吉林省东北师大附中、长春十一高中、吉林一中、四平一中、松原实验中学21届高三2月联合模考文科数学答案

2022届吉林省东北师大附中、长春市十一高中、吉林一中、四平一中、松原实验中学高三上学期联合模拟考试数学（文）试题一、单选题1．已知集合{}23,N A x x x =<∈，则A 的真子集共有（ ）个．A ．3B ．4C ．6D ．7【答案】A【分析】解一元二次不等式求集合A ，根据所得集合的元素个数判断其真子集的个数.【详解】由题设，{|N}{0,1}A x x x =<∈=， ∴A 的真子集共有2213-=个. 故选：A.2．已知i 为虚数单位，复数i3iz =+，则复数z 在复平面内对应的点位于( ) A ．第二象限 B ．第三象限 C ．直线30x y -=上D ．直线30x y +=上 【答案】D【分析】根据复数的运算法则求出z 和z ，根据复数的几何意义即可求解. 【详解】()()()i 3i i 13i 13==i 3i 3i 3i 101010z -+==+++-， 13i 1010z =-对应的点13,1010⎛⎫- ⎪⎝⎭在第四象限，且在直线30x y +=上. 故选：D.3．已知x ∈R ，则“1x >”是“1x >”的（ ） A ．充分非必要条件 B ．必要非充分条件 C ．充要条件 D ．既非充分又非必要条件【答案】B【分析】解不等式转化条件，结合充分必要性定义即可作出判断. 【详解】由1x >得1x <-或1x >， ∴“1x >”是“1x >”的必要非充分条件． 故选：B ．4．工厂生产某种产品的月产量y 与月份x 满足关系式0.5x y a b =⋅+，现已知该工厂今年1月份､2月份生产该产品分别为1万件､1.5万件，则此工厂4月份生产该产品的产量为（ ） A ．1.275万件 B ．1.750万件C ．1.875万件D ．2.725万件【答案】C【分析】运用代入法，通过解方程组进行求解即可.【详解】因为该工厂今年1月份､2月份生产该产品分别为1万件､1.5万件，所以1210.5220.52221.50.52x x a b a y y a b b -=⋅+=-⎧⎧⇒⇒=-⋅+⇒=-+⎨⎨=⋅+=⎩⎩， 此工厂4月份生产该产品的产量为1422 1.875--+=万件， 故选：C5．在棱长为4的正方体内任取一点，则这个点到该正方体的中心距离不超过1的概率为（ ）A ．π6B ．π16C ．π32D ．π48【答案】D【分析】利用几何概型的概率公式求解，先求出以正方体的中心为球心，1为半径的球的体积，再求出正方体的体积，两体积相比可得答案【详解】因为棱长为4的正方体的体积为3464=，以正方体的中心为球心，1为半径的球的体积为43π，所以在棱长为4的正方体内任取一点，则这个点到该正方体的中心距离不超过1的概率为436448ππ=，故选：D6．已知π0,2α⎛⎫∈ ⎪⎝⎭，若3cos 5α=，则πcos 6α⎛⎫+ ⎪⎝⎭等于（ ）ABCD【答案】B【分析】利用同角三角函数的基本关系可得4sin 5α，再利用两角和的余弦公式，即可得到答案.【详解】0,2πα⎛⎫∈ ⎪⎝⎭，3cos 5α=，∴4sin 5α=，∴341cos +cos cos sin sin 666552⎛⎫=-=⨯= ⎪⎝⎭πππααα故选：B.7．已知向量a ，b 满足2a =，3b =，且a 与b 的夹角为6π，则()()2a b a b +⋅-=（ ） A ．6 B ．8 C ．10 D ．12【答案】B【分析】应用平面向量数量积的运算律展开目标式，根据已知向量的模和夹角求值即可.【详解】由题设，()()2222838a b a b a a b b +⋅-=+⋅-=+=. 故选：B.8．一个圆柱的侧面展开图是一个边长为2的正方形，则这个圆柱的体积为（ ） A ．π B ．2πC ．1πD ．2π【答案】D【分析】根据圆柱的侧面展开图确定圆柱的底面半径和高，即可求出其体积. 【详解】设圆柱的底面半径为r ，高为h ， 因为圆柱的侧面展开图是一个边长为2的正方形， 所以22r π=，2h =， 所以12,h r ==π，所以圆柱的体积为22r h ⋅=ππ.故选：D.9．已知()f x 是R 上的奇函数，且当0x >时，()2=-af x x x，若()()201f f +=，则()6f -=（ ）A ．4-B ．7-C ．11-D ．15-【答案】C【分析】由奇函数性质有(0)0f =，结合已知条件及函数解析式求参数a ，再利用奇函数性质求(6)f -的值.【详解】由题设，(0)0f =则()2412af =-=，故6a =， 所以0x >时，6()2f x x x=-， 则(6)(6)(121)11f f -=-=--=-. 故选：C10．已知线段MN 是圆()22:18C x y -+=的一条动弦，且23MN =，若点P 为直线260x y -+=上的任意一点，则PM PN +的最小值为（ ）A ．8525- B ．855C ．6525- D ．655【答案】D【分析】过圆心C 作CD MN ⊥，将问题转化为min PD ，结合点到直线距离公式来进行求解.【详解】圆22:(1)8C x y -+=的圆心为()1,0C ，半径为22r =，P 为直线260x y -+=上的任意一点，过C 作CD MN ⊥，垂足为D ，则D 是MN 的中点.由23MN =||5CD =min min ||5C PDP ∴=-，则min2063555PD-+=||2||PM PN PD +=， ||PM PN ∴+65故选：D.11．已知数列{}n a 的首项是11a =，前n 项和为n S ，且1231n n S S n +=++（*N n ∈），设()2log 3n n c a =+，若存在常数k ，使不等式()116n nc k n c -≥+（*N n ∈）恒成立，则k 的最小值为（ ） A ．19B ．116C ．125D ．136【答案】C【分析】根据数列的递推式可得到123n n a a +=+，构造等比数列求出123n n a +=-，继而求出n c ，再利用基本不等式求得()116n n c n c -+的最大值，则可得答案.【详解】当2n ≥ 时，由1231n n S S n +=++可得-123-2n n S S n =+， 两式相减得：123n n a a +=+ ，即132(3)n n a a ++=+，而134a +=，2121224,5a a S S a +==+=，故2132(3)a a +=+ ， 所以{3}n a + 是以134a +=为首项，2q为公比的等比数列，则11342,23n n n n a a -++=⨯=- ，故()122log 3log 21n n n c a n +=+==+，所以()111616(16)(1)17n n c n n c n n n n-==+++++， 而16N ,8n n n*∈+≥ ，当且仅当4n = 时取等号， 故()11116162517n n c n c n n-=≤+++，当且仅当4n = 时取等号， 所以若存在常数k ，使不等式()116n nc k n c -≥+（*N n ∈）恒成立，则k 的最小值为125， 故选：C12．已知1F ，2F 分别是双曲线E ：22221x y a b -=（0a >，0b >）的左､右焦点，直线y kx=与E 交于A ，B 两点，且1260F AF ∠=︒，四边形12F AF B 的周长C 与面积S满足2C =，则E 的渐近线方程是（ ） A．y x = B．y x = C．y = D．y =【答案】C【分析】不妨设(),AF m AF n m n ==>12，结合双曲线定义和余弦定理可得()m n c a +=-2221612，再由四边形12F AF B 的周长与面积关系求得,a c 关系，由222c a b =+得出,a b 关系即可求双曲线的渐近线.【详解】不妨设(),AF m AF n m n ==>12，由双曲线的定义可知，2m n a -=，即m n mn a +-=22224①， 又1260F AF ∠=︒，所以由余弦定理可得m n mn c +-=2224②，由②-①可得 2244mn c a =-， 由⨯②2-①可得222284m n c a +=-， 所以()m n c a +=-2221612.由题意可知，四边形12F AF B 为平行四边形，所以四边形12F AF B 的周长()C m n =+=2四边形12F AF B 的面积)S c a =⨯=-2212442.因为2C =，所以)()c a c a -=-22224441612，整理得223c a =，由222c a b =+，得222b a =，即ba=所以E 的渐近线方程是y =. 故选：C.二、填空题13．已知{}n a 是公差为3的等差数列.若1a ，2a ，4a 成等比数列，则{}n a 的前6项和6S =___________.【答案】63【分析】由等比中项的性质及等差数列通项公式求得13a =，进而写出{}n a 通项公式，利用等差数列前n 项和公式求6S .【详解】由题设，2214a a a =，则2111(3)(9)a a a +=+，解得13a =，所以3n a n =，则66(318)632S ⨯+==. 故答案为：6314．已知函数()π2sin 6f x x ω⎛⎫=- ⎪⎝⎭的最小正周期为π，则函数()y f x =在区间π0,3⎡⎤⎢⎥⎣⎦上的最大值与最小值的和是___________. 【答案】1或3-3-或1【分析】由最小正周期可得2ω=±，讨论ω并结合正弦函数的性质求()f x 的值域，即可求最大值与最小值的和. 【详解】由题设，2||T ππω==，则2ω=±，在π0,3⎡⎤⎢⎥⎣⎦上，当2ω=则2[,]662x πππ-∈-，故()[1,2]f x ∈-；当2ω=-则52[,]666x πππ--∈--，故()[2,1]f x ∈--；综上，最大值与最小值的和为1或3-. 故答案为：1或3-15．某三棱锥的三视图如图所示，则该三棱锥的体积为___________.【答案】16【详解】首先将三视图的直观图放入长方体里，再计算其体积即可. 【点睛】将三棱锥V ABC -放入长方体中，如图所示：121222ABC S ==，三棱锥的体积1111326V =⨯⨯=.故答案为：1616．已知函数()333sin x x x f x =+-，若对任意的()0,x ∈+∞，不等式()()ln 20f x f ax -+≤恒成立，则实数a 的取值范围为___________. 【答案】31,e ⎛⎤-∞ ⎥⎝⎦-【分析】利用函数奇偶性的定义可判断()f x 为奇函数，由导数判断()f x 为R 上的增函数，则所求不等式等价于ln 2x ax -≤-，分离参数可得ln 2x a x--≥，构造函数ln 2()x g x x-=(0)x >，利用导数求()g x 的最大值即可求解.【详解】因为()()()()()()3333sin 33sin f x x x x x x x f x -=-+---=-+-=-，所以()f x 为奇函数，因为()()22333cos 331cos 0x x x x f x '=+-=+-≥，所以()f x 为R 上的增函数，由(ln 2)()0f x f ax -+≤得(ln 2)()()f x f ax f ax -≤-=-，则ln 2x ax -≤-， 因为,()0x ∈+∞，所以ln 2x a x--≥． 令ln 2()(0)x g x x x-=>，则()23ln xg x x -'=，令()0g x '=，得3e x =， 当30e x <<时，()0g x '>，()g x 单调递增， 当3e x >时，()0g x '<，()g x 单调递减，故()()33max 1e e g x g ==，所以31e a -≥，即31ea ≤-， 所以实数a 的取值范围为31,e ⎛⎤-∞- ⎥⎝⎦.故答案为：31,e ⎛⎤-∞- ⎥⎝⎦三、解答题17．在ABC 中，角,,A B C 所对的边分别为,,a b c cos sin C c B =. (1)求角C ；(2)若2b =，ABC 的面积为c . 【答案】(1)3C π=(2)c =【分析】（1cos sin sin B C C B =，进而得tan C =求解即可得答案；（2）由面积公式得8ab =，进而根据题意得2b =，4a =，再根据余弦定理求解即可.【详解】(1)cos sin C c B =，cos sin sin B C C B =， 因为()0,,sin 0B B π∈≠，sin C C =，即tan C =因为()0,C π∈，所以3C π=.(2)解：因为ABC 的面积为23，3C π=，所以13sin 2324S ab C ab ===，即8ab =， 因为2b =，所以4a =，所以2222201cos 2162a b c c C ab +--===，解得23c =.