名校
1 . 设关于
的方程
的两根为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
(1)若
,求
的值;
(2)若方程至少有一根的模为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2914555b73c954e85aa05449d767eee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a39b9354d936ff25d65386d56671d56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)若方程至少有一根的模为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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2024高一下·全国·专题练习
2 . 如图,
不共面,且
且相等,
且相等,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0247b63d3f3cc573a24ba0036899c96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e725362f254f8f39245bd3ed311d6d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe2bb400160da0da6df6abbaa5fa0f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca5c70782c77be805de33099687afc07.png)
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名校
解题方法
3 . 如图,已知斜三棱柱
中,底面
是正三角形,
,点O是点A1在下底面内的正投影.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b41ee0f7694690b8ca82158bf210f49.png)
(2)若点O是
的中心,求高度A1O;
(3)在(2)的条件下求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40dcce3e3b82ddca1e9f021ef66b5684.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b41ee0f7694690b8ca82158bf210f49.png)
(2)若点O是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(3)在(2)的条件下求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32bbdf5dbf9df96742624ada95c36146.png)
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4 . 设
是方程
的一组解,计算:
(1)
;
(2)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad0086651e05c716cec1ba31cf09f1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c82e7d9f4f7ace849e09e9adcb786b7f.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/821dc5767ac7967687426e4b9c46c918.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b859b72b8127df011d0b3032d2c3ff3e.png)
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名校
5 . 某学校附近有条长500米,宽6米的道路(如图1所示的矩形ABCD),路的一侧划有100个长5米,宽2.5米的停车位(如矩形AEFG),由于停车位不足,放学时段道路拥堵,学校安保处李老师提出一个改造方案,在不改变停车位形状大小、不改变汽车通道宽度的条件下,可通过压缩道路旁边绿化带及改变停车位方向来增加停车位,记绿化带被压缩的宽度
(米),停车位相对道路倾斜的角度度
,其中
.
,求
和
的长;
(2)求d关于
的函数表达式
;
(3)若
,按照李老师的方案,该路段改造后的停车位比改造前增加多少个?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/990a3680645f22737907fc1b5b10d99a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eb60187001a6a7bbf887d2a233e5e26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09463dc187d5b2574170f1566ac4c382.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9088e31e00eaa4e95c1e6717b0c82cdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827be85b8f921007a5cf9edb64bd3491.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c495fd86e62c3090caba0702bbcffd7.png)
(2)求d关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3f74946c4e57365e0de85d277ba8d2a.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f2572192cc7ca046e9a3155ef3e56a.png)
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名校
6 . 在平面直角坐标系中,直线l与抛物线
交于
、
两点,点
在抛物线上,
是
轴上一动点.
(1)求直线l和抛物线
的解析式;
(2)如图1,
为抛物线
上位于直线l下方一动点,过
作
垂直于x轴交直线l于
,当线段
长度最大时,求
的最大值;
(3)如图2,
为抛物线
的顶点,y轴上是否存在点M,使得
?若存在,求出点M的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6e2f3a811787b14bcabdb6a4cd49da9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0898eaf39239999c25447d0512c95f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/585739fbbb741c51cdecdb39c30c0ebc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54e20ba7e4b42a007bf8c2846bcab995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/9/874cbdb3-fff4-4ddc-be75-e24328eb7b84.png?resizew=299)
(1)求直线l和抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)如图1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.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/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82bbfaf8a2aa02ec01210b8b6fa8d13.png)
(3)如图2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb02779236aff6a174a29dff0b2524f0.png)
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名校
7 . 如图,圆内接四边形ABCD中,G为对角线AC、BD的交点,过点D作
交AC于E,且
,F在线段GD上,且
,连接CF.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/7/4bd1187a-ba53-4a8b-9c6f-9dcef6fc6fb6.png?resizew=170)
(1)求证:
;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebe7eaf967808dad0a184eeedfa27721.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c16639200620d92e1a8c228b581dc57c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7953310f793ba2a127772d593b41aff0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/7/4bd1187a-ba53-4a8b-9c6f-9dcef6fc6fb6.png?resizew=170)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26f1b8f2d3052769ed15da4dcdaad203.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb4a99e41e4a30238de487e733e01ef.png)
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8 . 在平面直角坐标系中,直线
交曲线
于A、B两点,交x轴于点C,过点A作
轴于点D,且
,连接BD.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/7/add631ea-2585-4890-9d86-c47bcc8081d0.png?resizew=152)
(1)若A点的坐标为
,求线段AB的长;
(2)若
,且
的面积为3,求k的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1133fb905c01a6866c2e98e91d35d74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03517310ea1e913f709753592ac65ffd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5d3a0273d1f3046dfad2086d0df56c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a90966e8334b3288f4fd3589a6ec45d4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/7/add631ea-2585-4890-9d86-c47bcc8081d0.png?resizew=152)
(1)若A点的坐标为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bedaa50ecdc072fae79a33763780f52c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98fc4043b2b7fbb5a0608d504629b206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb961bd7db3adb76af2d4cedb611bd7.png)
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9 . 已知二次函数
(a为常数).
(1)当
,
时,求
的值;
(2)当
时,二次函数
的最小值为
,求a的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/822e1d5acaa60803c97eec89a903183d.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65397f11ea8af736f38debadf420c4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2a7df955fc17e92fd86302f8c34664a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f7f8363ae7e8fb0020856303ad61514.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55a12a30935c79ae769818043a496f53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/822e1d5acaa60803c97eec89a903183d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3edbd40e04e2a943051fa83d6e511add.png)
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10 . 如图①,在
中,
为
边上的中线(
),以
为直径的半圆分别交
于点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/31/3d2eef6d-ce51-4adf-a391-0f364092cbf5.png?resizew=288)
(1)求证:点
为
的内心;
(2)如图②,过点
作
的垂线交
的延长线于点
,试判断
与
的大小关系,并说明理由;
(3)在(2)的条件下,若
,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7157312ed5d2f11531b6707c3486db81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1811c0f86073a2de001cf9b741aacb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97bde5efa645a4c1ed6874088400d6a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/31/3d2eef6d-ce51-4adf-a391-0f364092cbf5.png?resizew=288)
(1)求证:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
(2)如图②,过点
![](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/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
(3)在(2)的条件下,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f4a01a1f3429f72fcdaeed62a5e2e63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
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