[{"id":1426,"subject":"数学","grade":"七年级","stage":"初中","type":"解答题","content":"某学校组织七年级学生参加数学实践活动，要求学生利用平面直角坐标系设计一个‘校园寻宝’路线。已知校园平面图上以正门为原点O(0,0)，向东为x轴正方向，向北为y轴正方向。第一个藏宝点A位于(3,4)，第二个藏宝点B位于(-2,6)，第三个藏宝点C位于(5,-3)。一名学生从正门出发，依次经过A、B、C三个点后返回正门。若该学生每走1个单位长度需要消耗2分钟，且在每个藏宝点停留整理数据的时间为5分钟。已知该学生总共用时不超过150分钟，问：该学生是否能在规定时间内完成整个寻宝任务？如果不能，最多可以跳过几个藏宝点（只能跳过B或C，不能跳过A），才能确保总时间不超过150分钟？请通过计算说明。","answer":"首先计算从原点O(0,0)到A(3,4)的距离：\n距离OA = √[(3-0)² + (4-0)²] = √(9+16) = √25 = 5\n\n从A(3,4)到B(-2,6)的距离：\n距离AB = √[(-2-3)² + (6-4)²] = √[(-5)² + 2²] = √(25+4) = √29 ≈ 5.385\n\n从B(-2,6)到C(5,-3)的距离：\n距离BC = √[(5+2)² + (-3-6)²] = √[7² + (-9)²] = √(49+81) = √130 ≈ 11.402\n\n从C(5,-3)返回原点O(0,0)的距离：\n距离CO = √[(5-0)² + (-3-0)²] = √(25+9) = √34 ≈ 5.831\n\n总行走距离 = OA + AB + BC + CO ≈ 5 + 5.385 + 11.402 + 5.831 = 27.618（单位长度）\n\n行走时间 = 27.618 × 2 ≈ 55.236（分钟）\n\n停留时间：共3个藏宝点，每个停留5分钟，总停留时间 = 3 × 5 = 15（分钟）\n\n总用时 ≈ 55.236 + 15 = 70.236（分钟）\n\n由于70.236 < 150，因此该学生能在规定时间内完成整个寻宝任务。\n\n但题目要求判断“是否能在规定时间内完成”，并进一步问“如果不能，最多可以跳过几个点”。然而根据计算，实际用时远小于150分钟，因此无需跳过任何点。\n\n但为严谨起见，我们验证是否存在理解偏差：题目中“总共用时不超过150分钟”是上限，而实际仅需约70分钟，远低于限制。\n\n因此结论是：该学生能在规定时间内完成整个寻宝任务，不需要跳过任何藏宝点。\n\n答案：能完成，不需要跳过任何点。","explanation":"本题综合考查了平面直角坐标系中两点间距离公式、实数的运算、近似计算以及实际问题的建模能力。解题关键在于正确运用距离公式√[(x₂−x₁)²+(y₂−y₁)²]计算各段路径长度，再结合时间与距离的关系（每单位2分钟）和停留时间进行总时间估算。虽然题目设置了‘是否超时’和‘跳过点’的复杂情境，但通过精确计算发现实际耗时远低于限制，体现了数学建模中数据验证的重要性。本题难度较高，因其融合了多个知识点并要求学生进行多步推理和实际判断，符合困难级别要求。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-06 11:34:57","updated_at":"2026-01-06 11:34:57","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":2412,"subject":"数学","grade":"八年级","stage":"初中","type":"选择题","content":"某学生在研究两个三角形时发现，△ABC 和 △DEF 中，∠A = ∠D，AB = DE，且 ∠B = ∠E。若他想证明这两个三角形全等，应使用以下哪个判定定理？此外，若 AC = 5 cm，BC = 7 cm，∠C = 60°，则根据全等性质，DF 的长度应为多少？","answer":"A","explanation":"题目中给出 ∠A = ∠D，AB = DE，∠B = ∠E，即两个角和它们的夹边分别相等，符合 ASA（角-边-角）全等判定定理。由于 AB 是 ∠A 与 ∠B 的夹边，对应边 DE 是 ∠D 与 ∠E 的夹边，因此 △ABC ≌ △DEF（ASA）。根据全等三角形的性质，对应边相等，AC 对应 DF，已知 AC = 5 cm，故 DF = 5 cm。因此正确答案为 A。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"中等","points":1,"is_active":1,"created_at":"2026-01-10 12:23:21","updated_at":"2026-01-10 12:23:21","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":"ASA，DF = 5 cm","is_correct":1},{"id":"B","content":"AAS，DF = 7 cm","is_correct":0},{"id":"C","content":"SAS，DF = 5 cm","is_correct":0},{"id":"D","content":"ASA，DF = 7 cm","is_correct":0}]},{"id":324,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"某学生在整理班级同学的课外阅读时间（单位：小时）时，记录了以下数据：2，3，5，3，4，3，6。这组数据的众数是多少？","answer":"B","explanation":"众数是一组数据中出现次数最多的数。观察数据：2，3，5，3，4，3，6。