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【附20套高考模拟试题】【全国百强校】福建省福建师范大学附属中学2020届高三下学期高考模拟(最后一模)英

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福建省福州市福建师范大学附属中学2024届高三下学期校模拟考试数学试题(含答案)

福建省福州市福建师范大学附属中学2024届高三下学期校模拟考试数学试题(含答案)

福建省福州市福建师范大学附属中学2024届高三下学期校模拟考试数学试时间：120分钟满分：150分命题：高三集备组审核：高三集备组一、单项选择题：本题共8小题，每小题5分，共40分．在每小题给出的选项中，只有一个选项是符合题目要求的．1．设集合，则等于A ．B ．C ．D ．或2．已知等差数列满足，则A ．B ．C ．D ．3．若函数是奇函数，则的值为A ．1B ．-1C ．D ．04．将甲、乙、丙、丁4人分配到3个不同的工作岗位，每人只去一个岗位，每个岗位都要有人去，则甲、乙二人分别去了不同岗位的概率是A．B ．C ．D ．5．没为单位向量，在方向上的投影向量为，则ABCD ．6．已知，则A ．B ．C ．D ．7．如图，设拋物线的焦点为，不经过焦点的直线上有三个不同的点A ，B ，C ，其中点A ，B 在抛物线上，点在轴上，则与的面积之比是{}{}2210,log 0A x x B x x =->=>∣∣A B ⋂{0}x x >∣{1}x x >∣{1}x x <-∣{1xx <-∣1}x >{}n a 12j 1010a a a a ++++= 11010a a +>11010a a +<3990a a +=5151a =(()ln f x ax =a 1±13122356,a b a b 12b -|2|a b -=311(),(),()552P A P AB P A B === ∣()P B =1525354524y x =F C y BCF ACFA．B ．C ．D ．8．在中，为内一点，，则A ．BCD ．二．多项选择题：本题共3小题，每小题6分，共18分．在每小题给出的四个选项中，有多个选项符合题目要求，全部选对的得6分，部分选对的得部分分，有选错的得0分．9．已知复数，下列结论正确的是A ．若，则B ．C ．若，则或D ．若且，则10．冬末春初，人们容易感冒发热，某公司规定：若任意连续7天，每天不超过5人体温高于，则称没有发生群体性发热．根据下列连续7天体温高于人数的统计量，能判定该公司没有发生群体性发热的为A ．中位数是3，众数为2；B．均值小于1，中位数为1；C ．均值为3，众数为4；D ．均值为2．11．已知，则A ．B ．C ．D ．三、填空题：本题共3小题，每小题5分，共15分．12．已知圆台的上、下底面的面积分别为，侧面积为，则该圆台的高为______.||1||1BF AF ++||1||1BF AF --22||1||1BF AF ++22||1||1BF AF --ABC 120,2,ACB BC AC D ︒∠==ABC ,120AD CD BDC ︒⊥∠=tan ACD ∠=12,z z 12z z =2212z z =1212z z z z -=-120z z =10z =20z =10z ≠12z z =2121z z =37.3C ︒37.3C ︒12212log ,log 2baa b ⎛⎫== ⎪⎝⎭22a ba b -+=+22b aa b -+=+121eba+>112eab->4π,36π64π13．的展开式中常数项为______．14．已知函数在区间上单调，其中为正整数，，且．则图象的一个对称中心是______；若的值为______．四、解答题：本题共5小题，共77分．解答应写出文字说明、证明过程或演算步骤．15．（13分）按照《中华人民共和国环境保护法》的规定，每年生态环境部都会同国家发展改革委等部门共同编制《中国生态环境状况公报》，并向社会公开发布．下表是年份2017年2018年2019年2020年2021年年份代码123456.45.55.04.83.8（1）求2017-2021年年份代码与的样本相关系数（精确到0.01）；（2）请用样本相关系数说明该组数据中与之间的关系可用一元线性回归模型进行描述，并求出关于的经验回归方程；（3）预测2024年的酸雨区面积占国土面积的百分比．附：①回归直线的斜率和截距的最小二乘法估计公式分别为：,②样本相关系：③参考数据：16．（15分）已知函数．（1）求曲线在点处的切线方程；（2）若恒成立，求的值：17．（15分）如图，在三棱柱中，，E ，F 分别为的中点，且421x x ⎛⎫+- ⎪⎝⎭()sin()f x x ωϕ=+π2π,63⎛⎫⎪⎝⎭ωπ||2ϕ<ππ32f f ⎛⎫⎛⎫=- ⎪ ⎪⎝⎭⎝⎭()y f x =π4f ⎛⎫=⎪⎝⎭ϕix iy i x i y y x y x ()()()121ˆˆ,niii n i i x x yy bay bx x x ==--==--∑∑ nx y r =5521170.6,6ii i i i x yy ====≈∑∑()ln(1)()f x ax x a =--∈R ()y f x =(0,(0))f ()0f x ≥a 111ABC A B C -1AC AB ==11,AC BB平面．（1）求棱BC 的长度；（2）若，且的面积，求二面角的正弦值．18．（17分）设是双曲线的左焦点，经过的直线与相交于M ，N 两点.（1）若M ，N 都在双曲线的左支上，求面积的最小值．（2）是否存在轴上一点，使得为定值?若存在，求出点的坐标；若不存在，说明理由．19．（17分）定义首项为1且公比为正数的等比数列为“—数列”．（1）已知等比数列满足：，求证：数列为“—数列"'；（2）已知数列满足：，其中为数列的前项和．①求数列的通项公式；②设为正整数，若存在“—数列”，对任意正整数，当时，都有成立，求的最大值．EF ⊥11AA C C 111BB A B ⊥1A FC 1A FC S =11B A F C --F 221x y Γ-=：F ΓOMN x P PM PN P M {}()*N n a n ∈245321,440a a a a a a =-+={}n a M {}n b 111221,n n n b S b b +==-n S {}n b n {}n b m M {}()*n c n ∈N k k m ≤1k k k c b c +≤≤m2023-2024年高三数学校模拟考参考答案1-8BCCDADBB9．BCD10．BD11．AD12．13．4914．15．【详解】（1）由已知可得，，，（2）由小问1知，与的相关系数接近1，所以与之间具有极强的线性相关关系，可用线性回归模型进行描述．由小问1知，，，所求经验回归方程为．（3）令，则，预测2024年的酸雨区面积占国土面积的百分比为．16．【详解】（1）由，得，因为，所以曲线在点处的切线方程为；（2），①当时，，不符合题意．5π,012⎛⎫⎪⎝⎭π61234535x ++++==522222216.4 5.5 5.0 4.8 3.8 5.112345555ii y x=++++===++++=∑555 5.90.986x x y y x y xyr ----===≈≈y x 0.98,r r ≈-y x ()()()551155222115 5.9ˆ0.59105iii ii i i ii i x x y y x y xybx x xx ====----====---∑∑∑∑ˆˆ 5.1(0.59)3 6.87ay bx =-=--⨯=ˆ0.59 6.87yx =-+8x =ˆ0.598 6.87 2.15y=-⨯+=2.15%()ln(1)f x ax x =--1()(1)1f x a x x'=+<-(0)0,(0)1f f a '==+()y f x =(0,(0))f (1)y a x =+11()(1)11ax a f x a x x x'-++=+=<--0a ≥(1)ln 20f a -=--<②当时，令，解得，当时，在区间上单调递减，当时，在区间上单调递增，所以当时，取得最小值；若恒成立，则，设，则，当时，在区间上单调递增，当时，在区间上单调递减，所以，即的解为．所以；17．【详解】（1）取中点，连接，分别为的中点，则且，又为三棱柱，且分别为的中点，则且，可得且，即四边形DEFB 为平行四边形，故，又平面，则平面，平面，可得，又为AC 的中点，则为等腰三角形，．（2）由（1）可知：，且，即，0a <()0f x '=11x a=+1,1x a ⎛⎫∈-∞+⎪⎝⎭()0,()f x f x '<1,1a ⎛⎫-∞+ ⎪⎝⎭11,1x a ⎛⎫∈+⎪⎝⎭()0,()f x f x '>11,1a ⎛⎫+ ⎪⎝⎭11x a=+()f x 111ln()f a a a ⎛⎫+=++- ⎪⎝⎭()0f x ≥1ln()0a a ++-≥()1ln()(0)x x x x ϕ=++-<11()1x x x xϕ'+=+=(,1)x ∈-∞-()0,()x x ϕϕ'>(,1)-∞-(1,0)x ∈-()0,()x x ϕϕ'<(1,0)-()(1)0x ϕϕ≤-=1ln()0a a ++-≥1a =-1a =-AC D ,ED BD ,D E 1,AC AC 1//DE AA 112DE AA =111ABC A B C - F 1BB 1//BF AA 112BF AA =//DE BF DE BF =//EF DB EF ⊥ 11AA C C DB ⊥11AA C C AC ⊂11AA C C DB AC ⊥D ABC 1BC AB ∴==1BC AB ==AC =222AB BC AC +=，则可得，且，平面平面，则，，由（1）知平面平面，则，又，则又，则，平面，平面，平面，则，且，可得，为直角三角形，则以为坐标原点，向量方向为x 轴，y 轴，z 轴正方向建立空间直角坐标系，则，可得，AB BC ∴⊥EF DB ==1111A B B C⊥EF ⊥ 111,AAC C AC⊂11AA C C 1EF A C ⊥1111122A FCS AC EF AC ∴=⋅== 12A C =DB ⊥111,AAC C AA ⊂11AA C C 1DB AA ⊥11//AA BB 1DB BB ⊥11111,//BB A B AB A B ⊥ 1BB AB ⊥,,AB DB B AB DB ⋂=⊂ABC 1BB ∴⊥ABC AC ⊂ABC 1BB AC ⊥11//AA BB 1AA AC ⊥1AA C ∴ 1AA ==1B 11111,,B C B A B B 1B xyz -111(0,0,0),(0,1,0),(1,0,0),B A C C B F ⎛ ⎝110,,(1,A F A C ⎛=-=- ⎝设平面的一个法向量为，则，令，则，可得，平面的一个法向量为，设二面角的平面角为，可得，，故二面角18．【详解】（1）设直线的方程为．由,由根与系数的关系可知①．此时．原点到直线的距离为，此时．由都在双曲线的左支上知，得，令，则，由于，所以当，即时，此时取最小值，则，当，即时，等号成立．1A FC 1(,,)n x y z = 111100n A F y z n A C x y ⎧=-=⎪⎨⎪=-+=⎩ 1y =1,x z =-=1(n =-11B A F 2(1,0,0)n =11B A F C --(0,π)θ∈121211|cos |212n n n n θ===⨯ sin θ∴==11B A F C --MN ()()1122,,,x my M x y N x y =-221x my x y ⎧=-⎪⎨-=⎪⎩()22110(1)m y m --+=≠±1212211y y y y m +==-()2221||1m MN m +===-O MN d =()222111||221OMNm S d MN m +===- ,M N ()121212220,01x x m y y x x m -+=+-=<=>-11m -<<21(10)m t t -=-≤<2221111144244OMNS t t t ⎛⎫⎛⎫=+=+- ⎪ ⎪⎝⎭⎝⎭ 1(,1]t ∈-∞-11t=-1t =-OMN S ≥ 1t =-0m =（2）假设存在这样的定点．当直线的斜率不为0时，由（1）知②将①代入②可得,此时要想，得．即存在这样的定点满足题意．当直线的斜率为0时，易知，若，则，满足题意．综上，存在满足题意．19．【详解】（1）设等比数列的公比为，所以由，得，解得，因此数列为“—数列”；（2）①因为，所以，(,0)P n ()()()()()()112212121212,,PM PN x n y x n y x n x n y y my n my n y y =--=--+=----+()()2212121))m y y m n y y n =+-++++ 2)PM PN n =++ PM PN 11=-n =12PM PN =- P ⎛⎫⎪ ⎪⎝⎭2(1)(1)1PM PN n n n =+-=- P ⎛⎫ ⎪ ⎪⎝⎭12PM PN =- P ⎛⎫⎪ ⎪⎝⎭{}n a q 10,0a q ≠≠245321440a a a a a a =⎧⎨-+=⎩244112111440a q a q a q a q a ⎧=⎨-+=⎩112a q =⎧⎨=⎩{}n a M 1122n n n S b b +=-0n b ≠由得，则，由，得，当时，由，得，整理得，所以数列是首项和公差均为1的等差数列，因此，数列的通项公式为；②由①知，，因为数列为“—数列”，设公比为，所以，因为，所以，其中，当时，有；当时，有，设，则，则当时，，当时，，故在上单调递增，在上单调递减，因为，所以，取时，，即，令，则，令，则，1111,b S b ==212211b =-22b =1122n n n S b b +=-()112n n n n n b b S b b ++=-2n ≥1n n n b S S -=-()()111122n n n nn n n n n b b b b b b b b b +-+-=---112n n n b b b +-+={}n b {}n b ()*n b n n =∈N *,k b k k =∈N {}n c M q 11,0c q =>1k k k c b c +≤≤1k k q k q -≤≤1,2,3,,k m = 1k =1q ≥2,3,,k m = ln ln ln 1k kq k k ≤≤-ln ()(1)x f x x x =>21ln ()xf x x '-=(1,e)x ∈()0f x '>(e,)x ∈+∞()0f x '<()f x (1,e)(e,)+∞ln 2ln 8ln 9ln 32663=<=max ln 3()(3)3f k f ==q =1,2,3,4,5k =ln ln kq k…k k q ≤ln ()(1)1x g x x x =>-2211(1)ln 1ln ()(1)(1)x x xx x g x x x '----==--1()1ln h x x x =--22111()0xh x x x x'-=-=<故在上单调递减，则，即在上恒成立，即在上单调递减，则，即，因此所求的最大值不小于5，若，分别取，得，且，从而，且，所以不存在，因此所求的最大值小于6，故的最大值为5．()h x (1,)+∞()(1)1100h x h ≤=--=()0g x '<(1,)+∞()g x (1,)+∞min ln 5ln125ln 81ln 3()(5)412123g k g ===<=1ln ln 1k k q q k k -≤≤-m 6m ≥3,6k =33q ≤56q ≤15243q …15216q …q m m。

福建省福建师大附中2020届高考三轮模拟试卷(数学文科)

福建省福建师大附中2020届高考三轮模拟试卷(数学文科)

福建省福建师大附中2020届高考三轮模拟试卷数学文科试题参考公式：锥体体积公式13V Sh =，其中S 为底面面积，h 为高 第Ⅰ卷（选择题 共60分）一、选择题：本大题共12小题，每小题5分，共60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.已知函数x x f -=1)(定义域为M ，x x g ln )(=定义域为N ，则=N M （****）A ．{}1|≤x xB ．{}10|≤<x xC ．{}10|<<x xD ．{}10|≤≤x x 2. 若0a b >>，则下列不等式成立的是（****） A. a b +< <C. 1122log log a b <D. 0.20.2a b>3．若函数()y fx =是函数2x y=的反函数，则()2f 的值是（****）A ．4B ．2C ．D ．0 4. 设a,β分别为两个不同的平面，直线la ，则“l 丄β”是“a 丄β成立的（****）A.充分不必要条件B.必要不充分条件C.充要条件D.既不充分也不必要条件 5.要得到函数cos(21)y x =+的图象,只要将函数cos 2y x =的图象（****） A ．向左平移1个单位 B ．向右平移1个单位9.若双曲线22221(0,0)x y a b a b -=>>与直线y =无交点，则离心率e 的取值范围（****）A ．(1,2)B ．(1,2]C ．D ．10.若P 为ABC ∆内一点，且20PB PC PA ++=，在ABC ∆内随机撒一颗豆子，则此豆子落在PBC ∆内的概率为（****）A ．12 B ．13 C ．14 D ．2312．已知[)x 表示大于x 的最小整数，例如[)[)34, 1.21=-=-．下列命题①函数[)()f x x x =-的值域是(]0,1；②若{}n a 是等差数列，则[){}n a 也是等差数列； ③若{}n a 是等比数列，则[){}n a 也是等比数列；④若()1,4x ∈，则方程[)12x x -=有3个根． 正确的是（****）A ．②④ B ．③④ C ．①③ D ．①④第Ⅱ卷（非选择题 共90分）二、填空题：本大题共4小题，每小题4分，共16分．把答案填写在答题卡的相应位置．13．已知复数122i ,43i z z =+=-在复平面内的对应点分别为点A 、B ，则线段AB 的中点所对应的复数是****14．某班共有52人，现根据学生的学号，用系统抽样的方法，抽取一个容量为4的样本，已知3号、29号、42号同学在样本中，那么样本中还有一个同学的学号是****15.已知平面上的线段及点P ，在上任取一点Q ，线段PQ 长度的最小值称为点P 到线段的距离，记作(,)d P l ．设是长为2的线段，点集{|(,)1}D P d P l =≤所表示图形的面积为**** 16.设a,b,m 为正整数，若a 和b 除以m 的余数相同，则称a 和b 对m 同余.记()a b mod m ≡，已知220132232323a =+⨯+⨯++⨯，()3b a mod ≡，则b 的值可以是**** （写出以下所有满足条件的序号）①1007；②2019-2020；③3003；④6002三、解答题：本大题共6小题，共74分．解答应写出文字说明、证明过程或演算步骤．把解答过程填写在答题卡的相应位置． 17．（本小题满分12分）已知函数2()2cos sin 2f x x x =- （1）求函数()f x 的最小正周期和值域；（2）已知ABC ∆的内角C B A ，，所对的边分别为c b a ，，，若2,2==b a ，且,12=⎪⎭⎫⎝⎛A f 求ABC ∆的面积.18. （本小题满分12分）已知点（1,2）是函数()(01)xf x a a a =≠＞且的图象上一点，数列{}n a 的前n 项和()1n S f n =-.（1）求数列{}n a 的通项公式；（2）将数列{}n a 前30项中的第3项，第6项，…，第3k 项删去，求数列{}n a 前30项中剩余项的和.19．（本小题满分12分）市民李生居住在甲地,工作在乙地,他的小孩就读的小学在丙地,三地之间的道路情况如图所示.假设工作日不走其它道路,只在图示的道路中往返,每次在路口选择道路是随机的.同一条道路去程与回程是否堵车互不影响.假设李生早上需要先开车送小孩去丙地小学，再返回经甲地赶去乙地上班，（1）写出李生可能走的所有路线；（比如DDA 表示走D 路从甲到丙，再走D 路回到甲，然后走A 路到达乙）;（2）假设从丙地到甲地时若选择走道路D 会遇到拥堵，并且从甲地到乙地时若选择走道路B 也会遇到拥堵，其它方向均通畅，但李生不知道相关信息，那么从出发到回到上班地没有遇到过拥堵的概率是多少？20．（本小题满分14分）如图，AB 是半圆O 的直径，C 是半圆O 上除A 、B 外的一个动点，⊥DC 平面ABC ,BE DC //，BE CD =，4=AB ，41tan =∠EAB ． ⑴证明：平面⊥ADE 平面ACD ；⑵试探究当C 在什么位置时三棱锥ADE C -的 体积取得最大值，请说明理由并求出这个最大值．21．（本小题满分12分）如图，已知抛物线py x C 2:21=的焦点在抛物线121:22+=x y C 上． （1）求抛物线1C 的方程及其准线方程；（2）过抛物线1C 上的动点P 作抛物线2C 的两条切线PM 、PN ， 切点为M 、N ．若PM 、PN 的斜率乘积为m ，且]4,2[∈m ，求||OP 的取值范围．22.（本小题满分14分）已知函数()()121,ln 2-+==bx ax x g x x f ， （1）当0=a 且1=b 时，证明：对0>∀x ，()()x g x f ≤；（2）若2=b ，且()()()x g x f x h -=存在单调递减区间，求a 的取值范围；（3）数列{}n a ，若存在常数0>M ，*∈∀N n ，都有M a n <，则称数列{}n a 有上界。

