名校
解题方法
1 . 某工厂有甲、乙两个生产车间,其污水瞬时排放量
(单位:
)关于时间
(单位:
)的关系均近似地满足函数
,其图象如图所示.
(1)根据图象求函数解析式;
(2)若甲车间先投产,1小时后乙车间再投产,求该厂两个车间都投产
时刻的污水瞬时排放量;
(3)由于受工厂污水处理能力的影响,环保部门要求该厂的两个车间任意时刻的污水排放量之和不超过
,若甲车间先投产,为满足环保要求,乙车间比甲车间至少需推迟多少小时投产?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1ff4266515b801477248b32e10f3ac4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f42a97e3ae09c48e1d587f59af3621bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/795ea8d61f4e4e138231819356f4073b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/22/f9c7e29c-132e-490f-a765-86601184cddc.png?resizew=182)
(1)根据图象求函数解析式;
(2)若甲车间先投产,1小时后乙车间再投产,求该厂两个车间都投产
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16dca1f69db5d45d7e0195b60f546a40.png)
(3)由于受工厂污水处理能力的影响,环保部门要求该厂的两个车间任意时刻的污水排放量之和不超过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d18b1b3fa4bc1f8e48a68b9f77e37e3b.png)
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解题方法
2 . 已知函数
的部分图象如图所示.
的解析式;
(2)求
的单调递增区间;
(3)若锐角
满足
,求
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0ef9d537e1849ed448318247dce9f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(3)若锐角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fba6ff07d9794cb287ad46e857b2669e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8dc4c63a548b91061528aa11058de75.png)
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解题方法
3 . 已知
的内角
,
,
的对边分别为
,
,
,且
.
(1)求
;
(2)若
的面积为
.
①已知
为
的中点,求
的最小值;
②求内角
的平分线
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3df022d820aeb54fdd2a9746a36a243f.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c181f86de3c96a7ef7a1a04c3a438f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de0e448610eac96bdd5f864f873e575d.png)
①已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
②求内角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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解题方法
4 . 已知数列
前
项和
满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b982b2d1d703e124c0f2c5894bf908c.png)
(1)求数列
的通项公式;
(2)求数列
的前
项和
.
![](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/1b982b2d1d703e124c0f2c5894bf908c.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba57c83d526ac308d1461e80fcca9f36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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名校
5 . 已知函数
.
(1)讨论函数
的单调性;
(2)设
,对任意
,且
,使
恒成立,求正实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5af60d06658c16bb2bb2dae7007c9da.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a84cb24c6b9d85d81f57999b275c86e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e694ba0d3ee2b92073f41ef5aad4eecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/684bcf84f0a266515bfafde0da903050.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a69c9837c55274b9aa0b311fbc18fbc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
解题方法
6 . 根据《国家学生体质健康指标》,高三男生和女生立定跳远单项等级如下(单位:
)
从某校高三男生和女生中各随机抽取12名同学,将其立定跳远测试成绩整理如下(精确到
):
男生:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f355627a8002d55f5e1ada790e87ec.png)
女生:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/791596115aed03bff2100918658a72fd.png)
假设用频率估计概率,且每个同学的测试成绩相互独立.
(1)分别估计该校高三男生和女生立定跳远单项的优秀率;
(2)从该校全体高三男生中随机抽取2人,全体高三女生中随机抽取1人,设
为这3人中立定跳远单项等级为优秀的人数,求
的分布列和数学期望
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
立定跳远单项等级 | 高三男生 | 高三女生 |
优秀 | 260及以上 | 194及以上 |
良好 | ![]() | ![]() |
及格 | ![]() | ![]() |
不及格 | 204及以下 | 149及以下 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/048b61a5fb5f420c6d7de88db5bc3aa5.png)
男生:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f355627a8002d55f5e1ada790e87ec.png)
女生:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/791596115aed03bff2100918658a72fd.png)
假设用频率估计概率,且每个同学的测试成绩相互独立.
