名校
解题方法
1 . 若
是定义在
上的增函数,其中
,存在函数
,
,且函数
图像上存在两点
,
图像上存在两点
,其中
两点横坐标相等,
两点横坐标相等,且
,则称
在
上可以对
进行“
型平行追逐”,即
是
在
上的“
型平行追逐函数”. 已知
是定义在
上的奇函数,
是定义在
上的偶函数.
(1)求满足
的
的值;
(2)设函数
,若不等式
对任意的
恒成立,求实数
的取值范围;
(3)若函数
是
在
上的“
型平行追逐函数”,求正数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8fd1e808e015f4cb43d2e3a0529ac6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3df718662452f53034b6f702e46dcdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/958a7330503a10fecf9b6b6f4a30ea8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc5ffb9d0b5dc940f53d7370419a08db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3859cd3de4d6d485df050b0f5321d6c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c33c52de7a2c9d148e44283eec3dbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098a3e7d1f1890863b7483a98b618119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbdb21011ea821b91d539cb763aac649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdeb8cfadb41d94aaa1ba534aa040dcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e257d683ceaca6c6c3ee5ada0e447b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d908675a3ce0661cf6b3d7823143d4b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
(1)求满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfb874a24bc2fa3d232070f16712aa05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86df4f1b31c51da3551a76606d553f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0bb6dbc79a74521af6338b0140b713b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b8c164755dc2d7cff80fb4c9cffc9be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a86644d5b1157b35cf7b825f108d4c39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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2 . 陀螺是中国民间最早的娱乐工具之一.图1是一种木陀螺,可近似地看作是一个圆锥和一个圆柱的组合体,其直观图如图2所示,其中
分别是上、下底面圆的圆心,且
,现有一箱这种的陀螺共重
(不包含箱子的质量),陀螺的密度为
(
取3)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/4/bd29717f-0d9e-4b4b-acb4-18885d91ec12.png?resizew=238)
(1)试问该箱中有多少个这样的陀螺?
(2)如果要给这箱陀螺的每个表面涂上一种特殊的颜料,试问共需涂多少
的颜料?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d6b1a601cfab0d96a659149a4c3fc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43d00798286f8e07b736da3cea3b471f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/009fb05cda538a62cf0b5b9398600a58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70f5389990c3a0c5373f3bd9fb2454c9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/4/bd29717f-0d9e-4b4b-acb4-18885d91ec12.png?resizew=238)
(1)试问该箱中有多少个这样的陀螺?
(2)如果要给这箱陀螺的每个表面涂上一种特殊的颜料,试问共需涂多少
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88ce13774b09ff2edddaf21a072cf60a.png)
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3 . 已知数列
的首项为
,且满足
.
(1)求证:数列
为等比数列;
(2)设
,记数列
的前
项和为
,求
,并证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6545b8eca1c4223ed701a199a85683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7643e8b7aa32ebf299048417a94432dc.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/213e22890204937a5dded4436369390f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a00bfec58504040151e3e2101be245a8.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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6828a1cf75f19bb74a0e0490bd65c168.png)
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372次组卷
|
2卷引用:广东省江门市第一中学2023-2024学年高二下学期第二次段考数学试题
名校
4 . 为提高学生的数学应用能力和创造力,学校打算开设“数学建模”选修课,为了解学生对“数学建模”的兴趣度是否与性别有关,学校随机抽取该校30名高中学生进行问卷调查,其中认为感兴趣的人数占
.
(1)根据所给数据,完成下面的
列联表,并根据列联表判断,依据小概率值
的独立性检验,分析学生对“数学建模”选修课的兴趣度与性别是否有关?
(2)若感兴趣的女生中恰有4名是高三学生,现从感兴趣的女生中随机选出3名进行二次访谈,记出高三女生的人数为
,求
的分布列与数学期望.
附:
,其中
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cecd2fa8749209ba1ef51a3865ec1024.png)
(1)根据所给数据,完成下面的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b72fcdc709e77910cd36a26369648b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0298d106f2b72aadf3cffce041a25da6.png)
感兴趣 | 不感兴趣 | 合计 | |
男生 | 12 | ||
女生 | 5 | ||
合计 | 30 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
附:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2187714e660234f0b72f2b47d3ea685a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356b05e46b10ee51c3e43546d73ec96c.png)
![]() | 0.10 | 0.05 | 0.01 | 0.005 | 0.001 |
![]() | 2.706 | 3.841 | 6.635 | 7.879 | 10.828 |
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544次组卷
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2卷引用:广东省四会中学、广信中学2023-2024学年高二下学期第二次联考数学试题
5 . 已知等差数列
的公差
,
与
的等差中项为5,且
.
(1)求数列
的通项公式;
(2)设
求数列
的前20项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ce64685821c3e55c07f151996ca8c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7da2f386b78cdf6489efaa2f5820d3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78d4d35ddfc27da37f4fa38ee424e508.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6907d5edb3b722809ce1d6ec272847d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2f710b0e6ccca316852bf3a94f68135.png)
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7日内更新
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814次组卷
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3卷引用:广东省江门市开平市开侨中学2023-2024学年高二下学期期末热身模拟数学试题
广东省江门市开平市开侨中学2023-2024学年高二下学期期末热身模拟数学试题黑龙江省双鸭山市第一中学等校2024届高三第四次模拟数学试题(已下线)高二数学下学期期末考点大通关真题必刷100题(2) --高二期末考点大串讲(人教B版2019选择性必修第二册)
6 . 如图,在正方体
中,求证:
平面
.
(2)平面
平面
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d2d64b329816072434bd1121b4dc0c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e86eec8526479272d15bb3b171a46de0.png)
(2)平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba0fbd88fdb064072eedd136e9cb41ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d9a8181f7a7fe7f3fac872ce9534f15.png)
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名校
7 . 如图,某运动员从
市出发沿海岸一条笔直的公路以每小时
的速度向东进行长跑训练,长跑开始时,在
市南偏东方向距
市
,且与海岸距离为
的海上
处有一艘小艇与运动员同时出发,要追上这位运动员.
(2)求小艇以最小速度行驶时的行驶方向与
的夹角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0017113f394bc42d821adeb220857cdb.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/8c373c35755378512192cca2317d169d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/016c442622f23a530b1913b157b35e87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)求小艇以最小速度行驶时的行驶方向与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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8 . 已知函数
.
(1)求
的单调区间;
(2)求
在区间
上的最大值和最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f98a6734718002961124ca219c348872.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e354f44c5841806fdc363073abdd052.png)
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解题方法
9 . 已知曲线
在点
处的切线与直线
垂直.
(1)求实数
;
(2)求函数
在
上的最大值与最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34323b0048b5c9be015e2bac5868952c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5922bf7445caf7b332ac48d4d835ae0a.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa66623cf54b42d6d12be4c8edaa7071.png)
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解题方法
10 . 学校要从5名男生和2名女生中随机抽取2人参加社区志愿者服务,若用X表示抽取的志愿者中女生的人数,
(1)求抽取的2人恰有1个女生的概率;
(2)请写出随机变量
的分布列、数学期望
与方差
.
(1)求抽取的2人恰有1个女生的概率;
(2)请写出随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90a0722562d03a0a55a6c63e5d4cc338.png)
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