2024年成考高起点《数学(理)》每日一练试题03月06日

<p class="introTit">单选题</p><p>1、若tanα=3，则<img src="https://img2.meite.com/questions/202303/2864225cb9b9476.png" /></p><ul><li>A：-2</li><li>B：<img src='https://img2.meite.com/questions/202303/2864225cc0f208a.png' /></li><li>C：2</li><li>D：-4</li></ul><p>答 案：A</p><p>解 析：<img src="https://img2.meite.com/questions/202303/2864225d63144f9.png" /></p><p>2、函数<img src="https://img2.meite.com/questions/202303/156411640a2cd90.png" />的定义域是（）</p><ul><li>A：{x|-3＜x＜-1}</li><li>B：{x|x＜-3或x＞-1}</li><li>C：{x|1＜x＜3}</li><li>D：{x|x＜1或x＞3}</li></ul><p>答 案：D</p><p>解 析：由对数函数的性质可知<img src="https://img2.meite.com/questions/202303/15641167889eb44.png" />，解得x＞3或x＜1，因此函数的定义域为{x|x＜1或x＞3}</p><p>3、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>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、在△ABC中，B=120°,BC=4,△ABC的面积为<img src="https://img2.meite.com/questions/202303/15641165ec55404.png" />，求AC.</p><p>答 案：由△ABC的面积为<img src="https://img2.meite.com/questions/202303/15641165ec55404.png" />得<img src="https://img2.meite.com/questions/202303/1564116bba3c98d.png" />所以AB =4.因此<img src="https://img2.meite.com/questions/202303/1564116be4bebd5.png" />所以<img src="https://img2.meite.com/questions/202303/1564116be967e8f.png" /></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、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、设函数f(x)=xlnx+x.(I）求曲线y=f(x）在点（(1,f(1))处的切线方程；<br />(II）求f（x）的极值．</p><p>答 案：(I）f(1)=1,f'(x)=2+lnx，故f'(1)=2.所以曲线y=f(x）在点（1,f(1))处的切线方程为y=2x-1.(II）令f'(x)=0，解得<img src="https://img2.meite.com/questions/202303/1564116d2d14a94.png" />当<img src="https://img2.meite.com/questions/202303/1564116d3d33026.png" />时，f'(x)<O；当<img src="https://img2.meite.com/questions/202303/1564116d6f6aec3.png" />时，f'(x)＞O.故f(x）在区间<img src="https://img2.meite.com/questions/202303/1564116db9a0764.png" />单调递减，在区间<img src="https://img2.meite.com/questions/202303/1564116dc99fc91.png" />单调递增．因此f(x）在<img src="https://img2.meite.com/questions/202303/1564116ddb842d0.png" />时取得极小值<img src="https://img2.meite.com/questions/202303/1564116de4f1b79.png" /></p><p class="introTit">填空题</p><p>1、lg(tan43°tan45°tan47°)=（）  </p><p>答 案：0</p><p>解 析：lg(tan43°tan45°tan47°)=lg(tan43°tan45°cot43°)=lgtan45°=lg1=0</p><p>2、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>