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1 . 利用反证法证明“已知
,求证:
中,至少有一个数大于20.”时,首先要假设结论不对,即就是要假设( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7267291073b77eab69d5d01383c045d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874781ab5711bff6ee8c9cbad5b3b3dc.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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2021-09-04更新
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2卷引用:陕西省咸阳市武功县普集高中2021-2022学年高二下学期第一次月考理科数学试题
2 . 2022年北京冬奥会仪式火种台(如图①)以“承天载物”为设计理念,创意灵感来自中国传统青铜礼器——尊(如图②),造型风格与火炬、火种灯和谐一致.仪式火种台采用了尊的曲线造型,基座沉稳,象征“地载万物”.顶部舒展开阔,寓意着迎接纯洁的奥林匹克火种.祥云纹路由下而上渐化为雪花,象征了“双奥之城”的精神传承.红色丝带飘逸飞舞、环绕向上,与火炬设计和谐统一.红银交映的色彩,象征了传统与现代、科技与激情的融合.现建立如图③所示的平面直角坐标系,设图中仪式火种台外观抽象而来的曲线对应的函数表达式为
.
(1)求函数
的图象在点
处的切线方程;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ad93dc19938b18b0a9a7dcfe3a7bf1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/20/74fa53e3-3998-4d0f-8f9e-266b5d590b43.png?resizew=388)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fab8a0cc6504aa4c3a38006f5394b4c2.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/714621c52d929e662febee72b9d68351.png)
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2023-10-30更新
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2卷引用:陕西省榆林市定边县第四中学2024届高三上学期第四次月考数学(文)试题
名校
解题方法
3 . “拐点”又称“反曲点”,是曲线上弯曲方向发生改变的点.设
为函数
的导数,若
为
的极值点,则
为曲线
的拐点.
已知函数
有两个极值点
,且
为曲线C:
的拐点.
(1)求a的取值范围;
(2)证明:C在Q处的切线与其仅有一个公共点;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85be63b7da7f1174c96176de8d1ecc9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85be63b7da7f1174c96176de8d1ecc9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35ce95d0450bc59111b516c56586cb78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661249bf6499017f9e5e03db3fcd93d0.png)
已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4d05faec455cea37e004e18cfb7e290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c12d99bdf82674ac9a1edceff81d54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
(1)求a的取值范围;
(2)证明:C在Q处的切线与其仅有一个公共点;
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e804ae37438267dd3a4b9c26d3d7c33.png)
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解题方法
4 . 如图1,山形图是两个全等的直角梯形
和
的组合图,将直角梯形
沿底边
翻折,得到图2所示的几何体.已知
,
,点
在线段
上,且
在几何体
中,解决下面问题.
平面
;
(2)若平面
平面
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dde327febef2331a4766a79b433cc02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dde327febef2331a4766a79b433cc02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10de2459bc376f9a3de90f74cc18ca7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cde387abe3ecba7cde65df9c58131b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74a39a7453e6994a580038828513c68c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a3b8602719b3d371bc9ec6c441bb9f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31c34b18525831f3eda7bb90be0199b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7513c5dc6e1d35f76020f8f60c95669.png)
(2)若平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3547a914468b082d8d8741b974a03190.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c897a54f2e36bc4b52fba74b41c89d2d.png)
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2023-11-24更新
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605次组卷
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8卷引用:陕西省榆林市府谷县第一中学2024届高三上学期第五次月考数学(理)试题
陕西省榆林市府谷县第一中学2024届高三上学期第五次月考数学(理)试题河北省部分高中2024届高三上学期11月联考数学试题山西省运城市盐湖区第五高级中学2024届高三上学期一轮复习成果检测数学试题江西省部分地区2023-2024学年高三上学期11月质量检测数学试题(已下线)热点6-1 线线、线面、面面的平行与垂直(6题型+满分技巧+限时检测)(已下线)第15讲 8.6.3平面与平面垂直(第2课时)-【帮课堂】(人教A版2019必修第二册)(已下线)13.2.4 平面与平面的位置关系(2)-【帮课堂】(苏教版2019必修第二册)(已下线)11.4.2平面与平面垂直-同步精品课堂(人教B版2019必修第四册)
5 . 已知抛物线
的方程为
.
