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2019届高考全国卷1冲刺最后一卷

2019届高考全国卷1冲刺最后一卷
2019届高考全国卷1冲刺最后一卷

2019届高考全国卷1冲刺最后一卷

理 科 数 学(一)

注意事项:

1、本试卷分第Ⅰ卷(选择题)和第Ⅱ卷(非选择题)两部分。答题前,考生务必将自己的姓名、考生号填写在答题卡上。

2、回答第Ⅰ卷时,选出每小题的答案后,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,再选涂其他答案标号。写在试卷上无效。

3、回答第Ⅱ卷时,将答案填写在答题卡上,写在试卷上无效。

4、考试结束,将本试卷和答题卡一并交回。

第Ⅰ卷

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.已知集合{}2log 2M x x =<,{}1,0,1,2N =-,则M N =( )

A .{}1,0,1,2-

B .{}1,1,2-

C .{}0,1,2

D .{}1,2

2.设1i

2i 1i

z +=+-,则z =( ) A .2

B .3

C .4

D .5

3.设等差数列{}n a 的前n 项和为n S ,若44a =,972S =,则10a =( ) A .20

B .23

C .24

D .28

4.我国古代数学名著《数学九章》有“米谷粒分”题:粮仓开仓收粮,有人送来米1534石,验得米夹谷,抽样取米一把,数得254粒夹谷28粒,则这批米中,谷约为( ) A .134石

B .169石

C .338石

D .454石

5. “1m >”是“方程22

115

y x m m +=--表示焦点在y 轴上的双曲线”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

6.某几何体的三视图如图所示,则该几何体的体积等于( )

A .

19π6 B .17π6 C .23π6 D .10π

3

7.函数()()2sin ππ1

x

f x x x =

-≤≤+的图象可能是( ) A . B .

C .

D .

8.若01a b <<<,b x a =,a y b =,log b z a =,则x ,y ,z 大小关系正确的是( ) A .y x z <<

B .x y z <<

C .z x y <<

D .z y x <<

9.执行如图所示程序框图,若输出的S 值为20-,在条件框内应填写( ) A .3?i >

B .4?i <

C .4?i >

D .5?i <

10.已知抛物线2:8C y x =的焦点为F ,直线)2y x =-与C 交于A ,B (A 在x 轴上方)两点,若AF mFB =,则实数m 的值为( )

A B .3

C .2

D .32

11.某几何体的三视图如图所示,该几何体表面上的点P 与点Q 在正视图与侧视图上的对应点分别为A ,B ,则在该几何体表面上,从点P 到点Q 的路径中,最短路径的长度为( )

A B C .D

12.设函数()()sin f x x ω?=+,()()

(){}

00,A x f x f x '=

=,()22,162x y B x y ????=+≤??????

,若

存在实数?,使得集合A B 中恰好有5个元素,则()0ωω>的取值范围是( )

A

.?

????

B

.????? C

.????? D

.?

????

第Ⅱ卷

二、填空题:本大题共4小题,每小题5分,共20分.

13.已知向量()3,0=a

,(2+=a b ,则a 与b 的夹角等于_________.

14

.若二项式6

21x x ?

+????的展开式中的常数项为m ,则21

3d m

x x =?

______. 15.数列{}n a 且21

,2πsin ,4n n n n

a n n ???+=?

???为奇数

为偶数

,若n S 为数列{}n a 的前n 项和,则

2018S =______.

16.长沙市为了支援边远山区的教育事业,组织了一支由13名教师组成的队伍下乡支教,记者采访队长时询问这个团队的构成情况,队长回答:“(1)有中学高级教师;(2)中学教师不多于小学教师;(3)小学高级教师少于中学中级教师;(4)小学中级教师少于小学高级教师;(5)支教队伍的职称只有小学中级、小学高级、中学中级、中学高级;(6)无论是否把我计算在内,以上条件都成立.”由队长的叙述可以推测出他的学段及职称分别是____.

