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1 . 已知函数
,点
、
是函数
图象上不同的两个点,设
为坐标原点,则
的取值范围是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/541ec913428703d4cae2476b147ce1a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d63711c480473cbc27a06cdb82ddd000.png)
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解题方法
2 . 古希腊数学家阿波罗尼奥斯用不同的平面截同一圆锥,得到了圆锥曲线,其中的一种如图所示.用过
点且垂直于圆锥底面的平面截两个全等的对顶圆锥得到双曲线的一部分,已知高
,底面圆的半径为4,
为母线
的中点,平面与底面的交线
,则双曲线的两条渐近线所成角的余弦值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae890f9e8b32aa53a54158f24f4a87bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab00e0cff0876c4183a47f1272cf9928.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2024·全国·模拟预测
3 . 已知椭圆
:
与双曲线
有相同的左,右顶点A,B,过点A的直线l交
于点P,交
于点Q.若
为等边三角形,则双曲线
的虚轴长为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/811cdc6e776b3fc15228af63eb03b7b0.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/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c495b8fd7f7bb21c177c9d50fbf6919.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
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4 . 如图1,某景区是一个以C为圆心,半径为
的圆形区域,道路
成60°角,且均和景区边界相切,现要修一条与景区相切的观光木栈道
,点
分别在
和
上,修建的木栈道
与道路
,
围成三角地块
.(注:圆的切线长性质:圆外一点引圆的两条切线长相等).
为正三角形时,求修建的木栈道
与道路
围成的三角地块
面积;
(2)若
的面积
,求木栈道
长;
(3)如图2,若景区中心
与木栈道
段连线的
.
①将木栈道
的长度表示为
的函数,并指出定义域;
②求木栈道
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78e3dce65f4583f209cc69eed1674341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4819c39c281427826e1b3f7a4c2b720.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4819c39c281427826e1b3f7a4c2b720.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c63cb79d595e99b19f17ad71de6eae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(3)如图2,若景区中心
![](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/414ef209e6ce6428bd358eafd74ddaa1.png)
①将木栈道
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
②求木栈道
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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2024高三·全国·专题练习
5 . 在
中,内角A,B,C对应的三边分别为a,b,c,且
.
(1)若
,
,求
的面积;
(2)若点D在线段CB上,
于点E,且
,当
最大时,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ab30ebcc181d35763900e73ec1783ba.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6de1d395e6c48c0676a1488a299479d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4ef6f920cf01e61596caa2243af1619.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(2)若点D在线段CB上,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32c38dfd14dde969702dff97ef2270f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90fbef4d8ce69016bc2e77783e617a97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a8698607a4ad7635bb22e99760314a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5201fc26d013f6fb889933c0e32f5c53.png)
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解题方法
6 . 如图,正方形
的边长为
分别为边
上的点,则以下错误的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c850a0ec73c3fba08a6764a7bc2209c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/630754333e7043c573d0ecdb64cf1246.png)
A.若![]() ![]() ![]() |
B.若![]() ![]() ![]() |
C.若点![]() ![]() ![]() ![]() |
D.![]() ![]() |
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7 . 以
为钝角的
中,
.
(1)若
,且
,
,求![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942a42bbc5abc30793245ed2965f4a56.png)
(2)若
,当角
最大时,求
的面积
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f8eeeea1c9652cacce976f8129cf520.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bd069b35ea935375e93da13b91b6b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d18d62f2b8c06e20eb31d2007ae66cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b751e33ead651932d2145090888e7b90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942a42bbc5abc30793245ed2965f4a56.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0958c17b0d01520cbfe4528f16b45bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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解题方法
8 . 设
,
,
,
,若满足条件的
与
存在且唯一,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59f07029c97c324d32f258f18a96654a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd61d59ee2dc6cc43b25b7ee39ca58d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ec9920ed72728ab3fc89c2e3e68f136.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/8640aafe54018ba013295b68e2946a7a.png)
A.![]() | B.1 | C.2 | D.4 |
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解题方法
9 . 同余定理是数论中的重要内容.同余的定义为:设
且
.若
,则称a与b关于模m同余,记作
(“|”为整除符号).
(1)解同余方程:
;
(2)设(1)中方程的所有正根构成数列
,其中
.
①若
,数列
的前n项和为
,求
;
②若
,求数列
的前n项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4991ae7c93a141bf73ce7f0b6b7fd7b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e34f42b3be15518c29e3689c9fe6d6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e709c6565f8241310b97af5e0c831778.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f608dd253088da169fb57ad5d1f525c.png)
(1)解同余方程:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/657b55d01c91799ec194df07eea1808e.png)
(2)设(1)中方程的所有正根构成数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c002c44d45907aad22da19859193270b.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e39834a599932f7f88a700cc36723a18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46a1a2133477cd27eed4a945a05d52c7.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8080993903f4969c2dac4a3e01b7123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307cd6a77de16aff5ab0defe75866ccc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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2024-02-28更新
|
2027次组卷
|
4卷引用:压轴题07三角函数与正余弦定理压轴题9题型汇总-1
(已下线)压轴题07三角函数与正余弦定理压轴题9题型汇总-1江西省红色十校2024届高三下学期2月联考数学试卷辽宁省沈阳市辽宁实验中学2024届高三下学期高考适应性测试(二)数学试题(已下线)第2套 全真模拟篇 【模块三】
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解题方法
10 . 已知双曲线
的离心率为2,左、右顶点分别为
,右焦点为
,
是
上位于第一象限的两点,
,若
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef66f4832adc43902055a7e6d258037.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00442d96d695db2c58bf1fb7165fca94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18290a33d540589705521b0c2bdd3914.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/570fa13a80c87d908c3b97987d45201e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b750c2c4bf7414890b336d4e70de1a5d.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2024-02-17更新
|
1426次组卷
|
5卷引用:专题07 双曲线与抛物线(讲义)
(已下线)专题07 双曲线与抛物线(讲义)(已下线)【讲】专题7 解三角形与其它知识的交汇问题河北省金科大联考2024届高三上学期1月质量检测数学试题吉林省通化市梅河口市第五中学2024届高三下学期开学考试数学试题安徽省合肥一六八中学2024届高三“九省联考”考后适应性测试数学试题(三)