所以23c =.18．如图，四面体ABCD 中，O 、E 分别是BD 、BC 的中点，2, 2.CA CB CD BD AB AD ======（Ⅰ）求证：AO ⊥平面BCD ； （Ⅱ）求点E 到平面ACD 的距离. 【答案】（Ⅰ）详见解析 （Ⅱ21【详解】试题分析：（Ⅰ）要证明AO ⊥平面BCD ，需要证明AO OC ⊥，AO BD ⊥，证明时主要是利用已知条件中的线段长度满足勾股定理和等腰三角形三线合一的性质（Ⅱ）中由已知条件空间直角坐标系容易建立，因此可采用空间向量求解，以O 为坐标原点，以,,OB OC OA 方向为x 轴，y 轴，z 轴正方向建立空间直角坐标系，求出平面ACD 的法向量(3,1,3)n =--和斜线的方向向量13(2EC =-，代入公式EC n d n⋅=计算试题解析：（Ⅰ）证明：,AB AD O =为BD 的中点，AO BD ∴⊥，2AD =,1OD =，1AO ∴=，2,3CB CD BD OC ===∴=又2,CA =222CA OA OC ∴=+，AO OC ∴⊥，BD OC O ⋂=，,BD OC 均在平面BCD 内，AO ∴⊥平面BCD（Ⅱ）方法一：以O 为坐标原点，以,,OB OC OA 方向为x 轴，y 轴，z 轴正方向建立空间直角坐标系，则13(0,0,1),(1,0,0),3,0),(1,0,0),(2A B C D E -，(0,3,1),(1,3,0)AC CD =-=--设n 为平面ACD 的法向量，则n AC ⊥，n CD ⊥ 30,{30,y z x y -=∴+=取n (3,1,3)=--，13(,,0)22EC =-，则点E 到平面ACD 的距离为32177EC n d n⋅===方法二：设点H 在CD 上，且14DH DC =，连AH ， 2,CB CD DB ===O 为BD 的中点，OH CD ∴⊥AO ⊥平面BCD ，CD ⊂平面BCD ，,AO CD ∴⊥,,AO OH O AO OH ⋂=⊂平面AOH ，CD平面AOHCD ⊂平面ACD ，∴平面AOH ⊥平面ACD ，且交线为AH过点O 作OP AH ⊥于点P ，则OP ∴⊥平面ACD,O E 分别为,BD BC 的中点，则//,OE CD OE ⊄平面ACD ，CD ⊂平面ACD ，//OE ∴平面ACD ，E ∴点到平面ACD 的距离即OP ，31372121,,,22772AO OH AO OH AH OP AH⨯⋅===∴===故点E 到平面ACD 的距离为217【解析】1.线面垂直的判定；2.点到面的距离19．为了了解某工厂生产的产品情况，从该工厂生产的产品随机抽取了一个容量为20的样本，测量它们的尺寸（单位：mm ），数据分为[)92,94，[)94,96，[)96,98，[)98,100，[)100,102，[)102,104，[]104,106七组，其频率分布直方图如图所示.(1)求上图中的x 值；(2)根据频率分布直方图，求200件样本尺寸在[)98,100内的样本数；(3)记产品尺寸在[)98,102内为A 等品，每件可获利5元；产品尺寸在[)92,94内为不合格品，每件亏损2元；其余的为合格品，每件可获利3元.若该机器一个月共生产3000件产品.以样本的频率代替总体在各组的频率，若单月利润未能达到11000元，则需要对该工厂设备实施升级改造.试判断是否需要对该工厂设备实施升级改造. 【答案】(1)0.12x =； (2)36（件）；(3)需要对该工厂设备实施升级改造.【分析】（1）根据频率分布直方图中所有小矩形的面积之和为1进行求解即可； （2）根据频率分布直方图中的数据进行求解即可；（3）根据题意，结合频率分布直方图中的数据求出单月利润，最后比较大小即可. 【详解】(1)因为()0.020.040.060.070.090.1021x ++++++⨯=， 解得0.12x =；(2)200件样本中尺寸在[)98,100内的样本数为2000.09236⨯⨯=（件） (3)由题意可得，这批产品中优等品有()30000.180.201140⨯+=（件）， 这批产品中不合格品有30000.04120⨯=件，这批产品中合格品有300011401201740--=300011401201740--=（件），1140517403120210680⨯+⨯-⨯=元.所以该工厂生产的产品一个月所获得的利润为10680元， 因为1068011000<，所以需要对该工厂设备实施升级改造.20．已知椭圆C ：22221x y a b +=（0a b >>）过点P ⎛ ⎝⎭，1F ､2F 分别为椭圆C 的左､右焦点，且焦距为(1)求椭圆C 的方程；(2)过椭圆左顶点A 的直线l 与椭圆C 交于另一点B （B 点在第二象限），AB 的垂直平分线交y 轴负半轴于E 点，D 为椭圆右顶点.设直线BD 的斜率为1k ，直线DE 的斜率为2k ，且满足12320k k =-，求直线l 的方程. 【答案】(1)2214x y +=(2)20x y -+=【分析】（1）利用椭圆的焦距求出c 的值，然后将P 的坐标代入椭圆方程，求得,a b 的值，即可得出椭圆C 的方程.（2）设直线AB 的方程为2x my =-，将此直线方程与椭圆的方程联立，求出点B 的坐标，求出线段AB 的中垂线方程，可求得点E 的坐标，由12320k k =-，求出m 的值，即可得出直线l 的方程【详解】(1)设椭圆C 的焦距为2c （0c >），则()1,0F c -､()2,0F c，c =由已知可得a ba b ⎧+=⎪⎨⎪-=⎩222213143,解得2,1a b ==， 所以椭圆C 的方程为2214x y +=；(2)设直线AB 的方程为2x my =-（0m >），设点()11,B x y ，联立x my x y =-⎧⎨+=⎩22244，可得()22440m y my +-=，故1244m y m =+，即点222284,44m m B m m ⎛⎫- ⎪++⎝⎭， 所以m m m m ⎧-<⎪⎪+⎨⎪>⎪+⎩2222804404，解得02m <<，因为点()2,0A -，所以，线段AB 的中点为2282,44m M m m ⎛⎫- ⎪++⎝⎭，因为点()2,0D ，21224428424m mm k m m +==---+，线段AB 的中垂线方程为222826444m m y m x mx m m m ⎛⎫=-++=-- ⎪+++⎝⎭，在直线方程264m y mx m =--+中，令0x =，可得264my m =-+，即点260,4m E m ⎛⎫- ⎪+⎝⎭，22263424m m m k m +==+， 所以，满足12320k k =-，有2334420m m m -⋅=-+， 可得2254m m =+，解得1m =，因此，直线l 的方程为2x y =-，即20x y -+=.21．已知函数()e xf x ax =-.(1)若1a =，求函数()f x 的单调区间及()f x 在1x =处的切线方程；(2)设函数()()222g x f x x a =--，若0a ≥时，()0g x ≥恒成立，求实数a 的取值范围.【答案】(1)在(),0∞-上单调递减，在()0,∞+上单调递增，切线方程为()e 1y x =-(2)2ln2a ≤-【分析】（1）求出函数的导数，判断导数的正负，可得函数的单调区间；根据导数的几何意义求得切线方程；（2）求函数的导数，判断导数的正负，确定函数的最值，将0a ≥时，()0g x ≥恒成立的问题转化为求函数的最小值问题，分类讨论函数的最小值情况，构造函数利用导数求得参数的取值范围.【详解】(1)1a =时，()()e ,e 1x xf x ax f x '=-=-由()e 10x f x '=->有0x >，由()e 10xf x '=-<有0x <，所以，()f x 在(),0∞-上单调递减，在()0,∞+上单调递增, 又()1e 1f =-，所以切点为()1,e 1-，切线斜率()1e 1k f '==-. 所以切线方程()()()e 1e 11y x --=--， 即切线方程为()e 1y x =-;(2)()()222222e 2x g x f x x a ax x a =--=---，()2e 22x g x x a ='--,设()2(2)e 2x g x x a x ϕ=--'= ，则()2e 2xx ϕ'=-()()0,2e 20,x x x g x ϕ'≥∴-≥'=在[)0,∞+上单调递增，()()min ()02221,g x g a a ==-=-''当10a -≥，即1a ≤时，()min ()210g x a '=-≥，()g x 在[)0,∞+上单调递增，则2min ()20,g x a a =-≥1a ≤. 当10a -<，即1a >时，()min ()210g x a '=-<，()00000,2e 220x x g x x a ∃>=--='，即00e xa x =-或00e x a x =+，00x x <<时，()()0,g x g x '<在()00,x 上单调递减， 0x x >时，()()0,g x g x '>在()0,x +∞上单调递增，则()()()()00000022min 00()2e 2e e e 2e 0,e 2x x x x x x g x g x x a ==-+=-=-≥∴≤，00ln2.x ∴<≤由00e xa x =-，今函数()e x h x x =-，且0ln2x <≤，()()e 10,e x x h x h x x ='-≥=-在(]0,ln2上单调递增，()12ln2h x <≤-，00e (0ln2),12ln2.x a x x a =-<≤∴<≤-综上，实数a 的取值范围是2ln2a ≤-.【点睛】本题综合考查了导数的几何意义和利用导数求函数的单调区间以及用导数解决恒成立问题，综合考查了学生灵活应用导数的知识解决问题的能力，有一定难度，解答的关键时将恒成立问题转化为函数的最值问题时，其中要注意分类讨论，构造函数等方法的运用.22．在直角坐标系xOy 中，已知曲线C 的参数方程为x y ϕϕ⎧=⎪⎨=⎪⎩（ϕ为参数），直线l 0y +-=.以坐标原点O 为极点，x 轴的正半轴为极轴建立极坐标系.（1）求曲线C 和直线l 的极坐标方程；（2）若点(),A x y 在直线l 上，且0y >，射线OA 与曲线C 相交于异于O 点的点B .求OA OB的最小值.【答案】（1）曲线C的极坐标方程为ρθ=，直线l的极坐标方程为cos sin0θρθ+-=；（2）43【分析】（1）首先将曲线C的参数方程转化为普通方程，再根据cossinxyρθρθ=⎧⎨=⎩将直角坐标方程转化为极坐标方程；（2）设点A的极坐标为()1,Aρθ，点B的极坐标为()2,Bρθ，其中23πθ<<，由（1）中的极坐标方程可得OA=inOBθ=，再根据三角恒等变换及正弦函数的性质计算可得；【详解】解：（1）因为曲线C的参数方程为xyϕϕ⎧=⎪⎨=⎪⎩（ϕ为参数），所以(()222212cos sin12x yϕϕ+-=+=，即220x y+-=，因为cossinxyρθρθ=⎧⎨=⎩，所以2sin0ρθ-=，所以曲线C的极坐标方程为ρθ=因为直线ly+-=，所以直线l的极坐标方程为cos sin0θρθ+-=（2）设点A的极坐标为()1,Aρθ，点B的极坐标为()2,Bρθ，其中23πθ<<，由（1）知1OAρ=，2OBρθ==，所以42sin216OAOBπθ==⎛⎫-+⎪⎝⎭因为23πθ<<，所以72666πππθ-<-<，所以1sin2126πθ⎛⎫-<-≤⎪⎝⎭，12sin226πθ⎛⎫-<-≤⎪⎝⎭所以当sin216πθ⎛⎫-=⎪⎝⎭，即3πθ=时，min43OAOB⎛⎫=⎪⎪⎝⎭23．已知12f x x x的最小值为M.(1)解关于x的不等式()1f x M x<+-；(2)若正实数a ，b 满足42M a b+=，求12a b +取最小值时a b -的值. 【答案】(1)(5,1)-； (2)6.【分析】（1）由绝对值的几何意义求M ，再应用公式法求不等式的解集.（2）由（1）可得2312a b +=，应用基本不等式“1”的代换求12a b +的最小值，并确定对应参数a 、b 的值，即可求a b -.【详解】(1)由绝对值几何意义知：()123f x x x =-++≥，当21x -≤≤时等号成立， ∴3M =，由()131f x M x x <+-=+-，可得|2|3x +<，解得51x -<<， ∴不等式解集为(5,1)-.(2)由（1）及题设知：2312a b+=，∴24312(12)()2020322232b a a b a b a b a b +=+=++++≥，当且仅当48a b ==时等号成立， ∴6a b -=.。

