1 . 等差数列
的前
项和为
,
(
且
),
.
(1)求
的通项公式与前
项和
;
(2)记
,当
,
时,试比较
与
的大小;
(3)若
,正项等比数列
中,首项
,数列
是公比为4的等比数列
,且
,求
的通项公式与
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20849c00c47cbdc43f18d53341b6c4e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec3c9f09e932ad3cc1f3720d93b34c56.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1dcf4e0dd72f254d9eab961b892ae9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15520cf5be7c2685975aac51bc99ac4f.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385275d29d8c8a7841eaeaa3dfab2cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7afef6271af7462ffa935a1846e3ec90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89ba85f74cda4ddd621278e558bc036f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ee3a47bf7dfe538ae367cc006a69b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ea2efac902bfb513a224c40faef2516.png)
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2 . 甲、乙两个人独立地破译一个密码,他们能译出密码的概率分别为
和
,求:
(1)两个人都译出密码的概率;
(2)恰有1个人译出密码的概率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
(1)两个人都译出密码的概率;
(2)恰有1个人译出密码的概率.
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3 . 如图,四棱锥
内,
平面
,四边形
为正方形,
,
.过
的直线
交平面
于正方形
内的点
,且满足平面
平面
.
的轨迹长度;
(2)当点
到面
的距离为
时,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c45fbffb9e2c7fa7c5006cde8da0cabe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbd051fedb6691e2183e658f1fe487ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/392469b357b12b998528499929366c02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a03203dd5ac79dd8c6707e4340773359.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6ebfed81c159368d135f211b9860f01.png)
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4 . 为丰富学生的课外活动,学校羽毛球社团举行羽毛球团体赛,赛制采取5局3胜制,即某队先赢得3局比赛,则比赛结束且该队获胜,每局都是单打模式,每队有5名队员,比赛中每个队员至多上场一次目上场顺序是随机的,每局比赛结果互不影响,经过小组赛后,最终甲乙两队进入最后的决赛,根据前期比赛的数据统计,甲队明星队员M对乙队的每名队员的胜率均为
,甲队其余4名队员对乙队每名队员的胜率均为
.(注:比赛结果没有平局)
(1)若求甲队明星队员M在前三局比赛中出场,记前三局比赛中,甲队获胜局数为X,求随机变量X的分布列及数学期望;
(2)已知甲乙两队比赛3局,若甲队以
获得最终胜利,求甲队明星队员M上场的概率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
(1)若求甲队明星队员M在前三局比赛中出场,记前三局比赛中,甲队获胜局数为X,求随机变量X的分布列及数学期望;
(2)已知甲乙两队比赛3局,若甲队以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ef414095084c4c5eb3be5b73e719b44.png)
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解题方法
5 . 已知四边形
内接于
,若![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1e0651a3b5438579eb8282dcbab0502.png)
的半径长.
(2)若
,求
面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1e0651a3b5438579eb8282dcbab0502.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aef77977ede96cb28287eb995cd7e0ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e04730c124543b0318ad074676d43ab.png)
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6 . 定义在
上的函数
,若对任意
,存在常数
,都有
成立,则称
是
上的有界函数,其中
称为函数
的上界.已知函数
.
(1)若
是奇函数,判断函数
是否为有界函数,并说明理由;
(2)若
在
上是以
为上界的函数,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aea232de27d21a2646fd4520ea0726bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a45e285f154675b459b2247f3682fb.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2d2a706da87c1775d9e89799e45b4df.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a2ec965488c7e1cea085463c7731285.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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7 . 用篱笆在一块靠墙的空地围一个面积为
的等腰梯形菜园,如图所示,用墙的一部分做下底
,用篱笆做两腰及上底,且腰与墙成
,当等腰梯形的腰长为多少时,所用篱笆的长度最小?并求出所用篱笆长度的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c8fbf64cd29a08ae703e9fa2035f3c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
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解题方法
8 . 已知函数
.
(1)若
,求不等式
的解集;
(2)若
时,
的最小值为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b23fabe4b8c9810a3e043dfc27ffc7.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a81e971f501d78f5560c0c3d42f0f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f1a722a2595fdc71462654b764ee49d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24aa16b780156e18f12baa2b8ee0f9a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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解题方法
9 . 已知函数
.
(1)求不等式
的解集;
(2)若
,不等式
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/276246854dd33691f6798db87faeea0b.png)
(1)求不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d37dafb12565279285111a5948d835b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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10 . 解答以下两个小题:
(1)求
的值;
(2)若
,求
的值.
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6306f753fe8458efe5cf9887b64c4422.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a8c9f990fc696c4b27d02cb9e92fe9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46c1f68a5bc3695f488c3c9249ac77ea.png)
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