2024年成考高起点《数学(理)》每日一练试题07月17日

聚题库

07/17

单选题1、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" /> 2、设甲：<img src="https://img2.meite.com/questions/202303/15641165369dc0c.png" />；乙：<img src="https://img2.meite.com/questions/202303/156411653e04dd6.png" />.则（）<ul><li>A：甲是乙的必要条件但不是充分条件</li><li>B：甲是乙的充分条件但不是必要条件</li><li>C：甲是乙的充要条件</li><li>D：甲既不是乙的充分条件也不是乙的必要条件</li></ul>答案：A解析：三角形相似不一定全等，但三角形全等一定相似，因此，甲是乙的必要条件但不是充分条件．3、已知全集U=R，A={x|x≥1}，B={x|-1<x≤2},则<img src="https://img2.meite.com/questions/202303/2864229bae18781.png" /><ul><li>A：{x|x≤2}</li><li>B：{x|x<2}</li><li>C：{x|-1<x≤2}</li><li>D：{x|-1<x<1}</li></ul>答案：A解析：补集运算应明确知道是否包括端点.A在U中的补集是x<1， <img src="https://img2.meite.com/questions/202303/2864229c436e804.png" /> <img src="https://img2.meite.com/questions/202303/2864229c50ac610.png" /><img src="https://img2.meite.com/questions/202303/2864229c574ae01.png" /><img src="https://img2.meite.com/questions/202303/2864229c5e8c7e7.png" /> 4、对满足a>b的任意两个非零实数，下列不等式成立的是（） <ul><li>A：<img src='https://img2.meite.com/questions/202303/2864228b1f8efca.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/2864228b2567c86.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/2864228b2a86110.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/2864228b30e50d4.png' /></li></ul>答案：D解析：A错误，例如-2>4,而<img src="https://img2.meite.com/questions/202303/2864228ba68a732.png" /> B错误，例如：-10>100,而<img src="https://img2.meite.com/questions/202303/2864228bbe163a2.png" /> C错误，例如：-1>-2,而<img src="https://img2.meite.com/questions/202303/2864228bd9de925.png" /> <img src="https://img2.meite.com/questions/202303/2864228be175c60.png" />主观题1、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" />2、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" /> 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" />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 4、已知抛物线C：y2=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m2=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" /> 填空题1、函数<img src="https://img2.meite.com/questions/202303/28642285dd68e2c.png" />的定义域是（）答案：<img src="https://img2.meite.com/questions/202303/28642285f367786.png" />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" />2、函数<img src="https://img2.meite.com/questions/202303/2864225857f0d64.png" />的图像与坐标轴的交点共有（） 答案：2解析：当x=0时，y=<img src="https://img2.meite.com/questions/202303/286422587f2a68b.png" />-2=-1，故函数与y轴交于(0,-1)点，令y=0,则有<img src="https://img2.meite.com/questions/202303/28642258b372965.png" />故函数与x轴交于(1,0) 点,因此函数 <img src="https://img2.meite.com/questions/202303/28642258c6c51c8.png" />与坐标轴的交点共有 2个.