2024年成考高起点《数学(文史)》每日一练试题03月28日
聚题库
03/28
<p class="introTit">单选题</p><p>1、函数f(x)=<img src="https://img2.meite.com/questions/202303/2964239ea542b2e.png" />的单调增区间是（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/2964239ebc3fa1e.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/2964239ec1364eb.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/2964239ec5da4ec.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/2964239eca429dd.png' /></li></ul><p>答 案：A</p><p>解 析：<img src="https://img2.meite.com/questions/202303/2964239f6c29fd7.png" />中的<img src="https://img2.meite.com/questions/202303/2964239f78240cc.png" /><img src="https://img2.meite.com/questions/202303/2964239f8391b8a.png" />的减区间就为f(x)的增区间，设u(x)=<img src="https://img2.meite.com/questions/202303/2964239fa51c8e3.png" />当x∈R时，u(x)>0,函数u(x)在<img src="https://img2.meite.com/questions/202303/2964239fd320fd2.png" />是减函数， <img src="https://img2.meite.com/questions/202303/2964239fdcba7f4.png" />上是增函数
故f(x)=<img src="https://img2.meite.com/questions/202303/2964239ff389b79.png" />的单调增区间为<img src="https://img2.meite.com/questions/202303/2964239ffd784d3.png" />
ps：关于复合函数的问题要逐步分清每一层次的函数的图像和性质,再结合起来考虑整体,有时也可画出部分函数的图像来帮助分析和理解.
</p><p>2、不等式|2x-3|≤1的解集为（）</p><ul><li>A：{x|1≤x≤2}</li><li>B：{x|x≤-1或x≥2}</li><li>C：{x|1≤x≤3}</li><li>D：{x|2≤x≤3}</li></ul><p>答 案：A</p><p>解 析：<img src="https://img2.meite.com/questions/202303/2964239aeb5ea0d.png" /><img src="https://img2.meite.com/questions/202303/2964239afad8c35.png" />故原不等式的解集为{x|1≤x≤2}</p><p>3、函数f(x)=<img src="https://img2.meite.com/questions/202303/296423a45fdd209.png" />在区间[1,4]上的最大值和最小值分别是（）</p><ul><li>A：2和-2</li><li>B：2，没有最小值</li><li>C：1和1</li><li>D：2和4</li></ul><p>答 案：A</p><p>解 析：f(x)=<img src="https://img2.meite.com/questions/202303/296423a53590b45.png" /> <img src="https://img2.meite.com/questions/202303/296423a53f169a9.png" />
</p><p>4、已知<img src="https://img2.meite.com/questions/202303/14641027b0e9511.png" />，则sin2α=（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/14641027d7193ea.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/14641027dcded6f.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/14641027e2be0ff.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/14641027e75615d.png' /></li></ul><p>答 案：D</p><p>解 析：<img src="https://img2.meite.com/questions/202303/1464102b8ba4635.png" />两边平方得<img src="https://img2.meite.com/questions/202303/1464102bb0eb6cc.png" /><img src="https://img2.meite.com/questions/202303/1464102bb4c5429.png" />，故<img src="https://img2.meite.com/questions/202303/1464102bc8067d1.png" /></p><p class="introTit">主观题</p><p>1、设函数<img src="https://img2.meite.com/questions/202303/1564111c65679ad.png" /><br />(I）求f'(2)；<br />(II）求f(x）在区间[一1,2]的最大值与最小值．</p><p>答 案：(I）因为<img src="https://img2.meite.com/questions/202303/1564111dd4eb139.png" />，所以f'(2)=3×2<sup>2</sup>-4=8.(II）因为x＜-1，f(-1)=3.<img src="https://img2.meite.com/questions/202303/1564111ea39de57.png" />f(2)=0.<br />所以f(x）在区间[一1,2]的最大值为3，最小值为<img src="https://img2.meite.com/questions/202303/1564111eb99e49a.png" /></p><p>2、已知三角形的一个内角是<img src="https://img2.meite.com/questions/202303/296423b4bf816f0.png" />，面积是<img src="https://img2.meite.com/questions/202303/296423b4cd02d83.png" />周长是20,求各边的长.
</p><p>答 案：设三角形三边分别为a,b,c,∠A=60°， <img src="https://img2.meite.com/questions/202303/296423b55f6ab61.png" />
</p><p>3、每亩地种果树20棵时,每棵果树收入90元,如果每亩增种一棵,每棵果树收入就下降3元,求使总收入最大的种植棵数.
</p><p>答 案：设每亩增种x棵,总收入味y元，则每亩种树(20+x)棵，由题意知增种x棵后每棵收入为（60-3x） 则有y=(90-3x)(20+x)
整理得y=<img src="https://img2.meite.com/questions/202303/286422a66eea349.png" />+30x+1800
配方得y=<img src="https://img2.meite.com/questions/202303/286422a688a5af1.png" />+1875
当x=5时，y有最大值，所以每亩地最多种25棵</p><p>4、已知等差数列<img src="https://img2.meite.com/questions/202303/296423eaf9717d6.png" />前n项和<img src="https://img2.meite.com/questions/202303/296423eb032d219.png" />
（Ⅰ）求通项<img src="https://img2.meite.com/questions/202303/296423eb1a4ebf5.png" />的表达式
（Ⅱ）求<img src="https://img2.meite.com/questions/202303/296423eb26c2214.png" />的值
</p><p>答 案：（Ⅰ）当n=1时，由<img src="https://img2.meite.com/questions/202303/296423eb432a645.png" />得<img src="https://img2.meite.com/questions/202303/296423eb5068b03.png" /> <img src="https://img2.meite.com/questions/202303/296423eb59a45cd.png" />
<img src="https://img2.meite.com/questions/202303/296423eb6100c03.png" />
也满足上式，故<img src="https://img2.meite.com/questions/202303/296423eb755b7df.png" />=1-4n（n≥1）
（Ⅱ）由于数列<img src="https://img2.meite.com/questions/202303/296423eb93e2df0.png" />是首项为<img src="https://img2.meite.com/questions/202303/296423eba5a3367.png" />公差为d=-4的等差数列，所以<img src="https://img2.meite.com/questions/202303/296423ebc29c045.png" />是首项为<img src="https://img2.meite.com/questions/202303/296423ebe5ba947.png" />公差为d=-8，项数为13的等差数列，于是由等差数列前n项和公式得：
<img src="https://img2.meite.com/questions/202303/296423ec1ac9811.png" /><img src="https://img2.meite.com/questions/202303/296423ec20a013e.png" />
</p><p class="introTit">填空题</p><p>1、不等式<img src="https://img2.meite.com/questions/202303/286422a55aeabe3.png" />的解集是（）
</p><p>答 案：<img src="https://img2.meite.com/questions/202303/286422a56c78cca.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202303/286422a578989db.png" />或<img src="https://img2.meite.com/questions/202303/286422a57faa8bc.png" />或<img src="https://img2.meite.com/questions/202303/286422a58777039.png" /></p><p>2、已知向量a=(3,2),b=(-4,x),且a⊥b，则x=（）
</p><p>答 案：6</p><p>解 析：∵a⊥b， ∴3×(-4)+2x=0
∴x=6.
</p>
