解题方法
1 . 设数列
的前
项和为
,若
.
(Ⅰ)证明
为等比数列并求数列
的通项公式;
(Ⅱ)设
,数列
的前
项和为
,求
;
(Ⅲ)求证:
.
![](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/9ca9605501cceb252348510d860f07c7.png)
(Ⅰ)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(Ⅱ)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b008e35e4367db818d464d31bd2248c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(Ⅲ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbfc690ace28306596f1fa5c88fa3c3d.png)
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2020-12-14更新
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2192次组卷
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8卷引用:上海师范大学附属中学闵行分校2023-2024学年高二上学期期中数学试题
上海师范大学附属中学闵行分校2023-2024学年高二上学期期中数学试题浙江省强基联盟2020-2021学年高二上学期期中数学试题(已下线)【新东方】415(已下线)专题08 数列的通项、求和及综合应用 第一篇 热点、难点突破篇(讲)-2021年高考数学二轮复习讲练测(浙江专用)(已下线)专题4.3 等比数列(B卷提升篇)-2020-2021学年高二数学选择性必修第二册同步单元AB卷(新教材人教A版,浙江专用)(已下线)第4章 等比数列(A卷·夯实基础)-2021-2022学年高二数学同步单元AB卷(苏教版2019选择性必修第一册)【学科网名师堂】(已下线)专题08 数列的通项、求和及综合应用(讲)--第一篇 热点、难点突破篇-《2022年高考数学二轮复习讲练测(浙江专用)》河南省南阳市邓州春雨国文学校2022-2023学年高二下学期3月考试数学试题
2 . 已知直线与抛物线交于两点.
(1)求证:若直线
过抛物线的焦点,则
;
(2)写出(1)的逆命题,判断真假,并证明你的判断.
(1)求证:若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/749b17e02ac5325dcfcac745a51b5170.png)
(2)写出(1)的逆命题,判断真假,并证明你的判断.
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3 . 已知
,满足
.
(1)求证:
;
(2)现推广:把
的分子改为另一个大于1的正整数
,使
对任意
恒成立,试写出一个
,并证明之;
(3)现换个角度推广:正整数
满足什么条件时,不等式
对任意
恒成立,试写出条件并证明之.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c0e9d1ad9561d693958756ee8398218.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce613eaa5df46a50174085ef5d1087fb.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3aec1994a01be9e9335a62177131ee4.png)
(2)现推广:把
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b32df45c5ee591bb2b763deacb26110c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942932aac23ed64c833aacaae02e66bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce613eaa5df46a50174085ef5d1087fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(3)现换个角度推广:正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b72c9d613a2c88e73a7451210b7a98a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3823cef58d924746e16b32155e3bc16d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce613eaa5df46a50174085ef5d1087fb.png)
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4卷引用:上海市七宝中学2017-2018学年高一上学期期中数学试题
上海市七宝中学2017-2018学年高一上学期期中数学试题上海市七宝中学2021-2022学年高一上学期10月月考数学试题(已下线)上海市复旦大学附属中学2014-2015学年高一上学期期中数学试题(已下线)专题21+期中复习-2020-2021学年新教材高一数学秋季辅导讲义(沪教2020)
4 . (1)若圆
的方程是
,求证:过圆
上一点
的切线方程为
.
(2)若圆
的方程是
,则过圆
上一点
的切线方程为_______ ,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3503d330608e7138d1b529aba4512fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3511cdc6a9b56bc1d9415d3d94ef0f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d340bd3f078b9261238d4fe59f1473c1.png)
(2)若圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e885d3743ae3d9f0fbd740b75f900083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3511cdc6a9b56bc1d9415d3d94ef0f67.png)
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2卷引用:上海市七宝中学2019-2020学年高二上学期期末数学试题
名校
解题方法
5 . (1)已知a,b,x均为正数,且
,求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3085495517cf7b77cc88e513fa874cc.png)
(2)已知a,b,x均为正数,且
,对真分数
,给出类似上小题的结论,并予以证明
(3)证明:
中,
,(可直接应用第(1)(2)小题的结论)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/432d77fe5ad3032d59a237dd94c8a638.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3085495517cf7b77cc88e513fa874cc.png)
(2)已知a,b,x均为正数,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a46e678bf9d2df5ad4c782b3dc22f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2122e3f1e76a635e58e4d54aa594c552.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a89521e1106f61e66c762b5eb66bb1a3.png)
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6 . 已知函数
,函数
是函数
的反函数.