(1)若M是
上的一点,点N在
的准线l上,
的焦点为F,且
,
,求
;
(2)设
为圆
外一点,过P作
的两条切线,分别与
相交于点A,B和C,D,证明:当P在定直线
上运动时,
四点的纵坐标乘积为定值的充要条件为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ac62b1ade07205ae2693ec1ab135def.png)
(1)若M是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb333b87ab3ecde430010b4dd8b371fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/097f96fa2acd52a77bd9e2d3c33f53fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8c5c639651fd0df8f041185e5c080b4.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/302981b9a0645b6439fb0febfb4b1caf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee2abc983789634e0e57db4576e45b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6f7b16d65f1b2b8bea8cf4a83fde925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bc8335f8a3b076ccd596452bad61541.png)
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2卷引用:陕西省、青海省部分学校2024届高三上学期9月联考理科数学试题
名校
解题方法
6 . 已知函数
,
,若对任意的x,y都有
.
(1)求
的解析式;
(2)设
,
(ⅰ)判断并证明
的奇偶性;
(ⅱ)解不等式:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e17afd02a58c3d3c25ac4f8cab171e24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac829d3069cf983b89b67c73544c8baf.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c15203cc22f37937619bc22b880f407.png)
(ⅰ)判断并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be55c8f2760d6db125f46691a3de48.png)
(ⅱ)解不等式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c954734b0cb6212c0e185cd910bb7338.png)
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2卷引用:陕西省西北工业大学附属中学2022-2023学年高一上学期第二次月考数学试题
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7 . 根据《国家学生体质健康标准》,高三男生和女生立定跳远单项等级如下(单位:cm):
从某校高三男生和女生中各随机抽取
名同学,将其立定跳远测试成绩整理如下(精确到
):
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
人,全体高三女生中随机抽取
人,设
为这
人中立定跳远单项等级为优秀的人数,估计
的数学期望
;
(3)从该校全体高三女生中随机抽取
人,设“这
人的立定跳远单项既有优秀,又有其它等级”为事件
,“这
人的立定跳远单项至多有
个是优秀”为事件
.判断
与
是否相互独立.(结论不要求证明)
立定跳远单项等级 | 高三男生 | 高三女生 |
优秀 | ![]() | ![]() |
良好 | ![]() ![]() | ![]() ![]() |
及格 | ![]() ![]() | ![]() ![]() |
不及格 | ![]() | ![]() |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8da45c443af7994a26ffa9d8894e7262.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37fd17f45e1149748e8b8c336f9b9ec7.png)
男生 | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
女生 | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
(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
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
(3)从该校全体高三女生中随机抽取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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2023-03-27更新
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1903次组卷
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8卷引用:陕西省宝鸡市千阳县中学2023届高三下学期第十三次模考理科数学试题
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8 . 证明题:
(1)借助向量证明余弦定理(余弦定理有三种书写形式,只证明其中一种即可);
(2)借助完全平方公式证明均值不等式:
(
和
均为正数).
(1)借助向量证明余弦定理(余弦定理有三种书写形式,只证明其中一种即可);
(2)借助完全平方公式证明均值不等式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/689f982af451283289255c87593ec338.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
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9 . 如图1,直角梯形
中,
,
,
,
为
的中点,现将
沿着
折叠,使
,得到如图2所示的几何体,其中
为
的中点,
为
上一点,
与
交于点
,连接
.请用空间向量知识解答下列问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/15/786e3828-c164-4ab2-b188-132893bc5e5f.png?resizew=403)
(1)求证:
∥平面
;
(2)若三棱锥
的体积为
,求平面
与平面
的夹角
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3753faebdc15d2d2e598d5ffc4487a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a11029ca6b4b9e7f777af0280cf163c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63b43490ca09467a4c8cd8cfe91c94e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/321f96c4f808afe67cf565ca74ae0351.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cad4595d5352b2884568a59d8d766a4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/15/786e3828-c164-4ab2-b188-132893bc5e5f.png?resizew=403)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/065f7ff90e26ff382aa7b709955ad1b9.png)
(2)若三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a3ad76c5b79648e73a91065ef847f17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6734b2bef8750392d3c5c08b5d878505.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9a814b70236a108be5d6e7ff271fe92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
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|
278次组卷
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3卷引用:陕西省榆林市神木中学2021-2022学年高二上学期第四次检测理科数学试题
名校
10 . 在
中,点
,
分别在边
和边
上,且
,
,
交
于点
,设
,
.
,试用
,
和实数
表示
;
(2)试用
,
表示
;
(3)在边
上有点
,使得
,求证:
,
,
三点共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
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