2013年普通高等學(xué)校夏季招生全國統(tǒng)一考試數(shù)學(xué)(廣東卷)理科數(shù)學(xué)第Ⅰ卷(共40分)一.選擇題：本大題共8題,每小題5分,共40分.在每小題給出的四個選項中,只有一項是符合題目要求的.1.設(shè)集合M＝{x|x2＋2x＝0,x∈R},N＝{x|x2－2x＝0,x∈R},則M∪N＝().A.{0}B.{0,2}C.{－2,0}D.{－2,0,2}2.定義域為R的四個函數(shù)y＝x3,y＝2x,y＝x2＋1,y＝2sinx中,奇函數(shù)的個數(shù)是().A.4B.3C.23.若復(fù)數(shù)z滿足iz＝2＋4i,則在復(fù)平面內(nèi),z對應(yīng)的點的坐標是().A.(2,4)B.(2,－4)C.(4,－2)D.(4,2)EMBEDEquation.DSMT44.已知離散型隨機變量X的分布列為X123P則X的數(shù)學(xué)期望E(X)＝().A.B.2C.D.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT45.某四棱臺的三視圖如圖所示,則該四棱臺的體積是().A.4B.C.D.6EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT46.設(shè)m,n是兩條不同的直線,α,β是兩個不同的平面.下列命題中正確的是().A.若α⊥β,mα,nβ,則m⊥nB.若α∥β,mα,nβ,則m∥nC.若m⊥n,mα,nβ,則α⊥βD.若m⊥α,m∥n,n∥β,則α⊥β7.已知中心在原點的雙曲線C的右焦點為F(3,0),離心率等于,則C的方程是().A.B.C.D.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT48.設(shè)整數(shù)n≥4,集合X＝{1,2,3,…,n},令集合S＝{(x,y,z)|x,y,z∈X,且三條件x＜y＜z,y＜z＜x,z＜x＜y恰有一個成立}.若(x,y,z)和(z,w,x)都在S中,則下列選項正確的是().A.(y,z,w)∈S,(x,y,w)SB.(y,z,w)∈S,(x,y,w)∈SC.(y,z,w)S,(x,y,w)∈SD.(y,z,w)S,(x,y,w)S第Ⅱ卷(共110分)二.填空題：本大題共7小題,考生作答6小題,每小題5分,滿分30分.(一)必做題(9～13題)9.不等式x2＋x－2＜0的解集為__________.10.若曲線y＝kx＋lnx在點(1,k)處的切線平行于x軸,則k＝__________.EMBEDEquation.DSMT4EMBEDEquation.DSMT411.執(zhí)行如圖所示的程序框圖,若輸入n的值為4,則輸出s的值為__________.12.在等差數(shù)列{an}中,已知a3＋a8＝10,則3a5＋a7EMBEDEquation.DSMT413.給定區(qū)域D：令點集T＝{(x0,y0)∈D|x0,y0∈Z,(x0,y0)是z＝x＋y在D上取得最大值或最小值的點},則T中的點共確定__________條不同的直線.EMBEDEquation.DSMT4(二)選做題(14～15題,考生從中選做一題)14.(坐標系與參數(shù)方程選做題)已知曲線C的參數(shù)方程為(t為參數(shù)),C在點(1,1)處的切線為l,以坐標原點為極點,x軸的正半軸為極軸建立極坐標系,則l的極坐標方程為__________.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT415.(幾何證明選講選做題)如圖,AB是圓O的直徑,點C在圓O上.延長BC到D使BC＝CD,過C作圓O的切線交AD于E.若AB＝6,ED＝2,則BC＝__________.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4三.解答題：本大題共6小題,滿分80分.解答須寫出文字說明.證明過程和演算步驟.16.(本小題滿分12分)已知函數(shù),x∈R.(1)求的值；(2)若cosθ＝,θ∈,求.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT417.(本小題滿分12分)某車間共有12名工人,隨機抽取6名,他們某日加工零件個數(shù)的莖葉圖如圖所示,其中莖為十位數(shù),葉為個位數(shù).(1)根據(jù)莖葉圖計算樣本均值；(2)日加工零件個數(shù)大于樣本均值的工人為優(yōu)秀工人.根據(jù)莖葉圖推斷該車間12名工人中有幾名優(yōu)秀工人？(3)從該車間12名工人中,任取2人,求恰有1名優(yōu)秀工人的概率.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT418.(本小題滿分14分)如圖(1),在等腰直角三角形ABC中,∠A＝90°,BC＝6,D,E分別是AC,AB上的點,CD＝BE＝,O為BC的中點.將△ADE沿DE折起,得到如圖(2)所示的四棱錐A′BCDE,其中A′O＝.圖(1)圖(2)(1)證明：A′O⊥平面BCDE；(2)求二面角的平面角的余弦值.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT419.(本小題滿分14分)設(shè)數(shù)列{an}的前n項和為Sn.已知a1＝1,,n∈N*.(1)求a2的值；(2)求數(shù)列{an}的通項公式；(3)證明：對一切正整數(shù)n,有.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT420.(本小題滿分14分)已知拋物線C的頂點為原點,其焦點F(0,c)(c＞0)到直線l：x－y－2＝0的距離為.設(shè)P為直線l上的點,過點P作拋物線C的兩條切線PA,PB,其中A,B為切點.(1)求拋物線C的方程；(2)當點P(x0,y0)為直線l上的定點時,求直線AB的方程；(3)當點P在直線l上移動時,求|AF|·|BF|的最小值.EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.DSMT4EMBEDEquation.