三、解答题:本大题共6个大题,共70分.解答应写出文字说明、证明过程或演算步骤.

17.(12分)在ABC △中,内角A ,B ,C 所对的边分别为a ,b ,c ,()2

cos π3

B -=

,1c =

,sin sin a B A .

(1)求边a 的值;

(2)求cos 23πB ?

?+ ??

?的值.

18.(12分)如图,四棱锥中P ABCD -,四边形ABCD 为菱形,60BAD ∠=?,

PA PD AD ==

,平面PAD ⊥平面ABCD . (1)求证:AD PB ⊥;

(2)求二面角A PC D --的余弦值.

19.(12分)“回文数”是指从左到右与从右到左读都一样的正整数,如22,121,3553等.显然2位“回文数”共9个:11,22,33,

,99.现从9个不同2位“回文数”中任取1

个乘以4,其结果记为X ;从9个不同2位“回文数”中任取2个相加,其结果记为Y . (1)求X 为“回文数”的概率;

(2)设随机变量ξ表示X ,Y 两数中“回文数”的个数,求ξ的概率分布和数学期望()E ξ.

20.(12分)已知椭圆()2222:10x y E a b a b +=>>经过点12P ?? ??

?,且右焦点)

2

F .

(1)求椭圆E 的方程;

(2)若直线:l y kx =+与椭圆E 交于A ,B 两点,当AB 最大时,求直线l 的方程.

21.(12分)已知()()2e x f x ax a =-∈R . (1)求函数()f x '的极值;

(2)设()()e x g x x f x =-,若()g x 有两个零点,求a 的取值范围.

请考生在22、23两题中任选一题作答,如果多做,则按所做的第一题记分. 22.(10分)【选修4-4:坐标系与参数方程】

在直角坐标系xOy 中,曲线1C 的参数方程为22cos 2sin x y ??=+=???,(?为参数).以原点O 为极

点,x 轴正半轴为极轴建立极坐标系,曲线2C 的极坐标方程为4sin ρθ=. (1)求曲线1C 的普通方程和2C 的直角坐标方程;

(2)已知曲线3C 的极坐标方程为()0π,θααρ=<<∈R ,点A 是曲线3C 与1C 的交点,点

B 是

曲线3C 与2C 的交点,且A ,B 均异于极点O ,且AB =a 的值.

23.(10分)【选修4-5:不等式选讲】 已知函数()241f x x x =-++. (1)解不等式()9f x ≤;

(2)若对于任意()0,3x ∈,不等式()2f x x a <+恒成立,求实数a 的取值范围.

2019届高考全国卷1冲刺最后一卷

理科数学答案(一)

一、选择题. 1.【答案】D

【解析】由题知{}04M x x =<<,故{}1,2M N =.故选D .

2.【答案】B 【解析】

()()()()1i 1i 1i 2i

i 1i 1i 1i 2

+++===--+,则3i z =,故3z =,故选B . 3.【答案】D

【解析】由于数列是等差数列,故41913493672a a d S a d =+==+=???,解得18a =-,4d =,

故101983628a a d =+=-+=.故选D . 4.【答案】B

【解析】由题意可知:这批米内夹谷约为28

1534169254

?≈石,故选B . 5.【答案】B

【解析】22

115y x m m +=--表示焦点在y 轴上的双曲线1050m m ->???

-

,解得15m <<,故选B .

6.【答案】A

【解析】由三视图可以看出,该几何体上半部是半个圆锥,下半部是一个圆柱,

从而体积2211119π

π1π13236

V =???+??=?,故选A .

7.【答案】A 【解析】因为()()

()()()22sin sin ππ1

1

x x

f x f x x x x --=

=-

=--≤≤+-+,可得()f x 是奇函数.排除

C ; 当π

3x =

时,0π3f ??

> ???

,点在x 轴的上方,排除D ; 当3π

x =-时,π103f ??

