2024年成考高起点《数学(理)》每日一练试题07月18日
聚题库
07/18
<p class="introTit">单选题</p><p>1、设集合A={0,1}，B={0,1,2}，则A∩B=（）
</p><ul><li>A：{1,2}</li><li>B：{0,2}</li><li>C：{0,1}</li><li>D：{0,1,2}</li></ul><p>答 案：C</p><p>解 析：<img src="https://img2.meite.com/questions/202303/286422489c9882b.png" /></p><p>2、若甲：x>1,乙：<img src="https://img2.meite.com/questions/202303/286422537c285e5.png" />则
</p><ul><li>A：甲是乙的必要条件,但不是乙的充分条件</li><li>B：甲是乙的充分必要条件</li><li>C：甲不是乙的充分条件，也不是乙的必要条件</li><li>D：甲是乙的充分条件,但不是乙的必要条件</li></ul><p>答 案：D</p><p>解 析：<img src="https://img2.meite.com/questions/202303/286422541d9ea18.png" />而<img src="https://img2.meite.com/questions/202303/2864225425630cc.png" />故甲是乙的充分条件，但不是必要条件</p><p>3、已知向量a=(3,4),向量 b=(0,-2),则cos的值为（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/2864228b79d4590.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/2864228b7f2183c.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/2864228b82e5ed4.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/2864228b86922f8.png' /></li></ul><p>答 案：B</p><p>解 析：求cos<a,b>可直接用公式cos<a,b><img src="https://img2.meite.com/questions/202303/2864228c5310ae5.png" /> a·b=(3,4)·(0,-2)=3×0+4×(-2)=8,<img src="https://img2.meite.com/questions/202303/2864228c9dda4eb.png" />
</p><p>4、在△ABC中,已知2B= A+C,<img src="https://img2.meite.com/questions/202303/2864224869b5896.png" />= ac,则B-A=（）
</p><ul><li>A：0</li><li>B：<img src='https://img2.meite.com/questions/202303/286422487fede7b.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/286422488835324.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/286422488c2a04f.png' /></li></ul><p>答 案：A</p><p>解 析：在△ABC中,A+B+C=π，A+C=π-B，① 因为2B=A+C，②
由①②得2B=π-B，<img src="https://img2.meite.com/questions/202303/28642249c382987.png" />
<img src="https://img2.meite.com/questions/202303/28642249ca97aad.png" /><img src="https://img2.meite.com/questions/202303/28642249d5a95b9.png" />
<img src="https://img2.meite.com/questions/202303/28642249df58afe.png" />
由③④得<img src="https://img2.meite.com/questions/202303/28642249f0592a4.png" />a=c。所以A=C，又<img src="https://img2.meite.com/questions/202303/2864224a1878921.png" />所以△ABC为等边三角形，则B-A=0
</p><p class="introTit">主观题</p><p>1、某工厂每月生产x台游戏机的收入为R(x)=<img src="https://img2.meite.com/questions/202303/28642259676bd4d.png" />+130x-206(百元)，成本函数为C(x)=50x+100(百元)，当每月生产多少台时,获利润最大?最大利润为多少?
</p><p>答 案：利润 =收入-成本， L(x)=R(x)-C(x)=<img src="https://img2.meite.com/questions/202303/28642259c06a284.png" />+130x-206-(50x+100)=<img src="https://img2.meite.com/questions/202303/28642259e2077b7.png" />+80x-306
法一：用二次函数<img src="https://img2.meite.com/questions/202303/28642259fb36916.png" />当a<0时有最大值
<img src="https://img2.meite.com/questions/202303/2864225a14ed5aa.png" />是开口向下的抛物线，有最大值
<img src="https://img2.meite.com/questions/202303/2864225a2c75330.png" />
法二：用导数来求解
<img src="https://img2.meite.com/questions/202303/2864225a43988b8.png" />
因为x=90是函数在定义域内唯一驻点
所以x=90是函数的极大值点，也是函数的最大值点，其最大值为L（90）=3294
</p><p>2、在正四棱柱ABCD-A'B'C'D'中，<img src="https://img2.meite.com/questions/202303/28642255fa50503.png" />
（Ⅰ）写出向量<img src="https://img2.meite.com/questions/202303/286422561b1d145.png" />关于基底{a,b,c}的分解式
（Ⅱ）求证：<img src="https://img2.meite.com/questions/202303/286422563d58cde.png" />
（Ⅲ）求证：<img src="https://img2.meite.com/questions/202303/28642256478aacd.png" />
</p><p>答 案：(Ⅰ)由题意知(如图所示) <img src="https://img2.meite.com/questions/202303/286422566983935.png" />
<img src="https://img2.meite.com/questions/202303/28642256740213a.png" /><img src="https://img2.meite.com/questions/202303/286422567c06c5d.png" />
(Ⅱ)<img src="https://img2.meite.com/questions/202303/2864225695c5fbd.png" /><img src="https://img2.meite.com/questions/202303/286422569cdc533.png" />
(Ⅲ)<img src="https://img2.meite.com/questions/202303/28642256a537b6d.png" />
由已知，a,c是正四棱柱的棱，a,b,c两两垂直
<img src="https://img2.meite.com/questions/202303/28642256d1c4379.png" />
</p><p>3、已知数列<img src="https://img2.meite.com/questions/202303/286422511c19556.png" />的前n项和<img src="https://img2.meite.com/questions/202303/2864225128dc6e0.png" />
求证：<img src="https://img2.meite.com/questions/202303/286422513318bbb.png" />是等差数列，并求公差和首项。
</p><p>答 案：<img src="https://img2.meite.com/questions/202303/286422514f41b7b.png" /> <img src="https://img2.meite.com/questions/202303/28642251563e39b.png" />
</p><p>4、设函数f(x)=<img src="https://img2.meite.com/questions/202303/28642286431b211.png" />
(Ⅰ)求f(x)的单调区间;
(Ⅱ)求 f(x)的极值</p><p>答 案：(Ⅰ)函数的定义域为<img src="https://img2.meite.com/questions/202303/28642286bee9cc3.png" /> <img src="https://img2.meite.com/questions/202303/28642286c7d68a9.png" />
(Ⅱ)<img src="https://img2.meite.com/questions/202303/28642286d3444c8.png" />
</p><p class="introTit">填空题</p><p>1、不等式<img src="https://img2.meite.com/questions/202303/28642289d6ca884.png" />的解集为（）
</p><p>答 案：<img src="https://img2.meite.com/questions/202303/28642289e5c9bcc.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202303/28642289efc4ba4.png" /><img src="https://img2.meite.com/questions/202303/28642289fa37c87.png" /><img src="https://img2.meite.com/questions/202303/2864228a0077853.png" /></p><p>2、过点（2，0）作圆x2+y2=1的切线，切点的横坐标为（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202404/20662380f056b37.png" /></p><p>解 析：本题主要考查的知识点为圆的切线.
设切点(x0,y0)则有<img src="https://img2.meite.com/questions/202404/20662380f985f55.png" />
即<img src="https://img2.meite.com/questions/202404/20662381033a418.png" /><img src="https://img2.meite.com/questions/202404/206623810c425f9.png" />所以<img src="https://img2.meite.com/questions/202404/2066238119981c7.png" />故切点横坐标为<img src="https://img2.meite.com/questions/202404/206623812263a88.png" />
</p>
