上海市交大附中2020届2020届高三数学期中试卷

交大附中高三期中数学试卷

2020.05

一. 填空题

1. 计算矩阵的乘积：()300c a b ??= ???

2. 计算：012393n n n

n n n C C C C ++++=L 3. 已知23sin cos 223

θ

θ

+=，则sin θ= 4. 若双曲线22

14x y m

-=的焦距为6，则该双曲线的虚轴长为 5. 在首项为21，公比为12

的等比数列中，最接近于1的项是第 项 6. 如图，二面角l αβ--的大小是3π，线段AB α，B l ∈，AB 与l 所成的角为6π，则 AB 与平面β所成的角是 （用反三角函数表示）

7. 已知a 、b 、c 分别为△ABC 三个内角A 、B 、C 的

对边，2a =，且(2)(sin sin )b A B +-=()sin c b C -，则

△ABC 面积的最大值为

8. 已知函数()lg(1)f x x =+，()g x 是以2为周期的偶函数，且当01x ≤≤时，有()g x = ()f x ，则函数()y g x = ([1,2]x ∈)的反函数是y =

9. 已知()y f x =是定义在R 上的函数，方程(2019)(2020-)0f x f x +?=恰好有7个解， 则这7个解的和为

10. 设0.ab ??

是一个循环节长度为两位的循环纯小数，其中a 和b 分别为10以内的非负整数， 且a b ≠，0b ≠，若集合??1{|0.,}A n ab n n *==∈N ，则A 中所有元素的和为 11. 已知数列{}n a 满足1312

n n n n n a a a a a ++??=???为奇数为偶数（*n ∈N ），127k a =?（k 是一个已知 的正整数），若存在*m ∈N ，当n m >且n a 为奇数时，n a 恒为常数p ，则p =

12. 若实数x 、y 满足222

(1)(1)22cos (1)1x y xy x y x y ++--+-=-+，则xy 的最小值为

二. 选择题

13. 已知函数()y f x =是R 上的增函数，则对任意12,x x ∈R ，“12x x <”是 “12()()f x f x <”的（ ）条件

A. 充分非必要

B. 必要非充分

C. 充分必要

D. 非充分非必要

14. 已知11z ≠-，111i 1z b z -=+（

b

∈R ），2141(+1)z z =-，则z 对应的点在（ ） A. 圆上 B. 抛物线上 C. 双曲线上 D. 椭圆上

15. 在平面直角坐标系中，O 是坐标原点，两定点A 、B 满足||||2OA OB OA OB ==?=uu r uu u r ，

则点集{|,||||1,,}P OP OA OB λμλμλμ==++≤∈R uu u r uu r uu u r 所表示的区域的面积是（ ）

A. 22

B. 23

C. 42

D. 43

16. 已知1234,,,{1,2,3,4}a a a a ∈，1234(,,,)N a a a a 为1234,,,a a a a 中不同数字的种类，如(1,1,2,3)3N =，(1,2,2,1)2N =，求所有的256个1234(,,,)a a a a 的排列所得1234(,,,)N a a a a 的平均值为（ ）

A.

8732 B. 114 C. 17764 D. 17564

三. 解答题

17. 如图所示，用一个半径为10厘米的半圆纸片卷成一个最大的无底圆锥，放在水平桌面上，被一阵风吹倒．

（1）求该圆锥的表面积S 和体积V ；

（2）求该圆锥被吹倒后，其最高点到桌面的距离d .

18. 已知函数()sin()f x A x b ω?=++（0A >，0ω>，||2π?<

）的图像如下图所示. （1）求出函数()f x 的解析式；

（2）若将函数()f x 的图像向右移动3π个单位长度再把所有点的横坐标变为原来的14（纵 坐标不变）得到函数()y g x =的图像，求出函数()y g x =的单调递增区间及对称中心.

19. 若函数()y f x =满足“存在正数λ，使得对定义域内的每一个值1x ，在其定义域内都存在2x ，使12()()f x f x λ=成立”，则称该函数为“依附函数”.

（1）分别判断函数①()2x f x =，②2()log g x x =是否为“依附函数”，并说明理由；

（2）若函数()y h x =的值域为[,]m n ，求证：“()y h x =是‘依附函数’”的充要条件是 “0[,]m n ?”.

20. 如图，已知点P 是x 轴下方（不含x 轴）一点，抛物线2:C y x =上存在不同的两点A 、

B 满足PD DA λ=uu u r uu u r ，PE EB λ=uur uu r ，其中λ为常数，且D 、E 两点均在

C 上，弦AB 的中点为M .

（1）若P 点坐标为(1,2)-，3λ=时，求弦AB 所在的直线方程；

（2）在（1）的条件下，如果过A 点的直线1l 与抛物线C 只有一个交点，过B 点的直线2l 与抛物线C 也只有一个交点，求证：若1l 和2l 的斜率都存在，则1l 与2l 的交点N 在直线PM 上；

（3）若直线PM 交抛物线C 于点Q ，求证：线段PQ 与QM 的比为定值，并求出该定值.

21. 设{}n a 是公差不为零的等差数列，满足6713a a a +=，2224967a a a a +=+，设正项数列

{}n b 的前n 项和为n S ，且423n n S b +=.

（1）求数列{}n a 和{}n b 的通项公式；

（2）在1b 和2b 之间插入1个数11x ，使1b 、11x 、2b 成等差数列；在2b 和3b 之间插入2个数 21x 、22x ，使2b 、21x 、22x 、3b 成等差数列；???；在n b 和1n b +之间插入n 个数1n x 、2n x 、???、nn x ，使n b 、1n x 、2n x 、???、nn x 、1n b +成等差数列. ① 求11212212n n n nn T x x x x x x =+++++++L L ；

② 对于①中的n T ，是否存在正整数m 、n ，使得12m n m a T a +=成立？若存在，求出所有的 正整数对(,)m n ；若不存在，请说明理由.

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