2024年高职单招《数学(中职)》每日一练试题09月29日

<p class="introTit">单选题</p><p>1、函数<img src="https://img2.meite.com/questions/202409/2166ee6d0ea9899.png" />的定义域是（）  </p><ul><li>A：<img src='https://img2.meite.com/questions/202409/2166ee6d1ca9398.png' /></li><li>B：<img src='https://img2.meite.com/questions/202409/2166ee6d23cc4ec.png' /></li><li>C：<img src='https://img2.meite.com/questions/202409/2166ee6d2c4d124.png' /></li><li>D：<img src='https://img2.meite.com/questions/202409/2166ee6d33c218e.png' /></li></ul><p>答 案：B</p><p>解 析：因为<img src="https://img2.meite.com/questions/202409/2166ee705bb6a58.png" />，所以<img src="https://img2.meite.com/questions/202409/2166ee7063481ff.png" />，所以<img src="https://img2.meite.com/questions/202409/2166ee706a29d52.png" />，所以f(x)的定义域为<img src="https://img2.meite.com/questions/202409/2166ee6d23cc4ec.png" />。</p><p>2、若<img src="https://img2.meite.com/questions/202409/1366e3944e977b3.png" />,则x的值为（）  </p><ul><li>A：3</li><li>B：6</li><li>C：9</li><li>D：3或6</li></ul><p>答 案：A</p><p>解 析：因为<img src="https://img2.meite.com/questions/202409/1366e394ab45e69.png" />,所以x+3+x=9,解得x=3.</p><p>3、已知平面直角坐标系上的两点A(-1,1),B(3,2),则<img src="https://img2.meite.com/questions/202409/2166ee786366f38.png" />=()</p><ul><li>A：√2</li><li>B：3</li><li>C：√6</li><li>D：√17</li></ul><p>答 案：D</p><p>解 析：由题意得<img src="https://img2.meite.com/questions/202409/2166ee786ce7e8c.png" /></p><p>4、已知集合M={0,1,2,3,4},N={3,4,5},则下列结论正确的是  </p><ul><li>A：<img src='https://img2.meite.com/questions/202409/0966de634faf0ac.png' /></li><li>B：<img src='https://img2.meite.com/questions/202409/0966de6356bb798.png' /></li><li>C：M∩N={3,4}</li><li>D：M∪N={0,1,2,5}</li></ul><p>答 案：C</p><p>解 析：因为M={0,1,2,3,4},N={3,4,5},所以M不包含N,N不包含M,M∩N={3,4},M∪N={0,1,2,3,4,5},故选C</p><p class="introTit">填空题</p><p>1、已知|a|=2,|b|=4,a·b=-4,则向量a与b的夹角为（）  </p><p>答 案：5</p><p>解 析：<img src="https://img2.meite.com/questions/202409/1966ebc583ea8e1.png" /><img src="https://img2.meite.com/questions/202409/1966ebc5d0ae256.png" /></p><p>2、若函数<img src="https://img2.meite.com/questions/202409/1266e29a85bbd26.png" />是对数函数，则a=（）</p><p>答 案：5</p><p>解 析：<img src="https://img2.meite.com/questions/202409/1266e29a8d74883.png" /></p><p>3、已知球的直径为2,则该球的体积是（）  </p><p>答 案：<img src="https://img2.meite.com/questions/202409/1866ea2dee84dd3.png" /></p><p>解 析：易得球的半径为1,故球的体积为<img src="https://img2.meite.com/questions/202409/1866ea2dfab0399.png" /></p><p>4、若(x-1)ⁿ的展开式中x²的系数是-10,则n的值为()  </p><p>答 案：5</p><p>解 析：因为(x-1)ⁿ的展开式中x²的系数是<img src="https://img2.meite.com/questions/202409/1266e2a816afaa9.png" />,所以n= 5.</p><p class="introTit">简答题</p><p>1、如图,正方体ABCD-A₁B₁C₁D₁的棱长为 2.(1)证明<img src="https://img2.meite.com/questions/202409/1366e3ec96a8814.png" /><br />(2)求 AD₁与 BD 所成角的大小. <img src="https://img2.meite.com/questions/202409/1366e3e3f06edd3.png" />  </p><p>答 案：(1)在正方体ABCD-A₁B₁C₁D₁中,易得<img src="https://img2.meite.com/questions/202409/1366e3ff49f3077.png" /> 又<img src="https://img2.meite.com/questions/202409/1366e3ff57a3def.png" /> <img src="https://img2.meite.com/questions/202409/1366e3ff6d67c1a.png" /> ∵四边形 ABCD 是正方形， <img src="https://img2.meite.com/questions/202409/1366e3ff91d4291.png" /> 又<img src="https://img2.meite.com/questions/202409/1366e3ffa619eac.png" /> <img src="https://img2.meite.com/questions/202409/1366e3ffb57d486.png" /> 又<img src="https://img2.meite.com/questions/202409/1366e3ffc59fc60.png" /> <img src="https://img2.meite.com/questions/202409/1366e3ffcf63532.png" /> (2)在正方体ABCD-A₁B₁C₁D₁中,易得A₁D₁// BC, ∴<img src="https://img2.meite.com/questions/202409/1366e400089b44e.png" />即AD₁与BD所成的角, 易知C₁B=BD=C₁D=<img src="https://img2.meite.com/questions/202409/1366e40027975ed.png" /> ∴<img src="https://img2.meite.com/questions/202409/1366e40038aa3af.png" />为等边三角形， <img src="https://img2.meite.com/questions/202409/1366e4003f7c51c.png" /> ∴AD₁与BD所成的角为<img src="https://img2.meite.com/questions/202409/1366e4005d07407.png" />  </p><p>2、已知等差数列{a_n}的前n项和为S_n且S₇=28,S₁₅=120 (1)求数列{an}的首项a₁和公差d; (2)证明数列<img src="https://img2.meite.com/questions/202409/1866ea728ca150a.png" /> 是等差数列，并求出<img src="https://img2.meite.com/questions/202409/1866ea7294997a6.png" />的前n项和T_n</p><p>答 案：<img src="https://img2.meite.com/questions/202409/1866ea729e7f2f4.png" /><img src="https://img2.meite.com/questions/202409/1866ea72a338d55.png" /> <img src="https://img2.meite.com/questions/202409/1866ea72a832b55.png" /><img src="https://img2.meite.com/questions/202409/1866ea72b4c64a8.png" /><img src="https://img2.meite.com/questions/202409/1866ea72b9d112a.png" /><img src="https://img2.meite.com/questions/202409/1866ea72c0c897c.png" /><img src="https://img2.meite.com/questions/202409/1866ea72c76d47a.png" /><img src="https://img2.meite.com/questions/202409/1866ea72ce267db.png" /></p>