最新高考文科数学广东卷试题与答案word解析版优秀名师资料.doc

2013年高考文科数学广东卷试题与答案word解析版2013年普通高等学校夏季招生全国统一考试数学文史类(广东卷) 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，只有一项是符合题目要求的( 221(2013广东，文1)设集合S,x|x，2x,0，x?R，T,x|x,2x,0，x?R，则S?T,( )( A(0 B(0,2 C(,2,0 D(,2,0,2 lg1，x2(2013广东，文2)函数的定义域是( )( y,x,1A(,1，?) B(,1，?) C(,1,1)?(1，?) D(,1,1)?(1，?) 3(2013广东，文3)若i(x，yi),3，4i，x，y?R，则复数x，yi的模是( )( A(2 B(3 C(4 D(5 51,4(2013广东，文4)已知，那么cos ,( )( ,sin，,25,2112,5555A( B( C( D( 5(2013广东，文5)执行如图所示的程序框图，若输入n的值为3，则输出s的值是( )( A(1 B(2 C(4 D(7 6(2013广东，文6)某三棱锥的三视图如图所示，则该三棱锥的体积是( )( 112633A( B( C( D(1 227(2013广东，文7)垂直于直线y,x，1且与圆x，y,1相切于第?象限的直线方程是( )( 22A(x，y,0 B(x，y，1,0 C(x，y,1,0 D(x，y，,0 8(2013广东，文8)设l为直线，是两个不同的平面(下列命题中正确的是( )( A(若l?，l?，则? B(若l?，l?，则? C(若l?，l?，则? D(若?，l?，则l? 19(2013广东，文9)已知中心在原点的椭圆C的右焦点为F(1,0)，离心率等于，则C的方程是( )( 222222222xyxyxyxy，,1，,1，,1，,143344243A( B( C( D( 10(2013广东，文10)设a是已知的平面向量且a?0.关于向量a的分解，有如下四个命题: ?给定向量b，总存在向量c，使a,b，c; ?给定向量b和c，总存在实数和，使a,b，c; ?给定单位向量b和正数，总存在单位向量c和实数，使a,b，c; ?给定正数和，总存在单位向量b和单位向量c，使a,b，c. 上述命题中的向量b，c和a在同一平面内且两两不共线，则真命题的个数是( )( A(1 B(2 C(3 D(4 2013 广东文科数学 第1页 二、填空题:本大题共5小题，考生作答4小题，每小题5分，满分20分( (一)必做题(11,13题) 11(2013广东，文11)设数列a是首项为1，公比为,2的等比数列，则a，|a|，a，|a|,n1234_. 212(2013广东，文12)若曲线y,ax,ln x在(1，a)处的切线平行于x轴，则a,_. xy,，,30,11,x13(2013广东，文13)已知变量x，y满足约束条件则z,x，y的最大值是_( ,y,1,(二)选做题(14,15题，考生只能从中选做一题) 14(2013广东，文14)(坐标系与参数方程选做题)已知曲线C的极坐标方程为,2cos .以极点为原点，极轴为x轴的正半轴建立直角坐标系，则曲线C的参数方程为_( 15(2013广东，文15)(几何证明选讲选做题)如图，在矩形ABCD中，AB,3，BC,3，BE?AC，垂足为E，则ED,_. 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(2013广东，文16)(本小题满分12分)已知函数，x?R. fxx()2cos,12,(1)求的值; f,3,33,2)若cos ,(，?，求. ,f,2,562,17(2013广东，文17)(本小题满分12分)从一批苹果中，随机抽取50个，其重量(单位:克)的频数分布表如下: 分组(重量) 80,85) 85,90) 90,95) 95,100) 频数(个) 5 10 20 15 (1)根据频数分布表计算苹果的重量在90,95)的频率; (2)用分层抽样的方法从重量在80,85)和95,100)的苹果中共抽取4个，其中重量在80,85)的有几个, (3)在(2)中抽出的4个苹果中，任取2个，求重量在80,85)和95,100)中各有1个的概率( 2013 广东文科数学 第2页 中，分别是，18(2013广东，文18)(本小题满分14分)如图(1)，在边长为1的等边三角形ABCDEABAC上的点，AD,AE，F是BC的中点，AF与DE交于点G.将?ABF沿AF折起，得到如图(2)所示的三棱锥A2,BCF，其中BC,. 2图(1) 图(2) (1)证明:DE?平面BCF; (2)证明:CF?平面ABF; 2(3)当AD,时，求三棱锥F,DEG的体积V. F,DEG3219(2013广东，文19)(本小题满分14分)设各项均为正数的数列a的前n项和为S，满足4S,annnn，1*,4n,1，n?N，且a，a，a构成等比数列( 2514(1)证明:aa,，45; 21(2)求数列a的通项公式; n1111(3)证明:对一切正整数n，有，,？. aaaaaa212231nn，2013 广东文科数学 第3页 的顶点为原点，其焦点(0，)(,0)到直线:20(2013广东，文20)(本小题满分14分)已知抛物线CFccl32x,y,2,0的距离为.设P为直线l上的点，过点P作抛物线C的两条切线PA，PB，其中A，B为切2点( (1)求抛物线C的方程; (2)当点P(x，y)为直线l上的定点时，求直线AB的方程; 00(3)当点在直线上移动时，求|?|的最小值( PlAFBF2013 广东文科数学 第4页 32(),(?R)( 21(2013广东，文21)(本小题满分14分)设函数fxx,kx，xk(1)当k,1时，求函数f(x)的单调区间; (2)当k,0时，求函数f(x)在k，,k上的最小值m和最大值M. 2013 广东文科数学 第5页 2013年普通高等学校夏季招生全国统一考试数学文史类(广东卷) 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，只有一项是符合题目要求的( 1( 答案:A 解析:?S,2,0，T,0,2，?S?T,0( 2( 答案:C x，,10,解析:要使函数有意义，则 ,x,10,解得x,1且x?1， 故函数的定义域为(,1,1)?(1，?)( 3( 答案:D 解析:?i(x，yi),y，xi,3，4i， x,4,? ,y,3.,?x，yi,4,3i. 2243，,，?|x，yi|,5. 