解题方法
1 . 我们知道,在平面中,给定一点和一个方向可以唯一确定一条直线.如点
在直线l上,
为直线l的一个方向向量,则直线l上任意一点
满足:
,化简可得
,即为直线l的方程.类似地,在空间中,给定一点和一个平面的法向量可以唯一确定一个平面.
(1)若在空间直角坐标系中,
,请利用平面
的法向量求出平面
的方程;
(2)试写出平面
(A,B,C不同时为0)的一个法向量(无需证明),并证明点
到平面
的距离为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1dab74e16403e8131f9f5b2a74f3a84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f41c46212d6f61fca9ce215a477ea1d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdfc3eef2f592a4e93a6968c7f31e32f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e463b86ed390c317de2383840fde5df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a24f3197942ff7bd44f44651dd9123b2.png)
(1)若在空间直角坐标系中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3ad64b23e508734de034ce16e1ebbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22ce50ba5e349425274f05d46d120a74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22ce50ba5e349425274f05d46d120a74.png)
(2)试写出平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a46e2fbcd9ba92ca62a67fef9d9652db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab9f353152c7f589c0caf5f964f803ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39f20004bf3d4eb52ec732d8acc65672.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e878d6f51b5830bd59f0d44aa5d8b38.png)
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2 . 为积极参与校运动会,某班要从A,B,C三位同学中任意抽取两位参加400米比赛.
(1)请写出不放回简单随机抽样的样本空间,并求出抽中A的概率;
(2)若抽中的两位同学参加400米预赛后能进入决赛的概率都是
,请求出两人中恰好一人进决赛的概率.
(1)请写出不放回简单随机抽样的样本空间,并求出抽中A的概率;
(2)若抽中的两位同学参加400米预赛后能进入决赛的概率都是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
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3 . 已知点
是双曲线
与椭圆
的公共点,直线
与双曲线
交于不同的两点
,
,设直线
与
的倾斜角分别为
,
,且满足
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/20/a8ccb6e8-8ffc-42f7-a3a8-3d79b2ebce40.png?resizew=212)
(1)求证:直线
恒过定点,并求出定点坐标;
(2)记(1)中直线
恒过定点为
,若直线
与椭圆
交于不同两点
,
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f13bf66fc845b115de4ec45b4be0e23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02d52ad74a134ca8a65cb9ad8ef5cbd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40499eaaf1e6ac9d8da30ccbbcaf5aac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.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/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f478abeeb4da23121b652cf907972d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/20/a8ccb6e8-8ffc-42f7-a3a8-3d79b2ebce40.png?resizew=212)
(1)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(2)记(1)中直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25a183c9577e8730b7962f75fc3f6e0c.png)
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4 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef1eba14fe54a00d1db2086b442a9117.png)
(1)若
,求函数
的单调区间;
(2)若存在
,使得
,设函数
的图像与
轴的交点从左到右分别为
,
,
,
,证明:点
,
分别是线段
和线段
的黄金分割点.(注:若线段上的点将线段分割成两部分,且其中较长部分与全长之比等于较短部分与较长部分之比,则称此点为该线段的黄金分割点)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef1eba14fe54a00d1db2086b442a9117.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070054c0b4182ab7399ed56925844e93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e9461464c917e18f35ac6eee68b800c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.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/8455657dde27aabe6adb7b188e031c11.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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
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5 . 已知
,若关于
的方程
恰有三个不同的解,则满足上述条件的
的值可以为_____________ .(写出一个即可)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d063529271d7c206a7a1e5c1d0ee12d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b028fadf04ca478c5bc16de707c3d6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
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名校
6 . 已知实数
,
,且
,若
,则
可能等于( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6455e38ff53ede2508e4d9cb23f0b86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91a871ef7bf13de3e15489d65b57a3cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2958030ec9d7543dda1f529593a915e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efdc0e0ca559f0f1af6127545f356fa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9f2416d1f75a45a314331146550832e.png)
A.0.5 | B.1 | C.2 | D.3 |
您最近一年使用:0次
2022-11-17更新
|
464次组卷
|
2卷引用:浙江省台州市2023届高三上学期11月第一次教学质量评估数学试题
名校
解题方法
7 . 水平放置的碗口朝上的半球形碗内,假设放入一根粗细均匀的筷子,在力的作用下,筷子在碗内及碗沿可无摩擦自由活动直到筷子处于平衡(即筷子质心
最低).此时若经过筷子作与水平面垂直的轴截面如图,其中半圆
(表示半球碗截面)半径为1,线段
(表示筷子)长为3,则线段
的中点
离碗口平面距离最大时,直线
与水平面夹角的余弦值为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/20/864cef99-e47a-488e-bd63-832c3d317440.png?resizew=184)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/20/864cef99-e47a-488e-bd63-832c3d317440.png?resizew=184)
A.![]() | B.![]() | C.![]() | D.![]() |
您最近一年使用:0次
2022-11-17更新
|
499次组卷
|
2卷引用:浙江省台州市2023届高三上学期11月第一次教学质量评估数学试题
名校
8 . 已知正方体
的棱长为1,点P为侧面
内一点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
A.当![]() ![]() |
B.当![]() ![]() |
C.当点P到平面ABCD的距离等于到直线![]() |
D.当![]() |
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2022-11-09更新
|
1143次组卷
|
6卷引用:浙江省台州市书生中学2022-2023学年高二上学期期末模拟数学试题
名校
解题方法
9 . 给出下列命题,其中正确的命题是( )
A.若![]() ![]() |
B.若![]() ![]() ![]() ![]() |
C.过点![]() ![]() ![]() |
D.直线![]() ![]() ![]() |
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2022-10-19更新
|
660次组卷
|
2卷引用:浙江省台州市玉环市玉城中学2022-2023学年高二上学期第一次月考数学试题
名校
10 . 已知二次函数
的图象上有两个不重合的点
,若
,则点
的轨迹是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bebe4effa790eb7d86d359aecbc051d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5ec6f5b958b3d00befdc2b2981774e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1292c47f62023b747f1a4bd615c75284.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82aaa597a5aa6176863eda3fdf83e181.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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