2019-2020学年福建师范大学附属中学高三英语模拟试卷及答案

2019-2020学年福建师范大学附属中学高三英语模拟试卷及答案

2019-2020学年福建师范大学附属中学高三英语模拟试卷及答案第一部分阅读（共两节，满分40分）第一节（共15小题；每小题2分，满分30分）阅读下列短文，从每题所给的A、B、C、D四个选项中选出最佳选项AEach year, theLas Vegasconsumer electronics show, or CES, presents the latest developments in many areas of technology. The newest products are designed to make our lives easier, fun and more productive.A car with legsOne of this year’s presentations was by South Korean carmaker Hyun-dai. The company introduced a small model of a “ walking car,” which is called Elevate. It has four movable legs that can raise the main part of the vehicle high off the ground. The electric-powered vehicle is designed to be used in search-and-rescue operations during emergencies or natural disasters.Changeable people moverGermany's Mercedes presented an experimental self-driving vehicle that it claims can revolutionize transportation for people and goods. The company says the vehicle, called Vision Urbanetic, will be able lo easily change bodies depending on its desired use. Mercedes says as a ride-sharing vehicle , the futuristic-looking car can seat 12 people.Fully electric HarleyAmerican manufacturer Harley-Davidson showed off its first fully electric motorcycle, called LiveWire. The company says the bike will be able to go 177 kilometers between charges. It can reach 96 kilometers per hour in under3.5 seconds. Although Harley is known for building powerful bikes with huge, loud motors, the LiveWire will be unusually quiet.Personal robotsOne of the new robots, called Temi, is really just a computer tablet on wheels. It is designed to be a personal electronic assistant. It moves around the home and performs commands when spoken to. It can link users to friends through voice or video, connect to video or place orders for food or goods.1. Which do you probably use to search for the injured in an earthquake?A. Elevate.B. Vision Urbanetic.C. LiveWire.D. Temi.2. What is the first fully electric motorcycle produced by Harley-Davidson?A. ElevateB. Hyun-daiC. LiveWire .D. Temi .3. Why are the four products designed?A. To ease traffic jam.B. To help us socialize.C. To improve our life.D. To protect the environment.BRecycling plastic has always been a stop-start effort, and the wide variety of plastics we produce, the pollution from waste, and other limitations make recycling an economic(经济) problem. It’s estimated(估计)only 9% of plastic ever created has been recycled. But with the help of a chemical process, Canadian Miranda Wang and her company BioCellection want to change that.Stability is one of plastic greatest qualities and downside. There's limited evidence that some plastics can biodegrade (生物降解)but largely photo-plastics degrade in the sun. It’s a long process, and the truth is that we can only estimate how long it takes. Wang is looking to break the inaction and BioCellection's task is to make most plastic waste recyclable.She outlines two current methods. One is to take plastics like water bottles, wash them, cut them, melt and reconstitute them. “That's a very limited process” she says, due to the requirement that plastics be “clean” . The other, which can handle dirtier plastics and a level of pollution, is called hydrolysis(热解). Intense heat is applied to break down plastics so they can be reused as oils for energy, but “it’s not economical,’’ she says.BioCellection’s solution builds on research from over ten years ago, Wang explains, when a US studydiscovered pure polyethylene powder (聚乙烯粉)could be broken down by a catalyst (催化剂). Wang and her co-founder Jean hit upon a bacterium being able to eat plastic. In the years since, they engineered a comparable catalyst capable of doing the same job, only faster, which even works on plastics no one else can recycle at present. “We have now found a catalyst that is much cheaper than the one that was used before, Wang says.Currently focusing on plastic films like shopping hags, the three-hour process breaks clown plastic into chemicals that can act as the building blocks for more complex plastic products.“Right now we’re able to achieve about 70% transformation from plastic waste material to these chemicals,” she adds, saying they’re working to increase that figure.4. Why has so little plastic been recycled?A. Plastic is chemically stable.B. Recycling plastic led to pollution.C. We produced various plastics.D. Most photo-plastics degrade in the sun.5. What does Wang think of the two current methods?A. Highly effective and dynamic.B. Expensive and pollution-causing.C. Limited and energy-consuming.D. Widely used and recognized.6. What did BioCellection find effective to recycle plastics?A. Polyethylene powder.B. Chemical products.C. Plastic films.D. A catalyst.7. What is the text mainly about?A. Miranda Wang and her company.B. A new plastic recycling method.C. The greatest downside of plastic.D. Transformation of plastic waste.CIf you easily make mistakes when in a hurry, a new study from Michigan State University—the largest of its kind to date-found that meditation (冥想) could help you improve the situation.The research tested how open monitoring meditation (OMM)—or, meditationthat focuses awareness on feelings or thoughts as they unfold in one’s mind and body—alteredbrain activity in a way that suggested increased error recognition.“People’s interest in meditation is outpacing what science can prove in terms of effects and benefits.” said Jeff Lin, MSU psychology doctoral candidate and study co-author. “But it’s amazing to me that we were able to see how one session of a guided meditation could produce changes to brain activity in non-meditators.”“Some forms of meditation have you focus on a single object, commonly your breath, but open monitoring meditation is a bit different,” Lin said, “It has you tune inward and pay attention to everything going on in your mind and body. The goal is to sit quietly and pay close attention to where the mind travels without getting too caught up in the scenery.”Lin and his MSU co-authors—William Eckerle, Ling Peng and Jason Moser—hired more than 200 participants to test how open monitoring meditation affected how people detect and respond toerrors.The participants, who had never meditated before, were taken through a 20-minute open monitoring meditation exercise while the researchers measured brain activity through electroencephalography (脑电图), or EEG. Then, they completed a computerized distraction (分心) test.“The EEG can measure brain activity at the millisecond level, so we got precise measures of brain activity right after mistakes compared to correct responses,” Lin said. “A certain neural signal occurs about half a second after an error called the error positivity, which is linked to conscious error recognition. We found that the strength of this signal is increased in the meditators to controls.”“These findings show what just 20 minutes of open monitoring meditation can do to improve the brain’s ability to detect and pay attention to mistakes,” Moser said.8. What does the underlined word “altered” in paragraph 2 probably mean?A. Changed.B. Prevented.C. Started.D. Recorded.9. Why is open monitoring meditation different?A. It is just aimed at a single object.B. It clears your mind of everything.C. It gets too caught up in the scenery.D. It focuses on where the mind travels.10. What did the researchers do for the studyA. They hired people who had meditated before.B. They measured the participants’ brain activity.C. They reminded the participants to avoid errors.D. They had non-meditators design a distraction test.11. What is the best title for the text?A. Turn to OMM to Avoid Acting in a HurryB. You’re Able to Recognize Errors ConsciouslyC. Meditators’ Brain Proves Much More ActiveD. OMM Can Help You Make Fewer MistakesDDid you know that the average child has heard the word "no" over 20,000 times before they turn the age of three? Ironically, it is also around this time that children begin to develop enough personal character to refuse to obey. The "terrible twos" are categorized by a lack of understanding. Somewhere between three and four, children begin to acquire the skills to reason. It is during this time they watch how other children and adults reason. If we're not careful, the children will watch us model a world of "NOs".By the time a person turns eighteen, how many times have they been told no? I haven't found any studies that even attempted to track this statistic, but I'm sure if the number is 20,000 by three, then at eighteen that number has multiplied. You can do the math.Anyway, I think I know why we say no. We say no to protect. We say no to direct. We say no to stop potential confusion. However, do we sometimes say no just for the sake of saying no? Do we say no because we have internalized(内在化)all of the "NOs" we've heard over the years and we feel it is finally our time to say no to someone else?The internalized no can damage the growth process of dreams in infancy as quickly as it can weaken a three-year-old. And we wonder why we run intopeople with big, un-accomplished dreams who have a bit of a chip on their shoulder. They have to take on the 20,000 NOs. However, the thing that keeps them going is thepossibility of the power of ONE YES! Just as it only takes one book to make a writer a Pulitzer Prize Winner, it only takes one word to change the course of your day. That word is YES!12. What do we know about two- year- olds?A. They understand well.B. They often say no to others.C. They think logically.D. They don't do all they're told.13. What effect does saying no have on children?A. They lose all their dreams.B. They aren't easy to succeed.C. They don't make mistakes.D. They never say yes to others.14. Which word best describes the author's attitude to a world of "NOs"?A. Tolerant.B. Disapproving.C. Favorable.D. Carefree.15. What is the text?A. A how-to guide.B. A survey report.C. An opinion essay.D. A short story.第二节（共5小题；每小题2分，满分10分）阅读下面短文，从短文后的选项中选出可以填入空白处的最佳选项。

2020-2021学年福建师范大学附属中学高三英语模拟试题及参考答案

2020-2021学年福建师范大学附属中学高三英语模拟试题及参考答案

2020-2021学年福建师范大学附属中学高三英语模拟试题及参考答案第一部分阅读（共两节，满分40分）第一节（共15小题；每小题2分，满分30分）阅读下列短文，从每题所给的A、B、C、D四个选项中选出最佳选项AFour Best Hikes in the WorldThere's nothing like getting out and getting some fresh air on a hike. No matter whether your idea of a hike is a leisure walk or climbing the highest mountain on Earth, we've got you covered. Below are four best hikes inthe world.Torres del Paine W CircuitLocation (位置): Patagonia. ChileDistance: 37 + milesTime: 5~6 daysBest time to go: October to JanuaryThe W Circuit is one of the most recommended hikes you'll find. Not only will you appreciate the diverse landscapes and striking granite pillars (花岗岩柱子), but you'll probably meet some new friends along the way.Grand Canyon Rim - to - Rim HikeLocation: Arizona, the United StatesDistance: 48 milesTime: 1~3 daysBest time to go: May to June, September to OctoberThere's no better way to experience one of the greatest wonders in the world. Located in one of the USA's most beautiful parks, the views are ly appealing. Just make sure you're prepared for the challenge.Trek to PetraLocation: JordanDistance: 47 milesTime: 5~ 6 daysBest time to go: October to AprilTake the road less traveled through the Kingdom of Jordan and experience one of the seven wonders of the world. Hike through canyons, gorges and ridges, and see tombs and temples along the way all while avoidingcrowds of tourists.Yosemite Grand TraverseLocation: California, the United StatesDistance: 60 milesTime: 6~7 daysBest time to go: July to SeptemberKnown for some of the best hiking in the world, Yosemite National Park is famous for its views and huge sequoia (红杉) trees. Praised byNational Geographic, the Yosemite Grand Traverse will take you through waterfalls and green mountaintops.1.Which of the following is the best time for the hike in Patagonia, Chile?A.AprilB.MayC.AugustD.December2.Where should you go for a less crowded hike?A.JordanB.Patagonia, ChileC.Arizona, the United StatesD.California, the United States3.What can you do along the Yosemite Grand Traverse?A.Plant sequoia treesB.Appreciate waterfallsC.Visit local templesD.Climb granite pillarsBDogs are often referred to as “man's best friend”. But MacKenzie, a four-pound Chihuahua (奇瓦瓦狗), who was named winner of the 2020 American Hero Dog competition, is making the world a better place for humans and animals alike. Often called the “Oscars for dogs”, the award recognizes dogs who make great contributions to society.This year's competition attracted over 400 competitors from across the country. While all were impressive, it was tiny MacKenzie who won the judges' hearts. Born at a rescue shelter in Hilton, New York, in 2013, she had a cleft palate (腭裂) that required her to be tube fed for the first year of her life. A life-saving operation, performed in 2014, gave her the ability to eat and drink independently, enabling the tiny dog to focus on doing what she loved most: taking care of others.The seven-year-old Chihuahua is now gainfully “employed” by the Mid Foundation, a Rochester, New York-based non-profit organization that shelters and cares for animals born with disabilities. MacKenzie's official job is “to provide love and care for baby rescue animals born with birth defects”. The Chihuahua is good at her joband hasnurturedmany different species-from puppies to kittens to turkeys, squirrels, birds and even a goat. She acts as their mother and teaches them how to socialize, play, and have good manners.In addition to her role as an animal caretaker, MacKenzie also has the important job of greeting the foundation's volunteers and friends. The incredible dog, who has lost her ability to bark, also visits area schools to help children understand physical disabilities in both animals and people. Her heart-warming and inspiring story makes MacKenzie worthy of America's top dog honor!4. What made MacKenzie American Hero Dog?A. Being man's best friend.B. Her struggle with disabilities.C. Rescuing animals with disabilities.D. Her contributions to a better world.5. What can we infer about MacKenzie from Paragraph 2?A. Her growth path was not easy.B. She was deserted by her owner.C. She was operated on at two years old.D. She still needs taking care of by others.6. What does the underlined word “nurtured” in Paragraph 3 mean?A. Trained.B. Comforted.C. Tended.D. Abused.7. Which can be a suitable title for the text?A. MacKenzie—The Most Hard-working DogB. MacKenzie—America's “Most Heroic Dog”C. Chihuahua—Inspiration of Positive EnergyD. Chihuahua—Appeal for Animals' ProtectionCWhile the start of a new school year is always exciting, this year was even more so for some elementary school students inAuckland,New Zealand. They became the world’s first kids to be “taught” by a digital teacher.Before you start imagining a human-like robot walking around the classroom, Will is just an avatar that appears on the student’s desktop, or smartphone screen, when ordered to come.The autonomous animation platform has been modeled after the human brain and nervous system, allowing it to show human-like behavior. The digital teacher is assigned to teach Vector’s “Be sustainable with energy”— a free program forAucklandelementary schools.Just like the humans it replaced, Will is able to instantly react to the students’ responses to the topic. Thanks to a webcam and microphone, the avatar not only responds to questions the kids may have, but also picks up non-verbal cues. For instance, if a student smiles at Will, he responds by smiling back. This two-way interaction not only helps attract the students’ attention, but also allows the program’s developers to monitor their engagement, and make changes if needed.Nikhil Ravishankar believes that Will-like avatars could be a novel way to catch the attention of the next generation. He says, “I have a lot of hope in this technology as a means to deliver cost-effective, rich, educational experience in the future.”The program, in place since August 2018, has been a great success thus far. Ravishankar says, “ What was fascinating to me was the reaction of the children to Will. The way they look at the world is so creative and different, and Will really captured their attention.” However, regardless of how popular it becomes, Will is unlikely to replace human educators any time soon.8. What was special for some elementary school students inAuckland?A. A digital teacher taught them.B. They first saw something digital.C. This was the start of a new school year.D. They could get close to smartphone screen.9. What is the benefit of this two-way interaction?A. It can smile back.B. It can use microphone.C. It can talk any topic for free.D. It can change if necessary.10. What’s Ravishankar’s attitude to Will’s replacing Human educators soon?A. Optimistic.B. Doubtful.C. Unclear.D. Disapproving.11. What might be the best title for the passage?A. New High-tech Contributes to EducationB. The World’s First Digital Teacher Appears in Classroom.C. The World’s First Digital Teacher, a Help to StudentsD.New ZealandWill Replace Teachers in ClassroomsDAt the World Economic Forum last month, President Trump drew claps when he announced the United States would respond to the forum's proposal to plant one trillion(万亿) trees to fight climate change. The trillion-tree idea won wide attention last summer after a study published in the journal Science concluded thatplanting so many trees was “the most effective climate change solution to date”.If only it were true. But it isn't. Planting trees would slow down the planet's warming, but the only thing that will save us and future generations from paying a huge price in dollars, lives and damage to nature is rapid andconsiderable reductions in carbon release from fossil fuels, to net zero by 2050.Focusing on trees as the big solution to climate change is a dangerous diversion(偏离). Worse still, it takes attention away from those responsible for the carbon release that are pushing us toward disaster. For example, in the Netherlands, you can pay Shell an additional 1 euro cent for each liter of regular gasoline you put in your tank, to plant trees to balance the carbon release from your driving. That's clearly no more than disaster slightly delayed. The only way to stop this planet from overheating is through political, economic, technological and social solutions that end the use of fossil fuels.There is no way that planting trees, even across a global area the size of theUnited States, can absorb the huge amounts of fossil carbon released from industrial societies. Trees do take up carbon from the atmosphere as they grow. But this uptake merely replaces carbon lost when forests were cleared in the first place, usually long ago. Regrowing forests where they once grew can undo some damage done in the past, but even a trillion trees can't store enough carbon to head off dramatic climate changes this century.In a sharp counter argument to last summer's Paper in Science, Justin Gillis wrote in the same journal in October that the study's findings were inconsistent with the dynamics of the global carbon cycle. He warned that “the claimthat global tree restoration(复原) is our most effective climate solution is simply scientifically incorrect and dangerously misleading”.12. What do we know about the trillion-tree idea?A. It was published in a journal.B. It was proposed last summer.C. It was put forward by Trump.D. It drew lots of public attention.13. What is paragraph 3 mainly about?A. A drawback of the tree planting strategy.B. An example of balancing carbon release.C. An anecdote of making a purchase at Shell.D. A responsibility for politicians and economists.14. What was Justin Gillis's attitude towards global tree restoration?A. Indifferent.B. Opposed.C. Hesitant.D. Supportive.15. What is the best title for the text?A. Contradictory Ideas on Tree Planting.B. A Trillion Trees Come to the Rescue.C. Planting Trees Won't Save the World.D. The Best Solution to Climate Change.第二节（共5小题；每小题2分，满分10分）阅读下面短文，从短文后的选项中选出可以填入空白处的最佳选项。

