习题练习

[{"id":1988,"subject":"数学","grade":"九年级","stage":"初中","type":"选择题","content":"某学生在纸上画了一个边长为6 cm的正方形ABCD，以顶点A为原点建立平面直角坐标系，AB边在x轴正方向，AD边在y轴正方向。若将正方形绕原点A逆时针旋转30°，则旋转后点B的坐标最接近以下哪一项？（结果保留两位小数，cos30°≈0.87，sin30°=0.5）","answer":"A","explanation":"本题考查旋转与坐标变换的综合应用，结合锐角三角函数知识。初始时点B坐标为(6, 0)。将点B绕原点A逆时针旋转30°，其新坐标可通过旋转公式计算：x' = x·cosθ - y·sinθ，y' = x·sinθ + y·cosθ。代入x=6，y=0，θ=30°，得x' = 6×0.87 - 0×0.5 = 5.22，y' = 6×0.5 + 0×0.87 = 3.00。因此旋转后点B的坐标约为(5.22, 3.00)，对应选项A。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"简单","points":1,"is_active":1,"created_at":"2026-01-07 15:06:54","updated_at":"2026-01-07 15:06:54","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":"(5.22, 3.00)","is_correct":1},{"id":"B","content":"(3.00, 5.22)","is_correct":0},{"id":"C","content":"(4.24, 4.24)","is_correct":0},{"id":"D","content":"(6.00, 0.00)","is_correct":0}]},{"id":625,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"某班级组织了一次环保知识竞赛，共收集了50名学生的成绩（单位：分），成绩分布如下表所示：\n\n| 分数段 | 人数 |\n|--------|------|\n| 60～70 | 8 |\n| 70～80 | 12 |\n| 80～90 | 18 |\n| 90～100| 12 |\n\n根据以上数据，该班级竞赛成绩的中位数所在的分数段是（ ）。","answer":"C","explanation":"本题考查数据的收集、整理与描述中的中位数概念。总人数为50人，中位数是第25和第26个数据的平均值。按分数从低到高累计人数：60～70分有8人，累计8人；70～80分有12人，累计20人；80～90分有18人，累计38人。第25和第26个数据均落在80～90分区间内，因此中位数所在分数段为80～90。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 21:52:02","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":"60～70","is_correct":0},{"id":"B","content":"70～80","is_correct":0},{"id":"C","content":"80～90","is_correct":1},{"id":"D","content":"90～100","is_correct":0}]},{"id":1018,"subject":"数学","grade":"初一","stage":"小学","type":"填空题","content":"某学生测量了学校花坛一周的温度变化，记录如下：早晨为-2℃，中午上升了7℃，傍晚又下降了3℃。那么傍晚的温度是___℃。","answer":"2","explanation":"早晨温度为-2℃，中午上升7℃，即 -2 + 7 = 5℃；傍晚又下降3℃，即 5 - 3 = 2℃。因此傍晚的温度是2℃。本题考查有理数的加减运算，结合生活情境，符合七年级学生对有理数应用的理解水平。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-30 05:35:37","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":2343,"subject":"数学","grade":"八年级","stage":"初中","type":"选择题","content":"某公园计划修建一个等腰三角形花坛，设计要求其周长为24米，且其中一条边长为9米。已知该三角形为轴对称图形，且满足三角形三边关系。若设底边为x米，两腰各为y米，则下列哪组方程能正确描述该三角形的设计条件？","answer":"D","explanation":"本题考查等腰三角形的性质、周长计算及三角形三边关系。已知花坛为等腰三角形，周长为24米，设底边为x，两腰为y，则周长公式为 x + 2y = 24。又因三角形任意两边之和大于第三边，任意两边之差小于第三边，即 |y - y| < x < y + y 可简化为 0 < x < 2y；同时需满足 |x - y| < y < x + y。由于 y > 0 且 x > 0，最关键的约束是两边之差小于第三边：|x - y| < y，即 -y < x - y < y，化简得 0 < x < 2y，这与三角形不等式一致。选项D中的 |x - y| < y < x + y 正确表达了以y为一边时，其余两边x与y需满足的不等关系，且结合 x + 2y = 24 可完整描述设计条件。其他选项要么逻辑错误（如A中|y−y|=0，表述冗余），要么不等式方向混乱。因此正确答案为D。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"中等","points":1,"is_active":1,"created_at":"2026-01-10 11:00:01","updated_at":"2026-01-10 11:00:01","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":"A","content":"x + 2y = 24 且 |y - y| < x < y + y","is_correct":0},{"id":"B","content":"x + 2y = 24 且 |y - x| < y < y + x","is_correct":0},{"id":"C","content":"x + 2y = 24 且 |y - y| < x < 2y","is_correct":0},{"id":"D","content":"x + 2y = 24 且 |x - y| < y < x + y","is_correct":1}]},{"id":509,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"某班级组织了一次环保活动，收集废旧纸张。第一周收集了总量的40%，第二周收集了30千克，此时已收集的与未收集的质量比为3:2。问这批废旧纸张的总质量是多少千克？","answer":"D","explanation":"设这批废旧纸张的总质量为x千克。