江苏省2018年普通高校对口单招数学试卷

江苏省2018年普通高校对口单招数学试卷
江苏省2018年普通高校对口单招数学试卷

江苏省2018年普通高校对口单招文化统考

数学试卷

一、单项选择题(本大题共10小题,每小题4分,共40分。在下列每小题中,选出一个正

确答案,将答题卡上对应选项的方框涂满、涂黑)

1.设集合{

}{}5,231+==a N M ,,,若{},3=?N M 则a 的值为 A.-1 B.1 C.3 D.5

2.若实系数一元二次方程02=++n mx x 的一个根为i -1,则另一个根的三角形式为

A 4sin 4cos ππi +

B )

(4

3sin 43cos 2π

πi + C

)(4sin 4cos 2ππi + D [])()(4

-sin 4-cos

πi + 3. 在等差数列{}n a 中,若20163,a a 是方程0201822=--x x 的两根,则2018133a a ?的值为

A

31

B 1

C 3

D 9 4. 已知命题p:()()102131101=和命题q:11=?A (A 为逻辑变量),则下列命题中为真命题 的是

A p ?

B q p ∧

C q p ∨

D q p ∧?

5. 用1,2,3,4,5这五个数字,可以组成没有重复数字的三位偶数的个数是

A 18

B 24

C 36 D48

6. 在长方体1111D C B A ABCD -中2==BC AB ,621=AA ,则对角线1BD 与底面ABCD 所成角是

A

6π B 4π C 3π D 2

π

7. 下图为某项工程的网络图。若最短总工期是13天,则图中x 的最大值为

A. 1

B.2

C.3

D.4

8. 若过点P (-1,3)和点Q(1,7)的直线1l 与直线2l :05)73(=+-+y m mx 平行,则m 的值为

A .2 B.4 C.6 D.8

9.设向量)(52,2cos θ=→

a ,)(6,4=→

b ,若5

3

)sin(=-θπ,则→→-b a 25的值为

A.5

3

B.3

C.4

D.5

10.若函数c bx x x f +-=2)(满足),1()1(x f x f -=+且,5)0(=f 则)(x b f 与)c (x f 的大小关系是

A.)()(x x c f b f ≤

B.)()(x x c f b f ≥

C.4)()(x x c f b f <

D.)()(x x c f b f >

二、填空题(本大题5小题,每小题4分,共20分)

11.设数组)4,2,1(-=a ,)2,,3(-=m b ,若1=?b a ,则实数___________m =。

12.若32sin -=θ,??

?

??∈2

3ππθ,

,则_______tan =θ。 13.题13图是一个程序框图,执行该程序框图,则输出的m 的值是____________。

14.若双曲线)0,0(122

2

2>>=-b a b y a x 的一条渐近线把圆???+=+=θ

θsin 32cos 31y x (θ为参数)

分成面积相等的两部分,则该双曲线的离心率是_____________.

15.函数?????>+--≤=2

,942

,)(2x a x x x x x f ,若关于x 的方程()1=x f 存在三个不相等的

实根,则函数解析式中a 的取值范围_________.

三、解答题(本大题共8小题,共90分) 16.(8分)。满足不等式设实数23<-a a

37

log 3

log 211

2a x a x a >+的不等式)解关于(的取值范围;)求(

17. (10分)已知)(x f 为R 上的奇函数,又函数)且(1011)(2

≠>+=-a a a x g x 恒过定点A 。

(1)、求点A 的坐标;

(2)、的值;点,求实数也过若函数时,当m A x f mx x x f x )(.)(02

+-=<

(3)、.)2

7

(,32)(10),()2(的值求时,且若f x x f x x f x f -=<<=-

18.(14分)已知各项均为正数的数列{}n a 满足。*

1222,log log 1,6N n a a a n n ∈=+=+

(1)、{}n a 求数列的通项公式及前n 项和n S ;

(2)、若)(9

log *2

2N n a b n

n ∈=,求数列{}n b 的前n 项和n T

19.(12分)某校从初三年级体育加试百米测试成绩中抽取100个样本,所有样本成绩全部在

11秒到19秒之间。现将样本成绩按如下方式分成四组:第一组[)1311,,第二组[)1513, , 第三组[)1715,

,第四组[)1917,,图是根据上述分组得到的频率分布直方图。 (1)若成绩小于13秒被认定为优秀,求该样本在这次百米测试中成绩优秀的人数; (2)试估算本次测试的平均成绩;

(3)若第四组恰有3名男生,现从该组随机抽取3名学生,求所抽取的学生中至多有一名

女生的概率。

20. (12分)已知正弦型函数),sin()(?ω+=x H x f 其中常数2

0,0,0π

?ω<

<>>H 。若函

数的一个最高点与其相邻的最低点的坐标分别是

),),(

,(3-12

7312

π

π

。 (1)求)(x f 的解析式;

(2)求)(x f 的单调增区间;

(3)在ABC ?中,A 为锐角,且0)(=A f 。若AB=3,BC=33,求ABC ?的面积S 。

21. (10分)某学校计划购买个x 篮球和y 个足球。

(1)若x ,y 满足的约束条件,7252??

?

??≤≤-≥-x y x y x 问该校计划购买这两种球的总数最多是多少

个?

(2) 若x ,y 满足的约束条件,7252??

?

??≤≤-≥-x y x y x 已知每个篮球100元,每个足球70元,求该

校最少要投入多少元?

22.(10分)某辆汽车以[]()120,60/∈x x 小时千米的速度在高速公路上匀速行驶,每小时的耗油量为??

?

??+

-x k x 360051升,其中k 为常数。若该汽车以120米/小时的速度速度匀速行驶时,每小时的耗油量是12升。 (1)求常数k 的值;

(2)欲使每小时的耗油量不超过8(升),求x 的取值范围;

(3)求该汽车匀速行驶100千米的耗油量y (升)的最小值和此时的速度。

23.(14分)已知椭圆C :22x +3

2y =1和直线l :y=x+m,直线l 与椭圆C 交于A,B 两点。

(1)求椭圆C 的准线方程;

(2)求ABO ?(O 为坐标原点)面积S 的最大值;

(3)如果椭圆C 上存在两个不同的点关于直线l 对称,求 m 的取值范围。

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