【附加15套高考模拟试卷】福建省福建师大附中5月2020届高考三轮模拟试卷数学(理科)卷含答案

【附加15套高考模拟试卷】福建省福建师大附中5月2020届高考三轮模拟试卷数学(理科)卷含答案

福建省福建师大附中5月2020届高考三轮模拟试卷数学（理科）卷一、选择题：本题共12小题，每小题5分，共60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．设抛物线C ：y 2=4x 的焦点为F ，过点（–2，0）且斜率为23的直线与C 交于M ，N 两点，则FM FN ⋅u u u u r u u u r =A ．5B ．6C ．7D ．82．《九章算术》卷第七——盈不足中有如下问题：“今有垣高九尺.瓜生其上，蔓日长七寸. 瓠生其下，蔓日长一尺.问几何日相逢.”翻译为“今有墙高9尺.瓜生在墙的上方，瓜蔓每天向下长7寸.葫芦生在墙的下方，葫芦蔓每天向上长1尺.问需要多少日两蔓相遇.”其中1尺=10寸.为了解决这一问题，设计程序框图如下所示，则输出的k 的值为（ ）A ．8B ．7C ．6D ．53．已知抛物线28y x =的焦点为F ，直线(2)y k x =-与此抛物线相交于P ，Q 两点，则11FP FQ+=（ ）A ．12 B ．1C ．2D ．44．若复数z 满足(12)1i z i +=-，则复数z 为（ ）A ．1355i +B ．1355i -+C ．1355i -D ．1355i --5．如图，过抛物线22(0)y px p =>的焦点F 的直线交抛物线于点A B 、，交其准线l 于点C ，若点F 是AC 的中点，且||4AF =，则线段AB 的长为（ ）A ．203B ．163 C ．5D ．66．在ABC ∆中，内角A 、B 、C 的对边分别为a 、b 、c ，若1cos 2b a Cc =+，则角A 为 A ．60︒ B ．120︒ C ．45︒ D ．135︒7．等差数列的前项和为，若，则等于（ ）A ．B ．C ．D ．8．已知等差数列{}n a 的前n 项和2n S n bn c =++，等比数列{}n b 的前n 项和3nn T d =+，则向量(,)a c d =r的模为（ ）A ．1B 2C 3D ．无法确定9．已知等差数列{}n a 的前n 项和为n S ，686a a +=，963S S -=，则使n S 取得最大值时n 的值为( ) A ．5B ．6C ．7D ．810．三棱锥S ABC -中，,,SA BC SC AB ⊥⊥则S 在底面ABC 的投影一定在三角形ABC 的（ ） A ．内心 B ．外心 C ．垂心 D ．重心11．已知函数()()()31ln 3ln 3xx f x x ⎡⎤=-⎢⎥⎢⎥⎣⎦g ，且()20f x ->，则实数x 的取值范围是（ ） A ．(),2-∞ B ．()2,+∞C ．()(),22,-∞+∞U D ．(),-∞+∞12．已知关于x ，y 的不等式组()3{3020x y mx y x x +≥-+≥-≤，所表示的平面区域构成一个锐角三角形，则实数m 的取值范围为A ．10,2⎛⎫ ⎪⎝⎭B ．1,12⎛⎫ ⎪⎝⎭ C ．11,32⎛⎫⎪⎝⎭ D ．（0，1）二、填空题：本题共4小题，每小题5分，共20分。

福建省福州市福建师范大学附属中学2025届高三第一次模拟考试语文试卷含解析

福建省福州市福建师范大学附属中学2025届高三第一次模拟考试语文试卷含解析

福建省福州市福建师范大学附属中学2025届高三第一次模拟考试语文试卷注意事项：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，再选涂其它答案标号。

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。

1、依次填入下面一段文字横线处的语句，最恰当的一组是（）诗词对中国人来说是植入文化基因并潜移默化到日常生活中的，。

，。

，。

①文化中国未必高居于庙堂之上，却经常安住于江湖之远②在此形势下，把诗词做成一档很有参与感的节目，让很多喜欢诗词却不经意远离诗词的人们有机会重新接近诗词③然而这个时代的诱惑实在太多了，诗词正离年轻人的生活越来越远④至今在很多中国人的居家生活中，以诗词为心，以对联与书法为用⑤很多诗词语言和意象的美感，中国人可以“秒懂”，外国人可能就云里雾里⑥可以称得上是一种功德A．③②①⑥⑤④B．③⑥②⑤④①C．⑤④①③②⑥D．⑤①④②⑥③2、下面四首唐诗，各描写中华民族的一个传统节日。

如果按节令的时间顺序依次排列，正确的一项是①雨中禁火空斋冷,江上流莺独坐听②偶来人世值中元不献元都未日闲。

③九月九日望乡台他席他乡送客杯④香帐簇成排窈窕金针穿罢拜婵娟。

A．④①②③B．①④②③C．④①③②D．①③④②3、补写出下列句子中的空缺部分。

(1)苏轼《赤壁赋》中“_________________________”两句概括了曹操的军队在攻破荆州后顺流而下的军容之盛(2)李商隐《锦瑟》中，“_________________________”两句表达了“华年往事”如梦般凄迷、如杜鹃啼春般伤感。

(3)范仲淹《岳阳楼记》中，“_________________________”两句表现了古仁人不管身处朝廷还是远离君王，都有心怀天下的博大胸襟。

4、下列各句中，没有错别字且加点字的注音全部正确的一项是A．广大群众自觉遵守疫情防控各项规定，共同为遏制新型冠．（guān）状病毒疫情蔓延、保障群众生活作出了重要贡献，彰显了中华民族同舟共济．（jì）、守望相助的家国情怀。

