名校
解题方法
1 . 对给定的实数a,b,q,其中
,
.如果函数
,
:满足(1)对任意的
,
且
;(2)对任意的
,
.则称
为在区间
上的一个“q-压缩函数”.区间
上所有“q-压缩函数”构成的集合记作
.
(1)判断下列函数,是否属于集合
?(直接写出结论)
①
②
③![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80e8d2d66b432fd2b8a2830ba21e5dd.png)
(2)设
,若
求实数a的取值范围.
(3)设
.若对任意的
,
,均有
,求M的最小值,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a46e678bf9d2df5ad4c782b3dc22f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ae99e050d0f1cfc0447304f06424d17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e207cf62e3a7e282eac4c4a3455bbf9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48318c1b3f9dffc48e3ce0626b277e4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/664c92988d61280ed12488f270151940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a8b4f05d2ba1a5514ebf390dbbd325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a293e0b4f751f4102c795b567a633194.png)
(1)判断下列函数,是否属于集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3c564b7a711c6ac5945a0d41b2564c8.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a740cdaca0afce73f68c890f86acc6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef72cf8297d91045117d50b4e9d3476.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80e8d2d66b432fd2b8a2830ba21e5dd.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db0e64d809acabedccf8a8516ff8b6a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c109bacd687055d3a6a79978dfa7c0e.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e39b23fb1290ed449f1700c42c3efc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfcd04b625189228b6d697edf095f7c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c97169936ca9785dfca39ca368c65c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b0aa28b363cb79e6f8988b9b2bad7dc.png)
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名校
解题方法
2 . 已知
和
都是直角三角形,
,E,F分别是边AB,AD的中点,现将
沿BD边折起到
的位置,如图所示,使平面
平面BCD.
平面BCD;
(2)求证:平面
平面
;
(3)请你判断,
与BD是否有可能垂直,做出判断并写明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661ff55b5ebbadfb600989af3cfce2fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ff103699fefbd87327b26b58fafd42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07c967c9b3f669ea78edd838e1d8b59e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba0fbd88fdb064072eedd136e9cb41ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57f9d682e5d3cc8573574d8d11636758.png)
(2)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21dee56b9f36ba8f76fe67b76383636b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74bca84ad86c648d3bb20c8909c8da3f.png)
(3)请你判断,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
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解题方法
3 . 诚信是立身之本,道德之基.某校学生会创设了“诚信水站”,既便于学生用水,又推进诚信教育,并用“
”表示每周“水站诚信度”.为了便于数据分析,以四周为一个周期,下表为该水站连续八周(共两个周期)的诚信度数据统计,如表1:
(1)计算表中八周水站诚信度的平均数
;
(2)从表中诚信度超过
的数据中,随机抽取2个,求至少有1个数据出现在第二个周期的概率;
(3)学生会认为水站诚信度在第二个周期中的后两周出现了滑落,为此学生会举行了“以诚信为本”主题教育活动,并得到活动之后一个周期的水站诚信度数据,如表2:
请根据提供的数据,判断该主题教育活动是否有效,并根据已有数据说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc891e44ebe17a1ffc956667a4112bf4.png)
第一周 | 第二周 | 第三周 | 第四周 | |
第一个周期 | ||||
第二个周期 |
(1)计算表中八周水站诚信度的平均数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfbe7f95b5d89f9409ec24536da9e826.png)
(2)从表中诚信度超过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4426df5be4e96b6385de735db9f69772.png)
(3)学生会认为水站诚信度在第二个周期中的后两周出现了滑落,为此学生会举行了“以诚信为本”主题教育活动,并得到活动之后一个周期的水站诚信度数据,如表2:
第一周 | 第二周 | 第三周 | 第四周 | |
第三个周期 |
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4 . 已知函数
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
(1)如果点
是角
终边上一点,求
的值;
(2)设
,求
的单调递增区间.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9688a68bd2d1b2c8dcd6568e1a5ac3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
(1)如果点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58e7c6b1e8d06e439c1ede19f99286c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/794f2c6bd63355105d179d11306a9cae.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72f3b2cc4fc44eef6a51e1a4ae89ecb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
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解题方法
5 . 已知函数
,
满足以下条件:
①
,
;
②
,
,
,
.
(1)求
,
的值.
(2)判断函数
,
的奇偶性,并说明理由.
(3)若
,
,试判断函数
的周期性,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/306984f0895ba32a7b3bb607065b1eaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e83e9f562c10762097469dea27c1e109.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df6593a700bf3e89107556454666b787.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a50ac027c6ebce491ae836524d89901c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ebeb1c1f2826da8a2e0761f2d2ba87d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98eb23b8c96a34dd720e00669aa8ed2b.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a078165d75cfb890141845324a6173b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60cf12a81b11e33356fe7e1c9e3d0b9.png)
(2)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfaff6cfe1d15bd64c1fa76af5e52831.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/158fbbd2cbedeb9a6fa1a900630369f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
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6 . 已知
,
,且函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/189575d7193fe6363865c2a664a89e9c.png)
(1)求
的最小正周期及单调递增区间;
(2)若
为锐角且
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf8edcf4850813f4c69fe6e9426207dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fdb3d88238ec6a94a7d1dedaa9b80d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/189575d7193fe6363865c2a664a89e9c.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4443d42de2db7505b7a901e51ac74d21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9146fc0a63e5c14a8fa46573e60c07ba.png)
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7 . 已知向量
,
.
(1)若
,求
及
在
上投影的数量;
(2)若
,求
与
的夹角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81c5293dfdbd72411423003ec100c910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecc2d0fb8a6b049073027ccfe1dce109.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5758ff4a1c7ba1a6ec0566c627b55f22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc7d8b42c85013ba07ba18e5e46d8716.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbfdaf6917dae9b2ca83be97c3631bba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
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8 . 已知函数
.
(1)求曲线
在
处的切线方程;
(2)证明:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a44f951eba35ed9b8e78e098fa7d6d8.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db2b74d89854116e411c089d053df053.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66d61d5f66d68b4c4a2a25fd7103621.png)
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解题方法
9 . 设
为等差数列
的前n项和,
,
.
(1)求数列
的通项公式;
(2)求
;
(3)若
,
,
成等比数列,求m的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5170584604571b5e1afd5ece941e2e73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23005bd2386f15812ce36833200d019d.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e985ada7a866d897f0d061aa965af671.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1752474698cd5466dd180df0a00ba9c.png)
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10 . 设函数
.
(1)求曲线
在点
处的切线方程;
(2)求
在区间
上的最大值和最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/762fe20fadca4c1c267af48cf35c88e9.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac9c64ba837387d640de4b8e2191b1b5.png)
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