浙江某高中高二数学下学期期中试题及解析8

浙江省东阳中学2019-2020学年高二数学下学期期中试题
一、选择题：本大题共10小题，每小题4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{1,2,3,4}A =，{2,4,6}B =，则A B I 的元素个数是 ( ) A .0个 B .1个 C .2个 D .3个
2．直线230x y ++=的斜率是 ( )
A .12-
B .1
2
C .2-
D .2
3．“2k =且1b =-”是“直线y kx b =+过点()1,1”的 ( )
A .充分而不必要条件
B .必要而不充分条件
C .充要条件
D .既不充分又不必要条件
4．函数π
()sin(2)3f x x =+()x ∈R 的最小正周期为 ( )
A .
π
2
B .π
C .2π
D .4π 5. 已知向量()1,2a =r ,(),4b x =r 且a b r r
∥，则实数x 的值是 ( )
A .2-
B .2
C .8
D .8-
6. 已知等比数列{}n a 中，123n n a -=⨯,则由此数列的偶数项所组成的新数列的前n 项和n S 的值为 ( ) A .31
n
- B .()
331
n - C .
91
4
n - D .
()
3914
n -
7. ABC ∆中,角,,A B C 所对的边分别为,,a b c ,若)
3cos cos b c A a C -=，则cos A =
( )A .
1
2
B 3
C 3
D .3
8．设椭圆1C 的离心率为5
13
，焦点在x 轴上且长轴长为26. 若曲线2C 上的点到椭圆1C 的两
个焦点的距离的差的绝对值等于8，则曲线2C 的标准方程为 ( ) A .2222143x y -= B .22221135x y -= C .2222134x y -= D .22
2211312
x y -=
9．设,x y 满足约束条件360,
20,0,0,x y x y x y --⎧⎪-+⎨⎪⎩
……
厖若目标函数(0,0)z ax by a b =+>>的最大值是12，则
22
94a b +的最小值为 ( ) A .1325 B .1
2 C .1 D .2
10. 定义域为R 的偶函数()f x 满足对任意的实数x ，有(2)()(1)f x f x f +=-，且当 []2,3x ∈时，2()21218f x x x -+-=，若函数()()log 1a y f x x =-+在()0,+∞上至少有
三个零点，则a 的取值范围是 ( ) A .20,⎛⎫ ⎪ ⎪⎝⎭ B .30,⎛⎫ ⎪ ⎪⎝⎭ C .50,⎛⎫ ⎪ ⎪⎝⎭ D .60,⎛⎫ ⎪ ⎪⎝⎭
二、填空题：本大题共7小题，多空题每小题6分，单空题每小题4分，共36分。

11. 已知3cos =
θ，则cos2=θ ，3πsin +2θ⎛⎫
= ⎪⎝
⎭ .
12. 若函数()()()()2R f x a x x a =+-∈是偶函数，则a = ，值域为 ． 13. 在等差数列{}n a 中，若1594πa a a ++=，则5a = ，()28tan +=a a .
14．一个几何体的三视图如图所示(单位：cm)， 则该几何体的表面积为 ， 该该几何体的体积为 ．
15．过点()2,1P 的直线与抛物线216y x =交于,A B 两点，且0PA PB +=uu r uu r r
则此直线的
方程为_________.
16．若函数3()2=f x x ax +-在区间()1,+∞内是增函数，则实数a 的取值范围是______ . 17．若对任意[]1,2x ∈且[]2,3y ∈,不等式222xy ax y +…恒成立, 则实数a 的取值范围是___________.
三、解答题：本大题共5小题，共74分，解答应写出文字说明、证明过程或演算步骤。

