1 . 如图是体育公园步道示意图.从A处测得点B在北偏东
,测得点C在北偏东
,在点C处测得点B在北偏西
,
米.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/19/ba937be0-418e-491a-9371-8d9f206d1e3f.png?resizew=173)
(1)求步道
的长度(结果保留根号);
(2)游客中心Q在点A的正东方向,步道
与步道
交于点P,测得
,小明和爸爸分别从B处和A处同时出发去游客中心,小明跑步的速度是每分钟
米,请计算说明爸爸的速度要达到每分钟多少米,他俩可同时到达游客中心.(结果精确到0.1)(参考数据:
,
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a4c9b697e3df2b5469b71d3b5e47a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f194bc021c012d8f373535be3c249e23.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/19/ba937be0-418e-491a-9371-8d9f206d1e3f.png?resizew=173)
(1)求步道
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
(2)游客中心Q在点A的正东方向,步道
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb6ede9761b5b90f8dc137708e1ee90f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9caf47639ff9218ef27068ee10c265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bda23637a152ef2a22245957e16cc5c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52881be613aa404e553da30d8987cfad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47bb3f35e3db7c1f3a3dd3eb20151b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dad09268b7cb8bfcbea010cb6d2a29e.png)
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2 . 已知函数
与
图象的交点为
,
,
,则不等式
的解为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/498bcfd1aac03057dcd0adef2dd113a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c81daf249bc88275baa32ffb4643db1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dc667996e571285a39c9df6c2260d8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af3fb7e3371de42aeef26f911d564cad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4fc75de69dcc72f51f5688c4ff10fdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca612f8a2677143f31befb7d87ac9bc6.png)
A.![]() ![]() | B.![]() ![]() ![]() |
C.![]() ![]() | D.![]() ![]() |
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3 . 如图,正方形
的对角线相交于点
,点
又是正方形
的一个顶点,而且这两个正方形的边长相等.下列结论正确的有( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/16/054e10da-36ce-424b-9df0-8a0c1089b351.png?resizew=156)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4635a4e561d862638f6818e85e74bfd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/16/054e10da-36ce-424b-9df0-8a0c1089b351.png?resizew=156)
A.![]() |
B.正方形![]() ![]() ![]() ![]() |
C.△BEF周长的最小值为![]() |
D.![]() |
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4 . 已知:如图,在平面直角坐标系中,直线
与x轴交于点A,与y轴交于点B.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/16/85ff9999-3e91-4014-9125-81bbc29a2173.png?resizew=288)
(1)若点C为直线AB上第二象限的点,且
,求点C的坐标;
(2)在(1)的条件下,直线AB上是否存在点Q,使得
OAQ与
BOC相似,若存在,求出点Q的坐标,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca621dc42304f844ac15dad7062d3959.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/16/85ff9999-3e91-4014-9125-81bbc29a2173.png?resizew=288)
(1)若点C为直线AB上第二象限的点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede577287237c6ce64ea220d6fb0ee0f.png)
(2)在(1)的条件下,直线AB上是否存在点Q,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
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5 . 如图,在平面直角坐标系中,已知抛物线
与直线AB相交于A,B两点,其中
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/15/0ee44fdc-b369-4357-8839-fc3f669d6dac.png?resizew=322)
(1)求该抛物线的函数表达式;
(2)点P为直线AB下方抛物线上的任意一点,连接PA,PB,求
面积的最大值
(3)将该抛物线向右平移2个单位长度得到抛物线
,平移后的抛物线与原抛物线相交于点C,点D为原抛物线对称轴上的一点,在平面直角坐标系中是否存在点E,使以点B,C,D,E为顶点的四边形为菱形,若存在,请直接写出点E的坐标;若不存在,请说明理由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e59da5115d0dafea24822245f92c48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bffe96f035d5b2e6865104a1541d25d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/326a6474089e6a774014906f4c1ba633.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/15/0ee44fdc-b369-4357-8839-fc3f669d6dac.png?resizew=322)
(1)求该抛物线的函数表达式;
(2)点P为直线AB下方抛物线上的任意一点,连接PA,PB,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
(3)将该抛物线向右平移2个单位长度得到抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a5720a643a2f9dae8e45002e3a75636.png)
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6 . 圆
内切于正三角形
,半径为R,圆
与圆
及
,
均相切,圆
的半径为r,则
等于( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c4f6f74444b2b7947fc6e35c8d62322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c4f6f74444b2b7947fc6e35c8d62322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dff3606c7bf728b4f539261461cde677.png)
A.4 | B.2 | C.3 | D.5 |
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7 . 如图,
中,
,
,
,P是
内部的一个动点,满足
,则线段CP长的最小值为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/15/5678b3e0-245d-4795-9a96-efe5982e84e0.png?resizew=130)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54275b7e571660d0a9e0370fbfe5050b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07160f14b3b453bebb64cb2bf96dc85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ab546e81ccaef75153d97e65b1ce978.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/15/5678b3e0-245d-4795-9a96-efe5982e84e0.png?resizew=130)
A.![]() | B.2 | C.![]() | D.![]() |
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8 . 阅读理解:高斯上小学时,有一次数学老师让同学们计算“从1到100这100个正整数的和”.许多同学都采用了依次累加的计算方法,计算起来非常烦琐,且易出错.聪明的小高斯经过探索后,给出了下面漂亮的解答过程.
解:设
①,则
②,
①+②,得
.
(两式左右两端分别相加,左端等于2S,右端等于100个101的和)
所以
,
③,所以
.
后来人们将小高斯的这种解答方法概括为“倒序相加法”
计算:
= _____ .
解:设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45d05f7125540086a961efd2afddb588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4663fd551144091fcd826a6ecd7a9603.png)
①+②,得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6800c25d59d4bf730f469ce16412a7fe.png)
(两式左右两端分别相加,左端等于2S,右端等于100个101的和)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46540f510d1f3537e0453ebb1bd6e9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da9c674c761493e544d7af9bb5046a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ac52232d822e91ac25df49702ba8c71.png)
后来人们将小高斯的这种解答方法概括为“倒序相加法”
计算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24d7c6e74c5501a04785b710ffe91ec6.png)
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9 . 如图,以
的两边
分别向外作等边
和等边
,
与
交于点P,已知
.
![](https://img.xkw.com/dksih/QBM/2024/1/22/3416677041340416/3419628471492608/STEM/2813bd9be9c047ab9beded363954ed75.png?resizew=156)
(1)求证:
;
(2)求
的度数及
的长;
(3)若点Q、R分别是等边
和等边
的重心(三边中线的交点),连接
,作出图象,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0d54431bbb28ebd98db5c1dc6083a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb49df05f2e31d005735c3f14a21d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97898c0102c6bc620abb5ab7442550bf.png)
![](https://img.xkw.com/dksih/QBM/2024/1/22/3416677041340416/3419628471492608/STEM/2813bd9be9c047ab9beded363954ed75.png?resizew=156)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3ccd24d1cdb7db9844a4c20c62970a2.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/add18fdf2049abff08a687b950f1cc1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
(3)若点Q、R分别是等边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb49df05f2e31d005735c3f14a21d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1168cd6865542e2a2efe6f6f21caa214.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d7b816eca15d4b7d060013df53edd53.png)
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10 . 已知实数
满足
且
,则
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32a004547477e9a36ccf8e7a4e65d74e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ead2d84f44d15afd79c6410078da36a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7260a18008048695970e768bb9a323dd.png)
A.![]() | B.![]() | C.![]() | D.2 |
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