名校
解题方法
1 . 通过平面直角坐标系,我们可以用有序实数对表示向量.类似的,我们可以把有序复数对
看作一个向量,记
,则称
为复向量.类比平面向量的相关运算法则,对于
,
,
、
、
、
、
,我们有如下运算法则:
①
;②
;③
;④
.
(1)设
,
,求
和
.
(2)由平面向量的数量积满足的运算律,我们类比得到复向量的相关结论:①
②
.试判断这两个结论是否正确,并说明理由.
(3)若
,集合
,
.对于任意的
,求出满足条件
的
,并将此时的
记为
,证明对任意的
,不等式
恒成立.根据对上述问题的解答过程,试写出一个一般性的命题(不需要证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbee4027127a0bce1cdc3fc50d28c5dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9ffed47d697d7a9965329d16654bde0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9ffed47d697d7a9965329d16654bde0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa0a749b475d60688fac80c38156eea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa224ed9be8766a4d0b5138bd57de0f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a67a742d2a43e907fb1c3a1bdf1d6a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b29a77cfdb8d2a0b684389921e1496c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7def0e6fc765f99565eaa1d498e291c.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d9562518a526e43b600f1b0c58df4da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/105a9c9b30276bbf76b1932836ee51aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/199ddc5bbc6de9f18be949468b376db9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede221f39dc56eb53d76d7e0de430741.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad6cc9ce836150c84f3c7b354e15057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9141558bd0831dd50720fab964afaa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e64c7d38ccc3b18052c3ad46daf6f133.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/532d596d507742a9d8598d3f7adb62fc.png)
(2)由平面向量的数量积满足的运算律,我们类比得到复向量的相关结论:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3f9dd01592a381d4566f3d855ece409.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b55031cf0985ff92dd0c16f1ad4d01b.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f943fd00c91acee53d2e9f4b31a5437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b802365ce6145a6caa1460cce745e2b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d1e98efe26c2c1442f6a73f09ec8d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40891013fa6a2a7ccee812efe7643e04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee952050f964cc9c4d877f9bb0b6618c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b452962126ea36badc6354f5e2b1d80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d1e98efe26c2c1442f6a73f09ec8d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96a948ca516eb624b5ec87f24aa187b1.png)
您最近一年使用:0次
2 . 已知函数
,其中
.
(1)讨论
的单调性;
(2)若函数
有两个不同的零点
,
.
①求实数a的取值范围;
②证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77f2e7f68e5c896506126ecc189ddbc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22dd8b3dc4c609bab82d356a5cc2208d.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e0a9e336769fba32ab7b516f52d0a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
①求实数a的取值范围;
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b4900c67f4b57fa430c4bd863f8e896.png)
您最近一年使用:0次
名校
解题方法
3 . 在
中,
,
,
.点
为
所在平面上一点,满足
(
、
且
).
(1)若
,用
,
表示
;
(2)若点
为
的外心,求
、
的值;
(3)若点
在
的角平分线上,当
时,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb5b12692517a39c320f99a479eb055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07160f14b3b453bebb64cb2bf96dc85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d81d732204a3c2384a27606f858677.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ec7ea9a9bf46fe5dfe7bdb028289bae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3276b5e12396fc4753eb3f8254f9fa68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9d100f6eb4bfc0b0191c12d5ab35d0d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e03698c9b972ce8c2a569195d00d0312.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbaeae7045ad94158cdf5ae97073bc17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a34bf00aeba15bce2cdee8ab487388dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef8337706c550bc095d7a2bd872221a1.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fabb884dc5f9609de491245463bbe9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84ce17f8f08bc11bd78885c80223466c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21306b0138363c1cb99ee8c1ecc5a486.png)
您最近一年使用:0次
名校
4 . 篮球运动是在1891年由美国马萨诸塞州斯普林尔德市基督教青年会训练学校体育教师詹姆士·奈史密斯博士,借鉴其他球类运动项目设计发明的.起初,他将两只桃篮钉在健身房内看台的栏杆上,桃篮上沿离地面约3.05米,用足球作为比赛工具,任何一方在获球后,利用传递、运拍,将球向篮内投掷,投球入篮得一分,按得分多少决定比赛胜负.在1891年的12月21日,举行了首次世界篮球比赛,后来篮球界就将此日定为国际篮球日.甲、乙两人进行投篮,比赛规则是:甲、乙每人投3球,进球多的一方获得胜利,胜利1次,则获得一个积分,平局或者输方不得分.已知甲和乙每次进球的概率分别是
和
,且每人、每次进球与否都互不影响.