(1)分别估计该校高三男生和女生立定跳远单项的优秀率;
(2)从该校全体高三男生中随机抽取2人,全体高三女生中随机抽取1人,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
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7 . 数学中有很多相似的问题,
材料一:十七世纪法国数学家,被誉为业余数学家之王的皮埃尔·德·费马提出了一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小”,他的答案是:“当三角形的三个内角均小于
时,所求的点为三角形的正等角中心,即该点与三角形的三个顶点的连线两两成角
,当三角形有一内角大于或等于
时,所求点为三角形最大内角的顶点”,在费马问题中所求的点称为费马点.
材料二:布洛卡点,也叫“勃罗卡点”,定义为:已知
内一点
满足
,则称
为
的布洛卡点,
为
的布洛卡角,1875年,三角形的这一特殊点,被一个数学爱好者——法国军官布洛卡重新发现,并用他的名字命名.
已知
,
,
分别是
的内角
,
,
的对边,且
.
(1)求
;
(2)若
为
的费马点,且
,求
的值;
(3)若
为锐角三角形,
为
的布洛卡点,
为
的布洛卡角,证明:
.
材料一:十七世纪法国数学家,被誉为业余数学家之王的皮埃尔·德·费马提出了一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小”,他的答案是:“当三角形的三个内角均小于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00e4979100d4078609e253e2f99eed0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00e4979100d4078609e253e2f99eed0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00e4979100d4078609e253e2f99eed0b.png)
材料二:布洛卡点,也叫“勃罗卡点”,定义为:已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c25d734ea37934683320c146c2c67a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/481b91aa00df0bf153f717d87d1b12f7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54728823efd2745d64ae9921f8807917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1424f6ac5e01f56e2d486c68a5be1a0.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26f61d98c51b9f0344cf7b4562680f45.png)
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8 . 图①是一块正四棱台
的铁料,上、下底面的边长分别为
和
,
,
分别是上、下底面的中心,棱台高
.
的表面积;
(2)若将这块铁料最大限度地打磨为一个圆台(如图②),求削去部分与圆台的体积之比.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbe9f7abf7bcf4e1aa2579cd191d7761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e57171806f407a98dd8a796d4d2d6bbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3668a3f3ce5b8a272ad92c2ebd233f5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
(2)若将这块铁料最大限度地打磨为一个圆台(如图②),求削去部分与圆台的体积之比.
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9 . 已知椭圆
短轴长为2,椭圆
上一点
到
距离的最大值为3.
![](https://img.xkw.com/dksih/QBM/2024/5/26/3504894981390336/3520242638839808/STEM/32c129353263403684caca44141869ef.png?resizew=151)
(1)求
的取值范围;
(2)当椭圆
的离心率达到最大时,过原点
斜率为
的直线
与
交于
两点,
分别与椭圆
的另一个交点为
.
①是否存在实数
,使得
的斜率
等于
?若存在,求出
的值;若不存在,说明理由;
②记
与
交于点
,求线段
长度的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9059420851d219dba2189975f8eb5178.png)
![](https://img.xkw.com/dksih/QBM/2024/5/26/3504894981390336/3520242638839808/STEM/32c129353263403684caca44141869ef.png?resizew=151)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7708c156249bb475564027f73313fa8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f2ec2e454afe8452a6f1714a986aa38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235557c7e459c7100787e436826c198e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f963b16dcebd808b41842febc329a6.png)
①是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7e08d5c04f0431fb57b33a01717b599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4b2e1a496f0df7918efed76a51f34c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
②记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
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7日内更新
|
43次组卷
|
2卷引用:辽宁省大连市部分学校2024届高三下学期联合模拟考试数学试题
2024高三·全国·专题练习
解题方法
10 . 已知函数
.
(1)若函数
有三个零点分别为
,
,
,且
,
,求函数
的单调区间;
(2)若
,
,证明:函数
在区间
内一定有极值点;
(3)在(2)的条件下,若函数
的两个极值点之间的距离不小于
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/077315c5a7b12294497294e536831d77.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f5cd91996571c9da95e6f26bc80661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23292eca257af6a97309ee40ce6cbf9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b37f19a2ad8f24cf63bff68be15faa67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/799f6009a476fa056e1af71f26dd2fd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b094cba781181aeb90752170e9ba6c94.png)
(3)在(2)的条件下,若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c6ce02259a85ea191541f4a708738f1.png)
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