江苏省连云港市2020-2021学年高二上学期期中考试数学试题及答案

x
轴垂线交椭圆于
P, 若
∠F1PF2 = 60° , 则该椭圆的离心率是
1
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A. 3
B. 3 2
C. 1 2
D. 3 3
8.数学著作《孙子算经》中有这样一个问题:“今有物不知其数,三三数之剩二(除以 3 余 2),五五数之剩三(除
以 5 余 3),问物几何?”现将 1 到 2020 共 2020 个整数中,同时满足“三三数之剩二,五五数之剩三”的数按
f (x) 2x2 4x k 2(x 1)2 k 2
g(x) x2 2x (x 1)2 1 x [ 3,3]
f (x)
f ( 1) , g(x)
g( 3) ················································ 10
min
max
f 1 g( 3)
2 ( 1)2 4 ( 1) k ( 3)2 2 ( 3) k 17 ·················································································· 12
18. (本小题满分 12 分)
已知等比数列{an} 中, a1 = 1, 且 2a2 是 a3 和 4a1 的等差中项. (1)求数列{an} 的通项公式; (2)若数列{bn}满足 bn = 2n + an2 (n ∈ N* ), 求{bn}的前 n 项和 Sn .
19. (本小题满分 12 分)
列:1,1,2,3,5,8,13,21,34,… 满足 an+2 = an+1 + an (n ≥ 1), 那么1+ a2 + a4 + a6 +L + a2020 ＝
A. a2021
B. a2022
C. a2023
D. a2024
5.焦点为(0,2)的抛物线标准方程是
A. x2 = 8y
B. x2 = 4 y
18 1
12
an
q
a1 1
a2 a1q q a3 a1q2 q2 ·······················································2
2a2 a3 4a1
4a2 a3 4a1 4q q2 4
q 2 . ·································4
(2)求证:
为定值.
| DF |
21. (本小题满分 10 分)
在 ①a1, a2 , a3 成等差数列, ②an+2 + an+1 + an−1 + an−2 = 4an (n ≥ 3) ，③ an+3 − an = 6(n ≥ 1) 这三个条件中任
选一个,补充到下面问题中.
问题:已知在数列 {an} 中,满足 an+1 − an−1 = 4(n ≥ 2), 且____________,若数列 {an} 等差数列,请证明;若数列 {an} 不是等差数列,请举例说明.
(1)求椭圆的方程; (2)试探究直线 AM 与直线 BN 的交点 P 是否落在某条定直线上?若是,请求出该定直线的方程;若不是,请说 明理由.
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2020 2021
8
5
40
1D 2C 3B 4A 5A 6D
4
5
20
5
3
0
9 AC 10 AB 11 ACD 12 BCD
4
5
20
7D
2
AF 2FB (1 x1, y1) 2(x2 1, y2) y1 2y2
y1 y2 4t, y1 2 y2 ,
y1 8t, y2 4t,
y1 y2 4
1
y1 0
t0
t 22
AB
2 2x y 2 2 0 ·············································6
2 AB
从小到大的顺序排成一列, 构成数列{an}, 则该数列共有
A.132 项
B.133 项
C.134 项
D.135 项
二､多项选择题:本题共 4 小题,每小题 5 分,共 20 分.在每小题给出的选项中,有多项符合题目要求.全部选对
的得 5 分,部分选对的得 3 分,有选错的得 0 分.
9.若 a>b>0,则
a 2b b 8 2b 9 2(b 1) 3 3 6 2 ····························· 10
b1
b1
9 2(b 1) b1
b 1 32 2
a 2b
3 6 2 ······························································· 12
16.若干个正整数之和等于 10,这些正整数乘积的最大值为____.
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旗开得胜 四､解答题:本大题共 70 分.请在答案卡指定区域内作答,解答时应写出必要的文字说明､证明过程或演算步骤. 17. (本小题满分 12 分) 已知实数 a>0,b>0 且 a+b+8=ab. (1)求 ab 的最小值 ; (2)求 a+ 2b 的最小值.
21
10
an 1 an 1 4 (n 2)
a2n 1
a2n
4
·········································· 2
a2n 1 a1 4(n 1) 4n a1 4 a2n a2 4(n 1) 4n a2 4 ·····························································4
C. y2 = 4x
D. y2 = 8x
6.已知数列{an} 中, a1 = 1,
a2
=
2,
对
∀n
∈
N
*
都有
2
a3 n +1
=
a3 n+2
+
an3 , 则 a10 等于
A.10
B. 3 10
C.64
D. 4
7. 已 知 椭 圆
x2 a2
+
y2 b2
= 1(a > b > 0) 的 左 ､ 右 焦 点 分 别 为 F1, F2 , 过 F1 作
注:如果选择多个条件分别解答,按第一个解答计分.