-<

-< ???

,排除B ;故选A . 8.【答案】B

【解析】取特殊值,令14

a =

,12b =,

则1

2

1142b x a ??=== ???,14

1122a y b ??==> ???,12

1

log log 24b z a ===,

14

1

1222??<< ???

,即x y z <<,可排除A 、C 、D 选项,故答案为B . 9.【答案】D

【解析】模拟执行程序,可得:1i =,10S =,

满足判断框内的条件,第1次执行循环体,11028S =-=,2i =, 满足判断框内的条件,第2次执行循环体,2824S =-=,3i =, 满足判断框内的条件,第3次执行循环体,3424S =-=-,4i =, 满足判断框内的条件,第4次执行循环体,44220S =--=-,5i =, 此时,应该不满足判断框内的条件,退出循环,输出的S 值为20-, 则条件框内应填写5?i <,故选D . 10.【答案】B

【解析】设A 、B 在l 上的射影分别是1A 、1B ,过B 作1BM AA ⊥于M .

由抛物线的定义可得出Rt ABM △中,得60BAE ∠=?,

1111

cos6012

AA BB AM AF BF m AB AF BF AF BF m ---?=

====+++,解得3m =,故选B . 11.【答案】C

【解析】由题,几何体如图所示

(1)前面和右面组成一面

此时PQ . (2)前面和上面在一个平面

此时PQ ,故选C . 12.【答案】A

【解析】()()sin f x x ω?=+的最大值或最小值,一定在直线1y =±上,又在集合B 中.

当1y =±时,22

162

x y +≤,得x ≤

23T T ?≤?∴?

>??

2π22π3ωω

??≤??∴???>??

,ω≤<,故选A .

二、填空题. 13.【答案】120?

【解析】已知向量()3,0=a

,(2+=a b ,

令(=c ,则(

)(

)(11

1022

=

-=-=-b c a , 设向量a 、b 的夹角是θ,于是

31031

cos 62

θ?-+?-=

=

=-

a b

a b

,故120θ

=?.

14.【答案】124

【解析】由题意,二项展开式的通项为66212316

61C C r

r

r

r r

r r T x x ---+???=??=?? ??

????

??

由1230r -=,得4r =,所以2

46C 5m =?=??,则5

223533

111

3d 3d |51124m x x x x x ===-=??

15.【答案】

3028

2019

【解析】数列{}n a 且21

,2πsin ,4

n n n n

a n n ???+=?

???为奇数

为偶数

①当n 为奇数时,21111222n a n n n n ??

=

=- ?++??

②当n 为偶数时,πsin

4

n n a =, 所以()()201813520172462018S a a a a a a a a =++++++++

+,

()11111110093028

1101012335

2017201920192019

??

=-+-++

-++-++=

+=

???

故答案为

3028

2019

. 16.【答案】小学中级

【解析】设小学中级、小学高级、中学中级、中学高级人数分别为a ,b ,c ,d , 则13a b c d +++=,1d ≥,c d a b +≤+,b c <,a b <, 所以()13a b a b -+≤+,7a b ∴+≥,6c d +≤,

若7a b +=,则6c d +=,a b <,3a ∴=,4b =,5c =,1d =,

若8a b +≥,则5c d +≤,1d ≥,4c ∴≤,b c <,3b ∴≤,5a b ≥>,矛盾, 队长为小学中级时,去掉队长则2a =,4b =,5c =,1d =, 满足11d =≥,64c d a b +=≤+=,45b c =<=,24a b =<=;

队长为小学高级时,去掉队长则3a =,3b =,5c =,1d =,不满足a b <; 队长为中学中级时,去掉队长则3a =,4b =,4c =,1d =,不满足b c <; 队长为中学高级时,去掉队长则3a =,3b =,5c =,0d =,不满足1d ≥; 综上可得队长为小学中级.

三、解答题.