4( 答案:C 5,解析:? ,，,，sinsin2,22,1,cos ,， ,sin，,52,1?cos ,. 55( 答案:C 解析:i,1，s,1，i?3，s,1，0,1，i,2; i?3，s,1，1,2，i,3; i?3，s,2，2,4，i,4; i,3，s,4. 6( 答案:B 解析:由俯视图知底面为直角三角形，又由正视图及侧视图知底面两直角边长都是1，且三棱锥的高为2，111故V,××1×1×2,. 三棱锥3237( 答案:A 解析:由于所求切线垂直于直线y,x，1，可设所求切线方程为x，y，m,0.由圆心到切线的距离等于半|mm,2径得，解得. ,12m,2又由于与圆相切于第?象限，则. 8( 答案:B 解析:如图，在正方体ABCD,ABCD中， 11112013 广东文科数学 第6页 对于A，设l为AA，平面BBCC，平面DCCD为，. 11111AA?平面BBCC，AA?平面DCCD， 111111而平面BBCC?平面DCCD,CC; 11111对于C，设l为AA，平面ABCD为，平面DCCD为.AA?平面ABCD，AA?平面DCCD， 1111111而平面ABCD?平面DCCD,DC; 11对于D，设平面AABB为，平面ABCD为，直线DC为l，平面AABB?平面ABCD，DC?平面AABB，1111111111而DC?平面ABCD( 11故A，C，D都是错误的( 而对于B，根据垂直于同一直线的两平面平行，知B正确( 9( 答案:D 解析:由中心在原点的椭圆C的右焦点F(1,0)知，c,1. 1c1又离心率等于，则,，得a,2. 2a2222由b,a,c,3， 22xy，,1故椭圆的方程为. C4310( 答案:B 解析:对于?，由向量加法的三角形法则知正确;对于?，由平面向量基本定理知正确;对于?，以a的终点作长度为的圆，这个圆必须和向量b有交点，这个不一定能满足，故?不正确;对于?，利用向量加法的三角形法则，结合三角形两边之和大于第三边，即必须|b|，|c|,，?|a|，故?不正确( 二、填空题:本大题共5小题，考生作答4小题，每小题5分，满分20分( (一)必做题(11,13题) 11(答案:15 n,1解析:由数列a首项为1，公比q,2，则a,(,2)，a,1，a,2，a,4，a,8，则a，|a|nn123412，a，|a|,1，2，4，8,15. 34112(答案: 21解析:由曲线在点(1，a)处的切线平行于x轴得切线的斜率为0，由y,2ax,及导数的几何意义得y|xx1,2a,1,0，解得a,. ,1213(答案:5 解析:由线性约束条件画出可行域如下图，平移直线l，当l过点A(1,4)，即当x,1，y,4时，z,5. 0max2013 广东文科数学 第7页 (二)选做题(14,15题，考生只能从中选做一题) x,，1cos,14(答案:(为参数) ,y,sin,解析:由曲线C的极坐标方程,2cos 知以极点为原点，极轴为x轴的正半轴建立直角坐标系知曲线x,，1cos,22C是以(1,0)为圆心，半径为1的圆，其方程为(x,1)，y,1，故参数方程为(为参数)( ,y,sin,15( 21答案: 2BC3,3解析:在Rt?ABC中，AB,，BC,3，tan?BAC,， AB13?则BAC,60?，AE,AB,. 22在?AED中，?EAD,30?，AD,3， 222ED,AE，AD,2AE?ADcos?EAD 2,332,，3,2××3×cos 30? ,22,333,，9,2××3× 42221,. 421?ED,. 2三、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( ,解:(1). f,2cos2cos1,33124,33,(2)?cos ,，?， ,2,52,2013 广东文科数学 第8页 42sin ,， ,1cos5,? ,f2cos,64,1,. ,，,2coscossinsin,445,17( 20解:(1)苹果的重量在90,95)的频率为,0.4; 505(2)重量在80,85)的有4×,1个; 515，(3)设这4个苹果中80,85)分段的为1，95,100)分段的为2,3,4，从中任取两个，可能的情况有:(1,2)，(1,3)，(1,4)，(2,3)，(2,4)，(3,4)，共6种(任取2个，重量在80,85)和95,100)中各有1个记为事31件A，则事件A包含有(1,2)，(1,3)，(1,4)，共3种，所以P(A),. 6218( (1)证明:在等边三角形ABC中， ADAE?AD,AE，?,. DBECADAE又,，在折叠后的三棱锥A,BCF中也成立， DBEC? DEBC(?平面，平面， DEBCFBCBCF,?DE?平面BCF. 1(2)证明:在等边三角形ABC中，?F是BC的中点，BC,1，?AF?CF，BF,CF,. 22?在三棱锥A,BCF中，BC,， 2222?BC,BF，CF.?CF?BF. ?BF?AF,F，?CF?平面ABF. (3)解:由(1)可知GE?CF，结合(2)可得GE?平面DFG. ,111111313?V,V,×?DG?FG?GE,. ，,F,DEGE,DFG,32323323324,19( 22(1)证明:当n,1时，4a,a,5，?a,4a，5. 1221aa,，45?a,0，?. n212(2)解:当n?2时，4S,a,4(n,1),1，? n,1n24S,a,4n,1，? nn，122由?,?，得4a,4S,4S,a,a,4， nnn,1n，1n222?a,a，4a，4,(a，2). n，1nnn?a,0，?a,a，2， nn，1n?当n?2时，a是公差d,2的等差数列( n?a，a，a构成等比数列， 251422?a,a?a，(a，6),a?(a，24)，解得a,3. 521422222由(1)可知，4a,a,5,4，?a,1. 121?a,a,3,1,2， 21?a是首项a,1，公差d,2的等差数列( n1?数列a的通项公式为a,2n,1. nn2013 广东文科数学 第9页 111(3)证明: ，？aaaaaa12231nn，1111, ，？1335572121，,，,，nn11111111，, ，,，,，,，,1？,2335572121nn,，,，111,. ，,1,2212n，,20( |02|32,c解:(1)依题意，解得c,1(负根舍去)( d,222?抛物线C的方程为x,4y. (2)设点A(x，y)，B(x，y)( 11221122由x,4y，即y,x，得y,x. 42x1?抛物线C在点A处的切线PA的方程为y,y,(x,x)， 112x121即y,x，y,x. 11221x21,，. ?yx?y,x,y11142?