习题练习

[{"id":438,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"某班级在一次数学测验中，收集了20名学生的成绩（单位：分），数据如下：68, 72, 75, 76, 78, 79, 80, 82, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 94, 98。如果将这些成绩按从小到大的顺序排列，那么中位数是多少？","answer":"B","explanation":"中位数是指将一组数据按从小到大（或从大到小）的顺序排列后，处于中间位置的数。如果数据个数为偶数，则中位数是中间两个数的平均数。本题共有20个数据，是偶数个，因此中位数是第10个和第11个数据的平均数。将数据排序后，第10个数是83，第11个数是85。计算中位数：(83 + 85) ÷ 2 = 168 ÷ 2 = 84。因此，中位数是84分。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 17:40:10","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":"83分","is_correct":0},{"id":"B","content":"84分","is_correct":1},{"id":"C","content":"85分","is_correct":0},{"id":"D","content":"86分","is_correct":0}]},{"id":581,"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，制作了如下统计表：阅读1本书的有5人，阅读2本书的有8人，阅读3本书的有10人，阅读4本书的有7人。若该学生想用扇形统计图表示这些数据，那么表示‘阅读3本书’这一类别的扇形圆心角的度数是多少？","answer":"B","explanation":"首先计算总人数：5 + 8 + 10 + 7 = 30人。阅读3本书的人数为10人，占总人数的比例为10 ÷ 30 = 1\/3。扇形统计图中，整个圆为360度，因此‘阅读3本书’对应的圆心角为360 × (1\/3) = 120度。故正确答案为B。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 20:10:11","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":"90度","is_correct":0},{"id":"B","content":"120度","is_correct":1},{"id":"C","content":"100度","is_correct":0},{"id":"D","content":"110度","is_correct":0}]},{"id":1963,"subject":"数学","grade":"七年级","stage":"初中","type":"选择题","content":"某学生在研究自家阳台盆栽植物的生长情况时，记录了连续6周每周植株的高度增长量（单位：厘米）：2.3, 3.1, 1.8, 2.9, 3.5, 2.7。为了评估这6周植株高度增长量的波动程度，该学生计算了这组数据的方差。已知方差是各数据与平均数之差的平方的平均数，请问这组数据的方差最接近以下哪个数值？","answer":"B","explanation":"本题考查数据的收集、整理与描述中方差的概念与计算。首先计算6周高度增长量的平均数：(2.3 + 3.1 + 1.8 + 2.9 + 3.5 + 2.7) ÷ 6 = 16.3 ÷ 6 ≈ 2.717。然后计算每个数据与平均数之差的平方：(2.3−2.717)²≈0.174，(3.1−2.717)²≈0.147，(1.8−2.717)²≈0.841，(2.9−2.717)²≈0.034，(3.5−2.717)²≈0.613，(2.7−2.717)²≈0.0003。将这些平方值相加：0.174 + 0.147 + 0.841 + 0.034 + 0.613 + 0.0003 ≈ 1.8093。最后求平均得方差：1.8093 ÷ 6 ≈ 0.3015，最接近选项B（0.35）。注意：虽然精确值略小于0.35，但在四舍五入和估算范围内，0.35是最合理的选项。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"中等","points":1,"is_active":1,"created_at":"2026-01-07 14:47:44","updated_at":"2026-01-07 14:47:44","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":"0.28","is_correct":0},{"id":"B","content":"0.35","is_correct":1},{"id":"C","content":"0.42","is_correct":0},{"id":"D","content":"0.50","is_correct":0}]},{"id":1786,"subject":"数学","grade":"七年级","stage":"初中","type":"选择题","content":"某学生在平面直角坐标系中绘制了一个四边形ABCD，已知点A的坐标为(0, 0)，点B的坐标为(4, 0)，点C的坐标为(5, 3)，点D的坐标为(1, 3)。该学生想判断这个四边形是否为平行四边形，并计算其面积。以下说法正确的是：","answer":"A","explanation":"首先判断四边形是否为平行四边形。根据坐标，可计算各边向量：向量AB = (4, 0)，向量DC = (5-1, 3-3) = (4, 0)，故AB与DC平行且相等；向量AD = (1, 3)，向量BC = (5-4, 3-0) = (1, 3)，故AD与BC也平行且相等。因此两组对边分别平行且相等，四边形ABCD是平行四边形。接着计算面积：可利用底乘高。以AB为底，长度为4，点D到AB（x轴）的垂直距离为3，故面积为4 × 3 = 12。或者用向量叉积法：|AB × AD| = |4×3 - 0×1| = 12。因此正确答案为A。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-06 15:56:29","updated_at":"2026-01-06 15:56:29","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":"四边形ABCD是平行四边形，面积为12平方单位","is_correct":1},{"id":"B","content":"四边形ABCD是平行四边形，面积为10平方单位","is_correct":0},{"id":"C","content":"四边形ABCD不是平行四边形，但面积为12平方单位","is_correct":0},{"id":"D","content":"四边形ABCD不是平行四边形，面积为10平方单位","is_correct":0}]},{"id":504,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"某班级进行了一次数学测验，老师将成绩整理后绘制成频数分布直方图，发现成绩在80分到90分之间的学生人数最多。这说明该分数段的什么统计量最大？","answer":"C","explanation":"题目中提到“成绩在80分到90分之间的学生人数最多”，这表示该分数段出现的次数最多。在统计学中，一组数据中出现次数最多的数值称为众数。因此，80分到90分这个区间对应的众数最大。平均数是所有数据的总和除以个数，中位数是数据排序后位于中间的数，极差是最大值与最小值之差，它们都不能直接由‘人数最多’得出。