名校
解题方法
1 . 已知椭圆
.
(1)若椭圆
的左右焦点分别为
为
的上顶点,求
的周长;
(2)设过定点
的直线
与椭圆
交于不同的两点
,且
为锐角(其中
为坐标原点),求直线
的斜率
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb4402aeb853b22f20992156957ef0fd.png)
(1)若椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76ed463bf16c78a4bbb9d3acff922afa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33d776753746914c2410a3946c357f35.png)
(2)设过定点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ed90ebf0061c8a79beed307fc1719a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ebaa32f4f1f4f807ca9aeb7fb29951.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d7b2fe01a33c4825f9974ed9663a99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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2 . 已知
,解关于
的不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f452035eb1caa44ecf978780feb490ca.png)
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解题方法
3 . 已知数列
的前n项和
满足
,
.
(1)求
的通项公式;
(2)若
表示不超过x的最大整数,如
,求
的值;
(3)设
,
,问是否存在正整数m,使得对任意正整数n均有
恒成立?若存在,求出m的最大值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5211cd2b4ebcdaad8d73cf999b275475.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2ab85825d4a002600ca41bd3cd2ee7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3406792cf683de07aa4371168ad65226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa192e136584c2abab136070a430b9e1.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b0ea9bbe51ee5a78c22ad18807ecf59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bae9df8b3c69acd594e155714263335a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0b71732f5f5fb0f70fbccc918948608.png)
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4 . 银行按规定每经过一定的时间结算存(贷)款的利息一次,结算后将利息并入本金,这种计算利息的方法叫做复利.现在某企业进行技术改造,有两种方案:
甲方案:一次性向银行贷款10万元,技术改造后第一年可获得利润1万元,以后每年比上年增加30%的利润;
乙方案:每年向银行贷款1万元,技术改造后第一年可获得利润1万元,以后每年比前一年多获利5000元.
(1)设技术改造后,甲方案第n年的利润 为
(万元),乙方案第n年的利润 为
(万元),请写出
、
的表达式;
(2)假设两种方案的贷款期限都是10年,到期一次性归还本息.若银行贷款利息均以年息10%的复利计算,试问该企业采用哪种方案获得的扣除本息后的净获利更多?(精确到0.1)(净获利=总利润-本息和)(参考数据
,
甲方案:一次性向银行贷款10万元,技术改造后第一年可获得利润1万元,以后每年比上年增加30%的利润;
乙方案:每年向银行贷款1万元,技术改造后第一年可获得利润1万元,以后每年比前一年多获利5000元.
(1)设技术改造后,甲方案第n年的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
(2)假设两种方案的贷款期限都是10年,到期一次性归还本息.若银行贷款利息均以年息10%的复利计算,试问该企业采用哪种方案获得的扣除本息后的净获利更多?(精确到0.1)(净获利=总利润-本息和)(参考数据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a178984a657a8fc6b53511319f19fdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2289b28fcda0a30feec45cc9092a3b66.png)
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解题方法
5 . 已知关于x的实系数一元二次方程
有一对共轭虚根
,
.
(1)当
时,求共轭虚根
和
;
(2)若
,求实数a的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3552a880182b6c7f54f320fe7483ace7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f498b6874410fb46e9807e04371e6e5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec9f93ddf1453ac95368dcc84e1e4342.png)
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6 . 已知函数
的最大值为2.
(1)求a的值,并求
的最小正周期;
(2)求
在
上的单调递增区间.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55c3be12eb0e15adc19a27df21bab201.png)
(1)求a的值,并求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5f8879c9994fe5c7681d7a754be0279.png)
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解题方法
7 . 设
.
(1)若
,求函数
的图象在
处的切线方程;
(2)若
在
上恒成立,求实数
的取值范围;
(3)若函数
存在两个极值点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab44240a0cb09a1b9f5966c63fa290f3.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae386699086599e0bcb0fbe59cffd9fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fd785bc48778543268feeca30728c34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03ca13a93b5f401c0d39ba52b0cffcb0.png)
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解题方法
8 . 已知数列
各项均为正数,且
,记其前
项和为
.
(1)若数列
为等差数列,
,求数列
的通项公式:
(2)若数列
为等比数列,
,求满足
时
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1dfe5b322577f02fd19caab8cf20170.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31a059268c7801f786e93a76d65b68f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a8bedfcdf5e750f5451e0e5183c0d80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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解题方法
9 . 已知向量
,
,
.
(1)若
,求
值;
(2)若向量
在
方向上的投影向量为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f4dcf415977dea53f52a85b6b82136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92b1305dff4fc89f3e8bc75fa7c258de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab00543b1f4647a7249fe5a4507e1ec1.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b610fe3872cd8c1009b6a3e0111bacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(2)若向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed492f7b29166ba5c1f0023b05a439c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73a0b19e69be46452425916a0fcb49c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cef9c6d3fc0c7d3ec11c782fd106511.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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名校
解题方法
10 . 已知复数
,
,
.
(1)若复数
在复平面内的对应点落在第四象限,求实数
的取值范围;
(2)若复数
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f68c2c5556f04d1fdce13f37076effc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0055868371dd09b74ca4cd76750f9172.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
(1)若复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34b88f343ca5a4c29057465541b9cf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ebc4b0d5565b30f8963cef1c8bd94a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fe615164ed2995bdeea0f5b0ba94231.png)
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7日内更新
|
250次组卷
|
2卷引用:上海市青浦高级中学2023-2024学年高一下学期期末考试数学试卷