2024年成考高起点《数学(理)》每日一练试题03月23日

<p class="introTit">单选题</p><p>1、中心在坐标原点，对称轴为坐标轴，且一个顶点（3，0），虚轴长为8的双曲线方程是（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/15641165746df3c.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/1564116579ce1fd.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/156411657f3914e.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/1564116582cc298.png' /></li></ul><p>答 案：B</p><p>解 析：双曲线有一个顶点为(3,0)，因此所求双曲线的实轴在x轴上，可排除A、C选项，又由于虚轴长为8，故b=4，即b²=16，故双曲线方程为<img src="https://img2.meite.com/questions/202303/15641169fee41e2.png" /></p><p>2、设双曲线<img src="https://img2.meite.com/questions/202303/2864229bffc903a.png" />的渐近线的斜率为k，则|k|=（）  </p><ul><li>A：<img src='https://img2.meite.com/questions/202303/2864229c1452370.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/2864229c18b1b1c.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/2864229c22758e7.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/2864229c1da4817.png' /></li></ul><p>答 案：D</p><p>解 析：双曲线渐近线的斜率为k<img src="https://img2.meite.com/questions/202303/2864229c7d57118.png" />故本题中k<img src="https://img2.meite.com/questions/202303/2864229c8e41a4f.png" /></p><p>3、已知空间向量i,j,k为两两垂直的单位向量，向量a=2i+3j+mk，若<img src="https://img2.meite.com/questions/202303/156411654c87f3a.png" />，则m=（）</p><ul><li>A：-2</li><li>B：-1</li><li>C：0</li><li>D：1</li></ul><p>答 案：C</p><p>解 析：由题可知向量a=(2,3,m），故<img src="https://img2.meite.com/questions/202303/15641169a04e100.png" />，解得m=0.</p><p>4、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 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、已知直线l的斜率为1，l过抛物线C：<img src="https://img2.meite.com/questions/202303/156411660ae04fb.png" />的焦点，且与C交于A,B两点.(I）求l与C的准线的交点坐标；<br />(II）求|AB|.</p><p>答 案：(I）C的焦点为<img src="https://img2.meite.com/questions/202303/1564116c40cf40a.png" />，准线为<img src="https://img2.meite.com/questions/202303/1564116c45024f5.png" />由题意得l的方程为<img src="https://img2.meite.com/questions/202303/1564116c5cf0409.png" />因此l与C的准线的交点坐标为<img src="https://img2.meite.com/questions/202303/1564116c7901a26.png" />(II）由<img src="https://img2.meite.com/questions/202303/1564116c9294ce9.png" />，得<img src="https://img2.meite.com/questions/202303/1564116c9d411f3.png" />设A（x1,y1），B（x2,y2），则<img src="https://img2.meite.com/questions/202303/1564116cd0bfaf7.png" />因此<img src="https://img2.meite.com/questions/202303/1564116ce1375a9.png" /></p><p>3、已知等差数列前n项和<img src="https://img2.meite.com/questions/202303/2864228a3204a03.png" /> （Ⅰ）求这个数列的通项公式;（Ⅱ）求数列第六项到第十项的和</p><p>答 案：<img src="https://img2.meite.com/questions/202303/2864228a568855e.png" /> <img src="https://img2.meite.com/questions/202303/2864228a63bc5a4.png" />  </p><p>4、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 class="introTit">填空题</p><p>1、<img src="https://img2.meite.com/questions/202303/2864228ce0438da.png" />的展开式是（）</p><p>答 案：<img src="https://img2.meite.com/questions/202303/2864228d0c480eb.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202303/2864228d4197eb2.png" /><img src="https://img2.meite.com/questions/202303/2864228d47da120.png" /><img src="https://img2.meite.com/questions/202303/2864228d5101aa0.png" /><img src="https://img2.meite.com/questions/202303/2864228d5bc2a57.png" /><img src="https://img2.meite.com/questions/202303/2864228d65510b6.png" /></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>