，)点P(xy在切线PA上， 00x1?y,x,y.? 0012x2同理，y,x,y.? 0022x综合?，?得，点A(x，y)，B(x，y)的坐标都满足方程y,x,y. 1122002?经过A(x，y)，B(x，y)两点的直线是唯一的， 1122x?直线AB的方程为y,x,y，即xx,2y,2y,0. 00002(3)由抛物线的定义可知|AF|,y，1，|BF|,y，1， 12?|AF|?|BF|,(y，1)(y，1) 12,y，y，yy，1. 12122,xy,4,联立 ,xxyy,220,00,222消去x得y，(2y,x)y，y,0， 00022?y，y,x,2y，yy,y. 1200120?点P(x，y)在直线l上，?x,y,2,0. 000022?|AF|?|BF|,x,2y，y，1 00022,y,2y，(y，2)，1 000219,2,2y，2y，5,. 2y，00,022,19,?当y,时，|AF|?|BF|取得最小值为. 02221( 2013 广东文科数学 第10页 2(),3，1， 解:fxx,2kx(1)当k,1时， 2f(x),3x,2x，1，,4,12,8,0， ?f(x),0，即f(x)的单调递增区间为R. k2(2)(方法一)当k,0时，f(x),3x,2kx，1，其开口向上，对称轴，且过(0,1)( x,32?当,4k,12,?0， 4(3)(3)kk，,即,3?k,0时，f(x)?0，f(x)在k，,k上单调递增( 从而当x,k时，f(x)取得最小值m,f(k),k; 333当,x,k时，f(x)取得最大值M,f(,k),kkk,2kk. 2,3?当,4k,12,0，即k,时， 4(3)(3)kk，,2令f(x),3x,2kx，1,0， 22kk，,3kk,3，注意到,解得:kx,x,0. x,x,21123312k(注:可用韦达定理判断x?x,，x，x,k，从而k,x,x,0;或者由对称结合图象判断) 12122133?m,minf(k)，f(x)，M,maxf(,k)，f(x)( 1232?f(x),f(k),x,kx，x,k 11112,(x,k)(x，1),0， 11?f(x)的最小值m,f(k),k. 323222?f(x),f(,k),x,kx，x,(,k,k?k,k),(x，k)(x,k)，k，1,0， 2222223?f(x)的最大值M,f(,k),2k,k. 3综上所述，当k,0时，f(x)的最小值m,f(k),k，最大值M,f(,k),2k,k. (方法2)当k,0时，对?x?k，,k，都有 32332f(x),f(k),x,kx，x,k，k,k,(x，1)(x,k)?0，故f(x)?f(k)( 32332222f(x),f(,k),x,kx，x，k，k，k,(x，k)(x,2kx，2k，1),(x，k)(x,k)，k，1?0. 3故f(x)?f(,k)(?f(k),k,0，f(,k),2k,k,0， 3?f(x),f(,k),2k,k，f(x),f(k),k. maxmin2013 广东文科数学 第11页 (英文版 ) easily blame, to prevent the broken window effect. Supervise the leading cadres to play an exemplary role, take the lead in the strict implementation of the < code > and < rule >, lead to safeguard the solemnity and authority of the party discipline, ensure that the party discipline and the laws and regulations for implementation in place. Throughout the discipline in the daily supervision and management , strengthen supervision and inspection, from the thorough investigation of violations of discipline behavior. Strengthen to key areas, key departments and key projects as well as the masses reflect the concentration of the units and departments for supervision. - strengthening supervision, discipline inspection and supervision of cadres to set an example for compliance with the < code > and < rule > is a man must be hexyl, blacksmith needs its own hardware. Discipline inspection organs as the executor of the party discipline, and supervisor of the defenders, for its supervision must be more strictly, discipline inspection and supervision of cadres to firmly establish the awareness of Party Constitution, sense of discipline and rules consciousness, politics loyalty, sense obey. Action speak Ji Ordinance to set an example of the regulations of the rule of law, strengthen supervision and accept the supervision of the firmness and consciousness, do comply with < > and < >. To firmly establish the discipline must first be disciplined, the supervisor will be subject to the 2013 广东文科数学 第12页 supervision of "concept, and consciously