18.已知向量()()3
sin ,cos ,cos ,sin ,m A A n B B m n ==⋅u r r u r r ,,A B C 分别是锐角三角形
ABC 三边,,a b c 所对的角.
（Ⅰ）求C ∠的大小；
第14题
俯视图
8
5
5
8
8
5
5
（Ⅱ）若,,a c b 成等比数列，且18CA CB ⋅=uu r uu r
，求c 的值.
19.设数列{}n a 是公差大于零的等差数列，已知2
132
210==a a a -,. (1) 求数列{}n a 的通项公式；
(2) 设数列{}n b 是以1为首项，以3为公比的等比数列，求数列{}n n a b -的前n 项和n S ．
20. 在四棱锥P ABCD -中，PA ⊥平面ABCD ，
AD BC ∥，24BC AD ==
，AB CD ==，
(Ⅰ) 证明：BD ⊥平面PAC ；
(Ⅱ) 若二面角A PC D --的大小为60︒，求AP
的值.
21.已知椭圆22
221x y a b
+=()0a b >>
的离心率e ，连接椭圆的四个顶点得到的菱形的面积
为4.
（Ⅰ）求椭圆的方程；
（Ⅱ）设直线l 与椭圆相交于不同的两点,A B ，已知点A 的坐标为(),0a -，
若AB =
，
(第20题图)
求直线l 的倾斜角.
22.设函数()ln f x x x =(0)x >. (1) 求函数()f x 的最小值；
(2) 设2()()F x ax f x '=+()R a ∈，讨论函数()F x 的单调性；
(3) 斜率为k 的直线与曲线()y f x '=交于11(,)A x y ,22(,)B x y 12()x x <两点，求证：
121
x x k
<
<.
东阳中学高二数学期中试卷参考答案
1-5 CAABB 6-10 DCABB 11. 33,31--
12.(]4,,2∞- 13.3,3
4-π 14． 3264,144cm cm 15.0158=--y x 16.[)+∞-,3 17.[)+∞-,1
18.解：（1）
sin cos cos sin 2A B A B +=
即
sin()2
A B += 所以
sin C =
又因为C ∠是锐角三角形内角，所以3
π
=∠C （2）因为ab c =2
又18=⋅ 所以 18cos =C ab
所以 36=ab 即362
=c 所以6=c .
19.解：（1）n a n 2= （2）2
312
n
n n n S -++=
20.(2)
11
22
3 21.（Ⅰ）解：由
e=
2
c a =，得2234a c =.再由222c a b =-，解得a=2b. 由题意可知
1
2242
a b ⨯⨯=，即ab=2. 解方程组2,
2,
a b ab =⎧⎨
=⎩得a=2，b=1.
所以椭圆的方程为2
214
x y +=. (Ⅱ)解：由（Ⅰ）可知点A 的坐标是（-2,0）.设点B 的坐标为11(,)x y ，直线l 的斜率为k.则直线l 的方程为y=k （x+2）.
于是A 、B 两点的坐标满足方程组22
(2),1.4
y k x x y =+⎧⎪
⎨+=⎪⎩消去y 并整理，得
2222(14)16(164)0k x k x k +++-=.
由212164214k x k --=+，得2122814k x k -=+.从而1
2
414k
y k =+.
所以||AB ==
由||AB =
=整理得42
329230k k --=，即2
2
(1)(3223)0k k -+=，解得k=1±.
所以直线l 的倾斜角为
4
π或34π.
22.(1) 解：()ln 1f x x '=+(0)x >，令()0f x '=，得1
x e
=
. …………2分 ∵当1(0,)x e ∈时，()0f x '<；当1(,)x e
∈+∞时，()0f x '>， …………3分 ∴当1x e =
时，min 111
()ln f x e e e
==-. …………4分 (2) 2
()ln 1F x ax x =++(0)x >，
2121
()2(0)ax F x ax x x x
+'=+=>. …………5分
① 当0≥a 时，恒有()0F x '>，()F x 在),0(+∞上是增函 …………6分 ② 当0<a 时，
令()0F x '>，得2
210ax +>
，解得0x <<
； …………7分 令()0F x '<，得2
210ax +<
，解得x >
. …………8分 综上，当0≥a 时，()F x 在),0(+∞上是增函数； 当0<a 时，()F x
在
上单调递增，在)+∞上单调递减. …………9分 (3) 证：2121
2121
()()ln ln f x f x x x k x x x x ''--=
=--.
要证121
x x k <<，即证211221ln ln x x x x x x -<
<-，等价于证2
12211
11ln x x x x x x -<<，令21x t x =， 则只要证1
1ln t t t
-<
<，由1t >知ln 0t >，故等价于证ln 1ln (1)t t t t t <-<> (*). ① 设()1ln (1)g t t t t =--≥，则1()10(1)g t t t
'=-≥≥，故()g t 在[1,)+∞上是增函数， ∴ 当1t >时，()1ln (1)0g t t t g =-->=，即1ln (1)t t t ->>.
② 设()ln (1)(1)h t t t t t =--≥，则()ln 0(1)h t t t '=≥≥，故()h t 在[1,)+∞上是增函数， ∴ 当1t >时，()ln (1)(1)0h t t t t h =-->=，即1ln (1)t t t t -<>.
由①②知(*)成立，得证. …………15分
赠送：初中英语代词
Ⅰ.词汇运用。

A)用括号内所给单词的适当形式填空。

1．I can't find my ruler.May I use yours(you)，Jenny? 2．—Is the woman who is dancing over there your teacher? —Yes ，she teaches us(we) music.
3．I'm sure I can improve my(I) spoken English in a year. 4．The little girl learned how to dress herself(she) with the help of her mother. 5．You can't put anything(something) more in the box because it is already full. 6．There are lots of foreigners in our city.And most of them(they) are very friendly.
7．Please come in and make yourselves(you) at home ，guys! 8．I have many photos about the friends of mine(I)． 9．She lost her(she) watch this morning.What bad news!
10．It's very dangerous for children to play by the river by themselves(they)． 11．Larry has put on too much weight because of his(he) unbalanced diet. 12．Your task is to draw a picture.Ours(we) is to make up a story.
13．Doing some chores at home helps to develop children's independence and make their(they) future lives easier.
14．My sister is waiting for me(I) in the hall.We are going to see a film together. 15．It's better to keep your(you) voice down in public even when you're with your friends.
B)根据句意，用方框中所给单词的适当形式填空。

both ，neither ，any ，little ，another
16．We have got two TV sets ，but neither works well.
17．If you want tickets for a round trip(往返旅行)，you'll have to pay another 100 yuan.
18．Keep together.I don't want any of you to get lost.
19．Come on！The school bus is coming.We have little time left.
20．I'll take both of the cameras.One for my brother，and the other for Jim.
Ⅱ.单项选择。