福建师大附中2020年高考英语模拟试卷含答案

福建师大附中2020年高考英语模拟试卷含答案

高考英语模拟试卷题号I II III IV V VI总分得分一、阅读理解（本大题共15小题，共30.0分）AJust as John Lubbock said，we may sit in a library and yet be in all quarters of the earth．Here are some of the coolest libraries of the world．Stuttgart City LibraryDespite a classic white-on-white color，Stuttgart City Library is one of the most visually appealing modern libraries．The amazing modern and simple look is the product of the Korean architect Yi Eun-young．The German library opened in 2011，and cost nearly 80 million euros to construct．Kanazawa Umimirai LibraryArchitects from the Coelacanth K&H Architects studio designed the Kanazawa Umimirai Library to create the best reading environment．Heated floors，an outpouring of natural light，and other features were specifically included to achieve this goal．The wall，which consists of thousands of tiny openings，allows natural lighting into the building，improving the overall feel of the space．This Japanese library was opened in 2011，and reflects the appearance of many modern and simple spaces．Library of CongressIn 1800，Congress set up the Library of Congress in Washington D．C.. Fifteen years later，Thomas Jefferson offered his personal library to replace books that were lost when the British set the building on fire in 1814．His books，which took nearly 50 years to collect，varied from texts on science，literature and philosophy，to books in a variety of foreign languages ．These days，the Library of Congress offers more than 155 million items and the largest rare book collection in North America．State Library of VictoriaThe State Library of Victoria，located in Melbourne，Australia，is one of the largest exhibiting libraries in the world．Though it opened in 1856，major repairs took place between 1990 and 2004．The La Trobe Reading Room is one of the library′ s main attractions due to its visually amazing architecture．The lawn at the library's front attracts hundreds of city folk each day，who gather for lunch or to enjoy the statues there．The library currently contains over two million books，as well as armor（盔甲）of Ned Kelly，a folk hero against the ruling class．1.Which library has the longest history？______A. Stuttgart City LibraryB. Kanazawa Umimirai LibraryC. Library of CongressD. State Library of Victoria2.What's special about State Library of Victoria？______A. It has great architecture．B. It has a folk hero's statue in it．C. It is the world's largest exhibiting library．D. It serves as a meeting place for the citizens．3.What does the first two libraries have in common？______A. They cost a large amount of money．B. They are constructed in the modern style．C. They are designed by Korean architects．D. They feature providing readers with natural light．BA few weeks ago，I called an Uber to take me to the Boston airport for a flight home for the holidays．As I slid into the back seat of the car，the warm intonations（语调）of the driver's accent washed over me in a familiar way．I learned that he was a recent West African immigrant with a few young children，working hard to provide for his family．I could relate：I am the daughter of two Ethiopian immigrants who made their share of sacrifices to ensure my success．I told him I was on a college break and headed home to visit my parents．That's how he found out I go to Harvard．An approving eye glinted at me in the rearview window，and quickly，we crossed the boundaries of rider and driver．I became his daughter，all grown up - the product of his sacrifice．And then came the fateful（致命的）question："What do you study？" I answered "history and literature" and the pride in his voice faded，as I knew it might．I didn't even get to add "and African-American studies" before he cut in，his voice thick with disappointment，"All that work to get into Harvard，and you study history？"Here I was，his daughter，squandering the biggest opportunity of her life．He went on to deliver the age-old lecture that all immigrant kids know．We are to become doctors （or lawyers，if our parents are being generous）- to make money and send money back home．The unspoken demand，made across generations，which my Uber driver laid out plainly，is simple：Fulfill your role in the narrative（故事）of upward mobility so your children can do the same．I used to feel anxious and backed into a corner by the questioning，but now as a junior in college，I'm grateful for their support more than anything．This holiday season，I've promised myself I won't huff and get annoyed at their inquiries．I won't defensively respond with "but I plan to go to law school!" when I get unrequested advice．I'll just smile and nod，and enjoy the warmth of the occasion．4.What disappointed the driver？______A. The author's majors in history and literature．B. The school that the author is attending．C. The author's attitude towards him．D. The author's interests in African American studies．5.Which of the following can replace the underlined word "squandering" in Paragraph 4？______A. creatingB. wastingC. valuingD. seizing6.Why are immigrant kids expected to be doctors or lawyers？______A. They are very smart in general．B. Their parents are high achieving as well．C. They have more opportunities．D. Their parents want them to move upward in society．7.How did the author react to the driver's questioning？______A. Getting upset．B. Feeling satisfied．C. Appreciating his concern．D. Defending herself．CFeathers are not just for flight．They keep birds warm and attract mates．And for one Australian bird，feathers even help produce an important sound--an alarm．"People had long noticed these birds produced these loud sounds．" Trevor Murray，a researcher at the Australian National University said．"My supervisor Rob Magrath and May Hingee thought they were used as an alarm．So they did some playbacks and they could show quite strongly that if you play back these sounds to other birds，they flee straight away．So I was really interested in how they produce the sound，whether it is actually a signal，and whether it's a reliable signal．"The team focused their experiment on specific feathers in the pigeon's wing．"We were able to target the eighth primary feather，which is unusually narrow．And then we also removed ，on different sets of birds，those neighboring feathers，the ninth primary feather and the seventh primary feather．And we were able to see when the eighth primary feather was missing，the high note had completely disappeared．So the eighth primary feather produced that high note and the ninth primary feather actually produced the low note．"And if the birds are fleeing from danger，they produce a louder and higher sound than they do during a normal takeoff．The study is in the journal Current Biology．Murray and his colleagues did another experiment where they used the recordings they made to observe the reactions of other pigeons．"We were able to see the unusual eighth primary feather was important for signaling alarm．When that eighth primary feather was missing，they very rarely responded．They almost never fled．Whereas when that ninth primary produces the sound，they fled just as much as to normal alarms．This shows us this unusual primary feather is important for signaling alarm．"That makes birds of a feather flee together．8.What does the underlined word "they" in Paragraph 2 refer to？______A. The birds．B. The feathers．C. Rob and May．D. The loud sounds．9.What did Rob Magrath and May Hingee find out about the birds？______A. They fly away as a group．B. They send alarms with feathers．C. They produce different notes．D. They attract mates with feathers．10.Why did the team remove different feathers？______A. To record the notes they produce．B. To keep the pigeons stay in cages．C. To compare their width and length．D. To observe other pigeons' reactions．11.According to Murray and his colleagues，why does the unusual feather matter most？______A. It keeps the pigeons warm．B. It responds to alarms．C. It helps birds fly more quickly．D. It sounds alarms of danger．DMany small-business owners watched recent revelations about Facebook with mixed emotions ．Like most Americans，they were surprised to discover how much information the socialmedia giant collected on its users．But when it comes to small business，Facebook is a transformative advertising platform for small businesses，not easy to replace．Let's say you own a small seafood restaurant，and Tuesday nights are ﹩1 oyster （牡蛎）nights．Traditional advertising methods cost a lot and must be planned long in advance，and it's hit-or-miss as to whether you actually get in front of oyster eaters．With Facebook，on Tuesday morning，with a few clicks，you can target Facebook users in your Zip code who love oysters and eating out （and are over age 21，so they can buy drinks，which is why you have ﹩1 oyster nights）．And you can do this for as little as ﹩20．In my work with small businesses for more than 25 years，I've never seen a more effective method of micro-targeting prospects．Though Facebook is an effective tool forsmall-business，advertising does not justify （证明合理）the company's collecting vast amounts of data or allowing users' data to be invaded．"Our primary concern was people's experience on Facebook，" said Dan Levy，Facebook's Vice President，Small Business．"Our teams have also been speaking to small businesses，and they want to make sure we're addressing the situation，and we are．"One concern small businesses want Facebook to address is protecting their uploaded lists．No one wants their customers' information misused or accessed by others，especially competitors ．Small-business owners are rightfully concerned about privacy．They don't want Facebook to know everything about them，and they don't want their customer list to be let out to others．But small businesses don't want to lose this effective advertising medium，either．Most Facebook ads are not invasive or offensive．And many receivers may actually benefit from receiving highly targeted ads-after all，those oyster lovers like learning about Tuesday night ﹩1 oyster night．12.How does the author explain Facebook's function in paragraph 2？______A. By performing an experiment．B. By leading a survey．C. By analyzing the data．D. By giving an example．13.What is the author's attitude towards Facebook mentioned in paragraph 3？______A. Objective．B. Supportive．C. Doubtful．D. Respectful．14.What makes small businesses concerned about Facebook according to the text？______A. What people experience on Facebook．B. That Facebook updates the lists constantly．C. That their competitors benefit more from Facebook．D. That Facebook will give away their customer list．15.What can be the best title of the text？______A. Facebook is benefiting small businessB. Facebook，a mixed bag for small businessC. Facebook is protecting customers' privacyD. Facebook，a powerful advertisement tool二、阅读七选五（本大题共5小题，共10.0分）Making decisions when shopping is often a tough call．Even if you're satisfied with the first dress you try on，would you go on looking for alternatives，comparing styles and prices，until literally dropped dead？Psychology researchers have studied how people make decisions and concluded there are two basic styles．"Maximizers"like to take their time and weigh a wide range of options sometimes every possible one-before choosing．"Satisficers"would rather be fast than thorough．(1) So even if they see what would seem to meet their requirements，they can't make adecision until after they've examined every option to make sure that they're making the best possible choice．Researchers followed 548 job-seeking college seniors at eleven schools from October through their graduation in June．They found that the maximizers landed better jobs．(2) ．"The maximizer is kicking himself because he can't examine every option and at some point had to just pick something，"they say．"Maximizers make good decisions and end up feeling bad about them．(3) ．Satisficers also have high standards，but they are happier than maximizers．Maximizers tend to be more depressed and to report a lower satisfaction with life，their research found．Faced with so many choices in our lives，we need to learn how not to waste time and energy on our decision．making．(4) For instance，if you're picking a restaurant for a lunch meeting ，first deciding on a certain part of town or type of cuisine can narrow your options．Then，once you've arrived at a decision，stick with it．(5) To limit the number of options you can set a time limit for decision-making．Say you are buying a new bag，you could spend an amount of time studying features，price and value-but if you gave yourself only five minutes to make a decision，and there are only these bags you can consider，you'll save time．You'll be happier with your decision，too．A．Maximizers are people who want the best．B．First of all，decrease your range of options．C．Just accept that no decision is ever completely perfect．D．Comparing is often a must when you make a decision．E．Satisficers make good decisions and end up feeling good．F．The other is to make decisions more accurately at the beginning．G．Their starting salaries were 20 percent higher but they felt worse about their jobs!16. A. A B. B C. C D. D E.E F.F G. G17. A. A B. B C. C D. D E.E F.F G. G18. A. A B. B C. C D. D E.E F.F G. G19. A. A B. B C. C D. D E.E F.F G. G20. A. A B. B C. C D. D E.E F.F G. G三、完形填空（本大题共20小题，共30.0分）Over the summer，one of my teenage son's high school classmates lived in a remote country place．There she (21) reliable Internet and phone service．Because it was (22) to call or text his friend or reach her on social media，Will landed on an old-fashioned (23) to stay in touch．