2024年成考高起点《数学(理)》每日一练试题08月15日
聚题库
08/15
<p class="introTit">单选题</p><p>1、下列函数中，为奇函数的是（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/156411642c7ba59.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/156411643238049.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/1564116437cc5f5.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/156411643c1ff56.png' /></li></ul><p>答 案：B</p><p>解 析：当f(-x)＝-f(x)，函数f(x)是奇函数，只有选项B符合.</p><p>2、已知向量a=(3,4),向量 b=(0,-2),则cos的值为（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/2864228b79d4590.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/2864228b7f2183c.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/2864228b82e5ed4.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/2864228b86922f8.png' /></li></ul><p>答 案：B</p><p>解 析：求cos<a,b>可直接用公式cos<a,b><img src="https://img2.meite.com/questions/202303/2864228c5310ae5.png" /> a·b=(3,4)·(0,-2)=3×0+4×(-2)=8,<img src="https://img2.meite.com/questions/202303/2864228c9dda4eb.png" />
</p><p>3、从点M(x,3)向圆<img src="https://img2.meite.com/questions/202303/2864224d89e6386.png" />作切线，切线的最小值等于（）
</p><ul><li>A：4</li><li>B：<img src='https://img2.meite.com/questions/202303/2864224da12068a.png' /></li><li>C：5</li><li>D：<img src='https://img2.meite.com/questions/202303/2864224da7648e6.png' /></li></ul><p>答 案：B</p><p>解 析：如图,相切是直线与圆的位置关系中的一种，此题利用圆心坐标、半径，求出切线长. 由圆的方程知,圆心为B(-2,-2),半径为1，设切点为A，<img src="https://img2.meite.com/questions/202303/2864224fb292668.png" />
由勾股定理得，
<img src="https://img2.meite.com/questions/202303/2864224fdde65a2.png" /><img src="https://img2.meite.com/questions/202303/2864224fe756e3a.png" /><img src="https://img2.meite.com/questions/202303/2864224feb9db09.png" /><img src="https://img2.meite.com/questions/202303/2864224ff1e7eb2.png" />
当x+2=0时，MA取最小值，最小值为<img src="https://img2.meite.com/questions/202303/28642250091e1b3.png" />
<img src="https://img2.meite.com/questions/202303/2864225047b55db.png" />
</p><p>4、从椭圆与x轴额右交点看短轴两端点的视角为60°的椭圆的离心率（）
</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/28642287ed710eb.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/28642287f182bba.png' /></li><li>C：1</li><li>D：<img src='https://img2.meite.com/questions/202303/28642287f85287f.png' /></li></ul><p>答 案：A</p><p>解 析：求椭圆的离心率,先求出a,c.(如图) <img src="https://img2.meite.com/questions/202303/286422892bd4fb5.png" /><img src="https://img2.meite.com/questions/202303/286422893b50a35.png" />，由椭圆定义知<img src="https://img2.meite.com/questions/202303/2864228950523ff.png" />
<img src="https://img2.meite.com/questions/202303/286422895c76294.png" /></p><p class="introTit">主观题</p><p>1、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>2、已知抛物线C：y<sup>2</sup>=2px(p＞0)的焦点到准线的距离为1。
（I）求C的方程；
（Ⅱ）若A(1，m）（m＞0）为C上一点，O为坐标原点，求C上另一点B的坐标，使得OA⊥OB</p><p>答 案：（I）由题意,该抛物线的焦点到准线的距离为<img src="https://img2.meite.com/questions/202404/1966222edee972e.png" />
所以抛物线C的方程为<img src="https://img2.meite.com/questions/202404/1966222ee6c66f9.png" />
（Ⅱ）因A(l,m)(m>0)为C上一点,故有m<sup>2</sup>=2,
可得<img src="https://img2.meite.com/questions/202404/1966222ef5c5007.png" />因此A点坐标为<img src="https://img2.meite.com/questions/202404/1966222efb949fc.png" />
设B点坐标为<img src="https://img2.meite.com/questions/202404/1966222f0a5cbbb.png" />则<img src="https://img2.meite.com/questions/202404/1966222f11e9340.png" />
因为<img src="https://img2.meite.com/questions/202404/1966222f17c1b05.png" />则有<img src="https://img2.meite.com/questions/202404/1966222f1dce70a.png" />
即<img src="https://img2.meite.com/questions/202404/1966222f27533ea.png" />解得x0=4
所以B点的坐标为<img src="https://img2.meite.com/questions/202404/1966222f308351e.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、已知函数f(x)=(x-4)(x<sup>2</sup>-a)
（I）求f"(x)；
(Ⅱ)若f"(-1)=8，求f(x)在区间[0，4]的最大值与最小值</p><p>答 案：<img src="https://img2.meite.com/questions/202404/1966222f3f733cc.png" /><img src="https://img2.meite.com/questions/202404/1966222f47c8e62.png" /></p><p class="introTit">填空题</p><p>1、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><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>