17.【答案】(1)53

;(2.

【解析】(1)由()2cos π3B -=

,得2cos 3

B =-,

因为1c =,由sin sin a B A ,得ab =,∴b =, 由余弦定理2222cos b a c ac B =+-,得234150a a +-=,

解得53a =或3a =-(舍),∴5

3

a =.

(2)由2cos 3B =-,得sin B =sin2B =1cos29

B =-,

∴cos 2cos 2cos sin 2sin 333πππB B B ?

?+=- ??

?.

18.【答案】(1)见解析;(2

【解析】(1)证明:取AD 中点O 连结PO ,BO ,

PA PD =,PO AD ∴⊥.

又四边形ABCD 为菱形,60BAD ∠=?,故ABD △是正三角形, 又点O 是AD 的中点,BO AD ∴⊥. 又PO

BO O =,PO 、BO ?平面BOP ,AD ∴⊥平面BOP ,

又PB ?平面BOP ,AD PB ∴⊥.

(2)PA PD =,点O 是AD 的中点,PO AD ∴⊥.

又平面PAD ⊥平面ABCD ,平面PAD

平面ABCD AD =,PO ?平面PAD ,

PO ∴⊥平面ABCD ,

又AO ,BO ?平面ABCD ,PO AO ∴⊥,PO BO ⊥.又AO BO ⊥, 所以OA ,OB ,OP 两两垂直.

以O 为原点,分别以OA ,OB ,OP 的方向为x 轴,y 轴,z 轴的正方向建立空间直角坐标系O xyz -.

设2AB =,则各点的坐标分别为()1,0,0A ,()B ,()

C -,()1,0,0

D -,()0,0,1P .

故()AC =-,()1,0,1AP =-,()

1PC =--,()1,0,1PD =--, 设()1111,,x y z =n ,()2222,,x y z =n 分别为平面PAC ,平面PCD 的一个法向量,

由1100

AC AP ?????==??n n

,可得1111300x x z -+?=-+=????,令11z =,则11x =

,1y =

()

11=n .

由220

PC PD ?=??????=n n

,可得22222200x z x z -+-=--=?????,令21z =,则21x =-

,2y =

故21,??

=- ? ???

n .

(

)

121,cos ,??

?- ? ?=

==n n .

又由图易知二面角A PC D --是锐二面角, 所以二面角A PC D --

19.【答案】(1)2

9

;(2)随机变量ξ的概率分布为

随机变量ξ的数学期望为()79

E ξ=

. 【解析】(1)记“X 是‘回文数’”为事件A .

9个不同2位“回文数”乘以4的值依次为44,88,132,176,220,264,308, 352,396.其中“回文数”有44,88.所以,事件A 的概率()2

9

P A =. (2)根据条件知,随机变量ξ的所有可能取值为0,1,2. 由(1)得()29

P A =

. 设“Y 是‘回文数’”为事件B ,则事件A ,B 相互独立. 根据已知条件得,()2

9C 2059

P B =

=. ()()()2528

0119981P P A P B ξ????===--=

???????

()()()()()252543

111999981P P A P B P A P B ξ????==+=-+-=

? ?????

; ()()()2510

29981

P P A P B ξ===?=.

所以,随机变量ξ的概率分布为

所以,随机变量ξ的数学期望为()28431070128181819

E ξ=?

+?+?=.

20.【答案】(1)2214x y +=;(2)y =+.

【解析】(1)设椭圆E 的左焦点()

1F ,则12242a PF PF a =+=?=,

又2

2

2

1c b a c ?=-=,所以椭圆E 的方程为2

214

x y +=.

(2)由()

2222144044

y kx k x x y ????=?+++=+=?,设()11,A x y ,()22,B x y ,

由()

2221

128161404Δk k k =-+>?>

,且12214x x k +=-+,122414x x k

=+,

AB ==

设2114t k =+,则10,2t ??∈ ???,AB ==,

当112

t =

,即k =AB ,此时:l y x =

21.【答案】(1)0a ≤时,()f x '没有极值,0a >时,()f x '有极小值22ln2a a a -; (2)()0,+∞.

