名校
解题方法
1 . 我国魏晋时期的数学家刘徽创造了一个称为“牟合方盖”的立体图形,如图1,在一个棱长为2r的立方体内作两个互相垂直的内切圆柱,其相交的部分就是牟合方盖(如图2),我国南北朝时期数学家祖暅基于“势幂既同则积不容异”这一观点和对牟合方盖性质的研究,推导出了球体体积公式.设平行于水平面且与水平面距离为
的平面为
,则平面
截牟合方盖所得截面的形状为______ (填“正方形”或“圆形”),设半径为r的球体体积为
,图2所示牟合方盖体积为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/625dbbd5d5f2617b7c53acdb936b1d07.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82d38e62ba27b42d838c51a6e0a88e40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4764374bd2fb78e59cd0b283637baeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63055a5d6916f99d07fede49120753f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/625dbbd5d5f2617b7c53acdb936b1d07.png)
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2 . 已知非零向量
,
,
满足:
,
,
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/498e31243c96cee074159c14530ce929.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d366d8fbb7258ee051f49977441e14a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2cc1f1ef9ad32eef62aefa4c3fc3d6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15f9df42552e714a5212708a04e160f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/156cf6cd7ae0948944d53a5f3eb53bb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93f05d10b1b3ab120364d061534e150b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/498e31243c96cee074159c14530ce929.png)
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解题方法
3 . 设
为复数,且
(
为虚数单位),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2d67321ace1e6b3be0fc0e5e8130022.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ff788f86edb479a0c1a75959f376bfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2d67321ace1e6b3be0fc0e5e8130022.png)
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4 . 已知平面向量
与
共线,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfd989ddfbed4a6aa18ea57734522153.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a93944d382b73bb691719b868fd69a54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f47f03ca8c235959f5dba54c6c83e7df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfd989ddfbed4a6aa18ea57734522153.png)
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解题方法
5 . 在边长为1的正六边形
中,记以A为起点,其余顶点为终点的向量分别为
,
,
,
,
,以D为起点,其余顶点为终点的向量分别为
,
,
,
,
.记
,
为
的两个三元子集,则
的最大值为______ ;
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d344174267f996c7cefecfd6985d380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2e900404ba71110c5861ced9634646f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67dbe2e19d8960789ec873b687998b58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99184832b21db4d92cb2b162679a4948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bb1edfbda49a6ef314578efd3224928.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65c2df57a4cdae59f1b703eff14d32b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e53874673ab14f92e915f7d4a988623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f09bac35174ec1507e3839eb54f815.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aad140abd985de3f688b0bc1fa240a00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f93809be489619bb80adc4790dda4d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edbe3d8318bb799cbef82a9b829cc325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85d544b8f6fdb0865c8cb0d4c8bcc46e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f37ab30f2e841f260e46be2714954d0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c9f3889ba128190cc756a237b3c1977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b966a03263abac15ba2db3477540fc46.png)
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6 . 已知平面向量
,
,
满足
,
,则
与
的夹角为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80d769ab21af5365d3c00dc617375ce4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/508e84534d4f30e7c5bcadeac1e298f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/203dade41d23f57cf5c045b9fc82d64c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bb91b944a28614d0977532e4d18903.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dd1f0e20049dcb0a3a817e4399048ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d366d8fbb7258ee051f49977441e14a2.png)
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7 . 复数
在复平面内对应的点的坐标为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8648b214f299170fcd754810f45ee82.png)
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8 . 若函数
在区间
上为单调函数,且图象关于直线
对称,则函数
的最小正周期为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4794be662f783b33ee48c6628e1d1628.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0efa0e8be1515d920b50fee0e96fc6fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7953aeb8547e4ecd956e5f0254c2a718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
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解题方法
9 . 已知平行四边形ABCD中,AC与BD交于点O,E为线段OD的中点,AE的延长线与CD交于点F.若
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ace1f0505dac8a8ea7bca12dc94eb41.png)
______ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e68028d33ab8a11b4cd40ecfdd253b1d.png)
______ (答案用含
,
的式子表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87919b376c1de16dc744b04d72877183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8682ba80b5d1e124e98c8661c753a19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ace1f0505dac8a8ea7bca12dc94eb41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e68028d33ab8a11b4cd40ecfdd253b1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
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10 . 若函数
的部分图像如图所示,
,则
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/288def5c6256ba51bd1e7d52b3ee7c59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d78decbdafa1659b49cd3cd1e4c9b53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
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