2016年华侨、港澳台联考数学真题 (含答案与详细解析)

绝密★启用前

2016年中华人民共和国普通高等学校 联合招收华侨、港澳地区、台湾省学生入学考试

数 学

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）设集合{|11}A x x =-<，{|22}x B x =<，则A

B =（ ）

（A ）{|01}x x <

< （B ）{|02}x x << （C ）{|

2}x x < （D ）?

（2）若02απ≤<，且2sin 1α≤，则α的取值范围是（ ）

（A ）[0,2)π

（B ）5[0,

)[

,2)3

3π

ππ （C ）5[,]66

ππ

（D ）5[0,)[,2)66πππ （3）平面向量(,3)a x =与(2,)b y =平行的充分必要条件是（ ）

（A ）0x =，0y

= （B ）3x =-，2y =- （C ）6xy =

（D ）6xy =-

（4）复数2

2

(12)(2)

i i -+的模为（ ）

（A ）1 （B ）2

（C （D ）5

（5）等比数列{}n a 的各项都为正数，记{}n a 的前n 项和为n S ，若31S =，524S S -=，则1a =（ ）

（A ）19 （B ）1

7

（C ）

15 （D ）1

3

（6）函数2

1

log ((1,))1

y

x x =∈+∞-的反函数是（ ）

（A ）21()x y x R -=+∈ （B ）12((1,))x y x -=-∈+∞

（C ）12

()x

y

x R -=∈ （D ）11

2

(,1)x y

x R x -=∈≠

（7）设直线24y x =-与双曲线C ：2

2

21y x b

-=的一条渐近线平行，则C 的离心率为（ ）

（A （B （C ）3 （D ）5

（8）若函数([1,1])x y a x =∈-的最大值与最小值之和为3，则22a a -+=（ ）

（A ）9 （B ）7

（C ）6 （D ）5

（9）从1，2，3，4，5，6中任取3个不同的数相加，则不同的结果共有（ ）

（A ）6种

（B ）9种 （C ）10种

（D ）15种

（10）正四棱锥的各棱长均为1，则它的体积是（ ）

（A ）

3

（B ）

6

（C ）6 （D ）16

（11）抛物线2

1

(1)4

y x =

-的准线方程为（ ）

（A ）0x = （B ）15

16

x =

（C ）1x = （D ）1716

x

=

（12）曲线1

11y x

=+

-的对称轴的方程是（ ）

（A ）y x =-与2y x =+ （B ）y x =与2y x =-- （C ）y x =-与2y

x =-

（D ）y x =与2y

x =-+

二、填空题：本大题共6小题；每题5分。 （13）定义域为R 的偶函数

()f x 为周期函数，其周期为8.当[4,0]x ∈-时，()1f x x =+，则

(25)f =______________.

（14）若tan 43πθ??+= ???，则tan 4πθ?

?-= ??

?___________.

（15）在空间直角坐标系中，若直线

111

32x y z c

---==与平面25x y z -+=平行，则c =________. （16）设函数

4()()(0)f x ax b b a =+>>.若(1)16(1)f f =-，则

a

b

=_____________. （17）多项式

()p x 除以1x -的余式为1，()p x 除以1x +的余式为3，则()p x 除以2

1x -的余式为_____.

（18）已知B AC D --为直二面角，Rt ABC Rt ADC ???，且A B

B C =，则异面直线AB 与CD 所

成的角的大小为_________.

三、解答题：本大题共4小题；每小题15分。解答应写出文字说明，证明过程或演算步骤。

（19）已知平面向量(

)

2sin ,1x α=

，()

1,2x β=.

（Ⅰ）当αβ⊥时，求αβ-；

（Ⅱ）求αβ+的最大值.

（20）已知数列{}n a 的前n 项和2n S n =.

（Ⅰ）求{}n a 的通项公式；

（Ⅱ）记n b =

，求数列{}n b 的前n 项和.

（21）某同学进行投篮训练，已知该同学每次投篮命中的概率都为3

4

，且各次投篮是否命中相互独立. （Ⅰ）求该同学在3次投篮中至少命中2次的概率； （Ⅱ）该同学在10次投篮中恰好命中k 次(0,1,2,...,10)k =的概率记为k P ，k 为何值时，k P 最大？

（22）过椭圆C ：22

1259

x y +=右焦点F 的直线l 交C 于两点11(,)A x y ，22(,)B x y ，且A 不在x 轴上.

（Ⅰ）求

12

y y 的最大值；

（Ⅱ）若1

4

AF FB =，求直线l 的方程.

2016年港澳台联考数学真题答案

一、选择题

1—12：ADCAB ABBCC BD 二、填空题

13．0 14

． 15．1 16．13 17．2x -+ 18．3

π

三、解答题

19．【解析】（Ⅰ）当α

β⊥时，

0x x +=，即sin cos 0x x +=.

(

)(

)

2

2

2

2114

cos )4x x

x x

αβ-=

-+=-+=，所以2αβ-=.

（Ⅱ）因为

(

)()

2

2

2

2114cos

)44sin()4

x x

x x x π

αβ+=

+++=++=++

，

所以2

[0,8]α

β+∈，所以αβ+的最大值为.

20．【解析】（Ⅰ）当1n =时，111a S ==；

当2n ≥时，221(1)21

n n n a S S n n n -=-

=--=-.

因为1

1a =

满足上式，所以21n a n =-.

（Ⅱ）记数列{}n b 的前n 项和为n T ，

n n

b b

=

===.

则11

...1)22

n

T =-+= 21．【解析】（Ⅰ）记“该同学在3次投篮中至少命中2次”为事件A ，则2

2333331327

()(

)()44433

P A C C =+= （Ⅱ）101010

10311()()()3444k k k k k

k P C C -==，假设k P 最大，则11

k k k k P P P P +-≥??≥?，

即10110110101011011010

11()3()34411()3()344k k k k k k k k C C C C ++--?≥????≥??，10!10!3!(10)!(1)!(9)!10!10!3!(10)!(1)!(11)!

k k k k k k k k ?≥??-+-????≥?---?，即13(10)3(11)k k k k +≥-??-≥?

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2016年考研管理类联考英语二真题.pdf

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