2024年成考高起点《数学(理)》每日一练试题03月29日

聚题库

03/29

单选题1、若<img src="https://img2.meite.com/questions/202303/286422574eab213.png" />则<img src="https://img2.meite.com/questions/202303/28642257543afd6.png" />（）<ul><li>A：<img src='https://img2.meite.com/questions/202303/286422575a7b0ce.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/286422575f5d2ea.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/28642257640e720.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/28642257690a458.png' /></li></ul>答案：B解析：首先做出单位圆,然后根据问题的约束条件,利用三角函数线找出满足条件的a角取值范围 <img src="https://img2.meite.com/questions/202303/28642257c3e01c3.png" /> <img src="https://img2.meite.com/questions/202303/28642257d4935f9.png" /> 2、已知复数z=a+bi,其中a,<img src="https://img2.meite.com/questions/202303/2864224cfda7e7d.png" />且b≠0，则（） <ul><li>A：<img src='https://img2.meite.com/questions/202303/2864224d5102575.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/2864224d5d6afe7.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/2864224d6343bc7.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/2864224d6ad6d12.png' /></li></ul>答案：C解析：注意区分<img src="https://img2.meite.com/questions/202303/2864224f432bb09.png" /> <img src="https://img2.meite.com/questions/202303/2864224f5bec577.png" /> 3、如果不共线的向量a和b有相等的长度,则(a+b)(a-b)=（） <ul><li>A：0</li><li>B：1</li><li>C：-1</li><li>D：2</li></ul>答案：A解析：(a+b)(a-b)=<img src="https://img2.meite.com/questions/202303/286422854d544bf.png" /><img src="https://img2.meite.com/questions/202303/286422855944b51.png" />4、（2-3i）2=（）<ul><li>A：13-6i</li><li>B：13-12i</li><li>C：-5-6i</li><li>D：-5-12i</li></ul>答案：D解析：<img src="https://img2.meite.com/questions/202303/15641169af99ac7.png" />主观题1、已知直线l的斜率为1，l过抛物线C：<img src="https://img2.meite.com/questions/202303/156411660ae04fb.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； (II）求|AB|.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" />2、在△ABC中，B=120°,BC=4,△ABC的面积为<img src="https://img2.meite.com/questions/202303/15641165ec55404.png" />，求AC.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" />3、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" /> 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" /> 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； (II）求f（x）的极值．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" />填空题1、椭圆的中心在原点，一个顶点和一个焦点分别是直线x+3y-6与两坐标轴的交点,则此椭圆的标准方程为（） 答案：<img src="https://img2.meite.com/questions/202303/286422989dd2b03.png" />解析：原直线方程可化为<img src="https://img2.meite.com/questions/202303/28642298bab2d76.png" />交点(6,0),(0,2). 当点(6,0)是椭圆一个焦点,点(0,2) 是椭圆一个顶点时,c=6,b=2,<img src="https://img2.meite.com/questions/202303/28642298d6bc461.png" />当点(0,2) 是椭圆一个焦点,(6,0) 是椭圆一个顶点时,c=2,b-6,<img src="https://img2.meite.com/questions/202303/28642298ef2aa6b.png" />2、若平面向量a=(x,1),b=(1,-2),且a//b,则x=（） 答案：<img src="https://img2.meite.com/questions/202303/286422508f0554f.png" />解析：由于a//b，故<img src="https://img2.meite.com/questions/202303/286422509cd5c14.png" />