求函数
的解析式,并写出定义域
;
设
,判断并证明函数
在区间
上的单调性:
若
中的函数
在区间
内的图像是不间断的光滑曲线,求证:函数
在区间
内必有唯一的零点(假设为
),且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/685b979275f63408d20543770df4f2ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe5853a3e36e55ccf04a974c6df2811.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf6c84731e5e1bd335ecfc2d36c3d81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f53190d6ead827a6338b9de847aeaf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abbcaa32b0525269d0cb445cabaa870.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f786a5701dc1a8a015e8843c3360151b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f62295c36d2e2174908c2bec0eb5b30f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f53190d6ead827a6338b9de847aeaf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f786a5701dc1a8a015e8843c3360151b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f786a5701dc1a8a015e8843c3360151b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a2a5e336b6bcba6354fd366c892dd06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60440d5dde56b026d8568075463a988a.png)
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7 . 我们知道一次函数、二次函数的图像都是连续不断的曲线,事实上,多项式函数的图像都是如此.
(1)设
,且
,若还有
,求证:
;
(2)设一个多项式函数有奇次项
(
),求证:总能通过只调整
的系数,使得调整后的多项式一定有零点;
(3)现有未知数为
的多项式方程
(其中实数
待定),甲、乙两人进行一个游戏:由甲开始交替确定
中的一个数(每次只能去确定剩余还未定的数),当甲确定最后一个数后,若方程由实数解,则乙胜,反之甲胜,问:乙有必胜的策略吗?若有,请给出策略并证明,若无,请说明理由.
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dceea9a267bf6a1a79a2b1be84dc8b9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292a756873e88b6e90ddc8d9711cc6da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6a02e169b7678c8b3741cb187299c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acf237f6c2170d7c7fb27acbafd16f64.png)
(2)设一个多项式函数有奇次项
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db98876d40d5afd3ba01c668e96e9d0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e10f2f74e201f77f853e9ed9078615c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db98876d40d5afd3ba01c668e96e9d0e.png)
(3)现有未知数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69d41e428667bede26795a0401ddcd77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52812f95d26eec5dcd489b076cd35718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1188fbc777615a17789b1fb54fcb7e34.png)
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8 . 给出集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d0c0d57080c83dfae371038b34fbc57.png)
(1)若
求证:函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6d4f7bcbafb423271f97e0d407c74ec.png)
(2)由(1)可知,
是周期函数且是奇函数,于是张三同学得出两个命题:
命题甲:集合M中的元素都是周期函数;命题乙:集合M中的元素都是奇函数,请对此给出判断,如果正确,请证明;如果不正确,请举出反例;
(3)设
为常数,且
求
的充要条件并给出证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d0c0d57080c83dfae371038b34fbc57.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca64afa00211df204a6302463890edbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6d4f7bcbafb423271f97e0d407c74ec.png)
(2)由(1)可知,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24d95da33526f7713ce2016bfa6efe0f.png)
命题甲:集合M中的元素都是周期函数;命题乙:集合M中的元素都是奇函数,请对此给出判断,如果正确,请证明;如果不正确,请举出反例;
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c99ac91fc1e9097126e4c2aa20cdeffe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cc1d1fd01b97f1f5414428bc0d711d0.png)
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9 . 若函数
对任意的
,均有
,则称函数具有性质
.
(1)判断下面两个函数是否具有性质
,并证明:①
(
);②
;
(2)若函数
具有性质
,且
(
,
),
①求证:对任意
,有
;
②是否对任意
,均有
?若有,给出证明,若没有,给出反例.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a276d86dfd71056a526b276e0dd091d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)判断下面两个函数是否具有性质
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da53929a8f67b9aa3827fdbd73ebd265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d33da711e50e96568facb18cef27165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c904567c3b3734e1eca8d042ef7a7b2d.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d9981433f6e748c8d1851dad9ef0db2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9153fb853cd99beec9e600a4eaf73fe8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
①求证:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dbe51c8c6fe7991af132f09819e2ed7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31c68cc539a11a3da15e71379e2879b4.png)
②是否对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de297f4cb93478adeaf9c1678613ea2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dc9ede2e55724383dd1093fc7fcdb59.png)
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10 . 如图所示,点
为斜三棱柱
的侧棱
上一点,
交
于点
,
交
于点
.
![](https://img.xkw.com/dksih/QBM/2016/10/22/1573089691934720/1573089698439168/STEM/7972e989-11cc-4b62-94c9-7f0e0d290d3f.png?resizew=194)
(1)求证:
;
(2)在任意
中有余弦定理:
.拓展到空间,类比三角形的余弦定理,写出斜三棱柱的三个侧面面积与其中两个侧面所成的二面角之间的关系式,并予以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f60b83e5a713c9d0409bf544c514f602.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c88d952630ddac66a1f077dcc9439990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://img.xkw.com/dksih/QBM/2016/10/22/1573089691934720/1573089698439168/STEM/7972e989-11cc-4b62-94c9-7f0e0d290d3f.png?resizew=194)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb0937dc905b06383bd34d5f9ae8384a.png)
(2)在任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e46534c1cb9de14c258eef9244272b5.png)
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2016-12-04更新
|
626次组卷
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