名校
1 . 已知某精密制造企业根据长期检测结果,得到生产的产品的质量差服从正态分布
,并把质量差在
内的产品称为优等品,质量差在
内的产品称为一等品,优等品与一等品统称为正品,其余范围内的产品作为废品处理.现从该企业生产的正品中随机抽取1000件,测得产品质量差的样本数据统计如下:
作为
的近似值,用样本标准差s作为
的估计值,记质量差服从正态分布
,求该企业生产的产品为正品的概率P;(同一组中的数据用该组区间的中点值代表)
参考数据:若随机变量服从正态分布
,则
,
,
.
(2)假如企业包装时要求把2件优等品和n(
,且
)件一等品装在同一个箱子中,质检员从某箱子中摸出两件产品进行检验,若抽取到的两件产品等级相同则该箱产品记为A,否则该箱产品记为B.
①试用含n的代数式表示某箱产品抽检被记为B的概率p;
②设抽检5箱产品恰有3箱被记为B的概率为
,求当n为何值时,
取得最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdfc26b8bdcd1fd3781c4593217c725e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed5a8175dd80373426244e9e9eb1caa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27c749dedc02a1c9cb70288055f8c518.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfbe7f95b5d89f9409ec24536da9e826.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0ad7e7853a069537387b5192f73844.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8471b1bd5c53256f122a0f57d6ecf628.png)
参考数据:若随机变量服从正态分布
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdfc26b8bdcd1fd3781c4593217c725e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe78b8a0f85687556d1efd3b16cd9f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f170ce0ebd2e15203fe97418abf7f976.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/700802c566f7db44ac51c086ecf8ee6c.png)
(2)假如企业包装时要求把2件优等品和n(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cca1d86c9f078347773f700fee49d1d8.png)
①试用含n的代数式表示某箱产品抽检被记为B的概率p;
②设抽检5箱产品恰有3箱被记为B的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19a0be4eebc5d70c51f72f28dbfc11e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19a0be4eebc5d70c51f72f28dbfc11e9.png)
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真题
解题方法
2 . 已知函数
.
(1)当
时,求
的极值;
(2)当
时,
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ed77e8a42241f8b693b9ad155171ced.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f22a4a0dd7307a1323d25331e60782d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58196b9e63ec00aa1119052b6de6ae12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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3 . 已知实数
满足
.
(1)证明:
;
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ae5648bcfccbe0b2f49c69a66793b0.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df63cb762aa1710337f49a3d086f09cf.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9c476055f2f44d1344c8bc117fba235.png)
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真题
解题方法
4 . 在直角坐标系
中,以坐标原点为极点,
轴正半轴为极轴建立极坐标系,曲线
的极坐标方程为
.
(1)写出
的直角坐标方程;
(2)设直线l:
(
为参数),若
与l相交于
两点,若
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bdf1e69c1d990633fe9b7706c5394ba.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设直线l:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02b2fc0d39350b7973b46dc6e2d0a95b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56387ff53874620addcb0b91a605a309.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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今日更新
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3819次组卷
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4卷引用:2024年高考全国甲卷数学(理)真题
真题
5 . 如图,在以A,B,C,D,E,F为顶点的五面体中,四边形ABCD与四边形ADEF均为等腰梯形,
,
,
,
为
的中点.
平面
;
(2)求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c7587fdb043dc96ed724386286c9941.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2231488d2261886446f5764fa559ba9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bc150deaf709d073034cd8d56817f46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f369bec2d5682bf6b8b317a08aff546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e736704191faaf440edf6e57c98fc56d.png)
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真题
解题方法
6 . 记
为数列
的前
项和,已知
.
(1)求
的通项公式;
(2)设
,求数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8550e5c94dfff15896583b430eb9d3e7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b0091df9a0ff8fa29cc9c6a55ab1efc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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解题方法
7 . 某射击运动员进行射击训练,已知其每次命中目标的概率均为
.
(1)若该运动员共射击6次,求其在恰好命中3次的条件下,第3次没有命中的概率;
(2)该运动员射击训练不超过n(
)次,当他命中两次时停止射击(射击n次后,若命中的次数不足两次也不再继续),设随机变量X为该运动员的射击次数,试写出随机变量X的分布列,并证明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(1)若该运动员共射击6次,求其在恰好命中3次的条件下,第3次没有命中的概率;
(2)该运动员射击训练不超过n(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2ab800bb4666f21dbe05ec239ca39ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2f0d793fc77a1befa103b46f0d5307b.png)
您最近一年使用:0次
真题
解题方法
8 . 如图,
,
,
,
,
为
的中点.
平面
;
(2)求点
到
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62144e91faeae6b634f7dc0a28d0f79a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92d65be32159e3b778677cddb989b28f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae16f0e7561e767d9c23f7b6b247df94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941eba4dbc1094107e1eeb02c8d8cd56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd0285afe567ca0b32f0ccafc30167cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a09d9d486b7f91ba933210dd013a7f2c.png)
(2)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
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解题方法
9 . 如图,已知四棱台
的上、下底面分别是边长为2和4的正方形,
,且
底面ABCD,点P、Q分别是棱
、
的中点.
内是否存在点M,满足
平面CPQ?若存在,请说明点M的位置,若不存在,请说明理由;
(2)设平面CPQ交棱
于点T,平面CPTQ将四棱台
,分成上、下两部分,求上、下两部分的体积比.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55a2310cbba5e050488cd9296eb195d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ccc0d735dc14c92a8926d9b1def6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632f2bf1cd0435041fa04b01901d1c8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0edb1508fc95765f3bb316bcb5252d.png)
(2)设平面CPQ交棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
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名校
10 . 已知椭圆C:
(
)过点
,右焦点为
.
(1)求椭圆C的方程;
(2)过点F的直线l(不与x轴重合)交椭圆C于点M、N,点A是右顶点,直线MA、NA分别与直线
交于点P、Q,求
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a571f86f21d6455bd867cd06b4562070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
(1)求椭圆C的方程;
(2)过点F的直线l(不与x轴重合)交椭圆C于点M、N,点A是右顶点,直线MA、NA分别与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c32805e421aec6bce3624baa0c954f1.png)
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