钢笔或签字笔作答，答案必须写在答题卡各题目指定区域内的相应位置上；如需改动，先划掉原来的答案，然后再写上新的答案；不准使用铅笔和涂改液.不按以上要求作答的答案无效.4. 本次考试不允许使用计算器.5. 考生必须保持答题卡的整洁，考试结束后，将试卷和答题卡一并交回.一、选择题：本大题共10小题，每小题5分，满分50分．在每小题给出的四个选项中，只有一个是符合题目要求的．1. 已知集合{}1,2A =, {}1,0,1B =-, 则A B 等于A .{}1 B. {}1,0,2- C. {}1,0,1,2- D. ∅ 2. cos120︒的值是A. 2- B. 12-C.12D.23. 不等式2230x x --<的解集是A . ()3,1- B. ()1,3- C. ()(),13,-∞-+∞ D. ()(),31,-∞-+∞ 4. 已知直线12:220,:410l x y l ax y +-=++=, 若12//l l , 则a 的值为 A . 8 B. 2 C. 12-D. 2-数学学业水平测试 第 2 页 （共 13 页）5. 函数sin 2y x =是A . 最小正周期为2π的偶函数 B. 最小正周期为2π的奇函数 C. 最小正周期为π的偶函数 D. 最小正周期为π的奇函数 6. 在等比数列{}n a 中, 若362459,27a a a a a ==, 则2a 的值为 A . 2 B. 3 C. 4 D. 97. 如果实数x 、y 满足条件1,210,10.y x y x y ≤⎧⎪--≤⎨⎪+-≥⎩则2x y +的最大值为A . 1 B.53C. 2D. 38. 已知某几何体的三视图如图1所示, 其中俯视图是腰长为2的等腰梯形, 则该几何体的体积为A. B.C.D.9. 已知向量=a ()1,n , =b (),1n , 其中1n ≠±, 则下列结论中正确的是A . ()()//-+a b a b B. ()//+a b bC. D. ()+⊥a b b10. 已知函数()1f x =, 则对任意实数12x x 、,且1202x x <<<, 都有A. B. C. D.二、填空题：本大题共4小题，每小题5分，满分20分. 11. 函数()ln 21y x =-的定义域是 .12. 在空间直角坐标系O xyz 中, 点()1,2,3-关于原点O 的对称点的坐标为 .()()1221x f x x f x <()()1122x f x x f x >()()-⊥+a b a b ()()1221x f x x f x>4正视图 侧视图俯视图图1()()1122x f x x f x <数学学业水平测试 第 3 页 （共 13 页）13. 某公司生产A 、B 、C 三种不同型号的轿车，产量之比依次为2:3:4，为了检验该公司的产品质量，用分层抽样的方法抽取一个容量为n 的样本，样本中A 种型号的轿车 比B 种型号的轿车少8辆，那么n = .14. 已知函数1(0x y a a -=>且1)a ≠的图象恒过点A . 若点A 在直线 上, 则12m n+的最小值为 .三、解答题：本大题共6小题，满分80分．解答应写出文字说明、演算步骤和推证过程. 15. （本小题满分12分）编号分别为12312,,,,A A A A 的12名篮球运动员在某次篮球比赛中的得分记录如下:（1）完成如下的频率分布表:（2）从得分在区间[)10,20内的运动员中随机抽取2人 , 求这2人得分之和大于25的概率.()100mx ny mn +-=>16．(本小题满分12分)在△ABC中，角A、B、C所对的边分别为a、b、c，已知13,2,cos3a b A===．（1）求sin B的值；（2）求c的值.17．(本小题满分14分)如图2，在三棱锥P A B C-中，5,4,3AB BC AC===，点D是线段P B的中点，平面P A C⊥平面ABC．（1）在线段AB上是否存在点E, 使得的位置,并加以证明；若不存在,（2）求证：P A B C⊥.