人称代词、物主代词及反身代词的辨析
(B)1.My bike is old，and I want to ask my parents to buy a new one for ________.
A．my B．me
C．mine D．I
(C)2.—Who is singing in the classroom?
—________ must be Susan.
A．You B．He
C．It D．She
(C)3.—Is this Kate's red pen?
—No，it's not ________.She only has a black pen.
A．ours B．theirs
C．hers D．his
(A)4.—Are Lucy and Lily ________ friends?
—Yes.And ________ are twin sisters.
A．your；they B．your；we
C．yours；they D．yours；we
(D)5.—What do you think of the new mobile phone?
—It looks cool and its screen is bigger than ________.
A．I B．me
C．my D．mine
(D)6.Jack played basketball with some friends of ________.
A．he B．him
C．himself D．his
(D)7.He is old enough to go to school by ________.
A．him B．he
C．his D．himself
(D)e on，boys！Nobody will help us.We have to depend on ________！
A．yourself B．myself
C．yourselves D．ourselves
(D)9.Kids，don't lose ________ in playing computer games for so long.It's bad for your eyes.
A．ourselves B．yourself
C．themselves D．yourselves
(B)10.Everyone may make mistakes in life.The most important thing is not to repeat ________.
A．it B．them
C．him D．her
(D)11.People like taking photos with friends.These photos are ________ good
memories.
A．they B．them
C．theirs D．their
不定代词
(C)12.I asked Mike for ________ paper，but he didn't have ________.
A．any；some B．any；any
C．some；any D．some；some
(D)13.Some young people like to play WeChat when they are free，but ________like to go to the cinema.
A．other B．the other
C．another D．others
(A)14.—Kate，which of the two skirts do you prefer?
—________ is OK.I don't like their styles.
A．Neither B．Both
C．None D．All
(D)15.—Wang Yang，can you speak English?
—Yes，but only ________.I have worked in New York for only three weeks.
A．few B．a few
C．little D．a little
(D)16.—Wow！You've got so many skirts.
—But ________ of them are in fashion now.
A．all B．both
C．neither D．none
(C)17.—There are four bedrooms in the house，________with its own shower.
—That's what I want.I've got a few kids.
A．every B．both
C．each D．either
(B)18.Jack and Alan play basketball very well.We hope________ of them can join our team.
A．all B．both
C．every D．some
(A)19.When I am in a western restaurant，I always use________ hand to hold the knife and________ hand to hold the fork.
A．one；the other B．the other；one
C．another；one D．the other；another
(A)20.At present，children mean________ to most parents in China.
A．everything B．nothing
C．anything D．something
(C)21.—Who helped you sweep the classroom yesterday?
—________.I did it all by myself.
A．Somebody B．Anybody
C．Nobody D．Everybody
(C)22.There must be ________ with my computer.It doesn't work.
A．anything wrong B．wrong anything
C．something wrong D．wrong something
指示代词
(B)23.The summer in Dalian is more beautiful than ________ in Jinan.
A．this B．that
C．these D．those
(A)24.—May I speak to Mr.Zhang，please?
—________is Zhang Ming speaking.Who's ________？
A．This；that B．That；he
C．That；that D．This；he
(D)25.—Do you know the girls over there?
—Yes.________ are members from the dance club.
A．This B．That
C．These D．Those
(C)26.—Excuse me.I want to buy some cakes.Where can I find a supermarket?
—I know________ not far from here.You can easily find ________.
A．it；one B．one；one
C．one；it D．it；it
(D)27.Things made by hand are usually more expensive than ________ produced in factories.
A．these B．this
C．that D．those
疑问代词
(C)28.—Wow！How beautiful these flowers are! ________ are they?
—Today is your birthday.They are for you，Lily.
A．What B．Which
C．Whose D．Who
(C)29.—________ is her daughter?
—The girl on the right wearing blue jeans.
A．Who B．What
C．Which D．Whose
(B)30.—________does your father do?
—He is a worker in a factory.
A．Which B．What
C．Who D．Whose
Ⅲ.语篇填空。

根据短文内容，用适当的代词填空。

Mount Heng(衡山) lies in the central part of Hunan Province.__1__ is known as the southern mountain of the Five Great Mountains in China.Although it is only 1300.2 meters high，__2__ is not easy to reach __3__ top.
Yesterday I climbed it with my classmates.Early in the morning，__4__ gathered at the school gate and started to walk towards Mount Heng.It is about seven kilometers from our school to the mountain，but __5__ took us only one hour to get to the foot of the mountain.After a short rest，we began to climb it.
On the half way，we saw a snake by the road and most of __6__ were afraid of
it，but Li Hua and Wang Lin weren't.__7__ even wanted to catch it.Half an hour later，__8__suddenly started to rain，but we went on walking.When we reached the top of the mountain，__9__ was eleven o'clock.__10__ was really great to stand on the top of the mountain.
1．It 2.it 3.its 4.we 5.it 7.They 8.it
9．it 10.It
11。