2024年成考高起点《数学(理)》每日一练试题03月31日

聚题库

03/31

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=（）<ul><li>A：-2</li><li>B：-1</li><li>C：0</li><li>D：1</li></ul>答案：C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=0.3、方程<img src="https://img2.meite.com/questions/202303/286422844e41cfc.png" />的图像是下图中的（） <ul><li>A：<img src='https://img2.meite.com/questions/202303/286422845f870bc.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/2864228467a7899.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/286422847066eef.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/286422847761b5b.png' /></li></ul>答案：D解析：本题属于读图题型，在寻求答案时,要着重讨论方程的表达式 <img src="https://img2.meite.com/questions/202303/28642284efc983b.png" /> <img src="https://img2.meite.com/questions/202303/286422851faebf4.png" /> <img src="https://img2.meite.com/questions/202303/2864228526ccc03.png" /> 4、已知向量a=(3,4),向量 b=(0,-2),则cos的值为（）<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>答案：B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" /> 主观题1、已知a,b,c成等差数列，a,b,c+1成等比数列．若b=6，求a和c．答案：由已知得<img src="https://img2.meite.com/questions/202303/1564116c009cc19.png" />解得<img src="https://img2.meite.com/questions/202303/1564116c0e039c1.png" />2、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。 答案：<img src="https://img2.meite.com/questions/202303/286422514f41b7b.png" /> <img src="https://img2.meite.com/questions/202303/28642251563e39b.png" /> 3、在正四棱柱ABCD-A'B'C'D'中，<img src="https://img2.meite.com/questions/202303/2864229a3bc3098.png" /> （Ⅰ）写出向量<img src="https://img2.meite.com/questions/202303/2864229a57ba174.png" />和<img src="https://img2.meite.com/questions/202303/2864229a5e46ac8.png" />关于基底{a,b,c}的分解式; （Ⅱ）求证：<img src="https://img2.meite.com/questions/202303/2864229a76ba56d.png" /> （Ⅲ）求证：<img src="https://img2.meite.com/questions/202303/2864229a7fdd541.png" /> 答案：（Ⅰ）由题意知(如图所示) <img src="https://img2.meite.com/questions/202303/2864229af6b1567.png" /> <img src="https://img2.meite.com/questions/202303/2864229afe90f50.png" /> <img src="https://img2.meite.com/questions/202303/2864229b08314c5.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、长方体的长、宽、高分别为2,3,6，则该长方体的对角线长为（）答案：7解析：由题可知长方体的底面的对角线长为<img src="https://img2.meite.com/questions/202303/1564116b1132166.png" />，则在由高、底面对角线、长方体的对角线组成的三角形中，长方体的对角线长为<img src="https://img2.meite.com/questions/202303/1564116b3811959.png" />2、lg(tan43°tan45°tan47°)=（） 答案：0解析：lg(tan43°tan45°tan47°)=lg(tan43°tan45°cot43°)=lgtan45°=lg1=0