福建省福州市2015届高中毕业班第二次质量检测数学理试题.pdf

·1· 2015 ?福州?高中?业班质?检测 理理科科数数学学能能力力测测试试 ??完完卷卷时时间间??120 ??钟钟??满满????150 ???? 注注意意??项项?? 1?本科考试?试题卷和答题卷?考生须在答题卷?作答?答题前?请在答题卷的密封线内填写 学校、班级、准考证号、姓?? 2? 本试卷??第?卷(选择题)和第?卷(非选择题)两部?? 全卷满? 150 ?? 考试时间 120 ?钟? 参参考考??式式?? 1?样本数据 12 ,,, n x xxL的标准差 ()()() 222 12 1 n sxxxxxx n  =−+−++−  L? 其中x?样本?均数? 2?球的表面?、体??式? 2 4SR= π, 3 4 3 VR=π? 其中R?球的半径? 第第??卷卷 ??选选择择题题 共共 50 ???? 一一、、选选择择题题??本本大大题题共共 10 小小题题????小小题题 5 ????共共 50 ????在在??小小题题所所给给的的四四个个选选项项中中有有 且且只只有有一一个个选选项项是是??确确的的??把把??确确选选项项涂涂在在答答题题卡卡的的相相??位位置置???? ?? 1? 已知全集U = R?集合 {} 22Mxx=−??任意的实数x?恒有()( ) f xTf x+= ?立? ? ?于任意给定的?数M?都?在实数 0 x?使得() 0 f xM… ? ? 函数( ) f x的图象?至少?在?个点?使得该函数在这些点处的?线重合? 其中?确结论的序号是 ★ ★ ★ (请把所有?确结论的序号都填?)? ??、、解解答答题题??本本大大题题共共6小小题题??共共80????解解答答??写写出出文文??说说明明、、证证明明过过程程或或演演算算??骤骤?? ?? 16???本本小小题题满满??13???? 已知函数( )3sincos(0)f xxxωωω=−的图象?直线2y=的相邻两个交点之间的 距离?π? ???求函数( )f x的单调递增?间? ????ABC∆的内角,,A B C所?的边?别是, ,a b c?若( )2,f A=3ab=?求角 第8题图 ·3· B的大小? 17???本本小小题题满满??13???? 调查表明?中?人的?就感?收入、学历、职业的满意度的指标有极强的相关性?现 将这?项的满意度指标?别记?, ,x y z?并?它们进行?化?0表示?满意?1表示基本满意?2表示 满意?再用综合指标wxyz=++的值评定中?人的?就感等级?若4w…?则?就感?一级?若 23w剟?则?就感??级?若01w剟?则?就感??级??了了解目前某群体中?人的?就感 情况?研究人员随机采?了该群体的10?中?人?得到如?结果? 人员编号 1 A 2 A 3 A 4 A 5 A (), ,x y z ()1,1,2 ()2,1,1 ()2,2,2 ()0,1,1 ()1,2,1 人员编号 6 A 7 A 8 A 9 A 10 A (), ,x y z ()1,2,2 ()1,1,1 ()1,2,2 ()1,0,0 ()1,1,1 ???在这10?被采?者中任取两人?求这两人的职业 ．． 满意度指标z相?的概率? ???从?就感等级是一级的被采?者中任取一人?其综合指标?a?从?就感等 级?是一级的被采?者中任取一人?其综合指标?b?记随机变?Xab=−?求X的??列及其数 学期望? 18???本本小小题题满满??13???? 已知一个空间几何体的直观图和?视图?尺?如图所示? ? ????点M?棱PD中点?求证?EM∥?面ABCD? ???线段PD?是否?在一点N?使得直线BN??面PCD所?角的?弦值等于 2 5 ？若?在?试确定点N的位置?若??在?请说明理由? ·4· 19???本本小小题题满满??13???? 如图?已知椭圆 22 22 :1 xy ab Γ+=?0ab?的离心率 1 2 e=?点,F A?别?椭圆Γ的 左焦点和右顶点?且3AF=? ???求椭圆Γ的方程? ???过点F作一条直线l交椭圆Γ于,P Q两点?点Q关 于x轴的??点?Q′?若PFAQ′∥?求证? 1 2 PFAQ′=? 20???本本小小题题满满??14???? 已知函数( ) x f xaa x=−⋅?ea…?e2.71828=⋅⋅⋅?自然?数的?数? ????ea=时?求函数( )f x在点(1,(1))f处的?线方程? ????*n∈N ??较 (1) ln 2 n n a + ?ln(1)ln(21)ln(31)ln(1)aaana−+−+−++−L的 大小?并??证明? 21?本本题题??有有??1?? ??2?? ??3????个个选选考考题题????题题 7 ????请请考考生生任任选选 2 题题作作答答??满满?? 14 ????若若多多做做?? 则则按按所所做做的的前前两两题题计计???? ?1? ?本小题满?7??选选修修 4??2??矩矩阵阵??变变换换 已知矩阵A的逆矩阵 1 22 22 22 22 A−    =  −  ? ???求矩阵A? ???求曲线1xy=在矩阵A所??的线性变换作用?所得的曲线方程? ?2? ?本小题满?7??选选修修 4??4??坐坐标标系系??参参数数方方程程 已知曲线 1 C的参数方程? 22cos , 2sin x y α α =+  =  ?α?参数? ?在?面直角坐标系中??坐 标原点?极点?x轴的非负半轴?极轴建立极坐标系?曲线 2 C的极坐标方程? π cos2 2 4 ρθ  +=   ? ???把 1 C的参数方程化?极坐标方程? ???求 1 C? 2 C交点的极坐标?0,02πρθ的最小值?3? ???求a的值? ???求?等式14xax−++ „的解集? x y P Q' Q A F O 第 19 题图 ·5· 2015 ??