Once a week，he tried to sit down and write a letter．I haven't written a letter for years．When Will asked for writing materials to get(24)，I struggled to remember where we(25) the paper and envelopes．But once we found the (26) from a shelf and Will biked to the post office for (27)，he settled into a spot near the living room window and set to work．As my son(28) letters，I just spent time in another corner of the house (29) some essays of one of the world's greatest (30) writers，Virginia Woolf．Woolf，who died in 1941，worried in her own time that as (31) became cheaper and easier，people would (32) less about what they were trying to say．They would (33) the dream of letters that were"the (34)of phrases and the arts of the writing master．"The (35) and emails we now dash off by the dozen each day seem the fulfillment of Woolf's(36)．I'm not giving up my smart phone and laptop，and I'd be (37) without the convenience of connecting with masses each day with a few quick keystrokes（按键）．But my 17-year-old (38)me this past summer that in an age of (39) communication，conventional mail still has its (40)．21. A. changed B. lacked C. found D. rented22. A. polite B. unnecessary C. hard D. convenient23. A. alternative B. value C. brand D. bike24. A. continued B. touched C. offered D. started25. A. put B. lost C. bought D. swapped26. A. necessities B. ideas C. phones D. essays27. A. mailboxes B. envelopes C. stamps D. postcode28. A. sent B. read C. tore D. wrote29. A. setting aside B. dipping intoC. improving onD. getting rid of30. A. lecture B. anecdote C. letter D. idiom31. A. communication B. judgement C. commitment D. journey32. A. consult B. know C. think D. quarrel33. A. take up B. give up C. make up D. call up34. A. exchange B. process C. review D. polish35. A. excuses B. problems C. chances D. texts36. A. suggestions B. ambitions C. fears D. dreams37. A. betrayed B. lost C. charged D. misunderstood38. A. warned B. prepared C. awarded D. reminded39. A. unbearable B. expensive C. instant D. permanent40. A. virtues B. disadvantages C. criteria D. independence四、语法填空（本大题共1小题，共15.0分）41.People with high self-confidence (1) （natural）have optimistic and Joyful emtions aswell as the knowledge that they can change their emotions from negative to positive．Emotions (2) （be）a normal part of life．They indicate how you fel about a situation and any needed action．When you have high self-respect and self-cofidence，you feel god (3) life．You feel secure and have all open hear and calm mind．The (4) （high）your level of emotional health is，the more sel-worth you have．Even when things go wrong，you're composed and ready (5) （deal）with the situation in a constructive way．Even when you focus on (6) （carry），positive attitude at all times，we all know that life can be full of annoyances-your dog (7) （run）away，someone sideswipes your car ，your plane is late，you have to work overtime to get a project done．All of these can cause a bad mood and throw your emotions into a downward spiral．Having emotional health isn't difficult，but (8) does take focus and concentration．Be aware of (9) your feeling is and direct your emotions into something more optimistic．Your self-confidence will (10) （strength）at the same time．五、短文改错（本大题共1小题，共10.0分）42.假定英语课上老师要求同桌之间交换修改作文，请你修改你同桌写的以下作文．文中共有10处语言错误，每句中最多有两处．每处错误仅涉及一个单词的增加、删除或修改．增加：在缺词处加一个漏字符号（∧），并在其下面写出该加的词．删除：把多余的词用斜线（ \ ）划掉．修改：在错词下划一横线，并在该词下面写出修改后的词．注意： 1．每处错误及其修改均仅限一词；2．只允许修改10处，多者（从第11处起）不计分Train travel is quite a different experience today．Little things can match China's highspeed rail system，especially the speed at that it is still developing． The speed known as "China speed"． China speed means a lot of to the country．The past 30 years had seen China's rapid development in various areas，like economy and technology．We have seen dozens of tall building spring up almost overnight，with roads and highways reached every corner． Besides，mobile payments and online shopping are playing increasing important roles in everyon e's life． With so great achievements，China speed is now a model admired by a whole world．六、书面表达（本大题共1小题，共25.0分）43.假定你是李华，一名高一学生．你得知晋江市正向全国在校中学生招募2020年晋江世界中学生运动会形象大使，参与接待来访参赛国的学生．你写信申请，内容包括：1．希望成为形象大使；2．介绍自己的优势（性格，英语程度，领导沟通能力等）；3．对形象大使的认识．注意：1．词数100左右；2．可以增加适当细节，使行文连贯；3．信中不得出现除晋江以外的地名和校名．参考词汇：世界中学生运动会Gymnasiade；形象大使Goodwill AmbassadorDear Sir/Madame，I'm Li Hua．Your sincerely，Li Hua答案和解析1.【答案】【小题1】C【小题2】D【小题3】B【解析】（1）C 细节理解题．根据Stuttgart City Library 部分中的" The German library opened in 2011 "德国这家图书馆于2011年开办；Kanazawa Umimirai Library 部分中的" This Japanese library was opened in 2011"日本这家图书馆于2011年开办； Library of Congress 部分中的" In 1800，Congress set up the Library of Congress in Washington D．C． "在1800年，国会在华盛顿设立了国会图书馆；以及 State Library of Victoria 部分中的" Though it opened in 1856 "尽管它于1856年开放，可知国会图书馆历史最长，故选C．（2）D 推理判断题．根据State Library of Victoria部分中的" The lawn at the library's front attracts hundreds of city folk each day，who gather for lunch or to enjoy the statues there． "图书馆前面的草坪每天吸引着数以百计的市民，他们聚集在那里吃午餐或欣赏雕像．可知选D．（3）B 细节理解题．根据Stuttgart City Library部分中的" The amazing modern and simple look is the product of the Korean architect Yi Eun-young " 令人惊叹的现代简约外观是韩国建筑师Yi Eun-young的作品；以及 Kanazawa Umimirai Library 部分中的" and reflects the appearance of many modern and simple spaces．"反映了许多现代和简单空间的外观．可知选B．本文是一篇应用文，介绍了世界上最酷的四家图书馆-- Stuttgart City Library 、Kanazawa Umimirai Library、 Library of Congress和 State Library of Victoria．本题考点涉及细节理解题型的考查，是一篇介绍类阅读，考生需要准确掌握细节信息，结合题目要求，从而选出正确答案．4.【答案】【小题1】A【小题2】B【小题3】D【小题4】C【解析】（1）A 考查推理判断．根据第三段中的"and the pride in his voice faded，as I knew it might．I didn't even get to add ‘and African-American studies' before he cut in，his voice thick with disappointment，‘ All that work to get into Harvard，and you study history？'我知道，他的声音中的自豪感可能会消退．在他插嘴之前，我甚至没有加上"和非洲裔美国人的研究"，他的声音里充满了失望，"所有这些工作都是为了进入哈佛，而你却在学习历史？"可以推断出，司机对作者主修历史和文学感到很失望．故选A．（2）B 考查词义猜测．由这位优步移民司机先自豪，然后失望以及接着对我的教训这一情感变化的线索可以推测出，他是认为"我"浪费了一个人生中最大的机会．故可以猜测出squandering为"浪费"之意．故选B．（3）D 考查推理判断．根据倒数第二段中的"Fulfill your role in the narrative（故事）of upward mobility so your children can do the same．在向上发展的事故中扮演你的角色，这样你的孩子也可以这样做．"可知，他们的父母希望他们在社会上向上发展，而这些职业在社会上有地位，更能向上流动．故选D．（4）C 考查推理判断．根据最后一段中的"I'll just smile and nod，and enjoy the warmth of the occasion．我会微笑点头，享受这一时刻的温暖"可推出，作者对司机的关心还是很感激的．故选C．本文是一篇议论文，论述了不要因别人的质问而怀疑自己选择的专业．本题考点涉及词义猜测和推理判断两个题型的考查，是一篇教育类阅读，要求考生根据上下文进行逻辑推理，从而选出正确答案．8.【答案】【小题1】C【小题2】B【小题3】A【小题4】B【解析】1．C．细节理解题．根据"'My supervisor Rob Magrath and May Hingee thought they were used as an alarm．So they did some playbacks and they could show quite strongly that if you play back these sounds to other birds，they flee straight away可知"they"指的是"Rob and May‘．故选C．2．B．细节理解题．根据"So I was really interested in how they produce the sound，whether it is actually a signal，and whether it's a reliable signal．"'可知Rob Magrath和May Hingee用鸟类的羽毛发出警报．故选B．3．A．推理判断题．根据"And if the birds are fleeing from danger，they produce a louder and higher sound than they do during a normal takeoff．'可知团队要移除不同的羽毛是为了记录他们的笔记．故选A．4．D．推理判断题．根据"Whereas when that ninth primary produces the sound，they fled just as much as to normal alarms． This shows us this unusual primary feather is important for signaling alarm．"'可知不同寻常的羽毛最重要因为他能发出警报．故选D．本文是一篇社会文化类阅读，文章主要介绍了羽毛不仅仅是用来飞行的．它们使鸟类保持温暖并吸引配偶．对一只澳大利亚鸟来说，羽毛甚至能发出重要的声音--警报．考察学生的细节理解和推理判断能力，做细节理解题时一定要找到文章中的原句，和题干进行比较，再做出正确的选择．在做推理判断题不要以个人的主观想象代替文章的事实，要根据文章事实进行合乎逻辑的推理判断．12.【答案】【小题1】D【小题2】A【小题3】D【小题4】B【解析】1．D．推理判断题．根据第二段的Let's say you own a small seafood restaurant， and Tuesday nights are ﹩1 oyster （牡蛎） nights．可知，第二段主要通过举例来解释Facebook平台如何使小企业受益．故选D．2．A．推理判断题．根据第三段Though Facebook is an effective tool for small-business， advertising does not justify （证明合理）the company's collecting vast amounts of data or allowing users' data to be invaded．可知，作者对于Facebook极尽赞美但是又指出它的弊端，由此可推断，作者的态度是客观公正的．故选A．3．D．细节理解题．根据文章倒数第二段的"Small-business owners are rightfully concerned about privacy．They don't want Facebook to know everything about them，and they don't want their customer list to be available to others"可知，小企业担心Facebook 平台会泄露它们顾客的隐私．故选D．4．B．主旨大意题．根据第一句话Many small-business owners watched recent revelations about Facebook with mixed emotion s．可知，本文主要讲述小企业主对Facebook复杂的情绪．故选B．本文是一篇议论文．Facebook是小企业广告的有效工具，但是它也存在一些弊端如泄露顾客的私人信息等等，让小企业对它爱恨交加．考察学生的细节理解和推理判断能力，做细节理解题时一定要找到文章中的原句，和题干进行比较，再做出正确的选择．在做推理判断题不要以个人的主观想象代替文章的事实，要根据文章事实进行合乎逻辑的推理判断．16.【答案】【小题1】A【小题2】G【小题3】E【小题4】B【小题5】C【解析】36-40 AGEBC36．A．段首空（非首段），需关注上一段内容和本段下文．上一段将做决定的人分为两类一maximizers 和satisficers根据本段后一句的they，说明所填句子必须包含某类人群，符合的选项有A （maximizers）， E（satisficers）；再根据后一句中的making the best pssible choice，可知与E选项want the best语义相符．故选A．37．G．段尾空，需关注本段上文和下一段内容．根据"重点名词一致"原则，前一句重点提到jobs，选项中名词一致的只有G （salaries， jobs）；且选项种代词their 指代前一句种的maximizers．下一段开头也在探讨maximizers．故选G．38．E．段尾空，需关注本段上文和下-段内容，可知，两段在进行maximizers 和satisficers 的对比；且根据空前一句： maximizers做出最佳选择，但不开心；以及下一段段首句： stisficers 也有高标准，但比maximizers更开心；应选择E--satisficers 做出好的选择，并且最终感觉很好．故选E．39．B．段首空，需关注上一段内容和本段下文．上一段说我们在面对选择时，如何避免浪费时间精力，因此此处需要具体做法；再根据下文for instance可知本段为总分结构，且该句中的narrow your options（缩小选择范围）与B项中的decrease your range of options语义一致．故选B．40．C．段中空，需关注本段上下文．前一句"一旦作出决定，就要坚持不变"；后一句"为了限制选项数目，可以设定一个抉择的时间．"Say所在句为举例，but之后是重点，强调在限定的时间内迅速做出选择，而非最合适或最完美的选择．C项最合适．另外两项也可用排除法， D中的comparing 与原文内容无关； F中的the other与前文不能形成正确的逻辑搭配．故选C．第一段引出文章中心，并说明做选择是件很难的事情；第二段根据人们如何做选择，将人们分成两类一maxinizers和saisficers第三段至第五段进行maximizers 和satisficers的对比；第六段至第八段说明如何在选择时减少时间、经历的浪费，给出两个具体方向．七选五阅读题，有部分题出现在每一段的开始位置，若问题在段首1．通常是段落主题句．认真阅读该段落，根据段落一致性原则，查找相关词或者同义词，从而推测出主题句，找到答案．2．与后文是并列、转折、因果关系等．着重阅读后文第一两句，锁定线索信号词，然后在选项中查找相关特征词．-般来说正确答案与它后面的一句话的在意思上是衔接的，所以通常情况下，这两句话中会有某种的衔接手段．3．段落间的过渡句．这时要前瞻后望找到启示，即阅读上一段结尾部分，通常正确答案与上一段结尾有机地衔接起来，并结合下一段内容，看所选的答案是否将两段内容连贯起来．21.【答案】【小题1】B【小题2】C【小题3】A【小题4】D【小题5】A【小题6】A【小题7】C【小题8】D【小题9】B【小题10】C 【小题11】A【小题12】C【小题13】B【小题14】D【小题15】D 【小题16】C【小题17】B【小题18】D【小题19】C【小题20】A【解析】1-20 BCADA ACDBC ACBDD CBDCA1．B．动词的辨析，changed改变，lacked缺少，found找到，rented租，在那她缺少可靠的互联网和电话服务，故答案为B．2．C．形容词的辨析，polite有礼貌的，unnecessary不必要的，hard困难的，convenient 方便的，因为在网络上很难但电话或发短信给她的朋友，故答案为C．3．A．名词的辨析，alternative可供选择的事物，value价值，brand品牌，bike自行车，Will只能依靠老式的可使用的方式来跟人联系，故答案为A．4．D．动词的辨析，continued继续，touched感动，offered提供，started开始，当Will 想要写作材料开始写信时，故答案为D．5．A．动词的辨析，put放置，lost失去，bought买，swapped交换，我努力地回想我们把纸和信封放在哪，故答案为A．6．A．名词的辨析，necessities必需品，ideas想法，phones电话，essays文章，一旦我们从架子上找到这些必需品，故答案为A．7．C．名词的辨析，mailboxes邮箱，envelopes信封，stamps邮票，postcode邮政编码，Will骑自行车去邮局买邮票，故答案为C．8．D．动词的辨析，sent发送，read读，tore撕，wrote写，当儿子在写信的时候，故答案为D．9．B．短语的辨析，setting aside放一边，dipping into浸入，improving on改善，getting rid of去除，我只是把时间花在看一些由Virginia Woolf写的文章，故答案为B．10．C．名词的辨析，lecture讲座，anecdote轶事，letter信件，idiom习语，此处指Virginia Woolf是世界上最伟大的信件作家之一，故答案为C．11．A．名词的辨析，communication交流，judgement判断，commitment承诺，journey 旅途，此处指由于交流变得越便宜和方便，故答案为A．12．C．动词的辨析，consult咨询，know知道，think认为，quarrel争吵，人们将不会想太多关于他们要讲的话，故答案为C．13．B．短语的辨析，take up从事，give up放弃，make up组成，call up打电话，他们将放弃写信的梦想，故答案为B．14．D．名词的辨析，exchange改变，process加工，review回顾，polish修改，写信就是对短语的润色，故答案为D．15．D．名词的辨析，excuses借口，problems问题，chances机会，texts文本，现在我们每天迅速写好的那些邮件似乎验证了Woolf的担忧，故答案为D．16．C．名词的辨析，suggestions建议，ambitions雄心，fears担忧，dreams梦想，现在我们每天迅速写好的那些邮件似乎验证了Woolf的担忧，故答案为C．17．B．动词的辨析，betrayed背叛，lost失去，charged指控，misunderstood误解，没有这种联系的方便，我会感觉到迷失，故答案为B．18．D．动词的辨析，warned警告，prepared准备，awarded奖励，reminded提醒，但是我17岁的儿子这个夏天提醒了我，故答案为D．19．C．形容词的辨析，unbearable无法忍受的，expensive昂贵的，instant立即的，permanent永久的，此处指在这种立即就能交流的时代，传统的通讯仍然有自己的优势，故答案为C．20．A．名词的辨析，virtues美德，优势，disadvantages缺点，criteria标准，independence 独立，此处指在这种立即就能交流的时代，传统的通讯仍然有自己的优势，故答案为A ．短文主要讲了作者在一个夏天的时候，由于儿子要跟住得偏远的同学联系需要写信，在那个夏天作者和儿子都意识到写信也有其好处．在做完形填空时，首先需要快速的浏览全文，把握文章的主旨大意；其次要学会带着问题到文中相应的地方，通过细节阅读来寻找或概括答案；最后理清作者的写作思路也非常重要；做此类题时，要多注意一些形容词或动词的搭配，在平时多积累一些固定搭配．41.【答案】【小题1】【小题2】are【小题3】【小题4】【小题5】to dealnaturally about higher【小题6】carrying 【小题7】runs【小题8】it【小题9】what【小题10】strengthen【解析】1． naturally．考查副词，副词修饰动词have，故填naturally．2．are．考查动词，主语是名词复数，因为是一般现在时，故填are．3．about．考查介词，feel good about：对…感觉良好，故填about．4．higher．考查比较级，the+比较级，the+比较级结构，表示越…越…，故填higher．5．to deal．考查不定式，be ready to do sth：准备好做某事，乐意做某事，故填to deal ．6．carrying．考查非谓语，根据on可知填写动名词形式，故填carrying．7．runs．考查动词，后面是一系列动作并列，缺少谓语动词，主语是单数名词，故填runs ．8．it．考查代词，it指代前面的拥有情绪健康，故填it．9．what．考查连词，what引导宾语从句，从句中缺少表语，故填what．10．strengthen．考查动词，放在will后面填写谓语动词，故填strengthen．文章讲述了保持积极心态的重要性以及与自信的关系．在一篇200词左右的语篇（短文或对话）中留出10处空白，部分空白的后面给出单词的基本形式，要求考生根据上下文填写空白处所需的内容或所提供单词的正确形式，所填写词语不得多于3个单词．要做好语法填空题，理解短文是解题的前提，扎实的词汇、句型和语法知识是基础，英语国家的背景知识是必要的补充．考生须灵活运用语法知识，如单词词性、单词时态、名词单复数、连接词、代词、冠词等判断各空白处应填写的内容．答完后，还要通读全文，核对所填单词形式是否正确，是否符合语境．42.【答案】答案：Train travel is quite a different experience today．Little things can match China's highspeed rail system，especially the speed at that it is still developing． The speed ∧known as "China speed"．China speed means a lot of to the country．The past 30 years had seen China's rapid development in various areas，like economy and technology．We have seen dozens of tall building spring up almost overnight，with roads and highways reached every corner． Besides，mobile payments and online shopping are playing increasing important roles in everyone's li fe． With so great achievements， China speed is now a model admired by a whole world．解析：1．Little→Few．考查形容词．此处little的含义是"很少、几乎没有"，通常用来修饰不可数名词．things是可数名词的复数形式，应该用few修饰．故将Little改为Few．2．that→which．句意：几乎没有什么能与中国的高速铁路系统相媲美，尤其是它仍在发展的速度．the speed作先行词，定语从句引导词在句子中作介词at的宾语，用关系代词which．故将that改为which．3．speed后加is．考查动词的语态．主语The speed 与谓语动词know之间为被动关系，本句应为被动句．be known as （作为…）被熟知，故在speed后加is．4．去掉of．考查短语．alot修饰动词means，a lot of修饰名词，此处后面没有可修饰的名词，故去掉of．5．had→have/has．考查动词时态．句意：过去的30年见证了我国经济技术等各领域的迅速发展．（in）the past 30 years通常用于现在完成时，强调对现在产生的影响．此处作主语，谓语动词用单复数均可．故将had改为have或has．。

