真题
1 . 中国某汽车公司坚持“技术为王,创新为本”的发展理念,凭借研发实力和创新的发展模式在电池、电子、乘用车、商用车和轨道交通等多个领域发挥着举足轻重的作用.2024年第一季度,该公司以
万辆的销售成绩稳居新能源汽车销量榜榜首,市场占有率高达
.将销售数据用科学记数法表示为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88169351407d18d82e18bbe6c10f0dd1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/288a5c69cfbf24abd0954177177e188a.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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真题
2 . 为响应国家政策,保障耕地面积,提高粮食产量,确保粮食安全,我市开展高标准农田改造建设,调查统计了其中四台不同型号的挖掘机(分别为
型,
型,
型,
型)一个月内改造建设高标准农田的面积(亩),并绘制成如图不完整的统计图表:
改造农田面积统计表
(1)①
______;
②扇形统计图中
的度数为______.
(2)若这四台不同型号的挖掘机共改造建设了960亩高标准农田,估计其中
型挖掘机改造建设了多少亩?
(3)若从这四台不同型号的挖掘机中随机抽调两台挖掘机参加其它任务,请用画树状图或列表的方法求出恰好同时抽到
,
两种型号挖掘机的概率.
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
改造农田面积统计表
型号 | ||||
亩数 | 16 | 20 | 12 |
(1)①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c57bbef89a37f1a3808c0ceeac0c22.png)
②扇形统计图中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(2)若这四台不同型号的挖掘机共改造建设了960亩高标准农田,估计其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(3)若从这四台不同型号的挖掘机中随机抽调两台挖掘机参加其它任务,请用画树状图或列表的方法求出恰好同时抽到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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真题
3 . 综合与实践
问题提出:在一次综合与实践活动中,某数学兴趣小组将足够大的直角三角板的一个顶点放在正方形的中心
处,并绕点
旋转,探究直角三角板与正方形
重叠部分的面积变化情况.
操作发现:将直角三角板的直角顶点放在点
处,在旋转过程中:
(1)若正方形边长为4,当一条直角边与对角线重合时,重叠部分的面积为______;当一条直角边与正方形的一边垂直时,重叠部分的面积为______.
(2)若正方形的面积为
,重叠部分的面积为
,在旋转过程中
与
的关系为______.
类比探究:如图1,若等腰直角三角板的直角顶点与点
重合,在旋转过程中,两条直角边分别角交正方形两边于
,
两点,小宇经过多次实验得到结论
,请你帮他进行证明.
角的顶点与点
重合,在旋转过程中,当三角板的直角边交
于点
,斜边交
于点
,且
时,请求出重叠部分的面积.
(参考数据:
,
,
)
问题提出:在一次综合与实践活动中,某数学兴趣小组将足够大的直角三角板的一个顶点放在正方形的中心
![](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/411b38a18046fea8e9fab1f9f9b80a5f.png)
操作发现:将直角三角板的直角顶点放在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(1)若正方形边长为4,当一条直角边与对角线重合时,重叠部分的面积为______;当一条直角边与正方形的一边垂直时,重叠部分的面积为______.
(2)若正方形的面积为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
类比探究:如图1,若等腰直角三角板的直角顶点与点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93379e452721cd23933d24ff2b74d901.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34ec1827a09872159663c8d1998a0ff2.png)
(参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b7bb1dd9e1bb49e95ecc1eb09b17f37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff421393cf5973dbed11a21ad22926cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ad0374e9dbe5da463291fde7e3ca7dc.png)
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真题
4 . 眉山是“三苏”故里,文化底蕴深厚.近年来眉山市旅游产业蓬勃发展,促进了文创产品的销售,某商店用
元购进的
款文创产品和用
元购进的
款文创产品数量相同.每件
款文创产品进价比
款文创产品进价多
元.
(1)求
,
两款文创产品每件的进价各是多少元?
(2)已知
,
文创产品每件售价为
元,
款文创产品每件售价为
元,根据市场需求,商店计划再用不超过
元的总费用购进这两款文创产品共
件进行销售,问:怎样进货才能使销售完后获得的利润最大,最大利润是多少元?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1139c55a2ab02b802c77bc0cb941befd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ebd36502bbc7e1493a0f9667434881.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.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/58b184c94e38f1e5dbe750b2168c2a37.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)已知
![](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/0efba7147f5b9ced8bc4a72f0a9fb8af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1f959e5f8d89390f0f136f6acc9f6fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca9196e540d9b22b0caeb958c747e03a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0efba7147f5b9ced8bc4a72f0a9fb8af.png)
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5 . 为落实阳光体育活动,学校鼓励学生积极参加体育锻炼.已知某天五位同学体育锻炼的时间分别为(单位:小时):1,1.5,1.4,2,1.5,这组数据的中位数和众数分别是( )
A.1.5,1.5 | B.1.4,1.5 | C.1.48,1.5 | D.1,2 |
您最近一年使用:0次
真题
6 . 眉山市东坡区永丰村是“天府粮仓”示范区,该村的“智慧春耕”让生产更高效,提升了水稻亩产量,水稻亩产量从2021年的670千克增长到了2023年的780千克,该村水稻亩产量年平均增长率为
,则可列方程为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
您最近一年使用:0次
7 . “直播带货”已经成为信息社会中商家的一种新型促销手段,2024年是中国农历甲辰龙年,某主播用3000元购进了一批“小金龙”布偶玩具在直播间销售,由于销售火爆,又用9900元购进了第二批这种玩具,所购数量是第一批购进数量的3倍,但每件的进价贵了3元.