一、单选题1. 已知实数a，b 满足（其中为虚数单位），则复数的共轭复数为（）A．B．C．D．2. 某人一周的总开支如图所示，这周的食品开支如图所示，则他这周的肉类开支占总开支的百分比为（）A．B．C．D．3. 设，，，都是不等于1的正数，函数在同一直角坐标系中的图象如图所示，则，，，的大小关系是（）A．B．C．D．4. 已知，则（）A．B．－1C．D．5. 根据下表中数据求得的线性回归方程是，则（）x456789y908483807568A．98B．107C．110D．1066. 已知四面体外接球的球心与正三角形外接圆的圆心重合，若该四面体体积的最大值为，则该四面体外接球的体积为（）A．B．C．D．7. 已知函数，实数，满足不等式，则下列不等式成立的是（）A．B．C．D．8.已知在上连续可导，为其导函数，且，则A．B．C．0D．吉林省(东北师大附中,长春十一高中,吉林一中,四平一中,松原实验中学)五校2023届高三上学期联吉林省(东北师大附中,长春十一高中,吉林一中,四平一中,松原实验中学)五校2023届高三上学期联二、多选题三、填空题四、解答题9. 函数的部分图象如图所示，则下列说法中正确的是（）A．B．的图象向右平移个单位长度后得到的新函数是偶函数C．的图象向右平移个单位长度后得到的新函数是奇函数D ．若方程在上有且只有6个根，则10. 函数与的定义域为，且.若的图像关于点对称.则（ ）A．的图像关于直线对称B．C．的一个周期为4D ．的图像关于点对称11.已知数列满足，，前n 项和为，则（ ）A．B．C．D．12. 如图，边长为2的正方形ABCD 中，E ，F 分别是AB ，BC 的中点，将，，分别沿DE ，DF ，EF 折起，使A ，B ，C 重合于点P ，则下列结论正确的是（）A．B ．三棱锥的外接球的体积为C ．点P 到平面DEF的距离为D ．二面角的余弦值为13.已知四面体中，，，为等边三角形，且平面平面，则四面体外接球的表面积为______.14.设且，则的最小值是______．15. 某个品种的小麦麦穗长度（单位：cm ）的样本数据如下：10.2、9.7、10.8、9.1、8.9、8.6、9.8、9.6、9.9、11.2、10.6、11.7，则这组数据的第80百分位数为______.16.如图，在直三棱柱中，，，，是线段上一点．（1）设，求异面直线与所成角的余弦值；（2）若平面，求二面角的正切值．17. 在中，底边上的中线，若动点满足.（1）求的最大值；（2）若为等腰三角形，且，点满足（1）的情况下，求的值.18. 某学生在上学路上要经过4个路口，假设在各路口是否遇到红灯是相互独立的，遇到红灯的概率都是，遇到红灯时停留的时间都是2min.（Ⅰ）求这名学生在上学路上到第三个路口时首次遇到红灯的概率；（Ⅱ）求这名学生在上学路上因遇到红灯停留的总时间的分布列及期望.19. 为了解温度对物质参与的某种化学反应的影响，研究小组在不同温度条件下做了四次实验，实验中测得的温度（单位：）与的转化率（转化率）的数据如下表所示：4555657523386574(1)求与的相关系数（结果精确到0.01）；(2)该研究小组随后又进行了一次该实验，其中的起始量为，反应结束时还剩余，若已知关于的线性回归方程为，估计这次实验是在多少摄氏度的温度条件下进行的.（结果精确到0.1）.参考数据：.参考公式：相关系数20. 在中，角所对的边分别为.(1)证明；(2)若，求的面积.21. 已知是实数，1和是函数的两个极值点．（Ⅰ）求和的值；（Ⅱ）设函数的导函数，求的极值点；（Ⅲ）设，其中，求函数的零点个数．。

