名校
1 . 已知函数
.
(1)若函数
在定义域上单调递增,求实数
的取值范围;
(2)讨论函数
的单调性.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bf1fd9e0bd3364d3d5ce2e16333d66c.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68072473a5106f93e3026d992859f7a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0980585997fdec11fc3ee06bdaaa3d06.png)
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2 . 在二项式
的展开式中,第5项和第6项的二项式系数相同,
(1)求所有偶数项的二项式系数的和;
(2)求各项系数绝对值之和.
(3)若记
,求展开式中
中取最大项时
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e68fb64e252a20d360cf890d9603a181.png)
(1)求所有偶数项的二项式系数的和;
(2)求各项系数绝对值之和.
(3)若记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f826552f5b0546132e7c0b72bfd4b8dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/857641f5cae585bd4655193270a7dae7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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3 . 如图,在四面体
中,
平面
,
,
,点
在线段
上.
是线段
中点时,求
与平面
所成角的正弦值;
(2)若二面角
的余弦值为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c4559d27e3905980d1a4f1856f07de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba924d427bf2df79144e2611f50ad00f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f571a1aac46c6d0cf440c0ec2846bf9.png)
(2)若二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad6a0cee8226e82cc57916e10d533369.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d8c176e0c2b435675a3b8efab0e498d.png)
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名校
解题方法
4 . 为普及安全知识,某单位举办了一场安全知识竞赛,经过初赛、复赛,有甲、乙两个代表队(每队三人)进入决赛,决赛规则如下:共进行三轮比赛,每轮比赛中每人各答一题,每答对一题得 10 分,答错不得分. 假设甲队每人答题正确的概率均为
,乙队三人答题正确的概率分别
.
(1)若决赛中三轮总得分大于70分就能获得特别奖,求乙队获得特别奖的概率;
(2)因两队在决赛中得分相同,现进行附加赛. 规则如下:甲,乙两队抽签决定谁先答题,每队每人各答题一次为一轮,有两人及以上答对就算成功答题,并继续下一轮答题,否则换另一队答题,连续两轮成功答题的队伍获胜,比赛结束. 求附加赛中甲队恰好在第5轮结束时获胜的概率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3783f11b078c7b7262f28e2e90bf334.png)
(1)若决赛中三轮总得分大于70分就能获得特别奖,求乙队获得特别奖的概率;
(2)因两队在决赛中得分相同,现进行附加赛. 规则如下:甲,乙两队抽签决定谁先答题,每队每人各答题一次为一轮,有两人及以上答对就算成功答题,并继续下一轮答题,否则换另一队答题,连续两轮成功答题的队伍获胜,比赛结束. 求附加赛中甲队恰好在第5轮结束时获胜的概率.
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5 . 如图,在四棱锥
中,
平面
,
,且
,
是
的中点.
;
(2)若
,直线
与直线
所成角的余弦值为
.
(ⅰ)求直线
与平面
所成角;
(ⅱ)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c753cb1eb73fd8d136d00462970797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2ffc6952e988d04f22f0fb2f7f0ab7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae1e04eeb4de72e5750dae77bcb6f88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25eb757d05fbff80d50c3bb8dbcb8657.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ce343ec5b0aa9ce4892fa682c614ba6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6164f6484b3b4acafcf1f3fd87ef196.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d97dc3b752832906de41447bb58a341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa42621cd6793e7f3673fdb49bc3123.png)
(ⅰ)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
(ⅱ)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c76abad7103e74e5613a802475f1c0f9.png)
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6 . 求值:
(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9574e1c06fd2dc9ccbae5d137fd0af5b.png)
(2)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dac4d516a6f25f72b7b5ec791fee421b.png)
(3)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9574e1c06fd2dc9ccbae5d137fd0af5b.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dac4d516a6f25f72b7b5ec791fee421b.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d1762dbbbf7d267b7332fe2706e754c.png)
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解题方法
7 . 如图在
中,
,
,设
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb95d089784702a0b6d459f18a4e1e72.png)
表示向量
.
(2)若
,
,
,求
.
(3)若
,
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41a84ab6721f8f7f510b54ae9ba5ef81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c4464d66a6064905e521a946843ceff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e923e4cdcbea6a029f5ba188a59229d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb95d089784702a0b6d459f18a4e1e72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b172cf8d898883d82e973f28c3c3a3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0531c57fe21c3c5ada307af8d49f3b0.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c278812c4813f6391c7993574e2f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b59ab6bf6111dda4a3d428454768b77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f2a8d0c77c54fdf9aa97978f8e5dc0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b7ab10c3b9021757e9a89ab47807484.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b47a08a25693bbfa01026573625ad15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d09a88dc7dc9cd668a57138e1ec71ea2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e811dee23eb88d9f05fbdc9fbabf77b.png)
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8 . 已知复数
在复平面内对应的点在虚轴的正半轴上,复数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0db5c1993bdf3e6ce4810533d62c90de.png)
(1)求
,
的值;
(2)求
,
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fa925e867d123a41814d16e77e9b59e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0db5c1993bdf3e6ce4810533d62c90de.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63fe57d4fbae536de2e641d9d349fcf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a17ca1e9283fc7ac1b1e427ef0310fc2.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c529ba8284ac452d88405f8dfc3b316.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8f798a9af75a091a8be0b71f2038260.png)
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9 . 已知
,
与
的夹角为
,
为
外接圆上一点,
与线段
交于点
.
,求
;
(2)设
.
(ⅰ)试用
的函数表示
;
(ⅱ)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89d64255de7162b1fa5f659732693daa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f605ec0729ce6d72237ad662a06862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc9656d8286c4d6fa309d6ae347c89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2b83beedb3438153e6f728545fe3e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a41a475407451ed8427d0e7a483568c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eff576db012cc8a793b1afcb0b47fbe.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f7983aade7a7c61f065c5bd3569fbd2.png)
(ⅰ)试用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72f14d13e081ed59bab413a9f30b62e3.png)
(ⅱ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5ef6d007c9539c08562fcc217ae131e.png)
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10 . 跑步是人们日常生活中常见的一种锻炼方式,其可以提高人体呼吸系统和心血管系统机能,抑制人体癌细胞生长和繁殖.为了解人们是否喜欢跑步,某调查机构在一小区随机抽取了40人进行调查,统计结果如下表.
(1)根据以上数据,判断能否有95%的把握认为人们对跑步的喜欢情况与性别有关?
(2)该小区居民张先生每天跑步或开车上班,据以往经验,张先生跑步上班准时到公司的概率为
,张先生跑步上班迟到的概率为
.对于下周(周一~周五)上班方式张先生作出如下安排:周一跑步上班,从周二开始,若前一天准时到公司,当天就继续跑步上班,否则,当天就开车上班,且因公司安排,周五开车去公司(无论周四是否准时到达公司).设从周一开始到张先生第一次开车去上班前跑步上班的天数为
,求
的概率分布及数学期望
.
附:
,其中
.
喜欢 | 不喜欢 | 合计 | |
男 | 12 | 8 | 20 |
女 | 10 | 10 | 20 |
合计 | 22 | 18 | 40 |
(2)该小区居民张先生每天跑步或开车上班,据以往经验,张先生跑步上班准时到公司的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
附:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2187714e660234f0b72f2b47d3ea685a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356b05e46b10ee51c3e43546d73ec96c.png)
0.100 | 0.050 | 0.025 | 0.010 | 0.001 | |
2.706 | 3.841 | 5.024 | 6.635 | 10.828 |
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