名校
1 . 已知关于
,
的方程组
其中
.
(1)当
时,求该方程组的解;
(2)证明:无论
为何值,该方程组总有两组不同的解;
(3)记该方程组的两组不同的解分别为
和
,判断
是否为定值.若为定值,请求出该值;若不是定值,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1685feba617e3d56860fe0a3a59804f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c36b234ba460321e811de1729eadd4b6.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5095a28bb1b91bf6bed9e2cfbd76bb18.png)
(2)证明:无论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)记该方程组的两组不同的解分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3396ead2a01ebd1d6134732541008a7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/734a03b0e1c4de970668548ebb944fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0af05cf4260c845bfb0675073bd81b6.png)
您最近一年使用:0次
2023-11-14更新
|
152次组卷
|
2卷引用:北京市西城区北京师范大学附属实验中学2023-2024学年高一上学期期中测验数学试题
名校
2 . 已知实数x,y满足方程
.
(1)求
的值;
(2)设
与
是方程组
两组不同的解,其中
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1beb6812158ca2a3082bd13ca07578f0.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c1afbc87ccffbc98b9ab58df8c69bee.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99307ab4373fbe72422ae5aa980db61c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41039d45e37899d233232de3d802b105.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccee8eb181dc117834582bc433eca559.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aab3cf6695638d5bcd26580174d7cbf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da3ff6f17be99ec311610efa08ba002.png)
您最近一年使用:0次
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3 . 根据多元微分求条件极值理论,要求二元函数
在约束条件
的可能极值点,首先构造出一个拉格朗日辅助函数
,其中
为拉格朗日系数.分别对
中的
部分求导,并使之为0,得到三个方程组,如下:
,解此方程组,得出解
,就是二元函数
在约束条件
的可能极值点.
的值代入到
中即为极值.
补充说明:【例】求函数
关于变量
的导数.即:将变量
当做常数,即:
,下标加上
,代表对自变量x进行求导.即拉格朗日乘数法方程组之中的
表示分别对
进行求导.
(1)求函数
关于变量
的导数并求当
处的导数值.
(2)利用拉格朗日乘数法求:设实数
满足
,求
的最大值.
(3)①若
为实数,且
,证明:
.
②设
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4a1d0dba29a77dd111efcde543d6c1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc4c14935585e8fa61d032730867d771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67b6f154c6b2de5695eb1807b98c2c63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/809615d1f91508e2c6c0cda7e592c479.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/244021f826099b18e31af1143597bba2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb5be11a5e6aaf00b2833930b198b4cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0203b006524305c3d8ee0b6c34cd872b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4a1d0dba29a77dd111efcde543d6c1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc4c14935585e8fa61d032730867d771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1c3c1ed4fb65ab9505ad8078d8d0fb5.png)
补充说明:【例】求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d7ca0caa9933b7afd4bed2683140a07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aebdee8d81b048b5aa520f7e8ba56ff2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1e15a54c6122c695239107dd0901bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/244021f826099b18e31af1143597bba2.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b3d9ab2fcf15b94f33cb64f84ed906c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)利用拉格朗日乘数法求:设实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c45d8122b61de13875003d00c002c5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de725a9fc66f67abbe0015131846a648.png)
(3)①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd24c686fbaaa68705d654b880481ffe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e778f95c72fec00bfbbc63e6dfd0c460.png)
②设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/497d269c30eec393e3f0e877ddbe2983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ade042c085bbad8aeaf111b9f4c33408.png)
您最近一年使用:0次
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4 . 已知参数k为非零实数,记
与
为关于x,y的方程组
的两组不同实数解;记
与
为关于x,y的方程组
的两组不同实数解.
(1)求证:
,
;
(2)求
的值;
(3)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3396ead2a01ebd1d6134732541008a7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/734a03b0e1c4de970668548ebb944fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a9c72eefda9b1192a4bc6982b52cc6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e78adf83070612501d95c80cd2f2bb45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e377ec55e3668e1daa9a58e0091930fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb66608b19763b0ebf13861149f2e10b.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8997257fd8e9f1286004489eb5ac4c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a21b23d1bddc1acc93debcf1c1747ca7.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b41a0384d42a7a75ea168f57262eaef1.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301dec441538499bc41fefcc53586841.png)
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5 . 《见微知著》谈到:从一个简单的经典问题出发,从特殊到一般,由简单到复杂:从部分到整体,由低维到高维,知识与方法上的类比是探索发展的重要途径,是思想阀门发现新问题、新结论的重要方法.
阅读材料一:利用整体思想解题,运用代数式的恒等变形,使不少依照常规思路难以解决的问题找到简便解决方法,常用的途径有:(1)整体观察;(2)整体设元;(3)整体代入;(4)整体求和等.
例如,
,求证:
.
证明:原式
.
波利亚在《怎样解题》中指出:“当你找到第一个藤菇或作出第一个发现后,再四处看看,他们总是成群生长”类似问题,我们有更多的式子满足以上特征.
阅读材料二:基本不等式
,当且仅当
时等号成立,它是解决最值问题的有力工具.
例如:在
的条件下,当x为何值时,
有最小值,最小值是多少?
解:∵
,∴
,即
,∴
,
当且仅当
,即
时,
有最小值,最小值为2.
请根据阅读材料解答下列问题
(1)已知如
,求下列各式的值:
①
___________.
②
___________.
(2)若
,解方程
.
(3)若正数a、b满足
,求
的最小值.
阅读材料一:利用整体思想解题,运用代数式的恒等变形,使不少依照常规思路难以解决的问题找到简便解决方法,常用的途径有:(1)整体观察;(2)整体设元;(3)整体代入;(4)整体求和等.