2022年吉林长春朝阳区东北师范大学附属中学高三上学期高考模拟英语试卷（长春十一高中、吉林一中、四平一中、松原实验）-学生用卷一、完形填空1、【来源】 2022年吉林长春朝阳区东北师范大学附属中学高三上学期高考模拟（长春十一高中、吉林一中、四平一中、松原实验）第1题I have to admit I’m an old-school reader. Put simply, I old books. Alone in a new city to my first daily newspaper job, I was feeling vaguely anxious as I dipped into a used bookstore to my mind.The shelves, full of the smell of the past, quickly me. One of the occupational hazards (职业病) of journalism, my chosen, is a desire to stay on top of the Next Big Thing. With their cracked spines (书脊) and yellowed, the titles in the shop usefully pointed me toward the longer view. “No to get too worked up over today’s headline,” they seemed to say. “ There is notmuch under the sun.”Francis Bacon’s essays are always a ready antidote (解药). “Prosperity (繁荣) is not withoutmany and dislikes,” Bacon wrote, and“ is not without comforts and hopes.” For me, that comfort and hope often come from a second-hand.Reading deeply from the past can be at leastas as following the day’s news.Don’t missing out on the gossip. All matters of fact, which a man did not before, are news to him.【小题1】【小题2】【小题3】【小题4】【小题5】【小题6】【小题7】【小题8】【小题9】【小题10】【小题11】【小题12】【小题13】【小题14】【小题15】二、阅读理解2、【来源】 2022年吉林长春朝阳区东北师范大学附属中学高三上学期高考模拟（长春十一高中、吉林一中、四平一中、松原实验）第2题The Dead Sea might be the most famous saline body in the world, but it’s by no means the only one. Here, we uncover some other salt lakes around the world.Dead SeaThe most famous of the lot, the Dead Sea—also known as the Salt Sea—is a natural place to start. It’s technically not a sea at all, but a land salt lake. Its waters are mainly drawn from the Jordan River, and the high salinity (盐度) means no life can survive here, so don’t expect any colorful fish. However, much amusement results from the sea’s buoyancy (浮力): Bring some reading material to the waters and lie back—nature will do the rest.Lake AssalSurrounded by black volcano rocks, Lake Assal takes the title of being the lowest point in Africa. Vast salt fields mark its edges; the lake has long been at the centre of the area’s salt trade, on which the region’s African locals still rely. Though the area might not be suitable to live in, the waters are rich in minerals and related to health benefits.Great Salt LakeThe biggest salt lake in the western hemisphere, and the one that gave Salt Lake City its name, the Great Salt Lake finds its home in northern Utah. What makes the lake well-known is not only the lake’s blue and green waters and white-sand beaches, but also Great Salt Lake State Park and Antelope Island State Park, both offering super hiking and biking trails, and great long views.Don Juan PondDon Juan Pond might not win any prizes for its beauty—at four inches deep, it’s little more than a pond, but it is worth visiting. Its salinity content of almost 40 percent makesit one of the saltiest bodies of water on Earth, meaning it doesn’t freeze even when Antarctic temperatures drop to -50℃. And with suggestions that water in a similar form could exist on Mars, there’s more than one reason for scientists to study this fascinating dry, cold environment.【小题1】What can visitors probably do in Lake Assal?【小题2】What is the Great Salt Lake famous for?【小题3】Which lake is valuable for space exploration?3、【来源】 2022年吉林长春朝阳区东北师范大学附属中学高三上学期高考模拟（长春十一高中、吉林一中、四平一中、松原实验）第3题Leia was fresh out of college when she began working as a member of a business-development team at a mid- size tech company. Though her skills had earned her the job, she was the youngest person in the team. “Everybody else was pretty much twice my age,” she says. “I heard about comments being made behind my back. There were a couple of times when my superiors referred to my age right in front of me, saying I was too young, ‘What does a 23- year-old know about these things?’”Leia tried to change things by altering her appearance at work. “I changed the way I dressed. I tried to dress older, more ‘ladylike’. I changed my mannerisms and tried to act older,”she says. “It worked,to an extent.”The comments about her age and perceived ( 认定的 ) inexperience lessened, but Leia says she still felt like her growth potential was limited. She left the company soon after.What Leia experienced was ageism traditionally seen as something only older people face. For instance, older workers might be judged based on assumptions that they won’t fit into a progressive office dynamics or learn technology as quickly. A US study showed that nearly two-thirds of workers aged 45 and older had seen or experienced age discrimination. But younger workers face age discrimination, too. In fact, new research shows it may actually be the youngest team members who are bearing the impact of workplace ageism right now.Leia says removing ageism entirely will ultimately require a fundamental change to corporate culture, which has long tied seniority ( 资历 ) to skill. “We prize years of experience alittle too much,and I don’t think years of experience and skill are necessarily related,” she says. “Steve Jobs was 21 when he founded Apple. We don’t know how much younger people actually have to contribute. Hopefully, more employers are realizing it.”【小题1】What was the main topic of Leia’s superiors about her?【小题2】Why didn’t Leia stay any longer at the company?【小题3】Which statement agrees with the US study the most?【小题4】Why does Leia mention Steve Jobs?4、【来源】 2022年吉林长春朝阳区东北师范大学附属中学高三上学期高考模拟（长春十一高中、吉林一中、四平一中、松原实验）第4题Octopuses(章鱼) are some of the cleverest animals on the planet! So what makes them so smart? Let’s focus first on their defense mechanisms. Faced with multiple predators(捕食者)—including fish, birds and whales—octopuses are masters of disguise.They can imitate their environment by changing their skin colour.Without a shell, octopuses are weak and easily hurt physically, and always try to remain hidden in a shelter such as a hole or the space beneath a rock. Some prefer to wrap themselves in shells and small stones, while others transport their shelters in their arms. This is the case for the coconut octopus, which,true to its name, has been observed carrying coconut shells around to hide within in case of danger.Octopuses are also fearsome predators themselves, and their attack mechanisms are suited to the wide variety of prey(猎物) they consume, including seashells,fish and even other creatures like them. They can use their vision and camouflage skills to hunt, and their arms to explore,touch and taste their environment to seize every bit of food within reach.Octopuses can also use conditional discrimination, that is, they can change their choice depending on the situation. For example, they can learn to attack an object only in the presence of bubbles. They can alsouse space learning, and find a hidden shelter by remembering its position. Last but not least, octopuses can learn by watching other octopuses carry out tasks, such as choosing one specific object over another. This is surprising,because they are mainly solitary(独居的)creatures.Octopuses meet every criteria for the definition of intelligence: They show a great flexibility in obtaining information (using several senses and learning socially), in processing it (through discriminative and conditional learning), in storing it (through long- term memory) and in applying it toward both predators and prey.【小题1】Which are both predators and prey of octopuses?【小题2】Why do coconut octopuses carry coconut shells everywhere?【小题3】What’s the fifth paragraph mainly about?【小题4】What can be a suitable title for the text?5、【来源】 2022年吉林长春朝阳区东北师范大学附属中学高三上学期高考模拟（长春十一高中、吉林一中、四平一中、松原实验）第5题English is the language for international communication, especially in areas such as science and technology. Galileo Galilei said, “Mathematics is the language with which God has written the universe.” Using numbers and mathematical symbols, one can write “words” and “sentences”.When appropriate, several “sentences” together might form a “mathematical story”. Mathematics has also grammar — the mathematical logic which determines whether statements are valid or not.A math teacher compares math to English and says, “Math may be viewed as a language — a simpler, more consistent and more regular language than English. Numbers represent nouns, while operational signs (+, －,×, ÷, =) serve as verbs. Both components are governed by rules of syntax. Math facts, such as 2×3=6, may be thought of as math sentences. Students should be encouraged to speak in complete sentences to convey an entire thought. Math at the simple sentence fact level is a much easier language than English. Although math has an infinite number of nouns, it has only five verbs (+, －, ×, ÷, =) associated with four basic operations.”Nina is an English teacher in elementary school and has 12 years of teaching experience. She has no doubts about the logical connection between English and mathematics. She says, “They are definitely languages. In both languages there are patterns and rules. They are both taught throughvisualizations and illustrations.Teaching vocabulary is like teaching fractions. You need to illustrate in both languages and practice a lot.”However, English and mathematics as languages are similar and yet different. The language of mathematics, for example, is precise and less flexible—it cannot afford ambiguity,while natural languages contain constructive ambiguity derived from their cultural and contextual wealth of meanings. Mathematics as well as English as a second language does not develop naturally as a child develops a natural language; they need to be learned.【小题1】What can be inferred from the first paragraph?【小题2】What is mainly discussed in Paragraph 2?【小题3】What does the underlined word “ambiguity” probably mean?【小题4】What is the author’s purpose in writing the text?三、七选五6、【来源】 2022年吉林长春朝阳区东北师范大学附属中学高三上学期高考模拟（长春十一高中、吉林一中、四平一中、松原实验）第6题In fact, this isn’t a new phenomenon. After all, people have complainedabout“kids these days”for decades. But is there really any truth in the idea that millennials (千禧一代) and Gen Z are weaker than Boomers or Gen X?People have complained about younger generations for thousands of years. 【小题1】.“The tendency for adults to look down upon the character of youth has been happening for centuries,” says Peter O’ Connor, a professor of management.He points out the fixed concept remains alive and well, with research showing thousands of Americans believe that “kids these days” lack positive qualities that participants associate with older generations. 【小题2】.The researchers argued that this was because we project our current selves onto our past selves.By doing this, older people are unconsciously comparing who they are today to today’s young people, giving an impression that today’s youth is somehow on the decline, no matter which times we’re living in. 【小题3】.The truth is that older generations have been accusing the young of being lazy and self-obsessed for centuries.【小题4】. Jason Dorsey, president of the Center for Generational Kinetics, believes that there is a way around this—but that general context is key to debunking (揭穿) the persisting weakness myth.“It’s awareness of what these different generations have gone through, why they are the way they are.”he says.“【小题5】, which simply does not exist right now. Instead of having frank conversations across generations, we always say that younger people are snowflakes and older people are dinosaurs. But the truth is that we’re all human.” he adds.四、用单词的适当形式完成短文7、【来源】 2022年吉林长春朝阳区东北师范大学附属中学高三上学期高考模拟（长春十一高中、吉林一中、四平一中、松原实验）第7题阅读下面短文，在空白处填入 1 个适当的单词或括号内单词的正确形式。

吉林省东北师大附中、长春市十一高中、吉林一中、四平一中、松原实验中学2023-2024学年高三上一、听力选择题1.A．She is also a fan of Argentina.B．She is also working very hard.C．She loves American football so much.D．She works for the World Cup.2. What does the man’s country export to Europe?A．Copper and coal.B．Coal and wood.C．Wood and copper.3. What problem does the man encounter?A．A time conflict.B．A troublesome accountant.C．An undecided marketing plan.4.A．Differences in customs.B．Differences in pronunciations.C．Differences in words.D．Differences in accents.5. What is the relationship between the speakers?A．Professor and student.B．Doctor and patient.C．Employer and employee.二、听力选择题6. 听下面一段较长对话，回答以下小题。

1. What did David do on his most recent holiday?A．He had a boat trip.B．He took a balloon flight.C．He climbed a mountain.2. What is on David's list?A．Gifts he needed to buy.B．Things he wished to do.C．Dates he had to remember.3. Where does the woman suggest David go for his next holiday?A．Australia.B．California.C．The Amazon.7. 听下面一段较长对话，回答以下小题。