第一周收集了40%即0.4x千克，第二周收集了30千克，因此已收集的总量为0.4x + 30千克。未收集的部分为x - (0.4x + 30) = 0.6x - 30千克。根据题意，已收集与未收集的质量比为3:2，可列方程：(0.4x + 30) \/ (0.6x - 30) = 3 \/ 2。交叉相乘得：2(0.4x + 30) = 3(0.6x - 30)，即0.8x + 60 = 1.8x - 90。移项整理得：60 + 90 = 1.8x - 0.8x，即150 = x。因此总质量为150千克，正确答案为D。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 18:14:45","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":"75千克","is_correct":0},{"id":"B","content":"100千克","is_correct":0},{"id":"C","content":"120千克","is_correct":0},{"id":"D","content":"150千克","is_correct":1}]},{"id":568,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"总人数40人，百分比55%","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 19:40:39","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":309,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"某班级在一次数学测验中，收集了30名学生的成绩（单位：分），并将数据整理如下：90分以上有8人，80～89分有12人，70～79分有6人，60～69分有3人，60分以下有1人。请问这次测验中，成绩在80分及以上的学生所占的百分比是多少？","answer":"D","explanation":"首先确定80分及以上的学生人数：90分以上有8人，80～89分有12人，因此80分及以上共有8 + 12 = 20人。总人数为30人。所求百分比为(20 ÷ 30) × 100% ≈ 66.7%。因此正确答案是D。本题考查数据的收集、整理与描述中百分比的计算，属于简单难度。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 15:35:32","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":"40%","is_correct":0},{"id":"B","content":"50%","is_correct":0},{"id":"C","content":"60%","is_correct":0},{"id":"D","content":"66.7%","is_correct":1}]},{"id":230,"subject":"数学","grade":"初一","stage":"小学","type":"填空题","content":"某学生在计算一个数减去5时，错误地算成了加上5，得到的结果是12。那么正确的计算结果应该是____。","answer":"2","explanation":"根据题意，某学生将‘减去5’误算为‘加上5’，得到12。说明原数加上5等于12，因此原数为12 - 5 = 7。正确的计算应是7减去5，即7 - 5 = 2。所以正确答案是2。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 14:41:00","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":2540,"subject":"数学","grade":"九年级","stage":"初中","type":"选择题","content":"某学生在学习投影与视图时，观察一个由两个相同立方体竖直叠放组成的不透明几何体。他从正面、左面和上面分别观察该几何体，得到的视图如下：正面和左面看到的都是上下排列的两个正方形，上面看到的是一个正方形。若将该几何体绕其竖直中心轴顺时针旋转90°，则旋转后从正面看到的视图是以下哪种？","answer":"B","explanation":"原几何体由两个立方体竖直叠放，因此其正面和左面视图均为上下两个正方形，上面视图为一个正方形。当绕竖直中心轴顺时针旋转90°后，几何体的左右侧面变为新的正面。但由于两个立方体是沿竖直方向堆叠的，旋转后高度方向不变，左右宽度也未改变，因此从新的正面观察，仍然看到的是上下排列的两个正方形。旋转不改变竖直堆叠关系，只改变水平朝向，故视图形状不变。因此正确答案为B。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"简单","points":1,"is_active":1,"created_at":"2026-01-10 16:43:03","updated_at":"2026-01-10 16:43:03","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":"一个正方形","is_correct":0},{"id":"B","content":"上下排列的两个正方形","is_correct":1},{"id":"C","content":"左右排列的两个正方形","is_correct":0},{"id":"D","content":"三个正方形排成一列","is_correct":0}]},{"id":533,"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，随机抽取了20名学生，记录了他们每周课外阅读的时间（单位：小时），数据如下：3, 5, 4, 6, 3, 7, 5, 4, 5, 6, 4, 3, 5, 6, 7, 4, 5, 6, 5, 4。为了分析这些数据，该学生制作了频数分布表。请问阅读时间为5小时的学生人数是多少？","answer":"C","explanation":"题目考查的是数据的收集、整理与描述中的频数统计。我们需要从给出的20个数据中，统计出数值为5的个数。原始数据为：3, 5, 4, 6, 3, 7, 5, 4, 5, 6, 4, 3, 5, 6, 7, 4, 5, 6, 5, 4。逐个数出5出现的次数：第2个是5，第7个是5，第9个是5，第13个是5，第17个是5，第19个是5，共出现6次。因此，阅读时间为5小时的学生有6人，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 18:45:20","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":"4人","is_correct":0},{"id":"B","content":"5人","is_correct":0},{"id":"C","content":"6人","is_correct":1},{"id":"D","content":"7人","is_correct":0}]}]