【解析】(1)()e 2x f x ax ='-,()e 2x f x a '-'=.

①若0a ≤,显然()0f x ''>,所以()f x '在R 上递增,所以()f x '没有极值. ②若0a >,则()0ln2f x x a ?>'',

所以()f x '在(),ln2a -∞上是减函数,在()ln2,a +∞上是增函数. 所以()f x '在ln2x a =处取极小值,极小值为()()ln221ln2f a a a =-'. (2)()()()2e 1e x x g x x f x x ax =-=-+.函数()g x 的定义域为R , 且()()

2e e 2x x g x x ax x a ='=++.

①若0a >,则()00g x x <'?<;()00g x x >'?>.所以()g x 在(),0-∞上是减函数, 在()0,+∞上是增函数.所以()()min 01g x g ==-.

令()()1e x h x x =-,则()e x h x x '=.显然()00h x x <'?<, 所以()()1e x h x x =-在(),0-∞上是减函数. 又函数2y ax =在(),0-∞上是减函数,取实数0

<,

则()2

0110g h a

??>+?=-+= ??.

又()010g =-<,()10g a =>,()g x 在(),0-∞上是减函数,在()0,+∞上是增函数. 由零点存在性定理,()g x 在

??

??

?,()0,1上各有一个唯一的零点.所以0a >符合题意.

②若0a =,则()()1e x g x x =-,显然()g x 仅有一个零点1.所以0a =不符合题意.

③若0a <,则()()ln 2e e a

x g x x -'??=-??

(i )若()ln 20a -=,则1

2

a =-.此时()0g x '≥,即()g x 在R 上递增,至多只有一个零

点,

所以1

2

a =-不符合题意.

(ii )若()ln 20a -<,则1

02

a -<<,函数()g x 在()(),ln 2a -∞-上是增函数,

在()()ln 2,0a -上是减函数,在()0,+∞上是增函数,

所以()g x 在()ln 2x a =-处取得极大值,且极大值()()(){

}

2

ln 2ln 2110g a a a -=--+

所以()g x 最多有一个零点,所以1

02

a -<<不符合题意.

(iii )若()ln 20a ->,则1

2

a <-,函数()g x 在(),0-∞和()()ln 2,a -+∞上递增,

在()()0,ln 2a -上递减,所以()g x 在0x =处取得极大值,且极大值为()010g =-<,

所以()g x 最多有一个零点,所以1

2

a <-不符合题意.综上所述,a 的取值范围是()0,+∞.

请考生在22、23两题中任选一题作答,如果多做,则按所做的第一题记分. 22.【答案】(1)()2

21:24C x y -+=,()2

22:24C x y +-=;(2)7π12

α=或11

π12.

【解析】(1)()2

21:24C x y -+=,()2

22:24C x y +-=.

(2)1:4cos C ρθ=,联立极坐标方程θα=,得4cos A ρα=,4sin B ρα=,

π4A B ρρα?

?∴-=-= ??

?sin 4πα??∴-=

??? 0πα<<,∴7π12

α=

或11

π12.

23.【答案】(1)[]2,4-;(2)5a ≥.

【解析】(1)()9f x ≤,可化为2419x x -++≤,

即2339x x >-≤???

或1259x x -≤≤-≤???或1

339x x <--+≤???,

解得24x <≤或12x -≤≤或21x -≤<-;不等式的解集为[]2,4-. (2)2412x x x a -++<+在()0,3x ∈恒成立,

52412124133

a

x x x a x a x x a x a -?-++<+?--+<-<+-?

<<+, 由题意得,()50,3,33a a -??

?+ ???,所以5005335a a a a a -≤≥???≥?+≥≥???

?.

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