数学学业水平测试第 4 页（共13 页）数学学业水平测试 第 5 页 （共 13 页）18. （本小题满分14分）已知等差数列{}n a 的前n 项和为n S ，且1310a a +=, 424S =． （1）求数列{}n a 的通项公式； （2）令12111n nT S S S =+++，求证：34n T <.19. （本小题满分14分）已知圆C 的圆心坐标为()1,2, 直线:10l x y +-=与圆C 相交于M 、N 两点，M N =2.（1）求圆C 的方程;（2）若1t ≠, 过点(),0A t 作圆C 的切线, 切点为B ，记1d AB =, 点A 到直线l 的距离为2d , 求 的取值范围.121d d -数学学业水平测试 第 6 页 （共 13 页）20. （本小题满分14分） 已知113a ≤≤, 若函数()22fx ax x =-在[]1,3上的最大值为()M a ,最小值为()N a ,令()()()g a M a N a =-.（1）求()g a 的表达式;（2）若关于a 的方程()0g a t -=有解, 求实数t 的取值范围.数学学业水平测试 第 7 页 （共 13 页）2011学年度广州市高中二年级学生学业水平测试数学试题参考答案及评分标准一、选择题：本大题主要考查基本知识和基本运算．共10小题，每小题5分，满分50分．二、填空题：本大题主要考查基本知识和基本运算．共4小题，每小题5分，满分20分． 11. 1,2⎛⎫+∞⎪⎝⎭12. ()1,2,3-- 13. 72 14.3+三、解答题：本大题共6小题，共80分．解答应写出文字说明、演算步骤和推证过程. 15．本小题主要考查统计与概率等基础知识，考查数据处理能力．满分12分． (1) 解:频率分布表:………4分 (2)解: 得分在区间[)10,20内的运动员的编号为2A ,3A ,4A ,8A ,11A .从中随机抽取2人,所有可能的抽取结果有:{}23,A A , {}24,A A ,{}28,A A ,{}211,A A ,{}34,A A ,{}38,A A ,{}311,A A ,{}48,A A ,{}411,A A ,{}811,A A ,共10种. ………7分“从得分在区间[)10,20内的运动员中随机抽取2人，这2人得分之和大于25”(记为事件B )的所有可能结果有:{}24,A A ,{}211,A A ,{}34,A A ,{}38,A A ,{}311,A A ,{}48,A A ,{}411,A A ,{}811,A A ,共8种.………10分数学学业水平测试 第 8 页 （共所以()80.810P B ==.答: 从得分在区间[)10,20内的运动员中随机抽取2人, 这2人得分之和大于25的概率为 0.8. ………12分16．本小题主要考查解三角形、三角恒等变换等基础知识，考查运算求解能力．满分12分．（1）解：∵0A π<<，1cos 3A =，∴sin 3A ==． ………2分 由正弦定理得：sin sin a b AB=， ………4分∴2sin 3sin 39b A B a⨯===. ………6分（2）解：∵13,2,cos 3a b A ===，∴222123b c abc+-=. ………8分∴222231223c c+-=⨯，解得3c =. ………12分17．本小题主要考查直线与平面的位置关系的基础知识，考查空间想象能力、推理论证能力和运算求解能力．满分14分．（1）解：在线段AB 上存在点E , 使得//D E 平面PAC , 点E 是线段AB 的中点. …2分 下面证明//D E 平面PAC :取线段AB 的中点E , 连接D E ， ………3 ∵点D 是线段P B 的中点，∴D E 是△P A B 的中位线. ………4 ∴//D E P A . ………6 ∵PA ⊂平面PAC ，D E ⊄平面PAC ，∴//D E 平面PAC . (8)数学学业水平测试 第 9 页 （共 13 页）（2）证明：∵5,4,3AB BC AC ===,∴222AB BC AC =+.