其中数字2出现1次，3出现3次，4出现1次，5出现1次，6出现1次。因此，出现次数最多的是3，共出现3次。所以这组数据的众数是3。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 15:38:21","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":"2","is_correct":0},{"id":"B","content":"3","is_correct":1},{"id":"C","content":"4","is_correct":0},{"id":"D","content":"5","is_correct":0}]},{"id":1314,"subject":"数学","grade":"七年级","stage":"小学","type":"解答题","content":"某学生在研究城市公园的路径规划时，发现一个矩形花坛ABCD被两条互相垂直的小路EF和GH分割成四个区域，其中E、F分别在AB和CD上，G、H分别在AD和BC上。已知矩形ABCD的长为(3x + 2)米，宽为(2x - 1)米，小路EF平行于AD，小路GH平行于AB，且两条小路的宽度均为1米。若四个区域的总面积比原矩形花坛面积减少了17平方米，求x的值。","answer":"解：\n\n设矩形ABCD的长为 AB = CD = (3x + 2) 米，宽为 AD = BC = (2x - 1) 米。\n\n则原矩形花坛的面积为：\nS_原 = 长 × 宽 = (3x + 2)(2x - 1)\n\n展开得：\nS_原 = 3x·2x + 3x·(-1) + 2·2x + 2·(-1) = 6x² - 3x + 4x - 2 = 6x² + x - 2\n\n小路EF平行于AD，说明EF是横向小路，长度为AB = (3x + 2) 米，宽度为1米，因此其面积为：\nS_EF = (3x + 2) × 1 = 3x + 2\n\n小路GH平行于AB，说明GH是纵向小路，长度为AD = (2x - 1) 米，宽度为1米，因此其面积为：\nS_GH = (2x - 1) × 1 = 2x - 1\n\n但注意：两条小路在中心相交，重叠部分是一个1×1 = 1平方米的正方形，被重复计算了一次，因此实际减少的面积为：\nS_减少 = S_EF + S_GH - 1 = (3x + 2) + (2x - 1) - 1 = 5x\n\n根据题意，四个区域的总面积比原面积减少了17平方米，即：\nS_减少 = 17\n\n所以有方程：\n5x = 17\n\n解得：\nx = 17 ÷ 5 = 3.4\n\n答：x 的值为 3.4。","explanation":"本题综合考查了整式的加减、一元一次方程以及几何图形初步中的面积计算。解题关键在于理解两条互相垂直的小路将矩形分割后，其面积减少的部分等于两条小路面积之和减去重叠部分（避免重复计算）。通过设定变量、列代数式表示原面积和小路面积，建立一元一次方程求解。难点在于识别重叠区域的处理，以及正确展开和化简整式。题目情境新颖，结合实际生活中的路径规划，考查学生的建模能力和逻辑推理能力，符合七年级数学课程中关于整式运算和一元一次方程的应用要求。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-06 10:51:57","updated_at":"2026-01-06 10:51:57","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":496,"subject":"数学","grade":"初一","stage":"初中","type":"选择题","content":"在一次环保活动中，某班级收集了可回收垃圾的重量数据如下：纸类12.5千克，塑料8.3千克，金属4.7千克，玻璃6.5千克。老师要求将总重量四舍五入到个位后，再计算平均每种垃圾的重量（保留一位小数）。请问平均重量是多少千克？","answer":"C","explanation":"首先计算四种垃圾的总重量：12.5 + 8.3 + 4.7 + 6.5 = 32.0（千克）。题目要求将总重量四舍五入到个位，32.0四舍五入后仍为32千克。接着计算平均重量：32 ÷ 4 = 8.0（千克），保留一位小数即为8.0。因此正确答案是C。本题考查了有理数的加法、四舍五入规则以及平均数的计算，属于数据的收集、整理与描述知识点，难度为简单。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 18:08:17","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":"7.8","is_correct":0},{"id":"B","content":"7.9","is_correct":0},{"id":"C","content":"8.0","is_correct":1},{"id":"D","content":"8.1","is_correct":0}]},{"id":553,"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时间（单位：小时）时，记录了以下5个数据：2.5，3，3.5，4，4.5。如果他想用这组数据制作频数分布表，并将数据分为两组：3小时以下（不含3小时）和3小时及以上，那么这两组的频数分别是多少？","answer":"A","explanation":"首先明确分组标准：第一组是“3小时以下（不含3小时）”，即小于3；第二组是“3小时及以上”，即大于或等于3。原始数据为：2.5，3，3.5，4，4.5。其中，只有2.5小于3，属于第一组，频数为1；其余数据3、3.5、4、4.5均大于或等于3，属于第二组，共4个数据，频数为4。因此，两组的频数分别是1和4，正确答案为A。