东北师大附中 长春十一高中 吉林一中 四平一中 松原实验中学2022届高三联合模拟考试数学（理）科试题一、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．已知集合{}23,N A x x x =<∈，则A 的真子集共有（ ）个 A ．3B ．4C ．6D ．72．已知i 为虚数单位，复数3iz i=+，则复数z 在复平面内对应的点位于（ ） A ．第一象限B ．第三象限C ．直线30x y -=上D ．直线30x y +=上3．在二项式521x x ⎛⎫- ⎪⎝⎭的展开式中，含x 的项的系数是（ ）A ．10-B ．5-C ．10D ．204．数列{}n a 为等差数列，且2220202022044a a x dx π+=-⎰，则()2021201920212023a a a a ++=（ ）A ．1B ．3C ．6D ．125．长春54路有轨电车建成于上个世纪30年代，大概是现存最美的电车路线了，见证着这座城市的历史与发展．学生甲和学生乙同时在长影站上了开往西安大路方向的电车，甲将在创业大街站之前任何一站下车，乙将在景阳大路站之前任何一站下车，他们都至少一站再下车，则甲比乙后下车的概率为（ ）A ．310B ．518C ．12D ．356．已知向量a ，b 满足2a =，3b =，且a 与b 的夹角为6π，则()()2a b a b +⋅-=（ ） A ．6B ．8C ．10D ．127．哥特式建筑是1140年左右产生于法国的欧洲建筑风格，它的特点是尖塔高耸、尖形拱门、大窗户及绘有故事的花窗玻璃，如图所示的几何图形，在哥特式建筑的尖形拱门与大窗户中较为常见，它是由线段AB 和两个圆弧AC ，弧BC 围成，其中一个圆弧的圆心为A ，另一个圆弧的圆心为B ，圆O 与线段AB 及两个圆弧均相切，则tan AOB ∠的值是（ ）A ．43-B ．125-C ．247-D ．34-8．从某个角度观察篮球（如图甲），可以得到一个对称的平面图形，如图乙所示，篮球的外轮廓为圆O ，将篮球你表面的粘合线视为坐标轴和双曲线，若坐标轴和双曲线与圆O 的交点将圆的周长八等分，且AB BO OC CD ===，则该双曲线的渐近线的斜率为（ ）A ．1±B ．2±C ．2±D ．3±9．已知线段MN 是圆()22:18C x y -+=的一条动弦，且23MN =，若点P 为直线260x y -+=上的任意一点，则PM PN +的最小值为（ ） A ．8525- B ．855C ．6525- D ．65510．把方程194x x y y+=-表示的曲线作为函数()y f x =的图象，则下列结论正确的是（ ） ①()f x 在R 上单调递减；②()y f x =的图像关于原点对称；③函数()()32g x f x x =+不存在零点；④()y f x =的图象上的点到坐标原点的距离的最小值为2； A ．①②③B ．①②④C ．①③④D ．②③④11．已知数列{}n a 的首项是11a =，前n 项和为n S ，且()1231n n S S n n N *+=++∈，设()2log 3n n c a =+，若存在常数k ，使不等式()()116n n c k n N n c *-≥∈+恒成立，则k 的取值范围为（ ） A ．1,9⎡⎫+∞⎪⎢⎣⎭B ．1,16⎡⎫+∞⎪⎢⎣⎭C ．1,25⎡⎫+∞⎪⎢⎣⎭D ．1,36⎡⎫+∞⎪⎢⎣⎭12．已知三棱锥P ABC -三条侧棱PA ，PB ，PC 两两互相垂直，且6PA PB PC ===，M 、N 分别为该三棱锥的内切球和外接球上的动点，则线段MN 的长度的最小值为（ ） A ．233-B ．436-C ．623-D ．23二、填空题：本题共4小题，每小题5分，共20分．13．某三棱锥的三视图如图所示，则该三棱锥的体积为______．14．若将函数()g x 图象上所有的点向左平移6π个单位长度得到函数()f x 的图象，已知函数()()sin 0,0,2f x A x A πωϕωϕ⎛⎫=+>>< ⎪⎝⎭的部分图象如图所示，则()g x 的解析式为______．15．已知函数()322sin f x x x x =+-，则不等式()()2650f x f x -+≤的解集为______．16．已知点P 是曲线24x y =上任意一点，过点P 向x 轴引垂线，垂足为H ，点Q 是曲线ln y x =上任意一点，则PH PQ +的最小值为______．三、解答题：共70分．解答应写出文字说明、证明过程或演算步骤．第17-21题为必考题，每个试题考生都必须作答．第22、23题为选考题，考生根据要求作答． （一）必考题：60分． 17．（12分）如图，正方形ADEF 与梯形ABCD 所在的平面互相垂直，AD CD ⊥，AB CD ，2AB AD ==，4CD =，M 为CE 的中点．（1）求证：平面BDE ⊥平面BCE ； （2）求二面角M DB A --的正弦值．18．（12分）在我国抗疫期间，为了保证高中数学的正常进行，通过“钉钉、腾讯会议”等软件进行了线上教学，为抗疫起到了积极的作用，但一个优秀的视频除了需要有很好的素材外，更要有制作上的技术要求，小明同学学习利用“VB ”等软件将已拍摄的素材进行制作，每次制作分三个环节来进行，其中每个环节制作合格的概率分别为35，23，34，只有当每个环节制作都合格才为一次成功制作，该视频视为合格作品． （1）求小明同学进行3次制作，恰有一次合格作品的概率；（2）若小明同学制作15次，其中合格作品数为X ，求X 的数学期望与方差；（3）随着制作技术的不断提高，小明同学制作的小视频被某高校看中，聘其为单位制作教学软件，决定试用一段时间，每天制作小视频（注：每天可提供素材制作个数至多40个），其中前7天制作合格作品数y 与时间t 如下表：（第t 天用数字t 表示） 时间()t 1 2 3 4 5 6 7 合格作品数()y3434768其中合格作品数()y 与时间()t 具有线性相关关系，求y 关于t 的线性回归方程（精确到0.01），并估算第 15天能制作多少个合格作品（四舍五入取整）？（参考答案()()()1122211ˆnni iiii i nni i i i x y nxy x x y y bx nxx x====---==--∑∑∑∑，ˆˆa y bx =-，参考数据：71163i ii t y ==∑）．19．（12分）ABC △中，4AC =，43BC =，AC BC ⊥，点M ，N 是线段AB 上两点（包括端点），30MCN ∠=︒． （1）当2AM =时，求MNC △的周长；（2）设ACM θ∠=，当MNC △的面积为23时，求θ的值．20．（12分）已知椭圆()2222:10x y C a b a b+=>>的离心率与等轴双曲线的离心率互为倒数关系，直线:20l x y -+=与以原点为圆心，以椭圆C 的短半轴长为半径的圆相切． （1）求椭圆C 的方程；（2）设M 是椭圆的上顶点，过点M 分别作直线MA ，MB 交椭圆于A ，B 两点，设两直线的斜率分别为1k ，2k ，且125k k +=，求证：直线AB 过定点．21．（12分）已知函数()()()2231ln 1f x x x mx m ⎛⎫=+--++⎪⎝⎭，()()221ln 4a g x x x a x x =+++--．（1）当2m =时，求()f x 在()0,+∞的单调区间；（2）当8a ≤时，若对任意()0,m ∈+∞，总存在[]01,2x ∈，使得不等式()()01f x g m <+成立，求实数a 的取值范围．（ln 20.7≈）（二）选考题：共10分．请考生在第22、23题中任选一题作答．如果多做，则按所做的第一题计分．22．［选修4-4：坐标系与参数方程］（10分）在直角坐标系xOy中，已知曲线C的参数方程为xyϕϕ⎧=⎪⎨=⎪⎩（ϕ为参数），直线l的方程为0y+-=．以坐标原点O为极点，x轴的正半轴为极轴建立极坐标系．（1）求曲线C和直线l的极坐标方程；（2）若点(),A x y在直线l上，且0y>，射线OA与曲线C相交于异于O点的点E．求OAOB的最小值．23．［选修4-5：不等式选讲］（10分）已知()12f x x x=-++的最小值为M．（1）解关于x的不等式()1f x M x<+-；（2）若正实数a，b满足42Ma b+=，求12a b+取最小值时a b-的值．。

