解题方法
1 . 已知
,
是椭圆T.
上的两点,且A点位于第一象限.过A作x轴的垂线,垂足为点C,点D满足
,延长
交T于点
.
(1)设直线
,
的斜率分别为
,
.
(i)求证:
;
(ii)证明:
是直角三角形;
(2)求
的面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05193d9096bd9da9230acc14228aa4e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/817edbb8e01ced216a63c838c7b1a288.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c82e7d9f4f7ace849e09e9adcb786b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0617414b2ad7c96f1a3df4a6dd935395.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b60af8e12b6205f65f8cb0ecd870601d.png)
(1)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
(i)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30bf02b822ea9ded2e9fdc868d74ab96.png)
(ii)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
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解题方法
2 . 设函数
(a,
);
(1)若
,求证:函数
的图像必过定点;
(2)若
,证明:
在区间
上的最大值
;
(3)存在实数a,使得当
时,
恒成立,求实数b的最大值;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80027540415bd2b98c9be19e21b5f8d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d285a4c557fc9748105b62ccd94b7859.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b40b1544e62be8b9e9f4dc9f2c0c74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fedf88c1afae37dcb344708fa1918db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d09a2b7c019dae83e027830b82b3ee8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2aa311daf7a73f8c45de4462f9d92b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d5193a9a29c504059dcbecfb81ca496.png)
(3)存在实数a,使得当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42f54feac6ed738a868ecd53d3a85a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1d9926ad419c75a83ca90457a1e2fc1.png)
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2020-02-10更新
|
256次组卷
|
2卷引用:上海市第二中学2017届高三上学期9月初态测试数学试题
名校
3 . 已知函数
的图象上有两点
,
.函数
满足
,且
.
(1)求证:
;
(2)求证:
;
(3)能否保证
和
中至少有一个为正数?请证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c4753463638237ed69779fff0c66746.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b2d666c964eaf1879e069d971b0d636.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/431521d6928b08a6d75e17676219dd64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/196be101149acfb6a6c4ceca7fc96828.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5089dc8320c35b7068b66001e2b69f8.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe716b3a010661bc2ffd99a6263c5108.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90e8d5d7fed033f48270b1ff825fcd5.png)
(3)能否保证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe6ae35ea314b3b3c56dc3500e14aa8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90d145687ec9531a97ab689a0a17023f.png)
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解题方法
4 . 对于数列
,设
表示数列
前
项
,
,
,
中的最大项.数列
满足:
.
(1)若
,求
的前
项和.
(2)设数列
为等差数列,证明:
或者
(
为常数),
,
,
,
.
(3)设数列
为等差数列,公差为
,且
.
记
,
求证:数列
是等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/438bfa8871f500fb656cc1cddda21022.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdc2b6b23da3e065820c15cf6c675e20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce9249b2afe90ee5beb05e00026f5a8a.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b38dd51241aa6d3b84c9b9f9cfbd2ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8ca53096136f884c6de4fce1d470c5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ef984efd949e6e03e4a628f98bbdadc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5095a28bb1b91bf6bed9e2cfbd76bb18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdc2b6b23da3e065820c15cf6c675e20.png)
(3)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98c3cf26670636d90d6f07d58e374fbf.png)
记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdb8ddc17ecca1d572249a3d1aadf41.png)
求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
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5 . 已知数列
满足:
,
(
).
(Ⅰ)求证:
;
(Ⅱ)证明:
;
(Ⅲ)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dfb2f24a05eed2fc2cf15f2b2e34dd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
(Ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f678e7e04f2240adb433ed8e8ed40639.png)
(Ⅱ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2431de86fb44ded9047ffe8da6cf12ed.png)
(Ⅲ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/530d0f56fc9a4511948dd41161495d03.png)
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13-14高三下·上海虹口·阶段练习
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6 . 已知数列
和
满足:
,其中
为实数,
为正整数.
