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上海中学高一周练05

上海中学高一周练数学卷

2016.10.13

一. 填空题

1. 下列不等式的解为：①2560x x -+< ，②2560x x -++<

2. 写出命题：若2017x y +≠，则2016x ≠或1y ≠的等价命题

3. 已知：11a b -≤+≤，且13a b ≤-≤，则3a b -的取值范围为

4. 不等式20ax bx a ++<(0)ab >的解集是空集，则222a b b +-的取值范围是

5. 不等式20ax bx c ++>的解集是1

(,3)2

-，则不等式20cx bx a ++<的解集为 6. 已知12

a ≥，22()f x a x ax c =-++，对于任意[0,1]x ∈，()1f x ≤恒成立，则实数c 的取值范围是

7. 已知实数,x y 满足2241x y xy ++=，则2x y +的最大值为

8. 若不等式2051x px ≤++≤恰好有一个实数值为解，则p =

9. 若下列三个方程：24430x ax a +-+=，22(1)0x a x a +-+=，2220x ax a +-=中至少有一个方程有实根，则a 的取值范围是

10. 已知,,a b c 为互不相等的整数，则2222

4()()a b c a b c ++-++的最小值为

11. 已知,a b R ∈，关于x 的方程432210x ax x bx ++++=存在一个实根，则22a b +的最小值为

二. 选择题

1. 集合{|41,}A x x k k Z ==+∈，{|42,}B x x k k Z ==+∈，{|43,}C x x k k Z ==+∈ 若a A ∈，b B ∈，c C ∈，则（）

A. abc A ∈

B. abc B ∈

C. abc C ∈

D. abc A B C ? 2. 设a 和b 都是非零实数，则不等式a b >和

11a b

>同时成立的充要条件是（） A. 0a b >> B. 0a b >> C. 0a b >> D. 以上答案均不对 3. 假设n 是不小于3的正整数，n 个给定的实数12,,,n x x x ???具有如下性质：对任意一个二次函数()y f x =，数12(),(),,()n f x f x f x ???中至少有三个数相同，则下列对于12,,,n x x x ??? 的判断中，正确的是（）

A. 至少有三个数是相同的

B. 至少有两个数是相同的

C. 至多有三个数是相同的

D. 至多有两个数是相同的

4. 当一个非空数集F 满足“如果,a b F ∈，则,,a b a b ab F +-∈，且0b ≠时，a F b

∈” 时，我们称F 就是一个数域，以下四个关于数域的命题：① 0是任何数域的元素；② 若数域F 有非零元素，则2016F ∈；③ 集合{|3,}P x x k k Z ==∈是一个数域；④ 有理数集是一个数域；其中真命题有（）个。

A. 0

B. 1

C. 2

D. 3

三. 解答题

1. 解关于x 的不等式[(3)1](1)0m x x +-+>()m R ∈；

2.（1）是否存在实数p ，使得40x p +<是220x x -->成立的充分不必要条件？如果存

在，求出p 的取值范围，如果不存在，说明理由；

（2）是否存在实数p ，使得40x p +<是220x x -->成立的必要不充分条件？如果存在，

求出p 的取值范围，如果不存在，说明理由；

3. 已知集合22{|410813,,}A t t a ab b a b a Z b Z ==++--+∈∈，对于任意的x A ∈， y A ∈，判断元素xy 与集合A 的关系，并证明你的结论；

4. 已知二次函数()y f x =的二次项系数是1，并且一次项系数和常数项都是整数，若(())0f f x =有四个不同的实数根，并且在数轴上四个根成等距排列，试求二次函数()y f x =的解析式，使得其所有项的系数和最小；

参考答案

一. 填空题

1. (2,3)、(,1)(6,)-∞-+∞

2. 若2016x =且1y =，则2017x y +=

3. [1,7]

4. 4[,)5-+∞

5. 1(2,)3-

6. 34c ≤

8. 4p =±

9. 32a ≤-或1a ≥- 10. 2 11. 8

二. 选择题

1. B

2. A

3. B

4. D

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