2023-2024学年高三联合模拟考试数学试题东北师大附中 长春十一高中 吉林一中 四平一中 松原实验中学注意事项：1.答卷前，考生务必将自已的考生号､姓名､考场号填写在答题卡上，2.回答选择时，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.如需要改动，用橡皮擦干净后，再选涂其他答案标号.回答非选择题时，将答案写在答题卡上，写在本试卷上无效.一､选择题：本题共8小题，每小题5分，共40分.在每小题给出的四个选项中，只有一项是符合题目要求的.1.已知集合(){}{}22log 2,2x A xy x B y y -==-==∣∣，则A B ⋂=（ ）A.()0,2B.[]0,2C.()0,∞+D.(],2∞-2.已知复数iz 1i=-，则z 的虚部为（ ） A.12- B.1i 2- C.12 D.1i 23.将一枚质地均匀的骰子连续抛掷6次，得到的点数分别为1,2,4,5,6,x ，则这6个点数的中位数为4的概率为（ ） A.16 B.13 C.12 D.234.刍薨是《九章算术》中出现的一种几何体，如图所示，其底面ABCD 为矩形，顶棱PQ 和底面平行，书中描述了刍薨的体积计算方法：求积术曰，倍下袤，上袤从之，以广乘之，又以高乘之，六而一，即()126V AB PQ BC h =+⋅（其中h 是刍薨的高，即顶棱PQ 到底面ABCD 的距离），已知28,AB BC PAD ==和QBC 均为等边三角形，若二面角P AD B --和Q BC A --的大小均为120︒，则该刍薨的体积为（ ）A. B. D.48+5.中国空间站的主体结构包括天和核心舱､问天实验舱和梦天实验舱.假设中国空间站要安排甲，乙，丙，丁4名航天员开展实验，其中天和核心舱安排2人，问天实验舱与梦天实验舱各安排1人.若甲､乙两人不能同时在一个舱内做实验，则不同的安排方案共有（ ）种 A.8 B.10 C.16 D.20 6.已知πcos sin 6αα⎛⎫-+= ⎪⎝⎭，则5πsin 6α⎛⎫- ⎪⎝⎭的值是（ ）A. B.14- C.147.已知点F 为地物线2:4C y x =的焦点，过F 的直线l 与C 交于,A B 两点，则2AF BF +的最小值为（ ）A. B.4C.3+D.68.已的1113sin ,cos ,ln 3332a b c ===，则（ ） A.c a b << B.c b a <<C.b c a <<D.b a c <<二､多选题：本题共3小题，每小题6分，共18分.在每小题给出的选项中，有多项符合题目要求.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.已知数列{}n a 满足*1121,,N 1n n a na n a n +==∈+，则下列结论成立的有（ ） A.42a =B.数列{}n na 是等比数列C.数列{}n a 为递增数列D.数列{}6n a -的前n 项和n S 的最小值为6S10.已知正方体1111ABCD A B C D -的棱长为2,M 为空间中动点，N 为CD 中点，则下列结论中正确的是（ ） A.若M 为线段AN 上的动点，则1D M 与11B C 所成为的范围为ππ,62⎡⎤⎢⎥⎣⎦B.若M 为侧面11ADD A 上的动点，且满足MN ∥平面1AD C ，则点MC.若M 为侧面11DCC D上的动点，且3MB =，则点Mπ D.若M 为侧面11ADD A 上的动点，则存在点M满足MB MN +=11.已知()()()()1ln ,e 1xf x x xg x x =+=+（其中e 2.71828=为自然对数的底数），则下列结论正确的是（ ）A.()f x '为函数()f x 的导函数，则方程()()2560f x f x ⎡⎤-'+=⎣⎦'有3个不等的实数解 B.()()()0,,x f x g x ∞∃∈+=C.若对任意0x >，不等式()()2ln e x g a x g x x -+≤-恒成立，则实数a 的最大值为-1D.若()()12(0)f x g x t t ==>，则()21ln 21t x x +的最大值为1e三､填空题：本题共3小题，每小题5分，共15分.12.622x x ⎛⎫- ⎪⎝⎭展开式的常数项为__________.13.已知向量a ，b 为单位向量，且12a b ⋅=-，向量c 与3a b +共线，则||b c +的最小值为__________. 14.已知双曲线2222:1(0,0)x y C a b a b-=>>的左，右焦点分别为12,,F F P 为C 右支上一点，21122π,3PF F PF F ∠=的内切圆圆心为M ，直线PM 交x 轴于点,3N PM MN =，则双曲线的离心率为__________.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.（本小题13分）为了更好地推广冰雪体育运动项目，某中学要求每位同学必须在高中三年的每个冬季学期选修滑冰､滑雪､冰壶三类体育课程之一，且不可连续选修同一类课程若某生在选修滑冰后，下一次选修滑雪的概率为13：在选修滑雪后，下一次选修冰壶的概率为34，在选修冰壶后，下一次选修滑冰的概率为25. （1）若某生在高一冬季学期选修了滑雪，求他在高三冬季学期选修滑冰的概率：（2）苦某生在高一冬季学期选修了滑冰，设该生在高中三个冬季学期中选修滑冰课程的次数为随机变量X ，求X 的分布列及期望， 16.（本小题15分）在ABC 中，角,,A B C 的对边分别为,,a b c ，已知1,cos cos 2cos 0a C c A b B =+-=. （1）求B ；（2）若2AC CD =，且BD =c . 17.（本小题15分）如图，在四棱锥PABCD -中，底面是边长为2的正方形，且PB =，点,O Q 分别为棱,CD PB 的中点，且DQ ⊥平面PBC .（1）证明：OQ ∥平面PAD ； （2）求二面角P AD Q --的大小. 18.（本小题17分）已知椭圆2222:1(0)x y C a b a b +=>>的两焦点()()121,0,1,0F F -，且椭圆C 过P ⎛ ⎝⎭. （1）求椭圆C 的标准方程；（2）设椭圆C 的左､右顶点分别为,A B ，直线l 交椭圆C 于,M N 两点（,M N 与,A B 均不重合），记直线AM 的斜率为1k ，直线BN 的斜率为2k ，且1220k k -=，设A M N ，BMN 的面积分别为12,S S ，求12S S -的取值范围18.（本小题17分） 已知()2e2e xx f x a x =-（其中e 2.71828=为自然对数的底数）.（1）当0a =时，求曲线()y f x =在点()()1,1f 处的切线方程，（2）当12a =时，判断()f x 是否存在极值，并说明理由； （3）()1R,0x f x a∀∈+≤，求实数a 的取值范围.五校联合考试数学答案一､单选题1-8ACADB BCD二､多选题9.ABD 10.BC 11.AC三､填空题12.60 13.1414.75四､解答题15.解：（1）若高一选修滑雪，设高三冬季学期选修滑冰为随机事件A ， 则()3234510P A =⨯=. （2）随机变量X 的可能取值为1，2.()()323113221171,2.534320534320P X P X ==⨯+⨯===⨯+⨯=所以X 的分布列为：()137272.202020E X =+⨯= 16.解：（1）1,cos cos 2cos cos cos 2cos 0a C c A b B a C c A b B =∴+-=+-=.()sin cos sin cos 2sin cos sin 2sin cos 0.A C C A B B A C B B ∴+-=+-=又()1ππ,sin sin 0,cos 23A B C A C B B B ++=∴+=≠∴=∴=.（2）2AC CD =，设CD x =，则2AC x =，在ABC 中2222141cos ,1422c x B c x c c +-==∴+-=.在ABC 与BCD 中，22222142cos ,cos ,63042x c x BCA BCD x c x x∠∠+--==∴--=.2330,0c c c c c ∴--=∴=>∴=. 17.解：（1）取PA 中点G ，连接,GQ GD ∴点Q 为PB 中点，GQ ∴∥1,2AB GQ AB =. 底面是边长为2的正方形，O 为CD 中点，DO ∴∥1,2AB DO AB =. GQ ∴∥,OD GQ OD =∴四边形GQOD 是平行四边形.OQ ∴∥DG . OQ ⊄平面,PAD GD ⊂平面,PAD OQ ∴∥平面PAD .（2）DQ ⊥平面,PBC BC ⊂平面PBC DQ BC ∴⊥.又底面是边长为2的正方形，,,DC BC DQ DC D BC ∴⊥⋂=∴⊥平面DCQ .OQ ⊂平面,DCQ BC OQ ∴⊥.又CQ ⊂平面,DCQ BC CQ ∴⊥. 26,6,2,2PB QB BC QC =∴==∴=底面是边长为2的正方形，DB DQ DQ CQ ∴=∴==，O 为CD 中点，OQ DC ∴⊥.又,,BC OQ DC BC C OQ ⊥⋂=∴⊥平面ABCD .取AB 中点E ，以,,OE OC OQ 所在直线分别为,,x y z 轴建立如图所示的空间直角坐标系O xyz -， 则()()()()()()0,0,0,0,0,1,2,1,0,2,1,0,0,1,0,2,1,2O Q A B D P ----所以()()()4,0,2,2,0,0,2,1,1AP AD AQ =-=-=-， 设平面PAD 法向量为(),,m x y z =，则()4200,1,020m AP x z m m AD x ⎧⋅=-+=⎪∴=⎨⋅=-=⎪⎩设平面QAD 法向量为(),,n x y z =，则()200,1,120n AQ x y z n n AD x ⎧⋅=-++=⎪∴=-⎨⋅=-=⎪⎩ 2cos ,2m n m n m n⋅>==⋅ 又二面角P AD Q --范围为()0,π， 所以二面角P AD Q --的大小为π4. 18.解：（1）由题意可得：2222213314c ab c ab ⎧⎪=⎪-=⎨⎪⎪+=⎩，解得2,1a b c =⎧⎪=⎨⎪=⎩22143x y +=；（2）依题意，()()2,0,2,0A B -，设()()1122,,,M x y N x y ，直线BM 斜率为BM k .若直线MN 的斜率为0，则点,M N 关于y 轴对称，必有120k k +=，不合题意.所以直线MN 的斜率必不为0，设其方程为()2x ty m m =+≠±，与椭圆C 的方程联立223412,,x y x ty m ⎧+=⎨=+⎩得()2223463120t y tmy m +++-=，所以()22Δ48340t m=+->，且12221226,34312.34tm y y t m y y t ⎧+=-⎪⎪+⎨-⎪=⎪+⎩因为()11,M x y 是椭圆上一点，满足 2211143x y +=，所以2121111221111314322444BM x y y y k k x x x x ⎛⎫- ⎪⎝⎭⋅=⋅===-+---，则12324BM k k k =-=，即238BM k k -⋅=.因为()()1221222BM y y k k x x ⋅=--()()()()121222121212222(2)y y y y ty m ty m t y y t m y y m ==+-+-+-++-()()()()()22222222223123432334,4(2)42831262(2)3434m m m t m m t m t m m m t t --++====------+-++ 所以23m =-，此时22432Δ4834483099t t ⎛⎫⎛⎫=+-=+> ⎪ ⎪⎝⎭⎝⎭，故直线MN 恒过x 轴上一定点2,03D ⎛⎫-⎪⎝⎭. 因此()12222122264,343431232.34334tm t y y t t m y y t t ⎧+=-=⎪++⎪⎨-⎪==-++⎪⎩，所以12S S -=12121212222323y y y y ⎛⎫⎛⎫-------- ⎪ ⎪⎝⎭⎝⎭.1223y y =-====令122110,,344x S S t ⎛⎤=∈-= ⎥+⎝⎦当211344t =+即0t =时，12S S -12S S ⎛∴-=⎝⎦ 19.解：（1）当0a =时，()()()2,21x xf x xe f x x e =-=+'-.()14.f e =-∴'曲线()y f x =在点()()1,1f 处的切线方程为 ()41242.y e x e ex e =---=-+（2）当12a =时，()2122x xf x e xe =-，定义域为(),∞∞-+()()()22122,x x x x f x e x e e e x '=-+=--令()e 22xF x x =--，则()2xF x e '=-，当()(),ln2,0x F x ∞∈-'<；当()()ln2,,0x F x ∞∈+'>； 所以()F x 在(),ln2∞-递减，在()ln2,∞+上递增，()min ()ln222ln222ln20F x F ==--=-<()()2110,260F F e e-=>=-> 存在()11,ln2x ∈-使得()10F x =，存在()2ln2,2x ∈使得()20F x =，()1,x x ∞∈-时，()()()0,0,F x f x f x >'>单调递增； ()12,x x x ∈时，()()()0,0,F x f x f x <'<单调递减； ()1,x x ∞∈+时，()()()0,0,F x f x f x >'>单调递增；所以12a =时，()f x 有一个极大值，一个极小值. （3）()()()222121xx x x f x ae x e e ae x '=-+=--，由()()21111,0,00a x f x f a aa a a+∀∈+≤+=+=≤R ，得0a <，令()e 1xg x a x =--，则()g x 在R 上递减，0x <时，()()()e 0,1,e ,0,e 11x x xa a g x a x a x ∈∈∴=-->--，则()()1110g a a a ∴->---=又()110g ae --=<，()01,1x a ∃∈--使得()00g x =，即()000e 10x g x a x =--=且当()0,x x ∞∈-时，()0g x >即()0f x '>； 当()00,x x ∞∈+时，()0g x <即()0f x '<，()f x ∴在()0,x ∞-递增，在()0,x ∞+递减，()002max 00()2x x f x f x ae x e ∴==-，由()000001e 10,e xx x g x a x a +=--==，由max 1()0f x a +≤得()000000e 1e 201x x x x x e x +-+≤+即()()00011101x x x -++≤+， 由010x +<得20011,1x x -≤≤<-，01,e x x a +=∴设()1(1)e x x h x x +=≤<-，则()0x xh x e-=>'， 可知()h x 在)⎡⎣上递增，()((()()110h x h h x h ≥==<-=实数a的取值范围是()1⎡⎣.。