safeguard and implement party compasses party, take the lead in practicing" three strict real strict, so loyal, clean, play. To be good at learning, the Constitution and the < code > as morality, politics and brought to fruition; to implement < >, do not want to, dare not, not with disciplinary ruler to supervision; to discipline a ruler, often the control inspection, and consciously in the ideological red line to draw the row Ming Good accumulation is indeed the bottom line, so that the heart has fear, said to have quit, the line has ended. Attached: indifferent to heart, calmly to the table in our life, there are many unpredictable things will happen, some good, some bad things, we cannot control is powerless to stop, but with time, you will find in life sometimes turns out to be not good, some bad things finally turned out to be a good thing, but then we muddy however did not know, this is the life teach us things. 1, life can be complex, can also be simple. Want simple life of precipitation, to have enough time to reflect, to make Become more perfect. Life is the most important thing is not to win, but the struggle; not to have conquered, but to have fought well. 2, the plain is the background of life. Live a plain life, give up on themselves is not a coward, but the wise answers; not disillusioned after the heart, such as ashes, but experience the storm after the enlightenment; not unrewarding perfunctorily, but calm attitude of life of unrestrained self-confidence. Plain living, there is no noise noisy, no earthly troubles, more did not fill in the discontent of desire, 2013 广东文科数学 第13页 some just a calm, a calm. 3, memory of heart will not good things to erase the, life is a When no movie, pain is a beginning, the struggle is a kind of process, death is a kind of ending. Give up this giving up is the helpless, do not give up the abandoned, do not give up this giving up is ignorance, do not give up should not give up is persistent. 4, a thing figured is heaven, think impassability is hell. Since the living, to live better. Sometimes we because of too narrow-minded, too care around the chores and penny wise and pound foolish, not worth the candle. Some things to attract trouble and worry, completely depends on how we look at and deal with it. Don't always take everything back to things, and don't get into a blind alley, don't want to face, don't be narrow-minded. Poke to care, is a kind of open-minded, a free and easy. 5, I am not afraid of others behind me a knife, I afraid to look back and see stab me, is my intention to treat people; I am not afraid of the truth to tell the best friend, I'm afraid he turned to it as a joke to tell don't 6, when we are in a positive frame of mind, you will find many good things; and when we are in a negative state of mind, you will find many