(1)若
,求在进行一轮比赛后甲比乙多投进2球的概率;
(2)若
,且每轮比赛互不影响,乙要想至少获得3个积分且每轮比赛至少要超甲2个球,求:
①设事件
表示乙每轮比赛至少要超甲2个球,求
;(结果用含
的式子表示)
②从数学期望的角度分析,理论上至少要进行多少轮比赛?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e74b0aa7a6f6dcab7d9101b98504ae2a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa1a2dbc3af1e2dc756333b40841466f.png)
①设事件
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd85c4d2f793db97480144558d4951fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
②从数学期望的角度分析,理论上至少要进行多少轮比赛?
您最近一年使用:0次
5 . 已知函数
.
(1)当
时,求函数
的单调区间;
(2)若函数
有两个极值点
.
(i)求
的取值范围;
(ii)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d6ca8fb3232380bf835e16c8cb41f2a.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/200f24e682c93e02a87f3f9d57dc5d40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(ii)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d70a31c1e21548002b21b55015d361cf.png)
您最近一年使用:0次
6 . 若函数
在区间
上有定义,且
,
,则称
是
的一个“封闭区间”.
(1)已知函数
,区间
且
的一个“封闭区间”,求
的取值集合;
(2)已知函数
,设集合
.
(i)求集合
中元素的个数;
(ii)用
表示区间
的长度,设
为集合
中的最大元素.证明:存在唯一长度为
的闭区间
,使得
是
的一个“封闭区间”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64116336f1c736233883ae68ba882011.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea20aa0147d6477cc388c92cf8526d68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(1)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87d71f56ef6906bc37ca312051d97d4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ef4377354d5b6539bc3395c5e230f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d5b609211567a3d6dd45035a0281c8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea3d0c46c4f9267ce2d8905eff6118f6.png)
(i)求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(ii)用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13502d46b8563c54c09b29b20b3006a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5e23b42cdffe75e705cf0b24763e862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
您最近一年使用:0次
解题方法
7 . 某烟花厂准备生产一款环保、安全的迷你小烟花,初步设计了一个平面图,如图所示,该平面图由
,直角梯形
和以
为圆心的四分之一圆弧
构成,其中
,
,
,且
,
,
,将平面图形
以
所在直线为轴,旋转一周形成的几何体即为烟花.
(2)工厂准备将矩形
(该矩形内接于图形
,M在弧
上,N在线段
上,
在
上)旋转所形成的几何体用来安放燃料,设
(
).
①请用
表示燃料的体积V;
②若烟花燃烧时间t和燃料体积V满足关系
,请计算这个烟花燃烧的最长时间.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ca082df4cfac078e6b0e254000a5660.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b312dab930cbbb9a4bb1a99f044dab73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21037e170bdbb322558e79c40c00b454.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ed1321e4081133b1a8bbe994ecac48a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcbde67cc84757b10bb66c47cee22de1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be082aedbea135ea8fdcadca2cf427b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0761165f1176f3a5fe4f7b052832316d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc04ed25823067a16e7e32a43442cfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aaf670a0170e85aeffdc41d71cc975e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ecc1cb55a57dde481f8dd07ab150676.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
(2)工厂准备将矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cec780e86f57790cf88ec761e219bf5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f84f169e50dc59d4f7a8e1e36f5c847.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62fbf288f38b71a82d245dabcd8a057c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdd0dc4399d83c8d72e34c6a4785973c.png)
①请用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
②若烟花燃烧时间t和燃料体积V满足关系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f636adb7148a585f2817fd16ddf2107a.png)
您最近一年使用:0次
解题方法
8 . △ABC的内角A,B,C的对边分别为a,b,c,D为BC边上的一点.
,
.
①证明:
;
②求角B的最大值;
(2)若
,
,AD平分
,求AD的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bd5b9bbd3d22bd2cef53dd4b9691257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7daa2e5fdd2f3db55021436a884a73.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4037561c629fd07503c6803e1eb62fb6.png)
②求角B的最大值;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d11a5a756d8fdd7b294c4f5fd63467b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8120119749d4bc28067e73fca7d46cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
您最近一年使用:0次
名校
解题方法
9 . 已知
,平面内动点
满足直线
的斜率之积为
.
(1)求动点
的轨迹方程;
(2)过点
的直线交
的轨迹
于
两点,以
为邻边作平行四边形
(
为坐标原点),若
恰为轨迹
上一点,求四边形
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d7fae066efa772e21142aef5f764018.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ec8858389f4c3156a946ba8bf0d8a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/459c84c9addfbd1cdd0a877ba7c584e4.png)
(1)求动点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5a5e484dfef494d27bc35ae7b8cf75d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea16ceca816f7d3d50650af141baf42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea16ceca816f7d3d50650af141baf42.png)
您最近一年使用:0次
名校
10 . 已知函数
.
(1)讨论
的单调性;
(2)证明:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26b060a1a793fa7d536d5e733e5f82d9.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf1b1edc850b3a0aca5796830a6ce261.png)
您最近一年使用:0次