∴A C B C ⊥. ………10分 ∵平面PAC ⊥平面ABC ，且平面PAC 平面ABC A C =，B C ⊂平面ABC ， ∴B C ⊥平面PAC . ………12分 ∵PA ⊂平面PAC ，∴P A B C ⊥. ………14分18．本小题主要考查等差数列、数列求和、不等式等基础知识，考查运算求解能力和推理论证能力．满分14分．（1）解：设等差数列{}n a 的公差为d , ∵ 1310a a +=, 424S =,∴112210,43424.2a d a d +=⎧⎪⎨⨯+=⎪⎩ ………2分 解得13a =, 2d =. ………4分 ∴ ()32121n a n n =+⨯-=+. ………6分（2）证明：由（1）得()()()1321222n n n a a n n S n n +++===+， ………8分∴ 12111n nT S S S =+++()11111324352n n =++++⨯⨯⨯+=11111111111232435112n n n n ⎡⎤⎛⎫⎛⎫⎛⎫⎛⎫⎛⎫-+-+-++-+- ⎪ ⎪ ⎪ ⎪⎪⎢⎥-++⎝⎭⎝⎭⎝⎭⎝⎭⎝⎭⎣⎦ ………10分数学学业水平测试 第 10 页 （共 13 页）111112212n n ⎛⎫=+-- ⎪++⎝⎭ =31114212n n ⎛⎫-+⎪++⎝⎭………12分 34<. ………14分19．本小题主要考查直线与圆的方程、不等式等基础知识，考查运算求解能力及推理论证 能力．满分14分．(1)解: 设圆C 的半径为r ，圆C 的圆心()1,2到直线l的距离d ==.………2分 ∵ M N =2, ∴2=. ………3分∴2=. ………4分解得r =………5分∴ 所求的圆C 的方程为()()22123x y -+-=. ………6分 (2) 解:∵圆C ：()()22123x y -+-=的圆心()1,2C ,半径r =∴1d AB====.………8分 又点(),0A t 到直线l 的距离2d ==………9分∴121d d-== (10)分m =，则1t -= ………11分∵1t ≠，∴1m >.数学学业水平测试 第 11 页 （共 13 页）∴121d d-===. ………12分∵1m >, ∴12m +>. ∴2011m <<+.∴20111m <-<+. ………13分∴0<<.∴121d d -的取值范围是(. ………14分20．本小题主要考查二次函数的最值、方程等基础知识，考查运算求解能力，以及分类讨论的数学思想方法．满分14分．(1) 解: ()22f x ax x =-211a x a a ⎛⎫=-- ⎪⎝⎭. ………1分∵113a ≤≤, ∴113a≤≤. ① 当112a≤≤,即112a ≤≤时, 则3x =时, 函数()fx 取得最大值; 1x a=时, 函数()f x 取得最小值.∴()()396M a f a ==-，()11N a f a a ⎛⎫==- ⎪⎝⎭. ∴ ()()()g a M a N a =-=196a a+-. ………3分② 当123a <≤,即1132a ≤<时, 则1x =时, 函数()f x 取得最大值; 1x a=时, 函数()f x 取得最小值.∴()()12M a f a ==-，()11N a f a a ⎛⎫==-⎪⎝⎭.数学学业水平测试 第 12 页 （共 13 页）∴ ()()()g a M a N a =-=12a a+-. ………5分综上，得()g a =1112,,321196, 1.2a a aa a a ⎧+-≤<⎪⎪⎨⎪+-≤≤⎪⎩………6分（2）解：任取1211,,32a a ⎡⎫∈⎪⎢⎣⎭，且12a a <， ()()1212121122g a g a a a a a ⎛⎫⎛⎫-=+--+- ⎪ ⎪⎝⎭⎝⎭()()1212121a a a a a a --=. ………7分∵1211,,32a a ⎡⎫∈⎪⎢⎣⎭，且12a a <，∴1212120,0,10a a a a a a -<>-<. ∴()()12121210a a a a a a -->,即()()120g a g a ->.