解题方法
1 . 费马原理是几何光学中的一条重要定理,由此定理可以推导出圆锥曲线的一些性质,例如,若点
是双曲线
(
为
的两个焦点)上的一点,则
在点
处的切线平分
.已知双曲线
的左、右焦点分别为
,直线
为
在其上一点
处的切线,则下列结论中正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62180fb2b68724b7b0f4f8337496c12a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6030294837c740b4fe4bb00162137e38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83d58e6b21b696adb73c986b0b2cdb6a.png)
A.![]() ![]() |
B.若点![]() ![]() ![]() ![]() ![]() |
C.直线![]() ![]() |
D.延长![]() ![]() ![]() ![]() ![]() |
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名校
解题方法
2 . 已知正四面体
的棱长为4,点
是棱
上的动点(不包括端点),过点
作平面
平行于
,与棱
交于
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b971ff1b6f62b36cf0b9a641fc419458.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942daed6ad022ec1244412405250ec8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d5559854fa31f00c55a9fe8c66de6d9.png)
A.该正四面体可以放在半径为![]() |
B.该正四面体的外接球与以![]() ![]() |
C.四边形![]() |
D.四棱锥![]() ![]() |
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名校
3 . 设无穷等差数列
的公差为
,集合
.则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/040b2d17eb8cc9c82983da549afd2663.png)
A.![]() |
B.当且仅当![]() ![]() |
C.当![]() ![]() |
D.当![]() ![]() ![]() ![]() |
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解题方法
4 .
与
交于
为曲线
上的动点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f65a3fc617112ff98e769691a7a52da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34801f6a91fcd4d9ec73cecf21e3504c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fc44299a4ce82839a8bc589c3d706e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edeed0665fc1a8e99ceb7911fe137390.png)
A.![]() ![]() ![]() |
B.![]() |
C.存在点![]() ![]() |
D.![]() |
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名校
解题方法
5 . 已知椭圆
的左焦点为
,
为
的上顶点,
,
是
上两点.若
,
,
构成以
为公差的等差数列,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e88ab2f7f0cc281e62fbc7e920e0cf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33b315a25c667233befb2a1d542eb59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a052a4645628e5ba018f5e2f17c1138.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4428dc4e1db8f04c28ba1a4e3b3d2bc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
A.![]() ![]() |
B.当![]() ![]() |
C.当![]() ![]() ![]() ![]() ![]() |
D.当![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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6 . 如图,已知
,
分别为
两边上的点,
,
,过点
,
作圆弧,
为
的中点,且
则线段
长度的最大值为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d7b2fe01a33c4825f9974ed9663a99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56150248dd4b787a2013311e4737e93f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91f1c40ebcc993a5778478154c124f0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5a6b11637ea396e989ab32d780e745b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0284e9ec571566921b9071f678f28638.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84fb85c78522640bcaf889f89ad89de9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/14/a707c65d-04d1-4c0b-9f41-2e1f244c0da5.png?resizew=181)
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名校
7 . 如果函数
存在零点
,函数
存在零点
,且
,则称
与
互为“n度零点函数”.
(1)证明:函数
与
互为“1度零点函数”.
(2)若函数
(
,且
)与函数
互为“2度零点函数”,且函数
有三个零点,求a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1304d260fae136e84bf9178c25e4ced3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
(1)证明:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdc0bd852c2cacb2f553cc27d3717e36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cd0bc7729ae70587ce0e202f249436.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2237a15d514d2f506a6906dc8495242.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b70f8691af2a1d287aa5c476ede5e7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc00264fd5eee13605ebc24b77a3393b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6311536db2518323f2fee73089ea2325.png)
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8 . 将正方形
的边
绕点
逆时针旋转至
,记旋转角为
.连接
,过点
作
垂直于直线
,垂足为点
,连接
,
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/23/86367bb7-74ca-4052-a1c0-d24f5046afed.png?resizew=244)
(1)如图1,当
时,
的形状为 ,连接
,可求出
的值为 ;
(2)当
且
时,
①(1)中的两个结论是否仍然成立?如果成立,请仅就图2的情形进行证明;如果不成立,请说明理由;
②当以点
为顶点的四边形是平行四边形时,请直接写出
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/943712a5e96b16cc15d775cc4687237e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6655cc150ddc9deba2254780984d0024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6655cc150ddc9deba2254780984d0024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f85c133d0a322e4012f7c0eb7f4c626c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/23/86367bb7-74ca-4052-a1c0-d24f5046afed.png?resizew=244)
(1)如图1,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97cf714ffb3fd5917a76b191640b55fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c4851109ebad7d40fd6e8c0ed91cde6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbaeec6b78ac2dce4b20c3fee0139bc6.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/783d872617eed2bbd474035156b6caaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d15b7c09aac2543b8615aca6b62ee36d.png)
①(1)中的两个结论是否仍然成立?如果成立,请仅就图2的情形进行证明;如果不成立,请说明理由;
②当以点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96e723dd679f5659390e39777911c58c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ff7201050b9d841420f6389dfd4663f.png)
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解题方法
9 . 如图所示,圆锥的轴截面
是以
为直角顶点的等腰直角三角形,
,
为
中点.若底面
所在平面上有一个动点
,且始终保持
,过点
作
的垂线,垂足为
.当点
运动时,
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/31/091a2b5e-b4d4-4ebc-98f6-a796eb74fba6.png?resizew=172)
①点
在空间形成的轨迹为圆
②三棱锥
的体积最大值为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d7abf02717d6e59d8a64a65a87c412.png)
③
的最大值为2
④
与平面
所成角的正切值的最大值为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dee14db57f0c762aad845cf5b4a243c0.png)
上述结论中正确的序号为( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b80ee363635d73f601654339028daec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76491e815433aee93310af8066f45e76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/31/091a2b5e-b4d4-4ebc-98f6-a796eb74fba6.png?resizew=172)
①点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
②三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0525edf8863c1dde10c5cd83916c05c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d7abf02717d6e59d8a64a65a87c412.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afb717b62d5d4128157417ecb62f7f7e.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdae78f4d3b8d8213ac3ac9a9567eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dee14db57f0c762aad845cf5b4a243c0.png)
上述结论中正确的序号为( ).
A.①② | B.②③ | C.①③④ | D.①②③ |
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6卷引用:河南省济源市第一中学2022-2023学年高二下学期开学实验班数学试题
河南省济源市第一中学2022-2023学年高二下学期开学实验班数学试题安徽师范大学附属中学2021届高三下学期5月最后一卷理科数学试题(已下线)热点08 立体几何-2022年高考数学【热点·重点·难点】专练(新高考专用)(已下线)专题23 立体几何中的压轴小题-1辽宁省本溪市高级中学2023-2024学年高三上学期高考适应性测试(一)数学试题(已下线)第二章 立体几何中的计算 专题一 空间角 微点4 直线与平面所成角(二)【基础版】
名校
10 . 黎曼函数
是由德国数学家黎曼发现并提出的,在高等数学中有着广泛的应用,
在
上的定义为:当
(
,且
,
为互质的正整数)时,
;当
或
或
为
内的无理数时,
.已知
,
,
,则( )注:
,
为互质的正整数
,即
为已约分的最简真分数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9966dfe9109671c587892bd32f0b6699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9966dfe9109671c587892bd32f0b6699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ec4622d03afb89bddc6ae300753322d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/342b666f58972815306763d9ccc3bc6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18a15214d0110a87d0f56c802f6855b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df21bcb07cb594d6614230b2317942f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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