故正确答案为C。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 18:10:48","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":"平均数","is_correct":0},{"id":"B","content":"中位数","is_correct":0},{"id":"C","content":"众数","is_correct":1},{"id":"D","content":"极差","is_correct":0}]},{"id":998,"subject":"数学","grade":"初一","stage":"初中","type":"填空题","content":"某学生调查了班级同学最喜欢的运动项目，收集数据后制作了频数分布表。其中喜欢跳绳的有8人，喜欢踢毽子的有5人，喜欢跑步的有12人，喜欢打篮球的有15人。则喜欢打篮球的人数占总人数的百分比是______%。","answer":"37.5","explanation":"首先计算总人数：8 + 5 + 12 + 15 = 40（人）。喜欢打篮球的人数为15人，因此所占百分比为 (15 ÷ 40) × 100% = 37.5%。本题考查数据的收集、整理与描述中的百分比计算，属于简单应用。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-30 04:50:48","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":510,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"某学生在整理班级同学最喜欢的课外活动调查数据时，将结果绘制成扇形统计图。已知喜欢阅读的同学所占圆心角为72度，喜欢运动的同学所占圆心角为108度，喜欢绘画的同学所占圆心角为60度，其余同学喜欢音乐。如果全班共有60人，那么喜欢音乐的同学有多少人？","answer":"B","explanation":"扇形统计图中，整个圆代表全班人数，圆心角总和为360度。已知阅读、运动、绘画对应的圆心角分别为72度、108度、60度，三者之和为72 + 108 + 60 = 240度。因此，喜欢音乐的同学所占圆心角为360 - 240 = 120度。由于圆心角与人数成正比，可列比例计算：120 ÷ 360 = 1\/3，所以喜欢音乐的人数为60 × (1\/3) = 20人。故正确答案为B。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 18:15:29","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":"18人","is_correct":0},{"id":"B","content":"20人","is_correct":1},{"id":"C","content":"22人","is_correct":0},{"id":"D","content":"24人","is_correct":0}]},{"id":1942,"subject":"数学","grade":"七年级","stage":"初中","type":"填空题","content":"某学生调查了所在班级同学每天使用手机的时间（单位：小时），将数据分为5组并绘制频数分布直方图。已知前四组的频数分别为4、7、9、5，第五组的频率为0.2，则该班级共有___名学生。","answer":"30","explanation":"设总人数为x，第五组频数为0.2x。前四组频数和为4+7+9+5=25，故25+0.2x=x，解得x=30。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-07 14:12:04","updated_at":"2026-01-07 14:12:04","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":814,"subject":"数学","grade":"初一","stage":"小学","type":"填空题","content":"某学生在调查班级同学最喜欢的课外活动时，收集了以下数据：阅读、运动、绘画、音乐。他将这些数据整理成扇形统计图，其中表示‘运动’的扇形圆心角为108度。如果全班共有40名学生，那么喜欢‘运动’的学生人数是___人。","answer":"12","explanation":"扇形统计图中，每个部分的圆心角占整个圆（360度）的比例等于该部分数据占总数据的比例。‘运动’对应的圆心角是108度，因此喜欢运动的学生所占比例为108 ÷ 360 = 0.3。全班共有40名学生，所以喜欢运动的学生人数为40 × 0.3 = 12人。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-30 00:30:51","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":2392,"subject":"数学","grade":"八年级","stage":"初中","type":"选择题","content":"在一次数学实践活动中，某学生测量了一块四边形土地的四个顶点坐标分别为 A(0, 0)、B(4, 0)、C(5, 2) 和 D(1, 2)。他通过计算发现该四边形的一组对边平行且相等，另一组对边也平行且相等。若他想进一步验证这个四边形是否为平行四边形，并计算其面积，以下哪种方法最合理？","answer":"B","explanation":"本题考查平行四边形的判定与面积计算，融合了坐标几何、一次函数斜率、向量思想和数据分析能力。选项 B 是最科学合理的方法：首先，通过一次函数斜率判断 AB 与 CD 是否平行（k_AB = (0-0)\/(4-0) = 0，k_CD = (2-2)\/(1-5) = 0，故平行），同理 AD 与 BC 的斜率均为 2\/1 = 2，说明两组对边分别平行，符合平行四边形定义；其次，可进一步用距离公式验证对边长度相等，增强结论可靠性；最后，面积可通过向量 AB = (4,0) 与 AD = (1,2) 的叉积 |4×2 - 0×1| = 8 得到，或使用分割法、坐标法（如鞋带公式）计算，方法严谨且符合八年级知识范围。选项 A 虽部分正确，但未利用坐标优势，效率较低；选项 C 错误，因角度并非直角；选项 D 混淆了轴对称与平行四边形的关系，平行四边形不一定是轴对称图形。因此，B 为最佳方法。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"中等","points":1,"is_active":1,"created_at":"2026-01-10 11:52:06","updated_at":"2026-01-10 11:52:06","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":"利用勾股定理分别计算四条边的长度，若对边相等，则该四边形是平行四边形，再用底乘高计算面积。","is_correct":0},{"id":"B","content":"利用一次函数的斜率判断 AB 与 CD、AD 与 BC 是否分别平行，再通过向量法或距离公式验证对边相等，最后用向量叉积或分割法求面积。","is_correct":1},{"id":"C","content":"直接假设该四边形是矩形，用长乘宽计算面积，因为所有角看起来都是直角。","is_correct":0},{"id":"D","content":"将该四边形沿 y 轴对折，若两部分完全重合，则说明是轴对称图形，因此是平行四边形，面积可用对称性估算。","is_correct":0}]}]