(1)求商场购进第一批“小金龙”每件的进价;
(2)直播间在第二批“小金龙”布偶销售过程中发现,“小金龙”布偶每分钟的销量
(件)与销售单价
(元)满足一次函数关系
,设每分钟的销售利润为
元,求
与
之间的函数关系式,并求
最大值.
(1)求商场购进第一批“小金龙”每件的进价;
(2)直播间在第二批“小金龙”布偶销售过程中发现,“小金龙”布偶每分钟的销量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa3508efa0ddff8c2f63dc1840f161c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/595044a7750ab4f84519041979c3d780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/595044a7750ab4f84519041979c3d780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/595044a7750ab4f84519041979c3d780.png)
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8 . 已知
为等边三角形,
,
为
上一点,
,连接
.
;
(2)如图2,延长
交
于点
,在
上取点
,使
,连接
,
,求证:
;
(3)如图3,已知
,
为射线
上一点,连接
,
,
,连接
,若
的面积为
,
的面积为
,
的面积为
,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e557ac8c744f9961a6d544a75321e8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eaffcd2070bfcb1955e2cab66c26c06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62dcef170114cd71b99f6646d364ef0.png)
(2)如图2,延长
![](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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbc42e5e8ba0bdcf9e103ba110bcb2a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d3b925c571dce32fb8219894f3e9cd0.png)
(3)如图3,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f539c9639a377a1496630fdf9271c22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a253c7fdf589ee3dece13d5b5b5732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb6ede9761b5b90f8dc137708e1ee90f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ef34fc41ddc22e335604f2aff03a91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1411961d518f616ddbe13da1ba17941d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d0469336f71edd52dc9148c67db052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec26d81361e13a37bcf3c95db0ee89b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db609c0ddcb9af21ddbf443d3e527cfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/485fd8aa254b523bef30560500ccaf41.png)
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真题
9 . 数学活动课上,同学们将两个全等的三角形纸片完全重合放置,固定一个顶点,然后将其中一个纸片绕这个顶点旋转,来探究图形旋转的性质.已知三角形纸片
和
中,
,
,
.
【初步感知】
(1)如图1,连接
,
,在纸片
绕点
旋转过程中,试探究
的值.
【深入探究】
(2)如图2,在纸片
绕点
旋转过程中,当点
恰好落在
的中线
的延长线上时,延长
交
于点
,求
的长.
【拓展延伸】
(3)在纸片
绕点
旋转过程中,试探究
,
,
三点能否构成直角三角形.若能,直接写出所有直角三角形
的面积;若不能,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57c6b0a6cb307c4c02f503831862f7d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85b65dc7c0d2524dbcc76c946178d686.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21c2425db34d83dbe44f3ee3d2c5439b.png)
【初步感知】
(1)如图1,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9af91834644b94e1b023b33e15a0644.png)
【深入探究】
(2)如图2,在纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21037e170bdbb322558e79c40c00b454.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
【拓展延伸】
(3)在纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
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10 . 如图,某处有一座塔
,塔的正前方有一平台
,平台的高
米,斜坡
的坡度
:
,点
,
,
,
在同一条水平直线上
某数学兴趣小组为测量该塔的高度,在斜坡
处测得塔顶部
的仰角为
,在斜坡
处测得塔顶部
的仰角为
,求塔高
.(精确到
米)(参考数据:
,
,
,
,
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefb0dc1b957781b5890f87cd413f15a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bf233a6d9ce7dd32c93cc1052532616.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8da45c443af7994a26ffa9d8894e7262.png)
![](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/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54e1d0f65817ba32a732040518f41440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a834dd9adc2557361a6a03bb54d8b7e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31ba8299ff43abec7bb3e61776655234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a87796ee30e6c5d5e6b6285b32abe10c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ac6b6edbebaa4cd26ede022cc857bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d6748b28b0b62d2fa9c1b487aab4a61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f73c483d52918a7fde66e5b1e38dfbb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afeabff144c47d6416b5d239c762a479.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1efc920e7fe39ed8dc51081db8de6e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e5a302c41f4aa3ab84d7a3b875200e8.png)
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