depressed things; life happy and worry, all is you of life attitude, optimistic, good luck; loss of sink, Eritrea company. When you are in adversity, may wish to change a point of view to think everything over to the good Think, because good mentality decided the fate of the! 7, people are tired, rest; heart tired, calm. Grow up, mature, this society read. Tired and sad, squat down, to their a hug. Because the world no one can 2013 广东文科数学 第14页 sympathize with you, have mercy on you. You cry, tears is your own; you pain, no one can understand. Then you only tears to smile. 8, each people have youth, Each youth are a story, the life of the world never gets easier, I want what, wish the world all know, as has been the same; now want anything, for fear that others know, or like to lose the same. 9, the heart move, everything in the world is followed by birth, Rangrang, important thing is often the most difficult to open one's mouth, because words will reduce its importance; to let strangers people care about your life in the good things, the original is not easy 10, do not blame, do not laugh at who, also don't envy who. Like a person is a kind of feeling, not like a person is true. The truth is easy to explain, I feel Is unspeakable. The best travel life is that you in a strange place found a long lost touched. 11, happy life not in the bustling in, and in the peace of mind; no matter how many grievances, how uncomfortable, and ultimately to heal themselves or their own, others may got you to comfort, but never know your heart is how wanjianchuanxin. 12, ma'am, like a movie, learn to appreciate, learn to be grateful, learn tolerance, and goodness, helping others. Instead of accusing the society, as into one; and an exception is better to give than to what 13, don't envy him A sum of, don't lose your life and the life, respectively is: the former is a we experienced cannot escape in a day finally will last minute, while the latter is our persistent, we want to cherish the memory of those people and things. 2013 广东文科数学 第15页 14, learn to smile, learn to strong, the world you know so many people, so many people and you are, you cannot change also can't let everyone like you, so also do not want to do. Life is too short to go crazy to love to go to waste, to chase the dream to regret. 15, when temper, a blessing to go. A wounding elegant people, the key is to control their own emotions. With the mouth is the most stupid behavior. A control negative emotions than a can take a city more powerful water flow slow, language is expensive. People spent two years of time to learn to speak, but to spend a few years time to shut up. That is a kind of ability, that is a kind of wisdom. 16, life is not perfect, sometimes, growth is not a cry, not an eyeful of tears, there is no trace of emotion, there is no gleam of