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2018学年普陀区高三年级一模试卷

2018学年普陀区高三年级一模试卷
2018学年普陀区高三年级一模试卷

2018学年普陀区高三年级一模试卷

2018.12

一、填空题

1.函数(

)2

f x x

=的定义城为 . 2.若1sin 3α=

,则cos =2πα??

+ ???

. 3.设11,1,2,332α??∈--????

,,若()f x x α=为偶函数,则=α .

4.若直线l 经过抛物线2

C y x :=4的焦点且其一个方向向量为()1,1d =,则直线l 的方程

为 .

5.若一个球的体积是其半径的

4

3

倍,则该球的表面积为 . 6.在一个袋中装有大小、质地均相同的9只球,其中红色、黑色、白色各3 只,若从袋中随

机取出两个球,则至少有一个红球的概率为 .(结果用最简分数表示)

7.设()()5

236

0123611x x a a x a x a x a x -+=++++???+,则3a = . (结果用数值表

示)

8.设0a >且1a ≠,若()log sin cos 0a x x -= , 则8

8

sin cos x x += .

9.如图,正四棱柱1111ABCD A BC D -的底面边长为4, 记1111=AC B D F ?,11=E BC B C ?,若AE BF ⊥,

则此棱柱的体积为 .

10.某人的月工资由基础工资和绩效工资组成2010年每月的基础工资为2100元、绩效工资为2000元,从2011年起每月基础工资比上一年增加210元、绩效工资为上一年的110%.照此推算,此人2019年的年薪为 .万元(结果精确到0.1)

11.已知点()2,0A -,设B C 、是圆2

2

:+1O x y =上的两个不同的动点,且向量

()1O B t O A t O C =+- (其中t 为实数),则=AB AC ? . 12.设a 为常数,记函数()1log 2a x

f x a x

=

+- (0a >且1,0a x a ≠<< )的反函数为()1f x -,则11111232++=21212121a f f f f a a a a ----????????++??? ? ? ? ?++++????????

.

A

B

C

D

1

A 1

B 1

C 1

D F

E

二、选择题

13.下列关于双曲线22

163

x y Γ-

=: 的判断,正确的是( ) .A 渐近线方程为20x y ±= .B 焦点坐标为()30±, .C 实轴长为12 .D 顶点坐

标为()60±,

14.函数2cos 24y x π??

=+

??

?

的图像( ) .A 关于原点对称 .B 关于点3,08π??

-

???

.C 关于y 轴对称 .D 关于直线=

4

x π

轴对称

15.若a b c 、、 表示直线,αβ、 表示平面,则“//a b ”成立的一个充分非必要条件是( )

.A a b b c ⊥⊥, .B //,//a b αα .C a b ββ⊥⊥, .D //a c b c ⊥,

16.设()f x 是定义在R 上的周期为4的函数,且()2sin 2,01

2log ,14

x x f x x x π≤≤?=?<

()()g x f x a

=-,若1

02

a << ,则函数()g x 在区间[]-45,上零点的个数是( ) .A 5 .B 6 .C 7 .D 8

三、解答题

17.在ABC ?中,三个内角,,A B C 所对的边依次为,,a b c ,且1cos 4

C =

. (1)求2

2cos +2sin 22

A B

C +的值; (2)设2c =,求a b +的取值范围.

18.已知曲线2

2:

11612

y x Γ+=的左、右顶点分别为,A B ,设P 是曲线Γ上的任意一点. (1)当P 异于,A B 时,记直线,PA PB 的斜率分别为12,k k ,求证:12k k ?是定值; (2)设点C 满足()0AC CB λλ=>,且PC 的最大值为7,求λ的值.

19.如图所示,某地出土的一种“钉”是由四条线段组成,其结构能使它任意抛至水平面后,总有一端所在的直线竖直向上,并记组成该“钉”的四条线段的公共点为O ,钉尖为

()1,2,3,4i A i =.

(1)设()0i OA a a =>,当123,,A A A 在同一水平面内时,求1OA 与平面123A A A 所成角的大小(结果用反三角函数值表示);

(2

)若该“钉”的三个钉尖所确定的三角形的面积为2,要用某种线型材料复制100枚这种“钉”(损耗忽略不计),共需要该种材料多少米?

O

4

A 1

A 2

A 3

20.设数列{}n a 满足()1133

,52

n n n a a a n N a *+==∈+. (1)求23,a a 的值;

(2)求证:11n a ??-????是等比数列,并求12

11

1

lim +n n n a a a →∞??+

+

- ???

的值; (3)记{}n a 的前n 项和为n S ,是否存在正整数k ,使得对于任意的n (n N *∈且2n ≥)均有n S k ≥成立?若存在,求出k 的值;若不存在,说明理由.

21.已知函数()()2x f x x R =∈,记()()()g x f x f x =+-. (1)解不等式:()()26f x f x -≤;

(2)设k 为实数,若存在实数(]01,2x ∈,使得()()20021g x k g x =?-成立,求k 的取值范围;

(3)记()()()22h x f x a f x b =++?+(其中,a b 均为实数),若对于任意的[]0,1x ∈,均有

()1

2

h x ≤

,求,a b 的值.

参考答案

一、填空题

1.()(],00,1-∞?

2.13-

3.2-

4.1y x =-

5.4

6.7

12

7.0

8.1 9.10.4 11.3 12.2a

二、选择题

13.B 14.B 15.C 16.D

三、解答题

17.(1)

8;(2)23? ?

?, 18.(1)34-

;(2)7 或17

19.(1)(2)3

42006?

20.(1)23927

,1335

a a =

= (2)2 (3)1k = 21.(1)(]2,log 3-∞ (2)27119,2259??

??

??

(3)1712,2a b =-=

2 2018届普陀区高三英语一模(作文有范文)

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