∴()()12g a g a >. ∴函数()g a 在11,32⎡⎫⎪⎢⎣⎭上单调递减. ………8分 任取341,,12a a ⎡⎤∈⎢⎥⎣⎦，且34a a <，()()343434119696g a g a a a a a ⎛⎫⎛⎫-=+--+- ⎪ ⎪⎝⎭⎝⎭()()34343491a a a a a a --=. ………9分∵341,,12a a ⎡⎤∈⎢⎥⎣⎦，且34a a <，数学学业水平测试 第 13 页 （共 13 页）∴3434340,0,910a a a a a a -<>->. ∴()()343434910a a a a a a --<,即()()340g a g a -<.∴()()34g a g a <. ∴函数()g a 在1,12⎡⎤⎢⎥⎣⎦上单调递增. ………10分 当12a =时,()g a 取得最小值,其值为12g ⎛⎫=⎪⎝⎭12, ………11分 又13g ⎛⎫=⎪⎝⎭43, ()1g =4. ∴函数()g a 的值域为1,42⎡⎤⎢⎥⎣⎦. ………12分∵关于a 的方程()0g a t -=有解等价于()t g a =有解，∴实数t 的取值范围为函数()g a 的值域． ………13分∴实数t 的取值范围为1,42⎡⎤⎢⎥⎣⎦. ………14分。

4 1 23 5 ∞ n 0 -13.60-3.40 -1.51 -0.85 -0.54 E /eV2011年广州市普通高中毕业班综合测试(二)理科综合 物理部分一、单项选择题13、氢原子的部分能级如图所示，氢原子吸收以下能量可以从基态跃迁到n =2能级的是A ．10.2 eVB ．3.4 eVC ．1.89 eVD ．1.51 eV14、将电容为1.2×10-11F 的平行板电容器接在3V 的电池上，则电容器所带电荷量为A ．0.4×10-11CB ．1.8×10-11C C ．3.6×10-11CD ．7.2×10-11C15、如图所示，L 1、L 2是理想变压器的两个线圈。

如果把它当作降压器，则A ．L 1接电源，原、副线圈两端电压之比为1：5B ．L 1接电源，原、副线圈两端电压之比为5：1C ．L 2接电源，原、副线圈两端电压之比为5：1D ．L 2接电源，原、副线圈两端电压之比为1：516、如图所示，力F 垂直作用在倾角为α的三角滑块上，滑块没被推动，则滑块受到地面的静摩擦力大小为A ．0B ．F cos αC ．F sin αD ．F tan α二、双项选择题 17、下列说法正确的是A ．146C 经一次α衰变后成为147CB ．42He 核由两个中子和两个质子组成C ．温度升高不能改变放射性元素的周期D ．核反应方程应遵循质子数和中子数守恒18、如图所示，弹簧被质量为m 的小球压缩，小球与弹簧不粘连且离地面的高度为h ，不计空气阻力，将拉住小球的细线烧断，则小球 A ．直线运动 B ．曲线运动C ．落地时的动能等于mghD ．落地时的动能大于mgh 19、关于热现象和热学规律的说法中，正确的是A ．第二类永动机违背了能量守恒规律B ．当物体被拉伸时，分子间的斥力减小、引力增大C ．温度高的物体内能不一定大，但分子平均动能大D ．悬浮在液体中的固体微粒越小，布朗运动越明显20、如图所示，匀强电场中某电场线上的两点A 、B 相距0.2m ，正电荷q =10-6C 从A 移到B ，电场力做功为2×10-6J ，则A ．该电场的场强为10V/mB ．该电场的方向由A 向BC ．A 、B 间的电势差为10VD ．B 点的电势比A 点的高21、如图所示，光滑的水平面上，质量为m 1的小球以速度v 与质量为m 2的静止小球正碰，碰后两小球的速度大小都为12v ，方向相反，则两小球质量之比m 1：m 2化量之比△E k 1：△E k 2为A ．m 1：m 2=1：3B ．m 1：m 2=1：1C ．△E k 1：△E k 2=1：3D ．△E k 1：△E k 2=1：134、(18分)⑴完成以下“验证平行四边形定则”实验的几个主要步骤：①如图甲，用两只弹簧测力计分别钩住细绳套，互成角度地拉橡皮条，使橡皮条伸长，记下结点位置O 和两测力计的示数F 1、F2以及两细绳套的方向。