上海市黄浦区2018届高三一模数学

上海市黄浦区2018届高三一模数学
上海市黄浦区2018届高三一模数学

上海市黄浦区2018届高三一模数学试卷2018.1

一、填空题(本大题共12题,1-6每题4分,7-12每题5分,共54分)

1、已知全集R U =,集合{}11>-=x x A ,?

?????<+-=013x x x B ,则()=B A C U 2、已知角θ的顶点在坐标原点,始边与x 轴的正半轴重合,若角θ的终边落在第三象限内,且5

32cos =??? ??+θπ,则=θ2cos 3、已知幂函数的图像过点??? ??41,2,则该幂函数的单调递增区间是

4、若n S 是等差数列{}n a ()*∈N n :...8,5,2,1-的前n 项和,则=+∞→1

lim 2n S n n 5、某圆锥体的底面圆的半径长为2,其侧面展开图是圆心角为

32π的扇形,则该圆锥体的体积是

6、过点()2,1P -作圆522=+y x 的切线,则该切线的点法向式方程是

7、已知二项式展开式()7722107....21x a x a x a a x +++=-,且复数i a a z 128

2171+=,则复数z 的模=z

8、若关于y x ,的二元一次线性方程组???=+=+2

22111c y b x a c y b x a 的增广矩阵是???? ??n m 3210,且???-==11y x 是该线性方程组的解,则三阶行列式1

2301

1n m -中第3行第2列元素的代数余子式的值是 9、某高级中学欲从本校的7位古诗词爱好者(其中男生2人、女生5人)中随机选取3名同学作为诗词朗读比赛的支持人,若要求主持人中至少有一位是男同学,则不同选取方法的种数是(结果用数值表示)

10、已知ABC ?的三个内角A 、B 、C 所对边长分别为a 、b 、c ,

记A B C ?的面积为S ,若()22c b a S --=,则内角=A (结果用反三角函数值表示)

11、已知函数()1

1-=

x x f ,关于x 的方程()()02=++c x bf x f 有7个不同实数解,则实数b 、c 满足的关系式是

12、已知正六边形ABCDEF (顶点的字母依次按照逆时针顺序确定)的边长为,点P 是CDE ?(内含

边界)的动点,设→→→+=AF y AB x AP (R y x ∈,),则y x +的取值范围是

二、选择题(本大题共4题,每题5分,共20分)

13、已知βα,是空间两个不同的平面,则“平面α上存在不共线的三点到平面β的距离相等”是“βα//”的( )

A 、充分非必要条件

B 、必要非充分条件

C 、充要条件

D 、非充分非必要条件

14、为了得到函数()R x x x y ∈+=3cos 3sin 的图像,可以将函数x y 3sin 2=的图像( )

A 、向右平移4π个单位

B 、向左平移4

π个单位 C 、向右平移12π个单位 D 、向左平移12

π个单位 15、用数学归纳法证明()

*∈≥++++++++N n n n n n n 24111....312111,由k n =到1+=k n 时,不等式左边应添加的项是( )

A 、121+k

B 、1

1121+-+k k C 、221121+++k k D 、2

21121+-+k k 16、已知函数12

+=x y 的图像与函数()x f y =的图像关于直线0=+y x 对称,则函数()x f y =的反函数

是( ) A 、()x y --=2log 1 B 、()x y --=1log 2

C 、12+--=x y

D 、12+-=x y

三、解答题(14+14+14+16+18=76分)

已知正方体1111ABCD A BC D -的棱长为2,

点E 、F 分别是所在楞11A B 、AB 的中点,点1O 是面1111A B C D 的中心,如图所示.

(1)求三棱柱1O FBC -的体积1O FBC V -;

(2)求异面直线1A F 与CE 所成角的大小.(结果用反三角函数值表示).

18、已知函数()11cos 222f x x =+,()1cos ,2

g x x x x R =+?∈. (1)若()0f a =,求()2g a 的数值;

(2)若02x π≤≤

,求函数()()()h x f x g x =+的值域.

19、已知椭圆22

221(0)x y E a b a b

+=>>:的右焦点为()1,0F ,点(0,)B b 满足=2FB . (1)求实数a 、b 的值;

(2)过点F 作直线l 交椭圆于M 、N 两点,若BFM ?与BFN ?的面积之比为2,求直线l 的方程.

20、(本题满分16分)

定义:若函数()x f 的定义域为R ,且存在实数a 和非零实数k ,使得()()x f k x a f ?=-2对R x ∈都成立,则称函数()x f 是具有“理想数对()k a ,”的函数。比如:函数()x f 有理想数对()1,2-,即()()x f x f -=-4,()()04=+-x f x f ,可知函数图像关于点()0,2称中心对称图形。设集合M 是有具有理想数对()k a ,的函数的全体。

(1)已知函数()12-=x x f ,R x ∈,试判断函数()x f 是否为集合M 的元素,并说明理由;

(2)已知函数()x x g 2=,R x ∈,证明:()M x g ?;

(3)数对()1,2和()1,1-都是函数()x h 的理想数对,且当11≤≤-x 时,()21x x h -=,若正比例函数()0>=m mx y 的图像与函数()x h 的图像在区间[]12,0上有且仅有5个交点,求实数m 的取值范围。

21、定义运算“⊕”:对于任意R y x ∈,,()()*∈+-=⊕R b by x b y x 1(等式的右边是通常的加减乘运算).

若数列{}n a 的前n 项和为n S ,且n n n a S 3=⊕对任意*

∈N n 都成立,

(1)求1a 的值,并推导出用1-n a 表示n a 的解析式; (2)若3=b ,令()*∈=N n a b n n n 3

,证明数列{}n b 是等差数列; (3)若3≠b ,令()*∈=N n a c n n n 3,数列{}n c 满足()

*∈≤N n c n 2,求正实数b 的取值范围.

客观题——参考答案:

一、填空题

1、[]0,2;

2、725;

3、(),0-∞;

4、32;

5、83

π;6、()(+2)+10x y -+=;7

、 8、4;9、57;10、815arcsin arccos 1717?? ???或8arctan 15;11、121b c b c +=??<-??<-?

;12、[]3,4; 二、选择题:

13、B ;14、D ;15、D ;16、C ;

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