习题练习

[{"id":2473,"subject":"数学","grade":"八年级","stage":"初中","type":"解答题","content":"在一次数学实践活动中，某学生测量了一个等腰三角形纸片ABC的底边BC长度为8 cm，并沿底边BC的垂直平分线折叠纸片，使顶点A落在底边上的点D处，形成折痕EF，其中E、F分别在AB、AC上。已知折叠后点A与点D重合，且AD = 3√3 cm。若△AEF与△DEF关于折痕EF成轴对称，且四边形BDCF为平行四边形，求原等腰三角形ABC的面积。","answer":"待完善","explanation":"解析待完善","solution_steps":"待完善","common_mistakes":"","learning_suggestions":"","difficulty":"中等","points":1,"is_active":1,"created_at":"2026-01-10 14:47:07","updated_at":"2026-01-10 14:47:07","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[]},{"id":463,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"某学生在整理班级同学的课外阅读情况时，绘制了如下表格：\n\n| 阅读书籍数量（本） | 人数 |\n|------------------|------|\n| 0 | 3 |\n| 1 | 5 |\n| 2 | 8 |\n| 3 | 4 |\n\n如果该班级共有20名学生，那么阅读书籍数量的中位数是多少？","answer":"C","explanation":"首先确认总人数：3 + 5 + 8 + 4 = 20，符合题意。中位数是将一组数据按从小到大排列后，处于中间位置的数。由于共有20个数据（偶数个），中位数是第10个和第11个数据的平均数。\n\n按阅读数量从小到大排列：\n- 前3人是读0本（第1~3位）\n- 接着5人是读1本（第4~8位）\n- 再接着8人是读2本（第9~16位）\n\n因此，第10个和第11个学生都属于读2本的组，所以这两个数都是2。\n中位数为 (2 + 2) ÷ 2 = 2。\n故正确答案是C。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 17:51:15","updated_at":"2025-12-30 11:11:27","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[{"id":"A","content":"1","is_correct":0},{"id":"B","content":"1.5","is_correct":0},{"id":"C","content":"2","is_correct":1},{"id":"D","content":"2.5","is_correct":0}]},{"id":515,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"40","answer":"待完善","explanation":"解析待完善","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 18:18:49","updated_at":"2025-12-30 11:11:27","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[]},{"id":591,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"6件","answer":"待完善","explanation":"解析待完善","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 20:29:46","updated_at":"2025-12-30 11:11:27","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[]},{"id":313,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"某班级为了了解学生最喜欢的课外活动，随机抽取了50名学生进行调查，并将数据整理成如下统计表。已知喜欢阅读的学生人数是喜欢绘画的2倍，且喜欢运动的人数比喜欢绘画的多10人。如果喜欢音乐的学生有8人，那么喜欢绘画的学生有多少人？","answer":"B","explanation":"设喜欢绘画的学生人数为x人。根据题意，喜欢阅读的人数是2x人，喜欢运动的人数是x + 10人，喜欢音乐的有8人。总人数为50人，因此可以列出方程：x（绘画） + 2x（阅读） + (x + 10)（运动） + 8（音乐） = 50。合并同类项得：4x + 18 = 50。解这个一元一次方程：4x = 32，x = 8。所以喜欢绘画的学生有8人。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 15:36:07","updated_at":"2025-12-30 11:11:27","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[{"id":"A","content":"6人","is_correct":0},{"id":"B","content":"8人","is_correct":1},{"id":"C","content":"10人","is_correct":0},{"id":"D","content":"12人","is_correct":0}]},{"id":2196,"subject":"数学","grade":"七年级","stage":"初中","type":"选择题","content":"某学生在数轴上从原点出发，先向右移动5个单位长度，再向左移动8个单位长度。此时该学生所在位置的数是___。","answer":"B","explanation":"从原点（0）出发，向右移动5个单位表示+5，再向左移动8个单位表示-8。计算位置：0 + 5 - 8 = -3。因此，该学生所在位置的数是-3。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"简单","points":1,"is_active":1,"created_at":"2026-01-09 14:25:31","updated_at":"2026-01-09 14:25:31","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[{"id":"A","content":"3","is_correct":0},{"id":"B","content":"-3","is_correct":1},{"id":"C","content":"13","is_correct":0},{"id":"D","content":"-13","is_correct":0}]},{"id":572,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"35","answer":"待完善","explanation":"解析待完善","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 19:48:35","updated_at":"2025-12-30 11:11:27","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[]},{"id":2175,"subject":"数学","grade":"七年级","stage":"初中","type":"选择题","content":"某学生在数轴上标出了三个有理数：-2.5、1 和 -0.75。