例如,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e2764ccd2cfe6de0c53dce98e45b120.png)
证明:原式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87898da3367d13667477a10c9cc47ac2.png)
波利亚在《怎样解题》中指出:“当你找到第一个藤菇或作出第一个发现后,再四处看看,他们总是成群生长”类似问题,我们有更多的式子满足以上特征.
阅读材料二:基本不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a28514741f365301978e922fdca0fcc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
例如:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13f40c24c64bbb0645fcf585f4e66872.png)
解:∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c42b50f6f9e56ea5f222b0a40cb4a3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91bb4a7110c19cd10cb915e55438314b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d32ba3941cef6b1d549f050f0d314e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63af71b9e6f71cd26e6e97541154cd8c.png)
当且仅当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b6a593ef3641dbd11e324dbe78b4dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13f40c24c64bbb0645fcf585f4e66872.png)
请根据阅读材料解答下列问题
(1)已知如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f0dd92f322200ecabfb74ffd7cf3f4a.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af71e37295978173629004816b65791a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56667aabbe787eb1c3189d487d203e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d9093a255130a938a4d84595c0c56ce.png)
(3)若正数a、b满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ab1cbf887eca130c254f6e0cf3fdb2f.png)
您最近一年使用:0次
2021-10-29更新
|
532次组卷
|
3卷引用:江苏省南通中学2020-2021学年高一上学期开学考试数学试题
江苏省南通中学2020-2021学年高一上学期开学考试数学试题江西省南昌市第二中学2023-2024学年高一上学期月考数学试题(一)(已下线)第二章 等式与不等式(压轴题专练)-速记·巧练(沪教版2020必修第一册)
6 . 已知方程组![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267ad467cfcbcc2dff5b77665f13a8e2.png)
(1)求证:方程组恰有一解;
(2)求证:以方程的解
为坐标的点在一条直线上;
(3)求
的最小值,并求此时a的范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267ad467cfcbcc2dff5b77665f13a8e2.png)
(1)求证:方程组恰有一解;
(2)求证:以方程的解
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82a79a33a83a7ba57a34b5093d1d1d02.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dbd4d32dcc1445563887de5d3ab7fb7.png)
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解题方法
7 . 在组合恒等式的证明中,构造一个具体的计数模型从而证明组合恒等式的方法叫做组合分析法,该方法体现了数学的简洁美,我们将通过如下的例子感受其妙处所在.
(1)对于
元一次方程
,试求其正整数解的个数;
(2)对于
元一次方程组
,试求其非负整数解的个数;
(3)证明:
(可不使用组合分析法证明).
注:
与
可视为二元一次方程的两组不同解.
(1)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/114b84ba3234b9bb1bf9f64c172292d7.png)
(2)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa38e21db62123319c9557d1bc52825d.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d63a043e64f7ed5d168cd2c9384e953b.png)
注:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65fe832c0460e00120d4bc3636aebcaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa6c8fe63bb58df1c5a12422e9c9e291.png)
您最近一年使用:0次
2024-03-08更新
|
1118次组卷
|
3卷引用:广东省五粤名校联盟2024届高三第一次联考数学试题
名校
8 . (1)解关于x,y的方程组![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee94cad924f3bde7a583545b6ac84012.png)
(2)已知
和
是关于x,y的方程组
(k为参数)的两组不同实数解.
求证:①
,
;
②
;
③
(其中
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee94cad924f3bde7a583545b6ac84012.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3396ead2a01ebd1d6134732541008a7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/734a03b0e1c4de970668548ebb944fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/494c830fbe4b161a0d1506c1aaf15cfb.png)
求证:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fda10b954abfc6bcd2fa0fe54536bcfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa675d90df61bdb59aa45a3654c6a71.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d28790c9a69068d3ce4caafae10a967.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/681683ea78209722151377053b34d082.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2851fd014aec602364532b264691c271.png)
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9 . 已知
是关于的方程组
的解.
(1)求证:
;
(2)设
分别为
三边长,试判断
的形状,并说明理由;
(3)设
为不全相等的实数,试判断
是“
”的 条件,并证明.①充分非必要;②必要非充分;③充分且必要;④非充分非必要.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdec654cfa585b39236b840ca50352b8.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48c39aaf7b4158565b89da8ed1e28724.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f06a12c7641613467a5852239baa3f4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5677d408e10dd47640d383fc28eefd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/652c9549eefa14f4ab138adf8daa5546.png)
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解题方法
10 . 对于任意实数
,引入记号
表示算式
,即
,称记号
为二阶行列式.
是上述行列式的展开式,其计算的结果叫做行列式的值.
(1)求下列行列式的值:
①
;②
;
(2)求证:向量
与向量
共线的充要条件是
;
(3)讨论关于
的二元一次方程组
有唯一解的条件,并求出解.(结果用二阶行列式的记号表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa5440a1b5d9338efd6976a56432e100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43f9683760df4268272525c8082c7ee5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0c8894e0b37af5da23a1c1bffb32017.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa5440a1b5d9338efd6976a56432e100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43f9683760df4268272525c8082c7ee5.png)
(1)求下列行列式的值:
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c601a13b26ec4fe000e79cf189d9bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4c5a0d1545e308e320a49e1c305ea90.png)
(2)求证:向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7ef9b43b03c19f5616e31888f053915.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2502935b71dab102edbe6f162046943.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9069422cc832b478cd86186e5f22897.png)
(3)讨论关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f334249bbad594a5db5137164b79f1d.png)
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