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南京市2019届高三年级学情调研卷

南京市2019届高三年级学情调研卷
南京市2019届高三年级学情调研卷

(第4题图) 南京市2019届高三年级学情调研

数 学 2018.09

注意事项:

1.本试卷共4页,包括填空题(第1题~第14题)、解答题(第15题~第20题)两部分.本试卷满分为160分,考试时间为120分钟.

2.答题前,请务必将自己的姓名、学校写在答题卡上.试题的答案写在答题卡...上对应题目的答案空格内.考试结束后,交回答题卡. 参考公式:

锥体的体积公式:V =1

3Sh ,其中S 为锥体的底面积,h 为锥体的高.

样本数据x 1,x 2,…,x n 的方差s 2=1n i =1∑n (x i --x )2,其中-x =1n i =1

∑n

x i .

一、填空题:本大题共14小题,每小题5分,共70分.请把答案填写在答.题卡..相应位置....上. 1.已知集合A ={ x |1<x <5,x ∈R },B ={x |x =2n ,n ∈Z },那么集合A ∩B 中有 ▲ 个元素.

2.复数z =(1+b i)(2-i),其中b ∈R ,i 为虚数单位.若z 是

纯虚数,则实数b 的值为 ▲ .

3.已知某地连续5天的最低气温(单位:摄氏度)依次是18, 21,22,24,25,那么这组数据的方差为 ▲ . 4.执行右图所示的算法流程图,则最后输出的S 的值 为 ▲ .

5.若函数f (x )=a +1

2x -1 是奇函数,则实数a 的值为 ▲ . 6.在平面直角坐标系xOy 中,若抛物线y 2=4x 的准线与双曲线

x 2a 2

-y 2

b 2=1(a >0,b >0)的一条渐近线的交点的纵坐标为2,则 该双曲线的离心率是 ▲ .

7.不透明的盒子中有大小、形状和质地都相同的5只球,其中2只白球,3只红球,现从中随机取出2只球,则取出的这2只球颜色相同的概率是 ▲ .

8.已知函数f (x )=2sin(2x +φ) (-π2<φ<π2)的图象关于直线x =π

6

对称,则f (0)的值为 ▲ .

9.如图,在正三棱柱ABC —A 1B 1C 1中,AB =2,AA 1=3,则四棱锥A 1- B 1C 1CB 的体积是 ▲ . 10.在数列{a n }中,已知a 1=1,a n +1=a n +1n (n +1) (n ∈N *),则a 10 的值

为 ▲ .

11.已知△ABC 的面积为315,且AC -AB =2,cos A =-1

4,则 BC 的长

为 ▲ .

12.在菱形ABCD 中,∠ABC =60°, E 为边BC 上一点,且AB →·AE →

=6, AD →·AE →=32

,则AB →·AD →

的值为 ▲ .

13.在平面直角坐标系xOy 中,已知点A (1,1),B (1,-1),点P 为圆(x -4)2+y 2=4上任意一点,记△OAP 和△OBP 的面积分别为S 1和S 2,则

S 1

S 2

的最小值是 ▲ . 14.若函数f (x )=12ax 2-e x +1在x =x 1和x =x 2两处取到极值,且 x 2

x 1

≥2,则实数a 的取值范围

是 ▲ .

二、解答题:本大题共6小题,共90分.请在答题卡指定区域内作答,解答时应写出必要的文

字说明,证明过程或演算步骤. 15.(本小题满分14分)

如图,已知四边形ABCD 是矩形,平面ABCD ⊥平面BCE ,BC =EC ,F 是BE 的中点. (1)求证:DE ∥平面ACF ; (2)求证:平面AFC ⊥平面ABE .

A

B

C

A 1

B 1

C 1

(第9题图)

A

E

D

F

B

C

(第15题图)

16.(本小题满分14分)

已知α,β为钝角,且sin α=35,cos2β=-3

5.

(1)求tan β的值; (2)求cos(2α+β)的值.

17.(本小题满分14分)

销售甲种商品所得利润是P 万元,它与投入资金t 万元的关系有经验公式P =at

t +1,销售乙

种商品所得利润是Q 万元,它与投入资金t 万元的关系有经验公式Q =bt ,其中a ,b 为常数.现将3万元资金全部投入甲、乙两种商品的销售:若全部投入甲种商品,所得利润为 9

4 万元;若全

部投入乙种商品,所得利润为1万元.若将3万元资金中的x 万元投入甲种商品的销售,余下的投入乙种商品的销售,则所得利润总和为f (x )万元. (1)求函数f (x ) 的解析式;

(2)怎样将3万元资金分配给甲、乙两种商品,才能使所得利润总和最大,并求最大值.

18.(本小题满分16分)

在平面直角坐标系xOy 中,椭圆E :x 2a 2+y 2b 2=1(a >b >0)的离心率为2

2,且直线l :x =2被椭

圆E 截得的弦长为2.与坐标轴不垂直的直线交椭圆E 于P ,Q 两点,且PQ 的中点R 在直线l 上.点M (1,0).

(1)求椭圆E 的方程; (2)求证:MR ⊥PQ .

(第18题图)

19.(本小题满分16分)

已知函数f (x )=ln x ,g (x )=x 2.

(1)求过原点(0,0),且与函数f (x )的图象相切的直线l 的方程;

(2)若a >0,求函数φ(x )=|g (x )-2a 2f (x )|在区间[1,+∞) 上的最小值.

20.(本小题满分16分)

如果数列{a n }共有k (k ∈N *,k ≥4)项,且满足条件:

① a 1+a 2+…+a k =0; ② |a 1|+|a 2|+…+|a k |=1,

则称数列{a n }为P (k )数列.

(1)若等比数列{a n }为P (4)数列,求a 1的值; (2)已知m 为给定的正整数,且m ≥2.

① 若公差为正数的等差数列{a n }是P (2m +3)数列,求数列{a n }的公差;

② 若a n =?

??q n -

1

3

,1≤n ≤m ,n ∈N *,m -n 12

,m +1≤n ≤2m ,n ∈N *

,其中q 为常数,q <-1.判断数列{a n }是否

为P (2m )数列,说明理由.

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