福福州州??高高中中??业业班班质质??检检测测 理理科科数数学学能能力力测测试试参参考考答答案案及及评评分分细细则则 第第??卷卷 ??选选择择题题 共共 50 ???? 一一、、选选择择题题??本本大大题题共共 10 小小题题????小小题题 5 ????共共 50 ???? ?? 1? A 2? B 3? B 4? C 5? C 6?D 7?D 8? A 9? C 10? B 第第??卷卷 ??非非选选择择题题 共共 100 ???? ??、、填填空空题题??本本大大题题共共 5 小小题题????小小题题 4 ????共共 20 ???? ?? 11?34i− 12?2 13?10xy+− =或10xy−− = 14?3 15???? ??、、解解答答题题??本本大大题题共共 6 小小题题??共共 80 ????) 16?本小题?要考查?角函数的图象?性质???性、周期性、单调性? 、两角差的?弦?式、利用 ?弦定理解?角形等基础知识?考查运算求解能力?考查数形结合思想、化??转化思想、函数? 方程思想?满?13?? 解?(?)因?( )3sincos(0,)f xxxxωωω=−∈R ? 所? π ( )2sin() 6 f xxω=−? ··········································································· 1? 所?函数( )f x的最大值?2? ······································································ 2? 因?函数( )f x的图象?直线2y=的相邻两个交点之间的距离?π? 所?πT=? ····························································································· 3? 所? 2π π ω =?解得2,ω= ············································································· 4? 所? π ( )2sin(2). 6 f xx=− ? πππ 2 π22 π, 262 kxkk−−+∈Z剟? ······························································· 5? 解得 ππ ππ, 63 kxkk−+∈Z剟? 所?函数( )f x的单调递增?间是 ππ [ π, π], 63 kkk−+∈Z? ···································· 6? (?)由(?)知? π ( )2sin(2). 6 f xx=− 在ABC∆中?因?( )2,f A= 所? π 2sin(2)2, 6 A−= ················································································· 7? 所? π sin(2)1, 6 A−= 因?0πA?所?AB?所? π 0 3 B= ⋅ uuuv uv uuuv uv uuuvuv ····················································· 10? ·8· 2222 222 5 5(2 )(22 )( )5984λλλλλ === ⋅+−+⋅−+ ? ·························· 12? 所? 2 9810λλ−− =? 解得1λ=或 1 9 λ= −(舍去)? 因?? 线段PD??在一点N? ?N点?D点重合时? 直线BN??面PCD所?角的?弦值等于 2 5 ? ············································································································· 13? 19?本小题?要考查椭圆的标准方程?性质、直线?椭圆的位置关系、推理?证明等基础知识?考 查推理论证能力、运算求解能力?考查化??转化思想、数形结合思想等?满?13?? 解:????椭圆Γ的半焦距?c?依题意?得 222 1 , 2 3, . c a ac bac  =    +=   =−  ································ 2? 解得 2 2, 3. a b =   =  ····························································································· 3? 所?椭圆Γ的方程? 22 1 43 xy +=? ································································· 4? ???方法一?依题意得? PQ?坐标轴?垂直??()() 1122 ,,,P x yQ xy?因?点Q?点 Q′ 关于x轴 ??,所?() 22 ,Q xy′−?由???讨论可知?()()2,0 ,1,0AF −? 因?PF AQ′ ∥?所?直线FQ?直线 AQ′ 的斜率相等?故 22 22 12 yy xx − = +− ? ············· 6? 解得 2 1 2 x =? ···························································································· 7? 又因?点() 22 ,Q xy在椭圆Γ??所? 2 3 5 4 y =?或 2 3 5 4 y = −? ····。