(1)对任意实数
,求证:
不成等比数列;
(2)试判断数列
是否为等比数列,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83d9ea28ccc8c24eeafa2ce1caf71b1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5fffd330dd6b9241659d790bd2a7fb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)对任意实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5fffd330dd6b9241659d790bd2a7fb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c14d9ae06f864498048d55088ff4e6.png)
(2)试判断数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
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7 . 已知
为实数集的一个非空子集,称
是一个加法群,如果
连同其上的加法运算满足如下四条性质:
①
,
;
②
,
;
③
,
,使得
;
④
,
,使得
.
例如
是一个无限元加法群,
是一个单元素加法群.
(1)令
,
,分别判断
,
是否为加法群,并说明理由;
(2)已知非空集合
,并且
,有
,求证:
是一个加法群;
(3)已知非空集合
,并且
,有
,求证:存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd242f355d5128425429a83e4b6632c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8362f15e544684164f38ff9ad7c38ac7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ea5a550b5452df9abdbca776c2ff500.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/509a09a7391de2cc86e5e44ccccc981b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8236622218d4d4012d8637538ac9032.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bef35ae51107e991163ea418c8dec53a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf00e8864c86c3ce8118ea76bf69773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc118659264aca9e263cb8edc41e9c44.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf00e8864c86c3ce8118ea76bf69773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71f0119b6de9149150071fe7ed848aa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a065a5ddaa18900ee15a8b436f0fcb95.png)
例如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/178e8cc61b87b4dc63105ab4fca8680c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7cb4e4e98b375294dc1dccbeebbd6c2.png)
(1)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/809b2e00ab8e43a0f886c7f83846d3d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b113752e4f989a338747b95a40cf386.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d18f9bbb6b9feb166f7ecfb49013262d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64ee969e5c3d880e0209235bb9cfc49f.png)
(2)已知非空集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a02a810b3332821bc444f215183c9e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a7d3e3d84e1fdee95574817741d731e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08152bab36dca188978d125e4b7a935a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/489ea5a5f5b5de37e238cbfbb4a01143.png)
(3)已知非空集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d28cc3165eef94c22c442b2f30c87cc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4034552829008c1daaee2701d2afe8dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e0f9ba8419972cff845bfd91f64297.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ddc4872d58eaa6bcc432b7b94939f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd5eb4f6f84d264f3403eece1e7c37b7.png)
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解题方法
8 . 已知函数
(其中
是自然对数的底数).
(1)是否存在实数a,使得函数
在定义域内单调递增?
(2)若函数
存在极大值
,极小值
,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41fb7695549fcc34ac31e6e7420ba708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/594663e98b797cdc4efbd098cc15854f.png)
(1)是否存在实数a,使得函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f1a08b6e18cde8d946f9b6e6b428ebd.png)
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9 . 如图,在四棱锥
中,平面
平面
,
,底面
为等腰梯形,
,且
.
平面
;
(2)若点A到平面PBC的距离为
,求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbb05b8b630052ff544249ebd72d95d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a11029ca6b4b9e7f777af0280cf163c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852847ba02c2b62abf27e9cc11f596a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d077f6da8b2c00b152d4679aa2ed7f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
(2)若点A到平面PBC的距离为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
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10 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8dda8e0f05f65226b95a71bc0d75bc9.png)
(1)当
时,求函数
的极值;
(2)求函数
的单调区间;
(3)若对任意的实数
,函数
与直线
总相切,则称函数
为“恒切函数”.当
时,若函数
是“恒切函数”,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8dda8e0f05f65226b95a71bc0d75bc9.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若对任意的实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ceee0ff5c929d67de3c294e027c9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a60550d48fcf76d109f426149d8185.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15fb18163df0690365a0d2e7ee88f5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0780bba5832fe480a5fddd87bd1af36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6077c98670d416a38f736c11f3591966.png)
您最近一年使用:0次
2023-12-20更新
|
587次组卷
|
4卷引用:北京市海淀区中关村中学2024届高三上学期12月月考数学试题