2024年成考高起点《数学(理)》每日一练试题04月19日
聚题库
04/19
<p class="introTit">单选题</p><p>1、已知集合M =(2,3,5,a),N =(1,3,4,b),若M∩N=(1,2,3),则a,b的值为
</p><ul><li>A：a=2,b=1</li><li>B：a=1,b=1</li><li>C：a=1,b= 2</li><li>D：a=1,b=5</li></ul><p>答 案：C</p><p>解 析：M∩N={2,3,5,a} ∩{1,3,4,6} ={1,2,3} 又因为M中无“1”元素，而有“a”元素，只有a=1
而N中无“2”元素，而有“b元素”，只有b=2
</p><p>2、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>3、<img src="https://img2.meite.com/questions/202303/1564116591bffc0.png" />的展开式中，x<sup>2</sup>的系数为（）</p><ul><li>A：20</li><li>B：10</li><li>C：5</li><li>D：1</li></ul><p>答 案：C</p><p>解 析：二项展开式的第二项为<img src="https://img2.meite.com/questions/202303/1564116a225f0b5.png" />，故展开式中的x<sup>2</sup>的系数为5.</p><p>4、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 class="introTit">主观题</p><p>1、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>2、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>3、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>4、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 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、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>