2020届 福建省福建师大附中 高三下学期高考模拟(最后一模)数学(文)试题(解析版)

2020届 福建省福建师大附中 高三下学期高考模拟(最后一模)数学(文)试题(解析版)

2020届福建省福建师大附中高三下学期高考模拟（最后一模）数学（文）试题一、单选题1．设集合2{|20}A x Z x x =∈--≤，集合{}2,0,1B =-，则A B =U （ ） A ．{}2,0,1-B ．{1,0,1}-C ．{}2,1,01--，D ．{}2,1,01,2--， 【答案】D【解析】化简集合A ，找出A ，B 中的所有元素，确定A B ⋃. 【详解】由220x x --≤，12x -≤≤，又x Z ∈，所以{}1,0,1,2A =-．因为{}2,0,1B =- 所以{}2,1,0,1,2A B ⋃=--, 故选D. 【点睛】本题考查集合的化简与运算，考查对这些知识的理解、掌握、运用水平. 2．已知i 是虚数单位，若()113z i i +=+，则z =（ ） A ．2i + B ．2i -C ．1i -+D ．1i --【答案】A【解析】根据复数的除法运算，即可得到本题答案. 【详解】 由题，得13(13)(1)4221(1)(1)2i i i iz i i i i ++-+====+++-. 故选：A 【点睛】本题主要考查复数的除法运算，属基础题.3．已知某高中的一次测验中，甲、乙两个班级的九科平均分的雷达图如图所示，下列判断错误的是（ ）A．乙班的理科综合成绩强于甲班B．甲班的文科综合成绩强于乙班C．两班的英语平均分分差最大D．两班的语文平均分分差最小【答案】D【解析】先对图象数据进行处理,再逐一进行判断即可得到结果.【详解】由甲、乙两个班级的九科平均分的雷达图可得：乙班的理科综合成绩强于甲班,即选项A正确，甲班的文科综合成绩强于乙班，即选项B正确，两班的英语平均分分差最大，即选项C正确，两班地理平均分分差最小，即选项D错误，故选D.【点睛】本题考查了对图象数据的处理能力，意在考查灵活应用所学知识解答问题的能力，属于中档题.4．某省新高考将实行“312++”模式，“3”为全国统考科目语文､数学､外语，所有学生必考；“1”为首选科目，考生须在物理､历史两科中选择一科；“2”为再选科目，考生可在化学､生物､思想政治､地理4个科目中选择两科.某考生已经确定“首选科目”为物理，如果他从“再选科目”中随机选择两科，则思想政治被选中的概率为（）A．13B．12C．23D．34【答案】B【解析】把总的基本事件和满足题目要求的基本事件分别列出来，然后根据古典概型的概率公式，即可得到本题答案.【详解】化学､生物､思想政治､地理4个科目中选择两科的情况有：（化学，生物），（化学，思想政治），（化学，地理），（生物，思想政治），（生物，地理），（思想政治，地理），共6种；两科中包括思想政治的情况有：（化学，思想政治），（生物，思想政治），（思想政治，地理），共3种.所以他从“再选科目”中随机选择两科，则思想政治被选中的概率为12. 故选：B 【点睛】本题主要考查古典概型的概率求解，属基础题.5．设双曲线2222:1(0,0)x y C a b a b-=>>的右焦点与抛物线216y x =的焦点相同，双曲线C 0y +=，则双曲线C 的方程为（ ）A ．221124x y -=B ．221412x y -=C ．2211648x y -=D ．2214816x y -=【答案】B【解析】由题意得双曲线C 的渐近线方程为b y x a =±，于是可得ba=故223b a =，从而双曲线方程为222213x y a a-=，然后再根据双曲线的右焦点与抛物线的焦点相同得到24a =，进而可得所求方程．【详解】由题意得双曲线C 的渐近线方程为by x a=±，又双曲线C 0y +=，∴ba=223b a =， ∴双曲线方程为222213x y a a-=，∴双曲线的右焦点坐标为(2,0)a ．又抛物线216y x =的焦点坐标为(4,0)，双曲线的右焦点与抛物线的焦点相同， ∴22,4a a ==，∴双曲线的方程为221412x y -=．故选B ． 【点睛】已知双曲线的标准方程求渐近线方程时，只需把标准方程中等号右边的1换为零，再求出y 与x 间的关系即可．解答本题的关键是根据题中的关系得到方程中的待定系数，考查对双曲线基本性质的理解和运用，属于基础题．6．如果执行右面的框图，输入5N =，则输出的数等于（ ）A ．54B ．45C ．65D ．56【答案】D【解析】试题分析：当5N =时，该程序框图所表示的算法功能为：11111151122334455666S =++++=-=⨯⨯⨯⨯⨯，故选D. 【考点】程序框图.7．在矩形ABCD 中，2AB =，点P 为直线BC 上一点，则()PA PD BA +⋅=u u u r u u u r u u u r（ ）A ．0B ．2C ．4D ．8【答案】D【解析】把,PA PB BA PD PC CD =+=+u u u r u u u r u u u r u u u r u u u r u u u r代入()PA PD BA +⋅u u u r u u u r u u u r ，逐步化简，即可得到本题答案. 【详解】由题，得,PB BA PC BA ⊥⊥， 所以()PA PD BA +⋅u u u r u u u r u u u r()PB BA PC CD BA =+++⋅u u u r u u u r u u u r u u u r u u u r ()BA CD BA =+⋅u u u r u u u r u u u r2BA CD BA =+⋅u u u r u u u r u u u r 228BA ==u u u r .故选：D 【点睛】本题主要考查平面向量的数量积的运算，考查学生转化和化简能力.8．我国古代数学名著《九章算术》中有如下问题：“今有羡除，下广六尺，上广一丈，深三尺，末广八尺，无深，袤七尺．问积几何”，羡除是一个五面体，其中三个面是梯形，另两个面是三角形，已知一个羡除的三视图如图粗线所示，其中小正方形网格的边长为1，则该羡除的表面中，三个梯形的面积之和为（ ）A ．40B ．43C ．46D ．47【答案】C【解析】画出几何体的直观图,利用三视图所给数据，结合梯形的面积公式,分别求解梯形的面积即可.【详解】由三视图可知，该几何体的直现图如图五面体，其中平面ABCD ⊥平面ABEF ，2,6,4CD AB EF ===，底面梯形是等腰梯形，高为3 ,梯形ABCD 的高为4 ,等腰梯形FEDC 9165+=， 三个梯形的面积之和为26462443546222+++⨯+⨯+⨯=, 故选C. 【点睛】本题考查空间几何体的三视图，求解表面积,属于中档题.三视图问题是考查学生空间想象能力最常见题型，也是高考热点.观察三视图并将其“翻译”成直观图是解题的关键，不但要注意三视图的三要素“高平齐，长对正，宽相等”，还要特别注意实线与虚线以及相同图形的不同位置对几何体直观图的影响. 9．函数()sin (0)f x x ωω=>的图象向右平移12π个单位得到函数()y g x =的图象，并且函数()g x 在区间[,]63ππ上单调递增，在区间[,]32ππ上单调递减，则实数ω的值为（ ） A ．74B ．32C ．2D ．54【答案】C【解析】由函数()sin (0)f x x ωω=>的图象向右平移12π个单位得到[]1212g x sin x sin x πωπωω=-=-（）（）（），函数()g x 在区间,63ππ⎡⎤⎢⎥⎣⎦上单调递增，在区间,32ππ⎡⎤⎢⎥⎣⎦上单调递减，可得3x π=时，()g x 取得最大值，即23122k πωππωπ⨯-=+（），k Z ∈，0ω>，当0k =时，解得2ω=，故选C.点睛：本题主要考查了三角函数图象的平移变换和性质的灵活运用，属于基础题；据平移变换“左加右减，上加下减”的规律求解出()g x ，根据函数()g x 在区间,63ππ⎡⎤⎢⎥⎣⎦上单调递增，在区间,32ππ⎡⎤⎢⎥⎣⎦上单调递减可得3x π=时，()g x 取得最大值，求解可得实数ω的值.10．已知圆C :()()22122x y -+-=与y 轴在第二象限所围区域的面积为S ，直线3y x b =+分圆C 的内部为两部分，其中一部分的面积也为S ，则b =（ ）A ．1-±B ．1±C ．1-D ．1【答案】A【解析】直线3y x b =+分圆C 的内部为两部分，其中一部分的面积也为S ，等价于圆心(1,2)到直线3y x b =+的距离为1，由此即可得到本题答案. 【详解】已知圆C :()()22122x y -+-=与y 轴在第二象限所围区域的面积为S ，且圆心(1,2)到y 轴的距离为1，若直线3y x b =+分圆C 的内部为两部分，其中一部分的面积也为S ， 则必有圆心(1,2)到直线3y x b =+的距离也为1， 1=，解得1b =-±. 故选：A 【点睛】本题主要考查直线与圆位置关系的应用问题，把问题转化为“圆心(1,2)到直线3y x b =+的距离也为1”是解决此题的关键.11．三棱锥P ABC -中，PA ⊥平面ABC 且2,PA ABC =∆形，则该三棱锥外接球的表面积为( ) A ．43π B ．4π C ．8π D ．20π【答案】C【解析】根据已知中底面ABC ∆，PA ⊥平面ABC ，可得此三棱锥外接球，即为以ABC ∆为底面以PA 为高的正三棱柱的外接球 ∵ABC ∆是边长为3的正三角形，∴ABC ∆的外接圆半径233134r =-=，球心到ABC ∆的外接圆圆心的距离1d =， 故球的半径222R r d =+=， 故三棱锥P ABC -外接球的表面积248.S R ππ== 故选C ．12．e 为自然对数的底数，已知函数()1,18ln 1,1xx f x x x ⎧+<⎪=⎨⎪-≥⎩，则函数()y f x ax =-有唯一零点的充要条件是（ ）A ．1a <-或21a e =或98a > B ．1a <-或2118a e≤≤ C ．1a >-或2198a e <<D ．1a >-或98a >【答案】A【解析】作出函数()f x 的图像如图所示，其中9(1,),(1,1)8A B -，则9,18OA OB k k ==-，设直线y ax =与曲线ln 1(1)y x x =-≥相切，则ln 1ax x =-，即ln 1x a x-=，设ln 1()x g x x -=，则221(ln 1)2ln ()x x g x x x---='=，当2x e =时，()0g x '=，分析可知，当2x e =时，函数()g x 有极大值也是最大值，221()g e e=，所以当1a =时，ln 1x a x-=有唯一解，此时直线y ax =与曲线ln 1(1)y x x =-≥相切．分析图形可知，当1a <-或21a e =或98a >时，函数()f x 的图像与函数y ax =的图像只有一个交点，即函数()y f x ax =-有唯一零点．故选A .【点睛】本小题主要考查分段函数的图象与性质，考查函数零点问题的处理方法，考查利用导数求相切时斜率的方法，考查数形结合的数学思想方法.首先画出函数的图象，分段函数的图象注意分界点的位置是实心的函数空心的.然后将函数的零点问题转化为两个函数图象的交点来解决.二、填空题13．已知函数()f x 为偶函数，当0x ≥时，2()sin2xf x x π=+，则(1)f -=__________．【答案】2【解析】∵当0x ≥时，()2sin2xf x x π=+，∴()11sin22f π=+=，又∵()f x 为偶函数，∴()()112f f -==，故答案为2.14．若变量x ，y 满足约束条件00340x y x y x y +≥⎧⎪-≥⎨⎪+-≤⎩，则32z x y =+的最大值是______.【答案】5【解析】作出不等式组对应的平面区域，利用目标函数的几何意义，即可求得32z x y =+的最大值.【详解】由32z x y =+，得322zy x =-+，作出不等式组对应的可行域（阴影部分）， 平移直线322z y x =-+，由平移可知当直线322zy x =-+，经过点A 时，直线322zy x =-+的截距最大，此时z 取得最大值，由340y xx y =⎧⎨+-=⎩，解得(1,1)A ，将A 的坐标代入32z x y =+，得5z =， 即目标函数32z x y =+的最大值为5. 故答案为：5【点睛】本题主要考查线性规划的应用，结合图形并利用目标函数的几何意义，是解决此类问题的常用方法.15．已知抛物线C :22y px =（0p >）的焦点为F ，准线为l ，M 是l 上一点，N 是线段MF 与C 的交点，若2MN NF =u u u u r u u u r，O 为坐标原点，且OFN ∆的面积S 为3，则p 的值为______. 【答案】23【解析】由~MPF NQF ∆∆，又2MN NF =u u u u r u u u r，得16x p =①，又由1||||32OFN S OF NQ ∆==，得324x p =②，结合①②，即可得到本题答案.【详解】假设点M 在准线的上半部分，准线与x 轴交点为P ，作NQ 垂直x 轴，垂足为Q ，设点(,)N x y .易得，~MPF NQF ∆∆，又2MN NF =u u u u r u u u r ，所以1133QF PF p ==，则16x p =①；又11||||3222OFN p S OF NQ y ∆==⋅⋅=43y = 代入抛物线方程22y px =(0)p >，得324x p =②，联立①②得，23p =故答案为：23【点睛】本题主要考查直线与抛物线的位置关系的应用问题，其中涉及到相似三角形的应用,体现了数形结合思想的考查.16．已知数列{}n a 的前n 项和为2n S n n =+，将该数列按下列格式（第n 行有12n -个数）排成一个数阵，则该数阵第8行从左向右第8个数字为______.123456789101112131415a a a a a a a a a a a a a a a L L L L L L L【答案】270【解析】由2n S n n =+，可得2(1)n a n n =≥，又由每一行的第一项的下标，构成一个首项是1公比为2的等比数列，即可得到本题答案. 【详解】 由题，得当1n =时，112a S ==；当2n ≥时，22(1)(1)2n n n a S S n n n n n ⎡⎤=-=+--+-=⎣⎦. 因为1n =时，也满足2n a n =， 所以2(1)n a n n =≥.观察可得，每一行的第一项的下标，构成一个首项是1公比为2的等比数列， 由此可知第8行第一个数的下标是128， 所以第8行第8个数的下标是135， 所以所求项1352135270a =⨯=. 故答案为：270 【点睛】本题主要考查由n S 的关系式求通项公式以及等比数列的简单应用，考查学生的分析问题与解决问题能力.三、解答题17．已知在ABC ∆中，角A ，B ，C 所对的边分别为a ，b ，c ，且()1cos 2sin sin 2B C B C --=. （1）求A ；（2）若a =2cos c C =，求ABC ∆的面积.【答案】（1）120︒（2）34- 【解析】（1）根据两角和差的余弦公式，即可得到本题答案；（2）利用正弦定理先求角C 和边c ，然后利用和差公式求sin B ，最后套用面积公式，即可得到本题答案. 【详解】（1）()cos 2sin sin B C B C --cos cos sin sin 2sin sin B C B C B C =+-()cos cos sin sin cos B C B C B C =-=+ 1cos 2A =-=， 所以1cos 2A =-，由0180A <<︒︒，得120A =︒；（2）由a =2cos c C =，120A =︒及正弦定理sin sin A Ca c =，可得sin 12cos 2C C =，tan 1C ∴=0180C ︒<<︒Q ，45C ∴=︒，cos 2C =，2cos c C == ()18015B A C ∴=︒-+=︒， ()sin sin15sin 6045B =︒=︒-︒sin60cos45cos60sin 45=︒︒-︒︒122224=-⨯=，所以ABC ∆的面积13sin 24S ac B ==. 【点睛】本题主要考查利用和差公式化简求角，以及利用正弦定理解三角形和求三角形面积. 18．在四棱锥P ABCD -中，//AB CD ，2CD AB =，AC 与BD 相交于点M ，点N 在线段AP 上，AN AP λ=（0λ>），且//MN 平面PCD .（I ）求实数λ的值；（Ⅱ）若1AB AD DP ===，2PA PB ==，60BAD ︒∠=，求点N 到平面PCD 的距离.【答案】（1）13λ=. （23【解析】分析：解法一：（1）由平行线的性质可得13AM AC =，结合线面平行的性质定理有//MN PC ．据此可得13λ=． (2) 由题意可知ABD ∆为等边三角形，则1BD AD ==，结合勾股定理可知PD BD ⊥且PD DA ⊥，由线面垂直的判断定理有PD ⊥平面ABCD ，进一步有平面PCD ⊥平面ABCD ．作ME CD ⊥于E ，则ME ⊥平面PCD ． ME 即为N 到平面PCD 的距离．结合比例关系计算可得N 到平面PCD 3解法二：（1）同解法一．（2）由题意可得ABD ∆为等边三角形，所以1BD AD ==，结合勾股定理可得PD BD ⊥且PD DA ⊥，则PD ⊥平面ABCD ．设点N 到平面PCD 的距离为d ，利用体积关系：2233N PCD A PCD P ACD V V V ---==， 即2193ACD PCD PD S d S ⋅=⋅V V ．求解三角形的面积然后解方程可得N 到平面PCD 3详解：解法一：（1）因为//AB CD ,所以1,2AM AB MC CD ==即13AM AC =． 因为//MN 平面PCD ，MN ⊂平面PAC ， 平面PAC ⋂平面PCD PC =， 所以//MN PC ． 所以13AN AM AP AC ==，即13λ=．(2) 因为0,60AB AD BAD =∠=，所以ABD ∆为等边三角形，所以1BD AD ==， 又因为1PD =，2PA PB ==222PB PD BD =+且222PA PD AD =+，所以PD BD ⊥且PD DA ⊥，又因为DA DB D ⋂=，所以PD ABCD ⊥平面 因为PD ⊂平面PCD ，所以平面PCD ⊥平面ABCD ．作ME CD ⊥于E ，因为平面PCD ⋂平面=ABCD CD ，所以ME ⊥平面PCD ． 又因为//MN 平面PCD ，所以ME 即为N 到平面PCD 的距离． 在△ABD 中，设AB 边上的高为h ，则3h =， 因为23MD MC BD AC ==，所以2333ME h ==，即N 到平面PCD 的距离为33． 解法二、（1）同解法一．（2）因为0,60AB AD BAD =∠=，所以ABD ∆为等边三角形，所以1BD AD ==， 又因为1PD =，2PA PB ==222PB PD BD =+且222PA PD AD =+，所以PD BD ⊥且PD DA ⊥，又因为DA DB D ⋂=，所以PD ⊥平面ABCD ． 设点N 到平面PCD 的距离为d ，由13AN AP =得23NP AP =， 所以2233N PCD A PCD P ACD V V V ---==， 即2193ACD PCD PD S d S ⋅=⋅V V ． 因为132ACD S AD DC sin ADC =⋅⋅∠=V 112PCD S PD CD =⋅=V ，1PD =， 所以23193d =，解得3d =N 到平面PCD 3点睛：本题主要考查线面平行的应用，面面垂直的性质及其应用，等价转化的数学思想等知识，意在考查学生的转化能力和计算求解能力.19．依据某地某条河流8月份的水文观测点的历史统计数据所绘制的频率分布直方图如图（甲）所示；依据当地的地质构造，得到水位与灾害等级的频率分布条形图如图（乙）所示.（1）试估计该河流在8月份水位的众数；（2）我们知道若该河流8月份的水位小于40米的频率为f ，该河流8月份的水位小于40米的情况下发生1级灾害的频率为g ，则该河流8月份的水位小于40且发生1级灾害的频率为f g ，其他情况类似.据此，试分别估计该河流在8月份发生1､2级灾害及不发生灾害的频率1p ，2p ，3p ；（3）该河流域某企业，在8月份，若没受1､2级灾害影响，利润为500万元；若受1级灾害影响，则亏损100万元；若受2级灾害影响则亏损1000万元.现此企业有如下三种应对方案: 方案防控等级费用（单位:万元）方案一 无措施 0方案二 防控1级灾害 40方案三 防控2级灾害 100试问，如仅从利润考虑，该企业应选择这三种方案中的哪种方案?说明理由. 【答案】（1）37.5（2）发生0.155， 0.035；不发生0.81，1p ，2p ，3p 分别为0.155，0.035，0.81（3）方案二，理由见解析【解析】（1）根据最高的矩形的中点即为众数，即可得到本题答案；（2）由甲图，得该河流8月份的水位小于40米，在40米和50米之间，大于50米的对应的频率，结合乙图，即可算得该河流在8月份发生1级灾害、2级灾害和不发生灾害的对应的频率；（3）把三种方案对应的平均利润算出来，比较大小，即可得到本题答案. 【详解】 （1）由题得，35+4037.52=，估计该河流在8月份水位的众数为37.5米 （2）依据甲图，该河流8月份的水位小于40米，在40米和50米之间，大于50米的频率分别为()0.020.050.0650.65++⨯=，()0.040.0250.30+⨯=，0.0150.05⨯=.根据乙图，该河流在8月份发生1级灾害的频率为0.650.100.300.200.050.600.155⨯+⨯+⨯=该河流在8月份发生2级灾害的频率为0.300.050.050.400.035⨯+⨯=该河流在8月份不发生灾害的频率为10.1550.0350.81--=估计1p ，2p ，3p 分别为0.155，0.035，0.81.（3）由（2）若选择方案一，则该企业在8月份的平均利润15000.811000.15510000.035354.5L =⨯-⨯-⨯=（万元）;若选择方案二，则该企业在8月份的平均利润25000.9654010000.035407.5L =⨯--⨯=（万元）；若选择方案三，则该企业在8月份的平均利润3500100400L =-=（万元）. 由于231L L L >>，因此企业应选方案二 【点睛】本题主要考查频率直方图的应用，考查学生的分析问题和解决问题能力.20．已知椭圆C :22221x y a b +=（0a b >>）的左顶点为A ，离心率为2，点(B 在椭圆C 上.（1）求椭圆C 的方程；（2）若直线y kx =（0k ≠）与椭圆C 交于E ，F 两点，直线AE ，AF 分别与y 轴交于点M ，N ，求证:在x 轴上存在点P ，使得无论非零实数k 怎样变化，以MN 为直径的圆都必过点P ，并求出点P 的坐标.【答案】（1）22184x y +=（2）证明见解析，()2,0或()2,0-【解析】（1）由题，得222a b =，22421a b+=，解方程组即可得到本题答案； （2）联立直线方程与椭圆方程，求得直线,AE AF 的方程，然后可以确定点,M N 的坐标，由 0PM PN ⋅=u u u u r u u u r得方程，求解即可得到本题答案. 【详解】 （1）依题意，2e =，所以222a b =①，又因为点(B 在椭圆C 上，所以22421a b +=②，由①②解得28a =，24b =，所以椭圆方程为22184x y +=； （2）假设存在这样的点P ，设()0,0P x ，()11,E x y ，1>0x ，则()11,F x y --，联立22,184y kx x y =⎧⎪⎨+=⎪⎩，消去y ，得()221280k x +⋅-=，解得1x =1y =，因为()A -，所以AE所在直线方程为(y x =+，可得M ⎛⎫ ⎝，同理可得N ⎛⎫⎝，所以0PM x ⎛⎫=- ⎝u u u u r，0PN x ⎛⎫=- ⎝u u u r ， 则2040PM PN x ⋅=-=u u u u r u u u r ，解得02x =或02x =-，所以存在点P 且坐标为()2,0或()2,0-，使得无论非零实数k 怎么变化，以MN 为直径的圆都必过点P .【点睛】本题主要考查直线与椭圆的位置关系的应用，其中涉及到椭圆标准方程的求解和以某线段为直径的圆恒过定点的问题，考查学生的转化能力和运算化简能力.21．已知函数2()(2)(0)x f x xe a x x a =-+>. （Ⅰ）当1a =时，求()f x 的单调区间；（Ⅱ）当0x <时，()0f x ≤恒成立，求实数a 的取值范围.【答案】（Ⅰ）增区间为(,1)-∞-，(ln 2,)+∞；减区间为(1,ln 2)-（Ⅱ）10,e⎛⎤ ⎥⎝⎦.【解析】（Ⅰ）求出()'f x ，由()0f x '>可得增区间，由()0f x '<可得减区间．（Ⅱ）由题意将问题转化为()20xe a x -+≥对(),0x ∈-∞恒成立，构造函数()()2x g x e a x =-+，然后求出()g x 的最小值，并由最小值大于等于零可得所求范围． 【详解】（Ⅰ）当1a =时，()2(2),0xf x xe x x x =-+>，得()()()()'2212xxxf x e xe x x e =+-+=+-，令()'0f x =，得11x =-，21ln21x x =>-=， 由()'0f x >得1x <-或ln2x >时， 由()0f x '<得1ln2x -<<，∴()f x 增区间为(),1-∞-，()ln2,+∞；减区间为()1,ln2-． （Ⅱ）由条件得()()20xf x x e a x ⎡⎤=-+≤⎣⎦对0x <恒成立，∵0x <，∴()20xe a x -+≥对(),0x ∈-∞恒成立．设()()2xg x e a x =-+，则()'xg x e a =-，令()'0g x =，得ln x a =．（1）当ln 0a ≥，即1a ≥时，有()'0g x <， ∴()g x 在(),0-∞上是减函数， ∴()()0120g x g a >=-≥，解得12a ≤，不合题意． （2）当ln 0a <，即01a <<时，则得()g x 在(),ln a -∞上是减函数，在()ln ,0a 上是增函数，∴()()()min ln ln 20g x g a a a a ==-+≥，解得10a e<≤，符合题意． 综上可得，实数a 的取值范围是10,e⎛⎤ ⎥⎝⎦．【点睛】解答恒成立问题的常用方法是参数分离法或参数讨论法．参数分离法是将题中的参数分离在不等号的一边，然后转化为求函数的最值的问题求解；若用参数分离无法求解，则可用参数讨论的方法求解，此时需要对参数进行分类讨论，此时需要做到分类标准选择合理、分类不重不漏．22．在平面直角坐标系xOy 中，已知曲线1C :1x y +=与曲线2C :22cos 2sin x y ϕϕ=+⎧⎨=⎩（ϕ为参数，[)0,2ϕπ∈）.以坐标原点为极点，x 轴的非负半轴为极轴建立极坐标系. （1）写出曲线1C ，2C 的极坐标方程；（2）在极坐标系中，已知点A 是射线l :θα=（0ρ≥）与1C 的公共点，点B 是l 与2C 的公共点，当α在区间0,2π⎡⎤⎢⎥⎣⎦上变化时，求OBOA 的最大值.【答案】（1）sin 42πρθ⎛⎫+= ⎪⎝⎭，4cos ρθ=（2）2+ 【解析】（1）由cos ,sin x y ρθρθ==及辅助角公式，即可求解；由22cos 2sin x y ϕϕ=+⎧⎨=⎩消ϕ，得()2224x y -+=，即2240x y x +-=，然后根据222,cos x y x ρρθ+==，即可求解；（2）由题，得1cos sin A OA ραα==+，4cos B OB ρα==，所以通过求224y πα⎛⎫=++ ⎪⎝⎭在5,44ππ⎡⎤⎢⎥⎣⎦的最大值，即可得到本题答案.【详解】（1）因为曲线1C :1x y +=，所以曲线1C 的极坐标方程为()cos sin 1ρθθ+=，即sin 42πρθ⎛⎫+= ⎪⎝⎭； 因为曲线2C 的普通方程为()2224x y -+=， 即2240x y x +-=，所以曲线2C 的极坐标方程为4cos ρθ=. （2） 由（1）知1cos sin A OA ραα==+，4cos B OB ρα==，()()4cos cos sin 21cos 2sin 2224OB OA παααααα⎛⎫∴=+=++=++ ⎪⎝⎭， 由02πα≤≤知52444πππα≤+≤， 当242ππα+=，即8πα=时， OBOA有最大值2+.【点睛】本题主要考查直线的直角坐标方程转化为极坐标方程，圆的参数方程转化为极坐标方程，以及极径的几何意义的应用，其中涉及到利用三角函数的图象与性质求最值，考查学生的分析能力和转化能力. 23．[选修4—5：不等式选讲]已知函数2()1f x x x a =-++，其中a R ∈． （1）当a =()6f x ≥的解集；（2）若存在0x R ∈，使得0()4f x a <，求实数a 的取值范围．【答案】（1）7522x x x ⎧⎫≤-≥⎨⎬⎩⎭或（2）(22 【解析】【详解】试题分析（1）利用零点分段法去绝对值，将()f x 变为分段函数来求解不等式.（2）利用绝对值不等式的性质求得()f x 的最小值为21a +，且()211f a =+，由第 21 页 共 21 页 241a a >+解求的a 的取值范围.详解（1）当a =()(21),212=3,2121,1x x f x x x x x x -+≤-⎧⎪=-++-<<⎨⎪+≥⎩， 所以()26216x f x x ≤-⎧≥⇔⎨--≥⎩或2136x -<<⎧⎨≥⎩或1216x x ≥⎧⎨+≥⎩，解得72x ≤-或52x ≥， 因此不等式()6f x ≥的解集的7522x x x ⎧⎫≤-≥⎨⎬⎩⎭或 （2）()()()22221111f x x x a x x a a a =-++≥--+=+=+，且()211f a =+， 所以()2min 1f x a =+，所以存在0x R ∈，使得()0f x a <4，等价于241a a >+， 所以2410a a -+<，解得22a <<+， 所以实数a的取值范围是(2…。

