解题方法
1 . 桌面上两两相切地摆放着四个球,球心依次为点
,且半径相同的球与桌面相切,记它们的半径分别为
.已知
,则最上面一个球离桌面的距离![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c98c59cd4749afdd21e73529fc84323.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2218a80f8c5c7880090aafb28c760a04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c0b965595229c2fcbcd50541ea8569e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bce3d5ea2a2e7eb1aeae4bac5346332.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c98c59cd4749afdd21e73529fc84323.png)
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2 . 定义运算“
”满足:
为从向量
按逆时针方向到向量
的夹角
,向量
垂直于
所确定的平面,当
时,其垂直平面的方向向上;当
时,其垂直平面的方向向下,下列说法一定正确的有__________ .(填序号)
①
;②
;③
;④![](https://staticzujuan.xkw.com/quesimg/Upload/formula/872c64b1cc0dd66a0d4f4bb437d6d50b.png)
;⑤当
时,
;⑥
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ad8d792188dba9a934ff4e034e6eedd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df3dfb49ad8bb91cc8a7488ff6f2b48d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2624a5cff11216693d7ecc7a8661c945.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18020f8f5d3ec9988d05d367b9858fe7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c82ef71de70fda2db4b882e57bacbb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef7b23414e09377caa0d7fe2e6a5dec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/932413ffae6ac40f704ef9606bdc06a8.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99a5dd9c18ec68a4e1af7a86abb44d68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e14f57826c118e5772513c03354d526b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17be5a6799ad26c91a8fd58448a3ce03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/872c64b1cc0dd66a0d4f4bb437d6d50b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea925583bb5b08798bf5c0f9f43aad3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91b958a367fa2f08b6202a5a6ebf5e9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d94b1db1042e2b0dbe7efb3a6e33190.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fb176d7f8f34762a45717df116878ed.png)
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3 . 给定集合
,若集合
,且对集合
中任意两个元素
、
,不妨设
,都有
或
,则称集合
具有性质
.假定集合
满足形式
,则具有性质
的集合
中的最小元素![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bbb27a21cea9706f1d7688dc8c9f352.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9533c90de1b697cc9e78adaf134fb972.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9d0d79fd0efdb59c9ca65bfae4a0861.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b21208364124b5c477b2ff8df1c2e8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f68a3f6fa83a180dd6cc40e227df87b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7c9158f76cef11b05915874095d22c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acde89240882f0af5169847117b9df77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bbb27a21cea9706f1d7688dc8c9f352.png)
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解题方法
4 . 在平行四边形
中,
为坐标原点,
,
,点
在函数
的图象上,则实数
的值为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3e3be5041c6034f69dcee2614c80e88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f209e05f98c37ff59a3cc5499710e40f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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5 . 由单摆实验得到如图所示曲线,现用正弦函数模型
来拟合,其中
.已知
,则在实数范围内
的最大值为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3e115f70dedd6eb3739913849679b8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb2d7841114daf4fd38e14e2d2b591c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dff761780a5cacea697eead555a112b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/354e0fcc5ce9a3446aeef0eb9a1d3c24.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/24/45f9fd9b-67fc-4017-afac-16f11fe19983.png?resizew=148)
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6 . 记不超过
的最大整数为
,若集合
,集合
,当
或1或
内的无理数时,
;当
(
为既约真分数)时,
.若
(表示
中任意一个元素都大于
中任意一个元素),则
的取值范围是___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3e8275188a5c3cfd655740915d40d31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3395a7ea8093c65c8f6daaf0a67bf244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.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/d78670f8bfe3c43f69f205b4035cc5b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a507c062709cfe2f218896247461c7d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35888de80226a81c4c80fecb8f4a9337.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca51be437b1a97ca92aa1159ab71102c.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/0a6936d370d6a238a608ca56f87198de.png)
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解题方法
7 . 设
,
,则最接近于S的整数是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15e3c33e1cb283d72fe1a6cbc27eb462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0228777a871cf7f71d46fe39f6aa2fd7.png)
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8 .
的整数部分是______________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52f61f4f195149d7ff7eb304649c67a8.png)
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9 . 对
,定义符号函数
:当
时
;当
时
;当
时,
.记点集
,点集
,点集
围成的区域的面积为______________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5341631cc2c6c07404fd05cfd5f6542a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1896f977c8276c08e3cc881724a14256.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e541ea2f855f981c96207070683d388.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff04454d9182598682e1371d749ca35b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a6363a88ca7ac2a145b3d924d4b1379.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18bb3252086e273b9eb3f017811b2ae7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fb738374fe3bcb30d9d7f250298c8c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
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10 . 已知集合
,且
,给出下列命题:
①满足
的集合
的个数为
;
②满足
⫋
的集合
的个数为
;
③满足
⫋
的集合
的个数为
;
④满足
⫋
⫋
的集合
的个数为
.
其中正确的是______ .(填上你认为正确的所有命题序号)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ab20b8212340e97321f421a74de3af4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eae51f0310b87cde2e206643e9d25a5.png)
①满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b24b70c88f46920bc6c3b0ed8d900f90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aa3ccc967a75d0739e71f8df80b5b76.png)
②满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/366749e18358402f70feb02785251caf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c89ae078ee147741e42b344b3d20776b.png)
③满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22882310035db6d3d953e060b0c91733.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa3e84efcba619b04bc1a89e68f1c507.png)
④满足
![](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/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d67e9f11658eb229c3f75eb3ff71ac23.png)
其中正确的是
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