2024年成考高起点《数学(理)》每日一练试题01月19日
聚题库
01/19
<p class="introTit">单选题</p><p>1、函数<img src="https://img2.meite.com/questions/202303/15641164b43c663.png" />的反函数是（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/15641164d78e1fc.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/15641164de92884.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/15641164e5206d4.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/15641164e93e32e.png' /></li></ul><p>答 案：A</p><p>解 析：<img src="https://img2.meite.com/questions/202303/15641168660282b.png" />，由于x≤0，故<img src="https://img2.meite.com/questions/202303/1564116881a60a4.png" />把x与y互换，得所求反函数为<img src="https://img2.meite.com/questions/202303/15641168ab18774.png" /></p><p>2、在△ABC中，若b=<img src="https://img2.meite.com/questions/202303/2864228792c7c59.png" />，c=<img src="https://img2.meite.com/questions/202303/286422879bef613.png" /><img src="https://img2.meite.com/questions/202303/28642287a3c143e.png" />则a等于（）</p><ul><li>A：2</li><li>B：<img src='https://img2.meite.com/questions/202303/28642287b3e4835.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/28642287b8ca84c.png' /></li><li>D：无解</li></ul><p>答 案：B</p><p>解 析：此题是已知两边和其中一边的对角,解三角形时,会出现一解、两解、无解的情况,要注意这一点.用余弦定理<img src="https://img2.meite.com/questions/202303/286422888d98016.png" />可得<img src="https://img2.meite.com/questions/202303/28642288b870fd3.png" /><img src="https://img2.meite.com/questions/202303/28642288bcdcd26.png" /><img src="https://img2.meite.com/questions/202303/28642288cb0279a.png" /><img src="https://img2.meite.com/questions/202303/28642288d358a96.png" /><img src="https://img2.meite.com/questions/202303/28642288e0d440f.png" />解出<img src="https://img2.meite.com/questions/202303/28642288f8ec133.png" /><img src="https://img2.meite.com/questions/202303/286422890103ec2.png" /><img src="https://img2.meite.com/questions/202303/286422890874181.png" /><img src="https://img2.meite.com/questions/202303/2864228911473af.png" /></p><p>3、如果不共线的向量a和b有相等的长度,则(a+b)(a-b)=（）
</p><ul><li>A：0</li><li>B：1</li><li>C：-1</li><li>D：2</li></ul><p>答 案：A</p><p>解 析：(a+b)(a-b)=<img src="https://img2.meite.com/questions/202303/286422854d544bf.png" /><img src="https://img2.meite.com/questions/202303/286422855944b51.png" /></p><p>4、<img border="0" style="width: 58px; height: 37px;" src="https://img2.meite.com/zzpuce/2023-04/643679c8eed3a35616.jpeg">（ ）</p><ul><li>A：-2</li><li>B：<img border="0" style="width: 24px; height: 37px;" src="https://img2.meite.com/zzpuce/2023-04/643679c9008af39045.jpeg"></li><li>C：<img border="0" style="width: 14px; height: 37px;" src="https://img2.meite.com/zzpuce/2023-04/643679c90917e56279.jpeg"></li><li>D：2</li></ul><p>答 案：C</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/202303/2864229a3bc3098.png" />
（Ⅰ）写出向量<img src="https://img2.meite.com/questions/202303/2864229a57ba174.png" />和<img src="https://img2.meite.com/questions/202303/2864229a5e46ac8.png" />关于基底{a,b,c}的分解式;
（Ⅱ）求证：<img src="https://img2.meite.com/questions/202303/2864229a76ba56d.png" />
（Ⅲ）求证：<img src="https://img2.meite.com/questions/202303/2864229a7fdd541.png" />
</p><p>答 案：（Ⅰ）由题意知(如图所示) <img src="https://img2.meite.com/questions/202303/2864229af6b1567.png" />
<img src="https://img2.meite.com/questions/202303/2864229afe90f50.png" />
<img src="https://img2.meite.com/questions/202303/2864229b08314c5.png" />
</p><p>3、为了测河的宽,在岸边选定两点A和B,望对岸标记物C,测得<img src="https://img2.meite.com/questions/202303/2864228db8c0e49.png" />AB=120m,求河的宽
<img src="https://img2.meite.com/questions/202303/2864228dd64bdcb.png" /></p><p>答 案：如图， <img src="https://img2.meite.com/questions/202303/2864228df3f06d3.png" />
∵∠C=180°-30°-75°=75°
∴△ABC为等腰三角形，则AC=AB=120m
过C做CD⊥AB,则由Rt△ACD可求得CD=<img src="https://img2.meite.com/questions/202303/2864228e8a387f3.png" />=60m，
即河宽为60m
</p><p>4、已知a,b,c成等差数列，a,b,c+1成等比数列．若b=6，求a和c．</p><p>答 案：由已知得<img src="https://img2.meite.com/questions/202303/1564116c009cc19.png" />解得<img src="https://img2.meite.com/questions/202303/1564116c0e039c1.png" /></p><p class="introTit">填空题</p><p>1、长方体的长、宽、高分别为2,3,6，则该长方体的对角线长为（）</p><p>答 案：7</p><p>解 析：由题可知长方体的底面的对角线长为<img src="https://img2.meite.com/questions/202303/1564116b1132166.png" />，则在由高、底面对角线、长方体的对角线组成的三角形中，长方体的对角线长为<img src="https://img2.meite.com/questions/202303/1564116b3811959.png" /></p><p>2、函数<img src="https://img2.meite.com/questions/202303/28642285dd68e2c.png" />的定义域是（）</p><p>答 案：<img src="https://img2.meite.com/questions/202303/28642285f367786.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202303/28642285fcec513.png" /><img src="https://img2.meite.com/questions/202303/2864228601aa2da.png" />所以函数<img src="https://img2.meite.com/questions/202303/286422860d8135c.png" />的定义域是<img src="https://img2.meite.com/questions/202303/286422861644b62.png" /></p>