一、单选题二、多选题1. 将函数图象上的所有点向右平移个单位，然后把图象上所有点的横坐标缩短为原来的倍，纵坐标不变得到的图象，则等于（ ）A．B．C．D．2. 设函数f (x )＝若函数g (x )＝f (x )－b 有三个零点，则实数b 的取值范围是（ ）A ．(1，＋∞)B．C ．(1，＋∞)∪{0}D ．(0，1]3. 双曲线的离心率为2，焦点到渐近线的距离为，则的焦距等于（ ）A ．2B．C．D ．44.设，则（ ）．A．B．C．D．5. 已知直线与抛物线相交于两点，为的焦点，若，则直线的斜率为（ ）A．B．C．D．6. 中国古代数学名著《九章算术》中记载了公元前344年商鞅督造一种标准量器-商鞅铜方升，其三视图如图所示（单位：寸），若取3，其体积为（立方寸），则图中的为（ ）A．B．C．D．7.函数的值域是（ ）A．B．C．D．8.定义在上的奇函数，当时，，则关于的函数的所有零点之和为（ ）A．B．C．D．9. 下列结论正确的是（ ）A ．若，则B．若，则的最小值为2C ．若，则的最大值为2D ．若，则吉林省(东北师大附中,长春十一高中,吉林一中,四平一中,松原实验中学)五校2023届高三上学期联吉林省(东北师大附中,长春十一高中,吉林一中,四平一中,松原实验中学)五校2023届高三上学期联三、填空题四、解答题10. 设函数，则（ ）A．的最小值为，其周期为B．的最小值为，其周期为C ．在单调递增，其图象关于直线对称D ．在单调递减，其图象关于直线对称11.下面是关于公差的等差数列的四个命题，其中的真命题为（ ）.A．数列是递增数列B ．数列是递增数列C．数列是递增数列D ．数列是递增数列12.已知抛物线的焦点为，为坐标原点，倾斜角为的直线过点且与交于，两点，若的面积为，则（ ）A．B．C ．以为直径的圆与轴仅有个交点D ．或13. 在平面直角坐标系中，已知向量，，试写一个非零向量_________，使得.14.函数的最小正周期为________;若函数在区间上单调递增，则的最大值为________.15. 函数在点处的切线的方程为___________.16. 如图，在中，．(1)若，求的长度；(2)若，求．17. 随机变量X 的分布列如下表如示，若数列是以为首项，以为公比的等比数列，则称随机变量X 服从等比分布，记为(．现随机变量 (．X 12…n…（1）求的值并求随机变量X 的数学期望EX ；（2）一个盒子里装有标号为1，2，…，n 且质地相同的标签若干张，从中任取1张标签所得的标号为随机变量X ．现有放回的从中每次抽取一张，共抽取三次，求恰好2次取得标签的标号不大于3的概率．18. 已知函数R ）．（Ⅰ）求函数的定义域，并讨论函数的单调性；（Ⅱ）问是否存在实数，使得函数在区间上取得最小值3？请说明由．19. 的内角，，的对边分别为，，.已知，.(1)求的面积；(2)若，且，求.20. 今年月以来，世界多个国家报告了猴痘病例，我国作为为人民健康负责任的国家，对猴痘病毒防控提前做出部署.同时国家卫生健康委员会同国家中医药管理局制定了《猴痘诊疗指南（年版）》.此《指南》中指出：①猴痘病人潜伏期天；②既往接种过天花疫苗者对猴痘病毒存在一定程度的交叉保护力.据此，援非中国医疗队针对援助的某非洲国家制定的猴痘病毒防控措施之一是要求与猴痘病毒确诊患者的密切接触者集中医学观察天，在医学观察期结束后发现密切接触者中未接种过天花疫苗者感染病毒的比例较大.对该国家个接种与未接种天花疫苗的密切接触者样本医学观察结束后，统计了感染病毒情况，得到下面的列联表：感染猴痘病毒未感染猴痘病毒未接种天花疫苗接种天花疫苗(1)是否有的把握认为密切接触者感染猴痘病毒与未接种天花疫苗有关；(2)以样本中结束医学观察的密切接触者感染猴痘病毒的频率估计概率，现从该国所有结束医学观察的密切接触者中随机抽取人进行感染猴痘病毒人数统计，求其中至多有人感染猴痘病毒的概率.附：，其中.21. 将函数的图象上各点的横坐标缩短为原来的（纵坐标不变），再向左平移个单位后，得到的图象与函数的图象重合.（1）写出函数的图象的一条对称轴方程；（2）若A为三角形的内角，且，求的值.。

绝密★启用前
吉林省五校联考
(东北师大附中长春十一高中吉林一中四平一中松原实验高级中学) 2021届高三毕业班下学期2月联合模拟考试文综-政治试题参考答案解析
2021年2月一．选择题（每小题4分，共48分）
12 . D
A：商品的价值由社会必要劳动时间决定,直播带货并不能降低商品价值A不选。

B：直播带货不一定是理性消费，B不选。

C：商品质量不会降低,C不选。

D：直播带货会促进商品的流通,实现商品的价值，D入选。

故本题选D。

13 . A
①②：部分企业兼并重组,呈现出新的“聚合现象”,①②是企业兼并的意义,故①②入选。

③④是破产的意义,故③④不选。

故本题选A。

14 . C
①：农业是国民经济的基础,强化农业主导作用是错误的。

④：粮食完全依赖国内是绝对的,故①④不选。

开展种源“卡脖子”技术攻关,重视农业核心技术,提高农业自主核心创新能力,维护我国国家粮食安全,故选②③。

故本题选C。

15 . D
1。

吉林省东北师大附中、长春十一高中、吉林一中、四平一中、松原实验中学2021届高三地理下学期2月联合模拟考试试题本试卷共47 小题，满分300 分。

考试用时150 分钟.注意事项：1。

答卷前,考生务必将自己的姓名、考生号、考场号和座位号填写在答题卡上.用2B 铅笔将试卷类型填涂在答题卡相应位置上.将条形码横贴在答题卡右上角“条形码粘贴处"。

2. 作答选择题时，选出每小题答案后，用2B 铅笔在答题卡上对应题目选项的答案信息点涂黑;如需改动，用橡皮擦干净后，再选涂其他答案。

答案不能答在试卷上.3。

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

4. 考生必须保证答题卡的整洁.考试结束后，将试卷和答题卡一并交回。

第I卷（选择题共140分）一、选择题：本题共35小题,每小题4分，共140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.同一岩层顶部海拔相同点的连线称为构造等高线。

下图为某区域的某岩层构造等高线和地形等高线示意图，其中B处有一落差30米的瀑布，图中地形等高线的等高距是60米，山峰最高点海拔为1030米，完成下题。

1．图中A处坡面径流方向是A．自西北向东南B．自东北向西南C．自东南向西北D．自北向南2．下列叙述正确的是A．图中河流自北向南流B．若D处地下含有煤层，开采时不易发生瓦斯爆炸C．D处地下岩层具有良好的储油构造D．图中河流为地上河3．图中C处该岩层的最大埋藏深度可能是A．468米B．470米C．498米D．500米目前海岸侵蚀现象普遍存在，世界上大多数海岸线处于侵蚀状态，中国海岸侵蚀亦相当普遍.以秦皇岛海岸线为例，近50多年以来，该地海岸线已经有将近67%的岸段遭受到不同程度的侵蚀，给当地社会经济发展造成了显著影响.下图为秦皇岛某处一侵蚀海岸带，完成下题。

