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江苏省对口单招数学试卷

江苏省对口单招数学试卷
江苏省对口单招数学试卷

2017年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.已知集合M ={0,1,2},N ={2,3},则M ∪N 等于 ( )

A.{2}

B.{0,3}

C.{0,1,3}

D.{0,1,2,3}

2.已知数组a =(1,3,-2),b =(2,1,0),则a -2b 等于 ( )

A.(-3,1,-2)

B.(5,5,-2)

C.(3,-1,2)

D.(-5,-5,2) 3.若复数z =5-12i ,则z 的共轭复数的模等于 ( )

4.下列逻辑运算不.正确的是

( )

+B=B+A

+A B —

=A

C.0—

·0—

=0

+A =1

5.过抛物线y 2

=8x 的焦点,且与直线4x -7y +2=0垂直的直线方程为 +4y -44=0 +4y -14=0

=0

=0

6.“a =

4

”是“角α的终边过点(2,2)”的

A.充分不必要条件

B.必要不充分条件

C.充分必要条件

D.既不充分也不必要条件

7.若一个底面边长为23,高为2的正四棱锥的体积与一个正方体的体积相等,则该正方体的棱长为

8.将一枚骰子先后抛掷两次,所得的点数分别为m ,n ,则点(m ,n

(θ是参数)上的概率为 A.36

1 B.

1 C.

12

1 D.

6

1

9.已知函数f (x 是奇函数,则g (-2)的值为

10.设m >0,n >0,且4是2m

与8n

的等比中项,则m 3+n

4

的最小值为 3

B.

4

17 3

D.

4

27

二、填空题(本大题5小题,每小题4分,共20分)

11.题11图是一个程序框图,若输入x 的值为3,则输出的k 值是

.

12.题12图是某工程的网络图(单位:天),若总工期为27天,则工序F 所需的工时x (天)的取值范围为 .

13.设向量a =(cosα,sinα),b =(2,1),α∈ -

2π,2

π

,若a·b =1,则cos α等于 . 14.已知函数f (x )是R 上的奇函数,且f (x +4)=f (x ),当a <x ≤2时,f (x )=log 2(x +1),则f(11)等于 .

15.设实数x,y 满足(x -1)2

+y 2

=1,则

1

+x y

的最大值为 . 三、解答题(本大题共8小题,共90分)

16.(8分)已知复数z =(m 2

-2m -8)+(log 2m -1)i 所表示的点在第二象限,求实数m 的取值范围.

17.(10分)设函数f (x )=3x -m ·3-x

,m 是实数. (1)若f(x )是R 上的偶函数. ①求m 的值;

②设g (x )=)

(3x f x

,求证:g (x )+g (-x )=1;

(2)若关于x 的不等式f (x )≥6在R 上恒成立,求m 的取值范围.

18.(12分)已知函数f (x )=3sin x cos x -

2

1

cos2x , (1)求f (x )的最小正周期;

(2)在△ABC 中,三个角A ,B ,C 所对的边分别为a,b,c ,若f (A )=1,c =2a ·cos B 、b =6,求△ABC 的面积.

19.(12分)为了弘扬传统文化,某校举办了诗词大赛.现将抽取的200名学生的成绩从低到高依次分成六组:[40,50),[50,60),[60,70),[70,80),[80,90),[90,100),得到频率分布直方图(题19图).解答下列问题:

(1)求a的值;

(2)若采用分层抽样的方法从6个小组中随机抽取40人,则应从第1组和第2组各抽取多少人?

(3)从成绩不低于80分的学生中随机抽取2人,求所抽取的2名学生至少有1人来自第5组的概率.

题10图

20.(14分)已知{a n}是公差为2的等差数列,其前n项和S n=pn2+n.

(1)求首项a1,实数p及数列{a n}的通项公式;

(2)在等比数列{b n}中,b2=a1,b3=a2,若{b n}的前n项和为T n,求证:{T n+1}是等比数列.

21.(10分)某企业生产甲、乙两种产品,已知生产每吨甲产品需投资5万元,且要用A原料2吨,B原料3吨,生产每吨乙产品需投资3万元,且要用A原料1吨,B原料2吨,每吨甲产品售价14万元,每吨乙产品售价8万元.该企业在一个生产周期内,投资不超过34万元,消耗A原料不超过13吨,B原料不超过22吨,且生产的产品均可售出.问:在一个生产周期内生产甲、乙产品各多少吨时可获得最大利润,最大利润是多少?

22.(10分)某经销商计划销售某新型产品,经过市场调研发现,当每吨的利润为x (单位:千元,x >0)时,销售量q (x )(单位:吨)与x 的关系满足以下规律:若x 不超过4时,则

q(x)=

1

120

x ;若x 大于或等于12时,则销售量为零;当4≤x ≤12时,q (x )=a-bx(a,b 为常数).

(1)求a ,b ;

(2)求函数q (x )的表达式;

(3)当x 为多少时,总利润L (x )取得最大值,并求出该最大值.

23.(14分)已知椭圆E :22a x +22b

y =1的右焦点是圆C :(x -2)2+y 2

=9的圆心,且右准线方程为

x =4.

(1)求椭圆E 的标准方程;

(2)求以椭圆E 的左焦点为圆心,且与圆C 相切的圆的方程;

(3)设P 为椭圆E 的上顶点,过点M 0,-32

的任意直线(除y 轴)与椭圆E 交于A ,B 两点,求证:PA ⊥PB .

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