解题方法
1 . 截面惯性矩
是衡量截面抗弯能力的一个几何参数,若截面图形为矩形,则
,其中
为矩形的宽,
为矩形的高.某木器厂要加工如图所示的长方体实木梁,已知该实木梁的截面图形为矩形
,且矩形
外接圆的直径为
,要使该截面的惯性矩最大,则矩形
对应的高应为______
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910c6cc1a6e5f6f601f9336d5f08c7a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbe9f7abf7bcf4e1aa2579cd191d7761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
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2 . 事实上,对于两个函数
和
的商的导数,我们有如下法则:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bae54ea79b90e1710336975ad63f55cd.png)
_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5085887422e1722dd9bdfd14c7fc8ef9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bae54ea79b90e1710336975ad63f55cd.png)
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3 . 一个质量为4kg的物体做直线运动,该物体的位移y(单位:m)与时间t(单位:s)之间的关系为
,且
(
表示物体的动能,单位:J;m表示物体的质量,单位:kg;v表示物体的瞬时速度,单位:m/s),则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3316d1026d7da6206c5501df825d05ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/750cc462b426e796fc6266149689bf08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef1c7229e345be3a4857b91d27126c88.png)
A.该物体瞬时速度的最小值为1m/s | B.该物体瞬时速度的最小值为2m/s |
C.该物体在第1s时的动能为16J | D.该物体在第1s时的动能为8J |
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4 . 在数学中,由
个数
排列成的m行n列的数表
称为
矩阵,其中
称为矩阵A的第i行第j列的元素.矩阵乘法是指对于两个矩阵A和B,如果4的列数等于B的行数,则可以把A和B相乘,具体来说:若
,
,则
,其中
.已知
,函数
.
(1)讨论
的单调性;
(2)若
是
的两个极值点,证明:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e70354d6ca5ad9f6b4592fac0b5e559.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db452ec3c9e60109fdfe9fae8e456edf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc970ba32a45946c514e98eac1e80ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e70354d6ca5ad9f6b4592fac0b5e559.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a14c188b1c9d61aa237b137ba18023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16a9a7e6a7ff34bb72659677929bf9c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7cf279527982a84842a2d6a4f212892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b65887e38142a10f30be2296310d1a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9408afcaa76f52987ca43733b828f66a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01f44187cd898fb01a4f8fa76bdc6cdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b8d8e4e3f777270997845f7d9cfe85f.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca579894dad67bc82cb715fd48e0d70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b3db0fe99d90b9a693562dd988eca5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/def4d3923ea803696106f42140e83bf4.png)
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3卷引用:山东省泰安市2024届高三四轮检测数学试题
山东省泰安市2024届高三四轮检测数学试题江西省宜春市宜丰中学2023-2024学年高二下学期6月月考数学试题(已下线)高二数学期末模拟试卷01【好题汇编】-备战2023-2024学年高二数学下学期期末真题分类汇编(北师大版2019选择性必修第二册)
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解题方法
5 . 已知双曲线
的实轴长为2,顶点到渐近线的距离为
.
(1)求双曲线
的标准方程;
(2)若直线
与
的右支及渐近线的交点自上而下依次为
,证明:
;
(3)求二元二次方程
的正整数解
,可先找到初始解
,其中
为所有解
中的最小值,因为
,所以
;因为
,所以
;重复上述过程,因为
与
的展开式中,不含
的部分相等,含
的部分互为相反数,故可设
,所以
.若方程
的正整数解为
,则
的面积是否为定值?若是,请求出该定值,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef66f4832adc43902055a7e6d258037.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827ccf0c04aa941ba20d5f4c6068b46b.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/390f2c99d60abc83d9bda1a79995486f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd1af14f9a53cb0f07d5d28dceba30aa.png)
(3)求二元二次方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6ad120ce64035347eb7325fae9617c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb81c100a8985b5cfc606dc60cacd5ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56720e2f2b0ddd72156da495923698da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5acbd95efd8b0cb3e108fce6dc02af80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f4d959c570141afd7d0d6abc3969012.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81d350c9707efa6d8bb584395ccc07dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bd475f0c71e7e8c66fad3642779dc7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d694975be0ce869d210e18f85e583f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52e9c5a319966741ff9c3b52fb4de883.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c8e7b7827e1735c45c1e5ce59cdd624.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3460cd2f27a53941986606734a9b479a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3460cd2f27a53941986606734a9b479a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b66d595bfea3722fbc56e2fdccd548.png)
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6 . 正弦型函数被广泛运用于信号处理领域.将不同周期的正弦型函数叠加,就可以构建各种各样的信号.如
就能构建一种信号,关于该函数,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05f049267c4eff2f6028d6a83c04e43c.png)
A.![]() ![]() | B.![]() ![]() |
C.![]() ![]() | D.![]() ![]() |
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解题方法
7 . (1)已知函数
,证明:
,
,
.
(2)已知函数
,定义:若存在
,
,使得曲线
在点
与点
处有相同的切线
,则称切线
为“自公切线”.
①证明:当
时,曲线
不存在“自公切线”;
②讨论曲线
的“自公切线”的条数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff6838d84b68c6f0d3b93b196d9b08d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/673207f6b77b8192d25463d071737b7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1bf60c5e8996d138198fe74f30ce520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9da092efa74406128332df5a053685a8.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aba730d4e2ff4c9cc155446b3d12e96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85a93969738a9bb969f40cf587f1d5d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/878fd4af5b8fff01627f560767e19b73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4197070db34f0419b6d85eed4cec9fc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
①证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
②讨论曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
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8 . 定义:如果一个数列从第2项起,每一项与它前一项的和都等于同一个常数,那么这个数列就叫做等和数列,这个常数叫做等和数列的公和.设甲:数列
满足
;乙:数列
是公差为2的等差数列或公和为2的等和数列,则甲是乙的( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd2d0b2f36521429a74e93faf6dedc3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
A.充分不必要条件 | B.必要不充分条件 |
C.充要条件 | D.既不充分也不必要条件 |
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9 . 若函数
存在零点
,函数
存在零点
,使得
,则称
与
互为亲密函数.
(1)判断函数
与
是否为亲密函数,并说明理由;
(2)若
与
互为亲密函数,求
的取值范围.
附:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d35b13df9d8831bb4368e7036488675d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db18e638db2fb367cfe10bfaee37229.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13e60075f5d53066c03f106346dada26.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fe2f63cdc7606986d6250facf20ad1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbfd7245d512a98d9105f843c094c7e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
附:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c292239a48d1475428eeb9863d5dceb.png)
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4卷引用:江西省部分学校2023-2024学年高二下学期第二次月考(5月联考)数学试题
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10 . 已知函数
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93208bc770714ae8311ab2ba6274ea8d.png)
A.存在![]() ![]() ![]() |
B.对任意![]() ![]() ![]() |
C.对任意![]() ![]() ![]() |
D.存在![]() ![]() ![]() |
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5卷引用:江西省部分学校2023-2024学年高二下学期第二次月考(5月联考)数学试题