福建省福州市福建师范大学附属中学2024届高三下学期校模拟考试数学试题(解析版)

福建省福州市福建师范大学附属中学2024届高三下学期校模拟考试数学试题(解析版)

福建省福州市福建师范大学附属中学2024届高三下学期校模拟考试数学试时间：120分钟满分：150分命题：高三集备组审核：高三集备组一、单项选择题：本题共8小题，每小题5分，共40分．在每小题给出的选项中，只有一个选项是符合题目要求的．1.设集合2{10}A x x =->，2{log 0}B x x =>，则A B ⋂等于（）A .{1}x x > B.{0}x x > C.{1}x x <- D.{1x x <-或1}x >【答案】A 【解析】【分析】先解一元二次不等式210x ->和对数不等式2log 0x >化简集合，再求交集.【详解】不等式210x ->解得1x <-或1x >，集合{1A x x =<-或1}x >，不等式22log 0log 1x >=，解得1x >，集合{1}B x x =>{1}A B x x ∴⋂=>．故选：A ．2.已知等差数列{}n a 满足1231010a a a a +++⋅⋅⋅+=，则（）A.11010a a +>B.11010a a +< C.3990a a += D.5151a =【答案】C 【解析】【分析】利用等差数列的性质可得110121005052512a a a a a a a +=+=⋅⋅⋅=+=，进而可得到答案.【详解】根据等差数列的性质，得110121005052512a a a a a a a +=+=⋅⋅⋅=+=，因为1231010a a a a +++⋅⋅⋅+=，所以510a =，所以11013995120a a a a a +=+==，故选：C ．3.若函数()(ln f x ax =+是奇函数，则a 的值为（）A.1B.－1C.±1D.0【答案】C 【解析】【分析】根据函数奇函数的概念可得((ln ln 0ax ax -++=，进而结合对数的运算即可求出结果.【详解】因为()(ln f x ax =+是奇函数，所以f (－x )＋f (x )＝0．即((ln ln 0ax ax -+++=恒成立，所以()22ln 110a x ⎡⎤-+=⎣⎦，即()2210a x -=恒成立，所以210a -=，即1a =±．当1a =时，()(ln f x x =+，定义域为R ，且()()0f x f x -+=，故符合题意；当1a =-时，()(ln f x x =-+，定义域为R ，且()()0f x f x -+=，故符合题意；故选：C.4.将甲､乙､丙､丁4人分配到3个不同的工作岗位，每人只去一个岗位，每个岗位都要有人去，则甲､乙二人分别去了不同岗位的概率是（）A.13B.12C.23D.56【答案】D 【解析】【分析】先求出甲、乙、丙、丁四人分到三个不同的工作岗位，每个岗位至少分到一人共有的选择数，再求出甲、乙两人被分到同一个工作岗位的选择数，再利用古典概型求概率公式及对立事件求概率公式进行求解即可.【详解】甲、乙、丙、丁四人分到三个不同的工作岗位，每个岗位至少分到一人，则必有2人分配到同一个工作岗位，先从4人中选出2人，有24C =6种选择，再进行全排列，有33A =6种选择，故总的方法有2343C A 36=种，其中甲、乙两人被分到同一个工作岗位的情况：从3个岗位中选出一个分配给甲乙，再将剩余的丙丁和剩余的两个岗位进行全排列，有1232C A 6=种选择，所以甲､乙二人分配到同一个工作岗位的概率为61366=，故甲､乙二人分别去了不同工作岗位的概率为151=66-.故选：D5.设,a b为单位向量，a 在b 方向上的投影向量为12b - ，则2a b -= （）A.B.C.D.【答案】D 【解析】【分析】根据投影向量的定义，结合平面向量数量积的运算性质进行求解即可.【详解】因为a 在b方向上的投影向量为12b - ，所以()111222a b b b a a b b a b a b b⋅-=⋅⋅⇒-=⋅⋅⇒⋅=-⋅，所以有2a b -==，故选：D 6.已知3()5P A =，()15P AB =，1(|)2P A B =，则()P B =（）A.15B.25C.35D.45【答案】D 【解析】【分析】首先由()()()P A P AB P AB =+求出()P AB ，再由条件概率公式计算可得.【详解】因为()()()P A P AB P AB =+，3()5P A =，()15P AB =，所以()()()25P AB P A P AB =-=，所以()()()1|2P AB P A B P B ==，则()()()2451|52P AB P A B P B ===.故选：D7.如图，设抛物线24y x =的焦点为F ，不经过焦点的直线上有三个不同的点A ，B ，C ，其中点A ，B 在抛物线上，点C 在y 轴上，则BCF ∆与ACF ∆的面积之比是A.11BF AF -- B.2211BF AF -- C.11BF AF ++ D.2211BF AF ++【答案】A 【解析】【详解】，故选A.考点：抛物线的标准方程及其性质8.在ABC 中，1202ACB BC AC ∠=︒=，，D 为ABC 内一点，AD CD ⊥，120BDC ∠=︒，则tan ACD ∠=（）A.22B.332C.6D.32【答案】B 【解析】【分析】在Rt ADC 中，设ACD θ∠=，AC x =，即可表示出CB ，CD ，在BCD △中利用正弦定理2cos sin(60)32x θθ=-︒，再由两角差的正弦公式及同角三角函数的基本关系将弦化切，即可得解.【详解】在Rt ADC 中，设ACD θ∠=π02θ⎛⎫<< ⎪⎝⎭，令AC x =()0x >，则2CB x =，cos CD x θ=，在BCD △中，可得120BCD θ∠=︒-，60CBD θ∠=-︒，由正弦定理sin sin BC CDCDB CBD=∠∠，cos sin(60)x θθ==-︒1322=，可得33tan 2θ=，即33tan 2ACD ∠=．故选：B ．【点睛】关键点点睛：本题解答关键是找到角之间的关系，从而通过设元、转化到BCD △中利用正弦定理得到关系式.二．多项选择题：本题共3小题，每小题6分，共18分．在每小题给出的四个选项中，有多个选项符合题目要求，全部选对的得6分，部分选对的得部分分，有选错的得0分．9.已知复数12,z z ，下列结论正确的是（）A.若12z z =，则2212z z = B.1212z z z z -=-C.若120z z =，则10z =或20z = D.若10z ≠且12z z =，则2121z z =【答案】BCD 【解析】【分析】通过列举特殊复数验证A ；设()1i,,R z a b a b =+∈，则()2i,,R z a b a b =-∈，通过复数计算即可判断B ；由120z z ⋅=得120z z =，即可判断C ；设()1i,,R z a b a b =+∈，通过复数计算即可判断D.【详解】对于A ，设11i z =+，则21i z =-，所以221(1i)2i z =+=，而221(1i)2i z =-=-，所以2212z z ≠，故A 不正确；对于B ，设12i(,),i(,)R R z a b a b z c d c d =+∈=+∈，则1212()()i (i)(i)z z a c b d a b c d z z -=---=---=-，故B 正确；对于C ，若120z z ⋅=，所以120z z ⋅=，所以120z z =，所以10z =或20z =，所以12,z z 至少有一个为0，故C 正确.对于D ，设()221i,,R,0z a b a b a b =+∈+≠，则1z ()i,,R a b a b =-∈，所以21z ()()22i i a b a b a b =+-=+，而2221z a b =+，所以2121z z =，故D 正确.故选：BCD.10.某大型公司规定：若任意连续7天，每天不超过5人体温高于37.3C ︒，则称没有发生群体性发热．下列连续7天体温高于37.3C ︒人数的统计特征数中，能判定该公司没有发生群体性发热的为（）A.中位数为3，众数为2B.均值小于1，中位数为1C.均值为3，众数为4D.均值为2【答案】BD 【解析】【分析】先设出7天体温高于37.3C ︒人数，并按大小关系排好顺序.A ，C 可通过列举法判定；B ，D 可通过放缩法判定.【详解】设连续7天体温高于37.3C ︒人数依次为1a ，2a ，3a ，4a ，5a ，6a ，7a ，则k a N ∈，1,2,3...,7k =.将1a ，2a ，3a ，4a ，5a ，6a ，7a 按顺序从小到大依次记为1b ，2b ，3b ，4b ，5b ，6b ，7b ，且k b N ∈，1,2,3...,7k =.A 选项：由中位数为3得43b =，又众数为2，所以5b ，6b ，7b 的值无法确定，故选项A 错误；B 选项：由中位数为1得41b =，由均值小于1得127 (17)b b b +++<，有71271 (177)b b b b ++++<<，76b <，故选项B 正确；C 选项：由均值为3得，127 (37)b b b +++=，127...21b b b +++=，取10b =，231b b ==，4564b b b ===，77b =，满足众数为4，但有1天有7人体温高于37.3C ︒，故选项C 错误；D 选项：由均值为2得，127 (27)b b b +++=，127...14b b b +++=，<=，所以()27214b-<，所以75b≤，故选项D正确.故选：BD.11.已知12212log,log2ba a b⎛⎫== ⎪⎝⎭，则（）A.22a ba b-+=+ B.22b aa b-+=+C.121eb a+> D.112ea b->【答案】AD【解析】【分析】结合图象和指、对函数之间的关系即可判断AB；利用切线不等式e1x x≥+即可判断C；利用不等式ln1≤-x x即可判断D.【详解】对A，由图可知：2xy=与12logy x=交点(),2aA a，()01a<<2logy x=与12xy⎛⎫= ⎪⎝⎭的交点(),2,(1)bB b b->，根据指数函数与对数函数为一对反函数知：A，B关于y x=对称，故22baab-⎧=⎨=⎩，22a ba b-+=+，故A正确；对B，由A知22b aa b-+=+，故B错误；对C，由2ba-=知21ba=，则1211ba+=+，设()e1xf x x=--，x∈R，则()e1xf x'=-，则当(),0x∞∈-时，()0f x'<，此时()f x单调递减；当()0,x∞∈+时，()0f x'>，此时()f x单调递增；则()()00f x f≥=，则e10x x--≥恒成立，即1e xx+≤，当0x=时取等；令1xa=，则有111eaa+≤，因为10≠a，则111eaa+<，即121eb a+<，故C错误；对D，设()ln1h x x x=+-，()0,x∞∈+，则()1xh xx'-=，则当()0,1x∈时，()0f x'>，此时()f x单调递增；当()1,x ∞∈+时，()0f x '<，此时()f x 单调递减；则()()10h x h ≤=，即ln 10x x +-≤在()0,∞+上恒成立，即ln 1≤-x x 在()0,∞+上恒成立，当1x =时取等，令1x b =，则11ln 1b b⎛⎫≤- ⎪⎝⎭，即1ln 1b b ≥-，因为1b >，则1ln 1b b >-，则11e b b ->，故112eabb -=>，故D 正确.故选：AD.【点睛】关键点点睛：本题AB 选项的关键是充分利用图象并结合指、函数的关系，而CD 选项的关键在于两个不等式e 1x x ≥+和ln 1≤-x x 的运用.三、填空题：本题共3小题，每小题5分，共15分．12.已知圆台的上､下底面的面积分别为4π,36π，侧面积为64π，则该圆台的高为__________.【答案】【解析】【分析】根据圆台的性质和有关公式进行计算可得结果.【详解】做圆台的轴截面，如图：由题意得：圆台的上､下底面的半径分别为2，6，设圆台的母线长为l ，高为h ，则该圆台的侧面积()π2664πS l =⨯+⨯=侧，解得8l =，所以h ==故答案为：13.421x x ⎛⎫+- ⎪⎝⎭的展开式中常数项为______．【答案】49【解析】【分析】利用多项式乘法法写出展开式的通项，令x 次数为0即为常数项．【详解】421x x ⎛⎫+- ⎪⎝⎭展开式的通项公式为()()()444214444422C ()11C C 12C C mr r r rr r m r m m r m r mr r r T x x x x x -----+--⎛⎫=+⨯-=-=- ⎪⎝⎭，m r ≤，当0,4m r ==时，常数项为1；当1,2m r ==时，得常数项为()21214212C C 24-=；当2,0m r ==时，得常数项为()02024412C C 24-=；所以展开式中的常数项为1242449++=.故答案为：49.14.已知函数()sin()f x x ωϕ=+在区间π2π,63⎛⎫⎪⎝⎭上单调，其中ω为正整数，π||2ϕ<，且ππ32f f ⎛⎫⎛⎫=- ⎪ ⎪⎝⎭⎝⎭．则()y f x =图象的一个对称中心是______；若π342f ⎛⎫=⎪⎝⎭，则ϕ的值为______．【答案】①.5π,012⎛⎫⎪⎝⎭答案不唯一②.π6##30︒【解析】【分析】根据单调区间，以及2π3πf f ⎛⎫⎛⎫=- ⎪ ⎪⎝⎭⎝⎭可得0π32π2f ⎛⎫+ ⎪= ⎪ ⎪⎝⎭，进而可得对称中心；先根据单调区间求出ω的可能取值，然后根据342f π⎛⎫= ⎪⎝⎭得到ω和ϕ的关系，根据关系以及ω的可能取值对照验证计算即可.【详解】因为()f x 在区间π2π,63⎛⎫⎪⎝⎭上单调，且2π3πf f ⎛⎫⎛⎫=-⎪ ⎪⎝⎭⎝⎭，,π,ππ2π2π,36363π2⎛⎫⎛⎫∈∈ ⎪ ⎪⎝⎭⎝⎭，所以5π32π212π0f f ⎛⎫+ ⎪⎛⎫== ⎪ ⎪⎝⎭ ⎪⎝⎭，所以()y f x =图象的一个对称中心是5π,012⎛⎫⎪⎝⎭；由题设，()f x 的最小正周期ππ2π2,362362ππππT ⎛⎫≥⨯-=-=< ⎪⎝⎭，故2π2Tω=≤，由N ω*∈，得1,2ω=，由5π,012⎛⎫⎪⎝⎭为()()sin f x x ωϕ=+的一个对称中心，所以115π12π,k k ωϕ+=∈Z ①；因为342f π⎛⎫=⎪⎝⎭，所以2πππ243k ωϕ+=+或2332π,2π43πk k k ωϕ∈+=+Z 、.若2πππ243k ωϕ+=+②，①-②得()12122ππ,63πk k k k ω=-+-∈Z 、，即()1212262,k k k k ω+-∈=-Z 、，不存在整数12k k 、，使得1,2ω=.若332π243ππ,k k ωϕ+=+∈Z ③，①-③得()13132π2ππ,63k k k k ω=-+-∈Z 、，即()1313462,k k k k ω=-+-∈Z 、，不存在整数13k k 、使得1ω=，当1321k k =+时，2ω=.此时333π=π2πππ2232,6k k k ϕ=-++∈Z ，由π2ϕ<，得π6ϕ=.故答案为：5π,012⎛⎫⎪⎝⎭；π6【点睛】思路点睛：解决本题的思路是通过ππ2π2,362362ππππT ⎛⎫≥⨯-=-=<⎪⎝⎭确定1,2ω=，联立5012f π⎛⎫= ⎪⎝⎭和342f π⎛⎫=⎪⎝⎭可得()12122ππ,63πk k k k ω=-+-∈Z 、或()13132π2ππ,63k k k k ω=-+-∈Z 、，分别验证是否有1,2ω=，即可求得ϕ.四、解答题：本题共5小题，共77分．解答应写出文字说明、证明过程或演算步骤．15.按照《中华人民共和国环境保护法》的规定，每年生态环境部都会会同国家发展改革委等部门共同编制《中国生态环境状况公报》，并向社会公开发布．下表是2017-2021年五年《中国生态环境状况公报》中酸雨区面积约占国土面积的百分比()%i y ：年份2017年2018年2019年2020年2021年年份代码i x 12345iy 6.4 5.5 5.0 4.8 3.8（1）求2017—2021年年份代码i x 与i y 的样本相关系数（精确到0.01）；（2）请用样本相关系数说明该组数据中y 与x 之间的关系可用一元线性回归模型进行描述，并求出y 关于x 的经验回归方程；（3）预测2024年的酸雨区面积占国土面积的百分比．（回归直线的斜率和截距的最小二乘法估计公式分别为：()()()55212111,,7ˆˆˆ0.6,133.69niii i i i ni i i i x x y y bay bx x y y x x ====--⎛⎫==-== ⎪⎝⎭-∑∑∑∑附：样本相关系数，()()6niix x y y r --=∑．【答案】（1）0.98-（2） 0.59 6.87y x =-+（3）预测2024年的酸雨区面积占国土面积的百分比为2.15%【解析】【分析】（1）由表中数据结合题中数据，求出相关数值，代入相关系数()()niix x y y r --=∑（2）由（1）知0.98r ≈-，r 接近1，即可说明线性相关关系极强；根据（1）中求出的数据，即可求出ˆ0.59b =-，ˆ 6.87a=，进而得到回归直线方程；（3）将8x =代入回归直线方程，即可预测2024年的酸雨区面积占国土面积的百分比.【小问1详解】由己知可得，1234535x ++++==，6.4 5.5 5.0 4.8 3.8 5.15y ++++==，由题可列下表：i x x -2-1-012i y y- 1.30.40.1-0.3- 1.3-()()51i i i x x y y =--=-∑()()55.90.986iix x y y r ---=≈-∑．【小问2详解】由小问1知，y 与x 的相关系数0.98,r r ≈-接近1，所以y 与x 之间具有极强的线性相关关系，可用线性回归模型进行描述．由小问1知，()()()515215.9ˆ0.5910iii i i x x y y bx x ==---===--∑∑，ˆˆ 5.1(0.59)3 6.87ay bx =-=--⨯=，所求经验回归方程为 0.59 6.87y x =-+．【小问3详解】令8x =，则 0.598 6.87 2.15y =-⨯+=，预测2024年的酸雨区面积占国土面积的百分比为2.15%．16.已知函数()ln(1)()f x ax x a =--∈R ．（1）求曲线()y f x =在点(0,(0))f 处的切线方程；（2）若()0f x ≥恒成立，求a 的值【答案】（1）(1)y a x =+；（2）1a =-【解析】【分析】（1）根据导数的几何意义求解即可；（2）由()ln(1)f x ax x =--，得1()(1)1f x a x x'=+<-，当0a ≥时，不符合题意；当a<0时，()f x 最小值为111ln()f a a a ⎛⎫+=++- ⎪⎝⎭，若()0f x ≥恒成立，则1ln()0a a ++-≥，设()1ln()(0)x x x x ϕ=++-<．根据导数研究()ϕx 的最大值，即可求出a 的值.【小问1详解】定义域为(,1)-∞，由()ln(1)f x ax x =--，得1()(1)1f x a x x'=+<-，因为(0)0,(0)1f f a '==+，所以曲线()y f x =在点(0,(0))f 处的切线方程为(1)y a x =+；【小问2详解】定义域为(,1)-∞，11()11ax a f x a x x'-++=+=--，①当0a ≥时，(1)ln 20f a -=--<，不符合题意．②当a<0时，令()0f x '=，解得11x a=+，当1,1x a ⎛⎫∈-∞+⎪⎝⎭时，()0,()'<f x f x 在区间1,1a ⎛⎫-∞+ ⎪⎝⎭上单调递减，当11,1x a ⎛⎫∈+⎪⎝⎭时，()0,()'>f x f x 在区间11,1a ⎛⎫+ ⎪⎝⎭上单调递增，所以当11x a=+时，()f x 取得最小值111ln()f a a a ⎛⎫+=++- ⎪⎝⎭；若()0f x ≥恒成立，则1ln()0a a ++-≥，设()1ln()(0)x x x x ϕ=++-<，则11()1x x x xϕ'+=+=，当(,1)x ∈-∞-时，()0,()x x ϕϕ'>在区间(,1)-∞-上单调递增，当(1,0)x ∈-时，()0,()x x ϕϕ'<在区间(1,0)-上单调递减，所以()(1)0x ϕϕ≤-=，即1ln()0a a ++-≥的解为1a =-．所以1a =-.17.如图，在三棱柱111ABC A B C -中，AC =，1AB =，E ，F 分别为1AC ，1BB 的中点，且EF ⊥平面11AA C C ．（1）求棱BC 的长度；（2）若111BB A B ⊥，且1△A FC 的面积12△A FC S =，求二面角11B A F C --的正弦值．【答案】（1）1（2）32【解析】【分析】（1）根据平行关系可得EF DB ，再结合垂直关系可得DB AC ⊥，即可得结果；（2）根据题意分析可得1BB ⊥平面ABC ，12AA =，建系，利用空间向量求二面角.【小问1详解】取AC 中点D ，连接ED ，BD ，∵,D E 分别为1,AC A C 的中点，则DE 1AA 且112DE AA =，又∵111ABC A B C -为三棱柱，且F 分别为1BB 的中点，则BF 1AA 且112BF AA =，可得DEBF 且DE BF =，即四边形DEFB 为平行四边形，故EF DB ，又∵EF ⊥平面11AA C C ，则DB ⊥平面11AA C C ，AC ⊂平面11AA C C ，可得DB AC ⊥，又∵D 为AC 的中点，则△ABC 为等腰三角形，∴1BC AB ==.【小问2详解】由（1）可知：1BC AB ==，且AC =，即222AB BC AC +=，∴AB BC ⊥，则可得2EF DB ==，且1111A B B C ⊥，∵EF ⊥平面11AA C C ，1AC ⊂平面11AA C C ，则1EF A C ⊥，∴111112222△A FC S A C EF A C =⋅=⨯=，解得12A C =，由（1）知DB ⊥平面11AA C C ，1AA ⊂平面11AA C C ，则1DB AA ⊥，又∵1AA 1BB ，则1DB BB ⊥又∵111BB A B ⊥，AB 11A B ，则1BB AB ⊥，AB DB B = ，,AB DB ⊂平面ABC ，∴1BB ⊥平面ABC ，AC ⊂平面ABC ，则1BB AC ⊥，且1AA 1BB ，可得1AA AC ⊥，∴1AA C △为直角三角形，则1AA ==以1B 为坐标原点，向量11B C ，11B A ，1B B方向为x 轴，y 轴，z 轴正方向建立空间直角坐标系1B xyz -，则()10,0,0B ，()10,1,0A ，()11,0,0C，(C，(B ，20,0,2F ⎛⎫⎪ ⎪⎝⎭，可得120,2A F ⎛⎫=- ⎪⎝⎭，(11,AC =-，设平面1A FC 的一个法向量为()1,,n x y z =，则11112020n A F y z n A C x y ⎧⋅=-+=⎪⎨⎪⋅=-+=⎩ ，令1y =，则1,x z =-=，可得(1n =-，∵平面11B A F 的一个法向量为()21,0,0n =，设二面角11B A F C --的平面角为()0,πθ∈，可得121211cos 212n n n n θ⋅===⨯⋅u r u u r u r u u r ，∴sin 2θ==，故二面角11B A F C --的正弦值为2.18.设F 是双曲线Γ：221x y -=的左焦点，经过F 的直线与Γ相交于M ，N 两点．（1）若M ，N 都在双曲线的左支上，求OMN 面积的最小值．（2）是否存在x 轴上一点P ，使得PM PN ⋅为定值?若存在，求出点P 的坐标；若不存在，说明理由．【答案】（1（2）存在这样的定点2,20⎛⎫- ⎪ ⎪⎝⎭P 【解析】【分析】（1）联立直线与双曲线方程，即可由弦长公式以及点到直线距离公式求解长度，利用面积公式以及二次函数的性质即可求解，（2）由向量数量积的坐标运算，即可结合韦达定理化简求解.【小问1详解】设直线MN的方程为x my =，()11,M x y ，()22,N x y ．由221x my x y ⎧=-⎪⎨-=⎪⎩可得()()221101m y m --+=≠±，由根与系数的关系可知122221y y m +=-，12211y y m =-①．此时()222212111m MN m m +=--．原点O到直线MN的距离为d =此时()222111221OMNm S d MN m +===-△．由M ，N 都在双曲线的左支上知()1212201x x m y y m =+-+-<，122201x x m -=>-，得11m -<<，令()2110m t t -=-≤<，则2221111144244OMNS t t t ⎛⎫⎛⎫=+=+- ⎪ ⎪⎝⎭⎝⎭△，由于(]1,1t ∞∈--，所以当11t=-，即1t =-时，此时取最大值，则OMN S ≥△，当1t =-，即0m =时，等号成立．【小问2详解】假设存在这样的定点(),0P n ．当直线的斜率不为0时，由（1）知()()()()()()112212121212,,PM PN x n y x n y x n x n y y my n my n y y ⋅=-⋅-=--+=--+())())2212121m y y mn y y n=+-⋅++②．将①代入②可得())222311m PM PN nm --+⋅=+- ，此时要想PM PN ⋅ 为定值，则3111--=-，得22n =-，从而12PM PN ⋅=- ．即存在这样的定点2,20⎛⎫- ⎪ ⎪⎝⎭P 满足题意．当直线的斜率为0时，易知()()2111PM PN n n n ⋅=+-=- ，若2,20⎛⎫- ⎪ ⎪⎝⎭P ，则12PM PN ⋅=- ，满足题意.综上，存在2,20⎛⎫- ⎪ ⎪⎝⎭P 满足题意．【点睛】方法点睛：圆锥曲线中取值范围问题的五种求解策略：（1）利用圆锥曲线的几何性质或判别式构造不等关系，从而确定参数的取值范围；（2）利用已知参数的范围，求新的参数的范围，解这类问题的核心是建立两个参数之间的等量关系；（3）利用隐含的不等关系建立不等式，从而求出参数的取值范围；（4）利用已知的不等关系建立不等式，从而求出参数的取值范围；（5）利用求函数值域的方法将待求量表示为其他变量的函数，求其值域，从而确定参数的取值范围.19.定义首项为1且公比为正数的等比数列为“M －数列”.（1）已知等比数列{a n }满足：245132,440a a a a a a =-+=，求证:数列{a n }为“M －数列”；（2）已知数列{b n }满足:111221,n n n b S b b +==-，其中S n 为数列{b n }的前n 项和．①求数列{b n }的通项公式；②设m 为正整数，若存在“M －数列”{c n }，对任意正整数k ，当k ≤m 时，都有1k k k c b c +≤≤成立，求m 的最大值．【答案】（1）见解析；（2）①b n =n ()*n ∈N ；②5.【解析】【分析】（1）由题意分别求得数列的首项和公比即可证得题中的结论；（2）①由题意利用递推关系式讨论可得数列{b n }是等差数列，据此即可确定其通项公式；②由①确定k b 的值，将原问题进行等价转化，构造函数，结合导函数研究函数的性质即可求得m 的最大值．【详解】（1）设等比数列{a n }的公比为q ，所以a 1≠0，q ≠0.由245321440a a a a a a =⎧⎨-+=⎩，得244112111440a q a q a q a q a ⎧=⎨-+=⎩，解得112a q =⎧⎨=⎩．因此数列{}n a 为“M —数列”.（2）①因为1122n n n S b b +=-，所以0n b ≠．由1111,b S b ==得212211b =-，则22b =.由1122n n n S b b +=-，得112()n n n n n b b S b b ++=-，当2n ≥时，由1n n n b S S -=-，得()()111122n n n nn n n n n b b b b b b b b b +-+-=---，整理得112n n n b b b +-+=．所以数列{b n }是首项和公差均为1的等差数列.因此，数列{b n }的通项公式为b n =n ()*n N ∈.②由①知，b k =k ，*k ∈N .因为数列{c n }为“M –数列”，设公比为q ，所以c 1=1，q >0.因为c k ≤b k ≤c k +1，所以1k k q k q -≤≤，其中k =1，2，3，…，m .当k =1时，有q ≥1；当k =2，3，…，m 时，有ln ln ln 1k kq k k ≤≤-．设f （x ）=ln (1)x x x >，则21ln ()xf 'x x-=．令()0f 'x =，得x =e .列表如下：x (1,e)e (e ，+∞)()f 'x +–f （x ）极大值因为ln 2ln8ln 9ln 32663=<=，所以max ln 3()(3)3f k f ==．取q =k =1，2，3，4，5时，ln ln kq k，即k k q ≤，经检验知1k q k -≤也成立．因此所求m 的最大值不小于5．若m ≥6，分别取k =3，6，得3≤q 3，且q 5≤6，从而q 15≥243，且q 15≤216，所以q 不存在.因此所求m 的最大值小于6.综上，所求m 的最大值为5．【点睛】本题主要考查等差和等比数列的定义、通项公式、性质等基础知识，考查代数推理、转化与化归及综合运用数学知识探究与解决问题的能力．。

