2024年成考高起点《数学(文史)》每日一练试题06月22日
聚题库
06/22
<p class="introTit">单选题</p><p>1、一个袋子中装有标号分别为1，2，3，4的四个球，采用有放回的方式从袋中摸球两次，每次摸出一个球，则恰有一次摸出2号球的概率为（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202404/2066237f6fcf38b.png' /></li><li>B：<img src='https://img2.meite.com/questions/202404/2066237f76e958b.png' /></li><li>C：<img src='https://img2.meite.com/questions/202404/2066237f7c1c62b.png' /></li><li>D：<img src='https://img2.meite.com/questions/202404/2066237f81456d7.png' /></li></ul><p>答 案：C</p><p>解 析：本题主要考查的知识点为独立重复试验的概率。 所求概率为<img src="https://img2.meite.com/questions/202404/2066237f8658c10.png" /></p><p>2、若函数f(x)=1+<img src="https://img2.meite.com/questions/202303/296423a8d0db03d.png" />在(0,+∞)上是减函数，则（）</p><ul><li>A：a>1</li><li>B：a>2</li><li>C：1<a<2</li><li>D：0<a<1</li></ul><p>答 案：D</p><p>解 析：由已知条件函数f(x)=1+<img src="https://img2.meite.com/questions/202303/296423a8d0db03d.png" />在(0,+∞)上是减函数，及对数函数<img src="https://img2.meite.com/questions/202303/296423a9706abcb.png" />的性质可得底数0<a<1</p><p>3、已知<img src="https://img2.meite.com/questions/202303/296423e580b835d.png" />成等差数列，且<img src="https://img2.meite.com/questions/202303/296423e58b478b7.png" />为方程<img src="https://img2.meite.com/questions/202303/296423e5978a20a.png" />的两个根，则<img src="https://img2.meite.com/questions/202303/296423e5a3a2171.png" />的值为（）
</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/296423e5afbd31c.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/296423e5b427e6e.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/296423e5b93c781.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/296423e5bd84cbf.png' /></li></ul><p>答 案：D</p><p>解 析：由根与系数的关系得<img src="https://img2.meite.com/questions/202303/296423e65c0bb98.png" />由等差数列的性质得<img src="https://img2.meite.com/questions/202303/296423e66c81d64.png" /></p><p>4、函数<img src="https://img2.meite.com/questions/202404/2066237f2842c6d.png" />的定义域为（）。</p><ul><li>A：R</li><li>B：{1}</li><li>C：{x||x|≤1}</li><li>D：{x||xl≥1}</li></ul><p>答 案：A</p><p>解 析：本题主要考查的知识点为函数的定义域。 对于<img src="https://img2.meite.com/questions/202404/2066237f2fe936e.png" />
奇次根号下无要求，故函数的定义域为R。</p><p class="introTit">主观题</p><p>1、已知直线l的斜率为1，l过抛物线C：<img src="https://img2.meite.com/questions/202303/1564111c54b8a1f.png" />的焦点，且与C交于A,B两点.<br />(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/202303/1564111ce1ece45.png" />，准线为<img src="https://img2.meite.com/questions/202303/1564111cf077c15.png" />由题意得l的方程为<img src="https://img2.meite.com/questions/202303/1564111d17b9087.png" />因此l与C的准线的交点坐标为<img src="https://img2.meite.com/questions/202303/1564111d41e334d.png" />(II）由<img src="https://img2.meite.com/questions/202303/1564111d60deac4.png" />得<img src="https://img2.meite.com/questions/202303/1564111d70ccff6.png" />设A（x1,y1）.B（x2,y2），则<img src="https://img2.meite.com/questions/202303/1564111da85efe1.png" />因此<img src="https://img2.meite.com/questions/202303/1564111db86701f.png" /></p><p>2、在△ABC中,已知三边 a、b、c 成等差数列,且最大角∠A是最小角的2倍, a: b :c.
</p><p>答 案：<img src="https://img2.meite.com/questions/202303/296423b0ae0659d.png" /></p><p>3、设函数f(x)<img src="https://img2.meite.com/questions/202303/296423a66bbdb95.png" />且f'(-1)=-36
(Ⅰ)求m
(Ⅱ)求f(x)的单调区间</p><p>答 案：(Ⅰ)由已知得f'=<img src="https://img2.meite.com/questions/202303/296423a74f41b7f.png" /> 又由f'(-1)=-36得
6-6m-36=-36
故m=1.
(Ⅱ)由(Ⅰ)得f'(x)=<img src="https://img2.meite.com/questions/202303/296423a792d1aff.png" />
令f'(x)=0，解得<img src="https://img2.meite.com/questions/202303/296423a7b14cf3f.png" />
当x<-3时，f'(x)>0;
当-3<x<2时，f'(x)<0;
当x>2时，f'(x)>0;
故f(x)的单调递减区间为(-3,2)，f(x)的单调递增区间为(-∞,-3),(2,+∞)
</p><p>4、设椭圆的中心是坐标原点,长轴在x轴上,离心率<img src="https://img2.meite.com/questions/202303/2964239c71aeb01.png" />已知点P<img src="https://img2.meite.com/questions/202303/2964239c7e7d039.png" />到圆上的点的最远距离是<img src="https://img2.meite.com/questions/202303/2964239c8cba609.png" />求椭圆的方程
</p><p>答 案：由题意，设椭圆方程为<img src="https://img2.meite.com/questions/202303/2964239cdfdcc2f.png" /> 由<img src="https://img2.meite.com/questions/202303/2964239cef23e00.png" /><img src="https://img2.meite.com/questions/202303/2964239cf5cc660.png" />
设P<img src="https://img2.meite.com/questions/202303/2964239d01ba1aa.png" />点到椭圆上任一点的距离为 d,
<img src="https://img2.meite.com/questions/202303/2964239d0dd8af6.png" />
<img src="https://img2.meite.com/questions/202303/2964239d1e44fb3.png" />则在y=-b时，<img src="https://img2.meite.com/questions/202303/2964239d2c1a969.png" />最大，即d也最大。
<img src="https://img2.meite.com/questions/202303/2964239d43183b3.png" /><img src="https://img2.meite.com/questions/202303/2964239d48e58c6.png" />
<img src="https://img2.meite.com/questions/202303/2964239d5187168.png" />
</p><p class="introTit">填空题</p><p>1、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><p>2、设<img src="https://img2.meite.com/questions/202303/2964239c3b4ac2f.png" />则<img src="https://img2.meite.com/questions/202303/2964239c42d80d7.png" /></p><p>答 案：-1</p><p>解 析：<img src="https://img2.meite.com/questions/202303/2964239ca8a1cc9.png" /> <img src="https://img2.meite.com/questions/202303/2964239cb367f80.png" />
<img src="https://img2.meite.com/questions/202303/2964239cc4f078f.png" />
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