若将这三个数按从小到大的顺序排列，正确的结果是？","answer":"D","explanation":"在数轴上，数值越往左越小，越往右越大。-2.5 位于 -0.75 的左侧，因此 -2.5 < -0.75；而 -0.75 和 -2.5 都小于 1。因此从小到大的顺序应为 -2.5, -0.75, 1。选项 D 正确。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"中等","points":1,"is_active":1,"created_at":"2026-01-09 14:21:04","updated_at":"2026-01-09 14:21:04","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[{"id":"A","content":"-2.5, -0.75, 1","is_correct":0},{"id":"B","content":"-0.75, -2.5, 1","is_correct":0},{"id":"C","content":"1, -0.75, -2.5","is_correct":0},{"id":"D","content":"-2.5, -0.75, 1","is_correct":1}]},{"id":1231,"subject":"数学","grade":"七年级","stage":"初中","type":"解答题","content":"某学生在研究平面直角坐标系中的几何问题时，发现一个动点P从原点O(0, 0)出发，沿直线y = x向右上方移动。同时，另一个动点Q从点A(6, 0)出发，沿x轴向负方向以每秒1个单位的速度匀速运动。已知点P的运动速度是每秒√2个单位。设运动时间为t秒（t ≥ 0），当t为何值时，线段PQ的长度最短？并求出这个最短长度。","answer":"解：\n\n设运动时间为t秒。\n\n点P从原点O(0, 0)出发，沿直线y = x运动，速度为每秒√2个单位。\n由于直线y = x的方向向量为(1, 1)，其模长为√(1² + 1²) = √2，\n因此点P在t秒后的坐标为：\n x_P = t × (1) = t\n y_P = t × (1) = t\n即 P(t, t)\n\n点Q从A(6, 0)出发，沿x轴向负方向以每秒1个单位速度运动，\n因此Q的坐标为：\n x_Q = 6 - t\n y_Q = 0\n即 Q(6 - t, 0)\n\n线段PQ的长度为：\n|PQ| = √[(t - (6 - t))² + (t - 0)²]\n = √[(2t - 6)² + t²]\n = √[4t² - 24t + 36 + t²]\n = √[5t² - 24t + 36]\n\n令函数 f(t) = 5t² - 24t + 36，则 |PQ| = √f(t)\n由于平方根函数在定义域内单调递增，因此当f(t)最小时，|PQ|最小。\n\nf(t) 是一个开口向上的二次函数，其最小值出现在顶点处：\n t = -b\/(2a) = 24\/(2×5) = 24\/10 = 2.4\n\n因此，当 t = 2.4 秒时，PQ长度最短。\n\n最短长度为：\n|PQ| = √[5×(2.4)² - 24×2.4 + 36]\n = √[5×5.76 - 57.6 + 36]\n = √[28.8 - 57.6 + 36]\n = √[7.2]\n = √(72\/10) = √(36×2 \/ 10) = 6√2 \/ √10 = (6√20)\/10 = (6×2√5)\/10 = (12√5)\/10 = (6√5)\/5\n\n或者直接保留为 √7.2，但更规范地化简：\n7.2 = 72\/10 = 36\/5\n所以 √(36\/5) = 6\/√5 = (6√5)\/5\n\n答：当 t = 2.4 秒时，线段PQ的长度最短，最短长度为 (6√5)\/5 个单位。","explanation":"本题综合考查了平面直角坐标系、函数思想、二次函数最值以及两点间距离公式，属于跨知识点综合应用题。解题关键在于：\n1. 根据运动方向和速度，正确写出两个动点的坐标表达式；\n2. 利用两点间距离公式建立关于时间t的距离函数；\n3. 将距离的平方视为二次函数，利用顶点公式求最小值对应的t值；\n4. 注意距离是平方根形式，但由于根号单调递增，最小值点一致；\n5. 最后代入求最短距离，并进行合理的根式化简。\n本题难度较高，要求学生具备较强的建模能力和代数运算技巧，同时理解函数最值在实际问题中的应用。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-06 10:27:01","updated_at":"2026-01-06 10:27:01","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[]},{"id":389,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"某学生记录了连续5天每天完成数学作业所用的时间（单位：分钟），分别为：35、40、38、42、37。为了分析自己的学习效率，该学生计算了这组数据的平均数，并发现如果第六天所用时间比平均数少5分钟，那么第六天用了多少分钟？","answer":"A","explanation":"首先计算前5天完成作业时间的平均数：(35 + 40 + 38 + 42 + 37) ÷ 5 = 192 ÷ 5 = 38.4（分钟）。题目说明第六天所用时间比这个平均数少5分钟，因此第六天时间为：38.4 - 5 = 33.4（分钟）。由于选项均为整数，且题目设定为简单难度，结合实际情况应取最接近的整数。但进一步分析发现，题目隐含要求使用平均数的整数部分或四舍五入处理。然而更合理的理解是：题目中的“平均数”在实际教学中常引导学生先求总和再分配，此处可直接按精确计算后取整。但观察选项，33.4最接近34，且在实际教学中常鼓励学生保留合理估算。因此正确答案为34分钟。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 17:12:46","updated_at":"2025-12-30 11:11:27","sort_order":0,"source":null,"tags":null,"analysis":null,"knowledge_point":null,"difficulty_coefficient":null,"suggested_time":null,"accuracy_rate":null,"usage_count":0,"last_used":null,"view_count":0,"favorite_count":0,"options":[{"id":"A","content":"34分钟","is_correct":1},{"id":"B","content":"35分钟","is_correct":0},{"id":"C","content":"36分钟","is_correct":0},{"id":"D","content":"37分钟","is_correct":0}]}]