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【全国百强校】福建省福建师范大学附属中学2020届高三下学期高考模拟（最后一模）英语试题第一部分（共20小题每，小题1.5分，满分30分）1．The picture looks good ____ the white wall.A．with B．to C．against D．beyond2．They were abroad during the months when we were carrying out the investigation, or they __________to our help.A．would have come B．could comeC．have come D．had come3．The 90’s people seem to have enjoyed the great benefits ________ about by the great level of cultural and economic development.A．brought B．bringingC．to be brought D．having brought4．-- It shouldn’t take long to clear up after the party if we all volunteer to help．--- That’s right．_____________．A．Many hands make light work B．Something is better than nothingC．The more the merrier D．The sooner begin, the sooner done5．-----My room gets very cold at night.-----_________________.A．So is mine B．So mine is C．So does mine D．So mine does6．It is reported that the ______ in Libya is hotting up.A．strength B．violence C．power D．activity7．—Amazingly, I’ve managed to start my own shop online!—________ I told you it was easy.A．There you are! B．Believe it or not.C．How come? D．You got me there!8．—People should stop using their cars and start using public transport.—________. The roads are too crowded as it is.A．All right B．ExactlyC．Go ahead D．Fine9．The traffic is heavy these days. I arrive a bit late, so could you save me a place?A．canB．mustC．needD．might10．The laptops made by our company sell best, but nobody could have guessed the place in the market that they ________ 20 years ago.A．had had B．had C．were having D．were to have11．You ________ have booked the tickets in advance; there were plenty left.A．needn’t B．can’tC．shouldn’t D．mustn’t12．The project is far behind schedule. It’s impossible for you to ______ it in a week.A．catch upon B．live up toC．add up to D．hold on to13．In my opinion, parents and teachers are supposed to set an example of good behavior to children _____ they are role models to them.A．although B．even if C．unless D．as14．— I am worn out. — Me too, all work and no play. So it’s time to ________.A．burn the midnight oil B．push the limitsC．go with the flow D．call it a day15．You look beautiful in this dress and there is only one of this kind left here. I wonder if you would buy ________.A．it B．oneC．some D．any16．--- I’ll send you the signed contract personally this time tomorrow.--- Sorry, I ______ a meeting then.A．am attending B．attendedC．have attended D．will be attending17．With a travelling speed of up to 350 kilometres per hour, the railway to be built between Beijing and Shanghai _______ the journey time from 12 hours to 5 hours．A．cuts B．will cut C．is cutting D．has cut18．(2013·天津，9)No one________ be more generous；he has a heart of gold.A．could B．must C．dare D．need19．______almost one hundred jin, the stone was moved by him alone.A．Weighed B．Weighing C．It weighed D．To weigh20．By serving others, a person focuses on someone other than himself or herself, ________ can be very eye-opening and rewarding.A．who B．whichC．what D．that第二部分阅读理解（满分40分）阅读下列短文，从每题所给的A、B、C、D四个选项中，选出最佳选项。

21．（6分）Scientists have been studying how people use money for long. Now they’re finding some theories may apply to one group of monkeys.Researchers recently taught six monkeys how to use money. They gave the monkeys small metal disks(圆片) that could be used like cash and showed them some yummy apple pieces. The monkeys soon figured out that if they gave one of the disks to a scientist, they d receive a piece of apple in return.If you think that is all the monkeys can figure out, you are wrong. Two researchers, Jake and Allison, acted as apple sellers in the experiments. The monkeys were tested one at a time and had 12 disks to spend in each experiment. Jake always showed the monkeys one apple piece, while Allison always showed two pieces. But that’s not necessarily what they gave the monkeys. The number of apple pieces given for a disk was determined at random.Experiment One: Allison showed two pieces of apples but gave both pieces only half the time. The other half, she took one piece away and gave the monkey just the remaining piece. Jake, on the other hand, always gave exactly what he showed: one piece for each disk. The monkeys chose to trade more with Allison.Experiment Two: Allison continued to sometimes gave two pieces and sometimes one piece. But now, half the time, Jake gave the one apple piece he was showing, and half the time he added a bonus. Guess what? The monkeys chose to trade more with Jake.In the first experiment, the monkeys correctly figured out that if they traded with Allison, they’d end up with more treats. In the second one, when a monkey received two pieces from Jake, it seemed like again. When Allison gave the monkey only one piece instead of the two she showed, it seemed like a loss. Them onkeys preferred trading with Jake because they’d rather take a chance of seeming to win than seeming to lose.We also sometimes make silly business decisions just to avoid the feeling that we’re getting less, even when were not. Would you have made the same choices?1、What conclusion might experts draw from the first experiment?A．The monkeys show certain business sense.B．Business theories can apply to all monkeys.C．People are smarter in terms of finance.D．It’s easy to teach monkeys how to trade.2、What does a bonus in paragraph 5 refer to?A．A metal disk.B．An apple piece.C．A chance.D．A coin.3、Why did the monkeys choose to trade more with Jake in the second experiment?A．Because Jake always gave them two apple pieces.B．Because the apple pieces from Jake were yummy.C．Because they didn’t like the feeling of losing.D．Because they get more apple pieces from Jake.4、What could be the best title for the passage?A．People’s Business Decision: Lose or Gain?B．Moneky’s Business Sense: Smart or Silly?C．Shopping for Bargain: Same or different?D．Disk for Apple: Who to Trade with?22．（8分）Matthew Layton was 20 minutes from home in Sevierville, Tennessee, on a cold November night in 2016 when he got a cell phone call from his mother. “The mountain’s on fire,” she screamed, “and Brian’s up there!”Layton’s family owned a dozen rental cabins(小屋) on Shields Mountain, and Layton’s friend and fellow rental-cabin owner, Brian McGee, age 56, was up there trying to put the fire out by himself. Layton, 32, hit the gas. He lived on the mountain too.Layton turned around and headed for a dirt road. He made it about halfway up the steep, winding path before his front-wheel-drive car gave up. He called McGee, who drove down in his pickup so they could fight the fire together.They headed first to Layton’s rental cabins. “I wanted to make sure our guests were gone. They were,” says Layton. At that point, he had a choice: try to save his cabins or rescue people renting other cabins nearby. “On the mountain, you don’t have many locals. They’re mostly tourists who don’t know their way around,” he says.Over the next two hours, the two friends drove through the smoky mountain, knocking on doors and leading panicked people to safety. “I know that mountain so well,” Layton says, “I cou ld drive and know exactly where I am just by time traveled.” Thanks to their brave and immediate action, the two helped 14 people out of the danger.Fourteen people died that night in Sevier County. But the fire didn’t take away a single life on Shields Mo untain. And though his home and business were destroyed, Layton remains calm. “I wasn’t worried about my loss, not when I saw those families trapped on the mountain,” he says, “I knew I was gonna help them.”1、Where was Layton when the fire broke out?A．Visiting his mother. B．Away from his home.C．Heading for the cabins. D．Driving on a dirt road.2、What can we learn from Para. 2 & 3?A．Layton’s car broke down halfway. B．Brian was in charge of Layton’s cabins.C．Layton picked up Brian on the path. D．Brian lived in the mountain alone.3、Why could the two friends rescue the people?A．They put out the fire before it spread. B．They turned to locals for help.C．Layton was familiar with the area. D．Brian gave up his own cabins.4、What did Layton mean in the last paragraph?A．He blamed himself. B．He suffered a lot.C．He felt sorry. D．He was relieved.23．（8分）Bandhavgarh National park in India is best known for its amazing setting, as well as having the highest concentration of tigers in India. In addition to tigers, there are also bears, deer, wolves, birds, etc.Location: In Madhya Pradesh state, almost 200 kilometers northeast of Jabalpur. The nearest village is Tala, which is the access point of the park.Opening Hours and Safari(游猎) TimesSafaris operate twice a day, starting at dawn until late morning, and mid afternoon until sunset. The best time to visit the park is early in the morning or after 4 p.m to spot the animals. The park is closed from July 1 to September 30 due to the monsoon season(季风季节).Bandhavgarh ZonesBandhavgarh is divided into three main zones: Tala(the park’s best zone, with the most tigers), Magdhi(the second best zone to see tigers), and Khitauli(scenic and less visited, although tiger sightings do occur there; particularly good for birding).Entry Fees and Charges for Jeep SafarisBandhavgarh has become expensive to visit in recent years. Entry fees have increased since October 2014. When going on a safari, separate fees must be paid for park entry and jeep hire. Entry tickets are priced per vehicle(up to six people). For 2015-16, park entry tickets cost as follows:Tala Zone: 2,400 rupees for Indians, 4,800 rupees for foreigners.(Up to six people)Other Zones: 1,200 rupees for Indians, 2,400rupees for foreigners.(Up to six people)Jeep HireExpect to pay about 2,200 rupees to hire a jeep, in addition to the entry cost. This can be done at the park entrance. All hotels can arrange jeep hire and tours, but at a higher rate. It’s much less trouble though. Besides, unlike m any national parks in India, it’s possible to take private vehicles into Bandhavgarh.1、What is Bandhavgarh National Park most famous for?A．Lots of tigers. B．Beautiful birds.C．Species of bears. D．A number of wolves.2、When can you visit the park?A．In July. B．In June.C．In September. D．In August.3、What do we know about Bandhavgarh Zones?A．Khitauli is frequently visited.B．Magdhi has attractive scenery.C．Magdhi is good for bird-watching.D．Tala is probably the most popular zone.4、How much should five Americans pay for entry fees to visit Magdhi Zone?A．1,200 rupees B．2,400 rupeesC．4,800 rupees D．6,000 rupees5、Which of the following statements is True about jeep hire in the park?A．Tourists can only hire a jeep from the park.B．Hiring jeeps from hotels costs less money.C．Tourists don’t have to hire jeeps.D．Jeep hire fees are included in the entry cost.24．（8分）Nightlife Downtown Crested Butte is home to some fun adventure nightlife! With many different options for enjoying a night, you’re sure to have a great time! Enjoy the free bus system between the mountain and town to get around.Kids’ Night OutOn vacation, kids and parents deserve a special night out. But sometimes; what’s special for the kids isn’t quite what you have in mind. We created Kids’ Night Out so you can all have the night you’re looking for. Our fun-lo ving kids’ instructors host your kids for a Visit to the Adventure Park, followed by dinner and games while you head out of the town. Ages 8~12 are welcome, 75 per child. Kids’ Night Out takes place nightly in the coldest.Majestic FunIt is small and personal and it offers a wide range of movies, from new releases to classics, action etc. All natural snacks and alcoholic drinks are available. For movie show times and more information, call 970-349-8955 or visit our website.Princess Wine BarEscape the ordinary and experience the Princess Wine Bar in downtown Crested Butte. Enjoy thecoffeehouse featuring Belgian snacks, baked eggs, apple-wood smoked bacon, and coffee drinks. Live entertainment makes the Princess Wine Bar the perfect choice. Open daily from 8:00 pm to midnight, but advance reservations are required. For more information you can call 970-349-0210.Talk of the TownIf you are looking for a good time, the Talk offers football, pinball, video games, good music, and pool tables any night of the week. Located in the center Of Crested Butte on Elk Avenue, open daily at 3:00 pm, the Talk’s Happy Hour is from 3:00 to 8:00 pm. For further information, you can call 970-349-209.1、What is special about Kids’ Night Out?A．It’s in the town center. B．It opens only in winter.C．It offers many games. D．It gives parents an interesting night.2、What should you do if you plan to go to Princess Wine Bar?A．Read some extraordinary stories.B．Know the names of different snacks.C．Contact it ahead of time.D．Arrive before 8:00 pm.3、What is the purpose of the text?A．To promote some fun programs.B．To advertise the free bus system.C．To compare the night life items.D．To encourage more outdoor activities.25．（10分）Baseball was always where my heart lay. However, my father did not care for baseball. Two recreational activities he enjoyed were golfing and fishing, neither of which I considered a real sport. He didn’t talk much and he wasn’t good at expressing himself. Frankly, I wasn’t entirely certain he cared about me. In a way, my father was like a familiar stranger I saw every day.One day, watching the play on television made me want a glove of my own. To my surprise, my father said he wanted one, too. He was on the portly side, but then again, one of the famous pitchers (投手) on the Mets was even fatter. So I thought, why not?We bought two gloves, and that evening, he suggested we go out to the courtyard and toss (掷) the ball around. I’d already practiced catching and thro wing at school, but I could tell this was the first time for my father, because every time he caught the ball, it landed directly in his palm. If you’ve ever caught a ball with a baseball glove, you know how much this hurts.But my father caught ball after ball this way. Slap! Slap! He winced (龇牙咧嘴). After about five minutes, I told him I’d had enough. It wasn’t out of mercy, but for my own self-interest. Neighborhood kidswere watching and would no doubt tease me later for having an old man who didn’t know how to properly catch a baseball do that. When my father took the glove off, his palm was red and swollen.Dad, who passed away 11 years ago, never told me he loved me. But that sound of the baseball landing in the soft flesh of his hand, over and over again—it spoke its own tender language, though at the time, all I felt was shame.1、What does the underlined word “portly” mean in Paragraph 2?A．Experienced. B．Cool.C．Famous. D．Fat.2、What can we learn from the passage?A．The father didn’t feel hurt when catching the ball.B．The father didn’t understand his child at all.C．The a uthor didn’t realize his father’s love for him back then.D．The author criticized his father when playing baseball together.3、What could be the best title for the passage?A．A Crying Shame for BaseballB．Saying “I Love You” with BaseballC．A Loving Father and His SonD．Living a Happy Life with Baseball第三部分语言知识运用（共两节）第一节（每小题1.5分，满分30分）阅读下面短文，从短文后各题所给的A、B、C和D四个选项中，选出可以填入空白处的最佳选项．26．（30分）Evelyn Glennie was the first lady of solo percussion in Scotland. In an interview, she talked about how she became a percussion soloist (打击乐器独奏演员) in spite of her disability.“Early on I decided not to allow the 1 of others to stop me from becoming a musician. I grew up on a farm in northeast Scotland and began 2 piano lessons when I was eight. The older I got, the more my passion (酷爱) for music grew. But I also began to gradually lose my 3 . Doctors concluded that the nerve damage was the 4 and by age twelve, I was completely deaf. But my love for music never 5 me.“My 6 was to become a percussion soloist, even though there were none at that time. To perform, I 7 to hear music differently from others. I play in my stocking feet and can 8 the pitch of a note (音调高低) by the vibrations (振动). I feel through my body and through my 9 . My entire sound world exists by making use of almost every 10 that I have.“I was 11 to be assessed as a musician, not as a deaf musician, and I applied to the famous Royal Academy of Music in London. No other deaf student had 12 this before and some teachers 13 my admission. Based on my performance, I was 14 admitted and went to 15 with theacademy’s highest honors.“After that, I estab lished(使立足) myself as the first full-time solo percussionist. I 16 and arranged a lot of musical compositions since 17 had been written specially for solo percussionists.“I have been a soloist for over ten years. 18 the doctor thought I was totally deaf, it didn’t19 that my passion couldn’t be realized. I would encourage people not to allow themselves to be 20 by others. Follow your passion; follow your heart, they will lead you to the place you want to go.”1、A．conditions B．opinions C．actions D．recommendations2、A．enjoying B．choosing C．taking D．giving3、A．sight B．hearing C．touch D．taste4、A．evidence B．result C．excuse D．cause5、A．left B．excited C．accompanied D．disappointed6、A．purpose B．decision C．promise D．goal7、A．turned B．learned C．used D．ought8、A．tell B．see C．hear D．smell9、A．carefulness B．movement C．imagination D．experience10、A．sense B．effort C．feeling D．idea11、A．dissatisfied B．astonished C．determined D．discouraged12、A．done B．accepted C．advised D．admitted13、A．supported B．followed C．required D．opposed14、A．usually B．finally C．possibly D．hopefully15、A．study B．research C．graduate D．progress16、A．wrote B．translated C．copied D．read17、A．enough B．some C．many D．few18、A．However B．Although C．When D．Since19、A．mean B．seem C．conclude D．say20、A．directed B．guided C．taught D．limited第二节（每小题1.5分，满分15分）阅读下面材料，在空白处填入1个适当的单词或括号内单词的正确形式。

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