名校
1 . 当药品
注射到人体内,它在血液中的残余量会以每小时25%的速度减少.
(1)按照医嘱,护士给患者甲注射了
药品
两小时后,患者甲血液中药品
的残存量为
,求
的值;
(2)另一种药物
注射到人体内,它在血液中的残余量会以每小时10%的速度减少.如果同时给两位患者分别注射
药品
和
药品
,请你计算注射后几个小时两位患者体内两种药品的残余量恰好相等.(第(2)问计算结果保留2位小数)
参考值:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)按照医嘱,护士给患者甲注射了
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7428057b4757de6d1d510712f01465e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55354a714cdf174cf4ebd43a2f041bbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)另一种药物
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc2baea0d7b4cae52246ff279a41e460.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/618c8bb0c6e025b11eeb7da738183d1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
参考值:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a49dc2827f437b3a1e7cbc25c680093c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51988fef1f0ae3f710f36865834790e5.png)
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名校
2 . 已知函数
,
,
.
(1)当
时,判断函数
的奇偶性并证明;
(2)当
且
时,利用函数单调性的定义证明函数
在
上单调递增;
(3)求证:当
且
时,方程
在
内有实数解.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0cc9b1b321520eae2bf944a9c85c9ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19339e3904e9541ff26b30ae5f1242b2.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/657435e1fda84118e7f63c97505c8b75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d6243e93c41978871cb23d8e66148d.png)
(3)求证:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91a871ef7bf13de3e15489d65b57a3cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99caed81bfb141d6e7dac8f6fe9db069.png)
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名校
解题方法
3 . 已知集合
,
.
(1)若
,求实数
的取值范围;
(2)若
,求实数
的取值范围;
(3)若将题干中的集合
改为
,是否有可能使命题
:“
,都有
”为真命题,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4cfb5d93fa28c188aa01a7b920634f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7566e20eeaa2ca570931bacc749b9bb9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd9d50619b779c1056602f46b2a95e90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96f1ba0a1129741502600e47bf058c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若将题干中的集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d0297c69a68d6d7d8abd636ca0fdab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd170c506a8ce70f550f5751ae016ca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e23af61cd402b3789af2401bde9cbefe.png)
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4 . 已知函数
.
(1)求证函数
为奇函数;
(2)判断
在区间
上的单调性,并用定义进行证明;
(3)求
在区间[2,6]上的最大值与最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c47b2d73a4858fe5a169a0964c7e878e.png)
(1)求证函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
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5 . 设全集为R,集合
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbfa5ad99d60a473d94bb6811ddbfe8a.png)
.
(1)若a=3,求
,
;
(2)若
,求a的集合.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78de216524b124cc5e111f5cf1918b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbfa5ad99d60a473d94bb6811ddbfe8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8af84160f28f01d6721bf22e94a86530.png)
(1)若a=3,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da331a8c58b97333fcf642c92f089e0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3744e71abf4b43e128eabea9181b712.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dea9a4259cca10c1f5af28e621ebafd6.png)
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6 . 求值:
(1)
;
(2)
.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa9e63f9cb08df1ae2ef2ef8c586ccb.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8441441592ab28e0fba7a53ad1c13ae9.png)
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2024-03-06更新
|
200次组卷
|
2卷引用:北京市东城区中央工艺美术学院附属中学2023-2024学年高一上学期期中考试数学试卷
解题方法
7 . 已知
是定义在
上的奇函数,当
时
.
(1)求
的解析式;
(2)根据定义证明
在
上单调递减,并指出
在定义域内的单调性;
(3)若对任意的
,不等式
恒成立,求实数k的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5b0126f3c5830bdba570e5398d07a7b.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)根据定义证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ed2f490aac02631c2ed9e6b76354a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc44426437d3610fe7212db3ac12230f.png)
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解题方法
8 . 已知函数
.
(1)当
时,
①求
的值;
②求
的图象与直线
的交点坐标;
(2)若
的值域为R,求实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/136b8fa6ada0d37fc1d4a4fe5658402a.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3e244d20e919c19caa85d1a83bcf237.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/107babba45f110012183dc4dc54490f7.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
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9 . 已知函数
是定义域为
的奇函数.
(1)求函数
的解析式;
(2)用定义证明
在定义域上是增函数;
(3)求不等式
的解集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8717af5b57ca8eb3402b17118fec7a04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ad775cee3eee4b28888a73d398d3e1c.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)用定义证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)求不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1585f4f3a58269d969804181057f42c.png)
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10 . 已知函数
,若![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70dfd3b70aab0849a459a241d904aa73.png)
(1)求
的值;
(2)证明函数在定义域内的奇偶性;
(3)证明函数在
上为增函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3efdb4474748c4862b8098482a6ea9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70dfd3b70aab0849a459a241d904aa73.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)证明函数在定义域内的奇偶性;
(3)证明函数在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
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