2023届吉林省东北师大附中、长春市十一高中、吉林一中、四平一中、松原实验中学高三上学期联合模拟考试英语试题学校:___________姓名：___________班级：___________考号：___________一、阅读理解The Dead Sea might be the most famous saline body in the world,but it’s by no means the only one.Here,we uncover some other salt lakes around the world.Dead SeaThe most famous of the lot,the Dead Sea—also known as the Salt Sea—is a natural place to start.It’s technically not a sea at all,but a land salt lake.Its waters are mainly drawn from the Jordan River,and the high salinity(盐度)means no life can survive here,so don’t expect any colorful fish.However,much amusement results from the sea’s buoyancy(浮力): Bring some reading material to the waters and lie back—nature will do the rest.Lake AssalSurrounded by black volcano rocks,Lake Assal takes the title of being the lowest point in Africa.Vast salt fields mark its edges;the lake has long been at the centre of the area’s salt trade,on which the region’s African locals still rely.Though the area might not be suitable to live in,the waters are rich in minerals and related to health benefits.Great Salt LakeThe biggest salt lake in the western hemisphere,and the one that gave Salt Lake City its name,the Great Salt Lake finds its home in northern Utah.What makes the lake well-known is not only the lake’s blue and green waters and white-sand beaches,but also Great Salt Lake State Park and Antelope Island State Park,both offering super hiking and biking trails,and great long views.Don Juan PondDon Juan Pond might not win any prizes for its beauty—at four inches deep,it’s little more than a pond,but it is worth visiting.Its salinity content of almost40percent makesit one of the saltiest bodies of water on Earth,meaning it doesn’t freeze even when Antarctic temperatures drop to-50℃.And with suggestions that water in a similar form could exist on Mars,there’s more than one reason for scientists to study this fascinating dry,cold environment.1．What can visitors probably do in Lake Assal?A．Read books.B．Have a hike.C．Enjoy small fishes.D．Take a bath in the lake.2．What is the Great Salt Lake famous for?A．Its great parks.B．Its scientific value.C．Its highest salinity.D．Its specific location.3．Which lake is valuable for space exploration?A．Dead Sea.B．Lake Assal.C．Don Juan Pond.D．Great Salt Lake.Leia was fresh out of college when she began working as a member of abusiness-development team at a mid-size tech company.Though her skills had earned her the job,she was the youngest person in the team.“Everybody else was pretty much twice my age,”she says.“I heard about comments being made behind my back.There were a couple of times when my superiors referred to my age right in front of me,saying I was too young,‘What does a23-year-old know about these things?’”Leia tried to change things by altering her appearance at work.“I changed the way I dressed.I tried to dress older,more‘ladylike’.I changed my mannerisms and tried to act older,”she says.“It worked,to an extent.”The comments about her age and perceived (认定的)inexperience lessened,but Leia says she still felt like her growth potential was limited.She left the company soon after.What Leia experienced was ageism traditionally seen as something only older people face.For instance,older workers might be judged based on assumptions that they won’t fit into a progressive office dynamics or learn technology as quickly.A US study showed that nearly two-thirds of workers aged45and older had seen or experienced age discrimination. But younger workers face age discrimination,too.In fact,new research shows it may actually be the youngest team members who are bearing the impact of workplace ageism right now.Leia says removing ageism entirely will ultimately require a fundamental change to corporate culture,which has long tied seniority(资历)to skill.“We prize years of experience a little too much,and I don’t think years of experience and skill are necessarily related,”she says.“Steve Jobs was21when he founded Apple.We don’t know how much younger people actually have to contribute.Hopefully,more employers are realizing it.”4．What was the main topic of Leia’s superiors about her?A．Her job skills.B．Her young age.C．Her cute appearance.D．Her family background.5．Why didn’t Leia stay any longer at the company?A．She failed to dress more lady-like.B．She received more negative comments.C．She felt pressure from her work to an extent.D．She wanted more space for her growth potential.6．Which statement agrees with the US study the most?A．Only older people face age discrimination at work.B．Young workers can’t fit into a progressive office.C．Older workers are very slow technology learners.D．Young workers are seriously affected by ageism.7．Why does Leia mention Steve Jobs?A．To offer information about the Apple Company.B．To tell about Steve Jobs’unknown contribution.C．To say that skill does not always come with age.D．To show that skill alone matters to young people.Octopuses(章鱼)are some of the cleverest animals on the planet!So what makes them so smart?Let’s focus first on their defense mechanisms.Faced with multiple predators(捕食者)—including fish,birds and whales—octopuses are masters of disguise.They can imitate their environment by changing their skin colour.Without a shell,octopuses are weak and easily hurt physically,and always try to remain hidden in a shelter such as a hole or the space beneath a rock.Some prefer to wrap themselves in shells and small stones,while others transport their shelters in their arms.This is the case for the coconut octopus,which,true to its name,has been observed carrying coconut shells around to hide within in case of danger.Octopuses are also fearsome predators themselves,and their attack mechanisms are suited to the wide variety of prey(猎物)they consume,including seashells,fish and even other creatures like them.They can use their vision and camouflage skills to hunt,and their arms to explore,touch and taste their environment to seize every bit of food within reach.Octopuses can also use conditional discrimination,that is,they can change their choice depending on the situation.For example,they can learn to attack an object only in the presence of bubbles.They can also use space learning,and find a hidden shelter by remembering its st but not least,octopuses can learn by watching other octopuses carry out tasks,such as choosing one specific object over another.This is surprising,because they are mainly solitary(独居的)creatures.Octopuses meet every criteria for the definition of intelligence:They show a great flexibility in obtaining information(using several senses and learning socially),in processing it(through discriminative and conditional learning),in storing it(through long-term memory) and in applying it toward both predators and prey.8．Which are both predators and prey of octopuses?A．Fish.B．Birds.C．Whales.D．Seashells. 9．Why do coconut octopuses carry coconut shells everywhere?A．To change their skin color quickly.B．To avoid being discovered by predators.C．To observe the surroundings conveniently.D．To hide quickly in the shells when in danger.10．What’s the fifth paragraph mainly about?A．The excellent memory of octopuses.B．The strong learning ability of octopuses.C．The space discrimination ability of octopuses.D．The ability of octopuses to adapt to the environment.11．What can be a suitable title for the text?A．The terrible predators.B．The masters of disguise.C．Exactly how clever is an octopus?D．How strong is the learning ability of octopuses?English is the language for international communication,especially in areas such as science and technology.Galileo Galilei said,“Mathematics is the language with which God has written the universe.”Using numbers and mathematical symbols,one can write“words”and“sentences”.When appropriate,several“sentences”together might form a“mathematicalstory”.Mathematics has also grammar—the mathematical logic which determines whether statements are valid or not.A math teacher compares math to English and says,“Math may be viewed as a language —a simpler,more consistent and more regular language than English.Numbers represent nouns,while operational signs(+,－,×,÷,=)serve as verbs.Both components are governed by rules of syntax.Math facts,such as2×3=6,may be thought of as math sentences.Students should be encouraged to speak in complete sentences to convey an entire thought.Math at the simple sentence fact level is a much easier language than English.Although math has an infinite number of nouns,it has only five verbs(+,－,×,÷,=)associated with four basic operations.”Nina is an English teacher in elementary school and has12years of teaching experience. She has no doubts about the logical connection between English and mathematics.She says,“They are definitely languages.In both languages there are patterns and rules.They are both taught through visualizations and illustrations.Teaching vocabulary is like teaching fractions.You need to illustrate in both languages and practice a lot.”However,English and mathematics as languages are similar and yet different.The language of mathematics,for example,is preciseand less flexible—it cannot afford ambiguity,while natural languages contain constru ctive ambiguity derived from their cultural and contextual wealth of meanings.Mathematics as well as English as a second language does not develop naturally as a child develops a natural language;they need to be learned.12．What can be inferred from the first paragraph?A．English grammar can also be used in math.B．English and math are both general languages.C．Math is the medium of international communication.D．Both English and math have logic behind them.13．What is mainly discussed in Paragraph2?A．Math is much easier than English.B．Math is the simplest language.C．Both math and English can convey an entire thought.D．Math at the simple sentence fact level is much easier than English.14．What does the underlined word“ambiguity”probably mean?A．Unclarity.B．Evidence.C．Certainty.D．Similarity. 15．What is the author’s purpose in writing the text?A．To explain the logic behind English and math.B．To argue for the differences between math and English.C．To compare math and English as languages.D．To promote the development of science and technology.二、七选五In fact,this isn’t a new phenomenon.After all,people have complainedabout“kids these days”for decades.But is there really any truth in the idea that millennials(千禧一代)and Gen Z are weaker than Boomers or Gen X?People have complained about younger generations for thousands of years._______16_______.“The tendency for adults to look down upon the character of youth has been happening for centuries,”says Peter O’Connor,a professor of management.He points out the fixed concept remains alive and well,with research showing thousands of Americans believe that“kids these days”lack positive qualities that participants associate with older generations._____17_____.The researchers argued that this was because we project our current selves onto our past selves.By doing this,older people are unconsciously comparing who they are today to today’s young people,giving an impression that today’s youth is somehow on the decline,no matter which times we’re living in.______18______.The truth is that older generations have been accusing the young of being lazy and self-obsessed for centuries.____19____.Jason Dorsey,president of the Center for Generational Kinetics,believes that there is a way around this—but that general context is key to debunking(揭穿)the persisting weakness myth.“It’s awareness of what these different generations have gone through,why they are the way they are.”he says.“___20___,which simply does not exist right now.Instead of having frank conversations across generations,we always say that younger people are snowflakes and older people are dinosaurs.But the truth is that we’re all human.”he adds.A．It is actually not the truthB．We were also judged by the former generationsC．And the old generations are usually too stubborn to change themselvesD．But this isn’t because the youths of today actually do lack these qualitiesE．The best way to get the old to stop blaming the young is to create a dialogueF．We seem almost forced to judge people who grew up in a different time to usG．In fact,looking down on the generation that comes after you could simply be human nature三、完形填空I have to admit I’m an old-school reader.Put simply,I___21___old books.Alone in a new city to___22___my first daily newspaper job,I was feeling vaguely anxious as I dipped into a used bookstore to___23___my mind.The shelves,full of the smell of the past,quickly___24___me.One of the occupational hazards(职业病)of journalism,my chosen___25___,is a desire to stay on top of the Next Big Thing.With their cracked spines(书脊)and yellowed___26___,the titles in the shop usefully pointed me toward the longer view.“No___27___to get too worked up over today’s headline,”they seemed to say.“There is not much___28___under the sun.”Francis Bacon’s essays are always a ready antidote(解药).“Prosperity(繁荣)is not without many_29_and dislikes,”Bacon wrote,and“_30_is not without comforts and hopes.”For me,that comfort and hope often come from a second-hand__31__.Reading deeply from the past can be at least as___32___as following theday’s news.Don’t__33__missing out on the_34_gossip.All matters of fact,which a man did not_35_before,are news to him.21．A．buy B．love C．recommend D．hate 22．A．stop B．lose C．keep D．take 23．A．relieve B．change C．train D．feed 24．A．disappointed B．surprised C．calmed D．stressed 25．A．freedom B．profession C．company D．action 26．A．books B．words C．lines D．pages 27．A．need B．wonder C．doubt D．question 28．A．amazing B．interesting C．new D．dull 29．A．inspirations B．fears C．amusements D．strengths30．A．ownership B．relationship C．friendship D．hardship 31．A．book B．computer C．chair D．shop 32．A．mind-blowing B．doubtful C．instructive D．breathtaking 33．A．reflect on B．worry about C．take up D．dream of 34．A．silly B．common C．best D．latest 35．A．know B．hear C．research D．confirm四、用单词的适当形式完成短文阅读下面短文，在空白处填入1个适当的单词或括号内单词的正确形式。