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 19:11:16","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和4","is_correct":1},{"id":"B","content":"2和3","is_correct":0},{"id":"C","content":"3和2","is_correct":0},{"id":"D","content":"4和1","is_correct":0}]},{"id":2327,"subject":"数学","grade":"八年级","stage":"初中","type":"选择题","content":"某学生在研究轴对称图形时，发现一个四边形ABCD关于直线MN对称，其中点A与点C对称，点B与点D对称。若∠ABC = 70°，则∠ADC的度数为多少？","answer":"A","explanation":"由于四边形ABCD关于直线MN轴对称，且点A与点C对称，点B与点D对称，说明图形在对称轴两侧完全重合。因此，对应角相等。∠ABC与∠ADC是关于对称轴对应的角，故∠ADC = ∠ABC = 70°。本题考查轴对称图形的性质：对称点所连线段被对称轴垂直平分，且对称图形中对应角、对应边相等。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"简单","points":1,"is_active":1,"created_at":"2026-01-10 10:51:50","updated_at":"2026-01-10 10:51:50","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":"70°","is_correct":1},{"id":"B","content":"110°","is_correct":0},{"id":"C","content":"90°","is_correct":0},{"id":"D","content":"140°","is_correct":0}]},{"id":2459,"subject":"数学","grade":"八年级","stage":"初中","type":"填空题","content":"某学生在研究一组数据时发现，这组数据的平均数是12，若将每个数据都乘以2后再减去3，得到的新数据组的平均数是___。","answer":"21","explanation":"原平均数为12，每个数据乘以2后平均数变为24，再减去3，新平均数为24 - 3 = 21。数据线性变换后平均数按相同规律变化。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"中等","points":1,"is_active":1,"created_at":"2026-01-10 14:10:31","updated_at":"2026-01-10 14:10:31","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":495,"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时间数据时，发现一周内每天阅读时间（单位：分钟）分别为：30、45、0、60、35、50、40。如果去掉一个最低值和一个最高值后，剩余数据的平均数是多少？","answer":"C","explanation":"首先将数据按从小到大排列：0、30、35、40、45、50、60。最低值是0，最高值是60，去掉这两个值后，剩余数据为：30、35、40、45、50。计算这五个数的平均数：(30 + 35 + 40 + 45 + 50) ÷ 5 = 200 ÷ 5 = 42。因此，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 18:06:51","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":"A","content":"40","is_correct":0},{"id":"B","content":"41","is_correct":0},{"id":"C","content":"42","is_correct":1},{"id":"D","content":"43","is_correct":0}]},{"id":386,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"某班级为了了解学生对数学课的喜爱程度，随机抽取了30名学生进行调查，并将结果整理如下：非常喜欢12人，比较喜欢10人，一般5人，不太喜欢3人。若用扇形统计图表示这些数据，则“比较喜欢”这一类别对应的圆心角度数是多少？","answer":"A","explanation":"在扇形统计图中，每个类别的圆心角度数 = （该类别人数 ÷ 总人数）× 360度。本题中，“比较喜欢”的人数为10人，总人数为30人，因此对应的圆心角为 (10 ÷ 30) × 360 = (1\/3) × 360 = 120度。故正确答案为A。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2025-12-29 15:56:18","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":"120度","is_correct":1},{"id":"B","content":"100度","is_correct":0},{"id":"C","content":"90度","is_correct":0},{"id":"D","content":"80度","is_correct":0}]}]