22. (本小题满分 12 分)
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如图,在平面直角坐标系
xOy
中 ,A,
B
x2 是椭圆 a2
+
y2 b2
= 1(a > b > 0) 的左､右顶点, AB = 2
2, 离心率
e = 2 . F 是右焦点,过 F 点任作直线 l 交椭圆于 M,N 两点. 2
A. ac2 ≥ bc2
B. a2 < ab < b2
C. 2ab ab a+b
D. 1 > 1 ab
10.下列命题正确的是
A.∃x∈R, log2 x = −1 C. ∀x ∈ N , x3 > x2
B. x=1 是 x2 = 1的充分不必要条件 D.若 a>b,则 a2 > b2
11. 下列有关双曲线 2x2 − y2 = 8 的性质说法正确的是
x3 k 27
(x2 6x)max 27 ··············································································· 5
2
x1 x2 [ 3, 3]
f (x1 ) g(x2 )
[ f (x)]min [g(x)]max ·····································································8
已知函数 f (x) = 2x2 + 4x − k, g(x) = x2 − 2x
(1)若对任意 x∈[-3,3],都有 f(x)≤g(x)成立,求实数 k 的取值范围;
(2)若存在 x1, x2 ∈[−3, 3], 使 f (x1) ≤ g(x2 ) 成立,求实数 k 的取值范围.
20. (本小题满分 12 分)
N xN ,yN
yN
=
y1 +y2 2
=2t,
xN
=2t 2
1
N (2t2
1, 2t )
AB
l : y 2t t(x 2t2 1) D(2t2 3,0)
DF 2t2 3 1 2t2 2 ·································································9
A. ∀x ∈ R, x2 + 3ax +1 ≤ 0
B.∃x∈∴ R, x2 + 3ax +1 < 0
C. ∃x ∈ R, x2 + 3ax +1 > 0
D. ∃x ∈ R, x2 + 3ax +1 ≤ 0
2.双曲线 y2 − x2 = 1的渐近线方程是 4
A. y=4x
B. y = ± 1 x 4
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如图,过抛物线 y2 = 4x 的焦点 F 任作直线 l,与抛物线交于 A,B 两点,AB 与 x 轴不垂直,且点 A 位于 x 轴上方.
AB 的垂直平分线与 x 轴交于 D 点.
uuur uuur (1)若 AF = 2FB, 求 AB 所在的直线方程;
| AB |
AB = (x2 x1)2 ( y2 y1)2 = (ty2 ty1)2 ( y2 y1)2
1 t2 ( y1 y2 )2 4 y1y2 4t2 4
1 t2 16t2 16
AB DF
4t 2 2t 2
4 2
2(
································································· 12
1
an a1qn 1 1 2n 1 2n 1 ························································6
2 bn 2n an2 2n 4n 1
Sn b1 b2 b3
bn 2 40 4 41 6 42
2n 4n 1
246
2n 1 4 42
4n 1 n 2 2n 1 4n n2 n 4n 1
2
14
3
··································································································· 12
19
12
1
g(x) f (x) 0
x [ 3,3]
k x2 6x
x [ 3,3]
k (x2 6x)max , x 3,3
2020~2021 学年第一学期期中考试
旗开得胜
高二数学试题
用时: 120 分钟满分: 150 分
一､单项选择题: 共 8 小题,每小题 5 分,共 40 分.在每小题给出的四个选项中,只有一项是符合题目要求的,请
将正确选项前的字母代号填涂在答题卡相应位置上.
1.命题 ∀x ∈ R, x2 + 3ax +1 > 0 的否定是
13.已知 x>1,则 x + 1 + 3 的最小值是____. x −1
14.已知椭圆
x2 a2
+
y2 b2
= 1(a
>b
> 0) 过点 (1,
2 ), 其长轴长的取值范围是[4,6],则该椭圆离心率的取值范围
是____.
15. 等差数列{an} 的前 n 项和为 Sn , 公差为 d,满足 a1 = 3, ak = 9, k < d (k ∈ N *), 则 Sn = _____.
A.离心率为 3
B.顶点坐标为(0,±2) C.实轴长为 4
D.虚轴长为 4 2
12.已知数列{an} 是等差数列,前 n 项和为 Sn , 且 2a1 + 2a3 = S5 , 下列结论中正确的是
A. S7 最小
B. s13 = 0
C. S4 = S9
D. a7 = 0
三､填空题:共 4 小题,每小题 5 分,共 20 分.请把答案直接填写在答题卡相应位置上.
3.设 a∈R,则“ a2 > a ”是“a>1"的
C.y=±2x
D. y = ± 1 x 2
A.充分不必要条件
B.必要不充分条件
C. 充要条件
D.既不充分也不必要条件
4. 公 元 13 世 纪 意 大 利 数 学 家 斐 波 那 契 在 自 己 的 著 作 《 算 盘 书 》 中 记 载 着 这 样 一 个 数
20
12
1
l
0 F(1,0)
l : x ty 1
A(x1, y1), B(x2, y2 ). A x
y1 0, y2 0
x ty 1, y2 4ty 4 0
y2 4x,
y1 y2 4t, y1y2 4 ·····································································3
13 6
14
3, 3
32
15 3n2
16 36
6
70
17
12
1 a b 8 ab 8 + 2 ab
8D
ab 2 ab 8 0 ( ab 4)( ab 2) 0
ab 4 ab
16 a b 4
····················· 6
2 a b 8 ab
a b8 b1
a 0 b 0 b 1 ·····························8