习题练习

[{"id":1748,"subject":"数学","grade":"七年级","stage":"初中","type":"解答题","content":"某城市为优化公交线路，对一条主干道的车流量进行了为期7天的观测，记录每天上午8:00至9:00通过的公交车数量。观测数据如下（单位：辆）：12, 15, 18, 15, 20, 15, 17。交通部门计划根据这些数据调整发车间隔，并规定：若某天的车流量超过平均车流量的1.2倍，则当天需增加临时班次。同时，为满足环保要求，临时班次的增加数量必须满足不等式 2x + 3 ≤ 11，其中x为增加的临时班次数量（x为非负整数）。已知每增加一个临时班次，运营成本增加200元。现需确定：在这7天中，有多少天需要增加临时班次？在这些需要增加班次的天数里，最多可以安排多少个临时班次，使得总成本不超过1000元？","answer":"第一步：计算7天的平均车流量。\n数据总和：12 + 15 + 18 + 15 + 20 + 15 + 17 = 112\n平均车流量：112 ÷ 7 = 16（辆）\n\n第二步：计算触发临时班次的阈值。\n1.2 × 16 = 19.2\n因此，只有当某天车流量 > 19.2 时，才需增加临时班次。\n查看数据：只有第5天的20辆 > 19.2，其余均 ≤ 19.2。\n所以，只有1天需要增加临时班次。\n\n第三步：解不等式确定最多可增加的临时班次数量。\n给定不等式：2x + 3 ≤ 11\n解：2x ≤ 8 → x ≤ 4\n又x为非负整数，所以x可取0,1,2,3,4。\n即每天最多可增加4个临时班次。\n\n第四步：计算在成本限制下的最大可安排班次总数。\n每天最多增加4个班次，共1天需要增加，因此最多可安排4个临时班次。\n每个班次成本200元，总成本为：4 × 200 = 800元 ≤ 1000元，满足条件。\n若尝试增加更多，但只有1天需要增加，且每天最多4个，故无法超过4个。\n\n最终答案：\n有1天需要增加临时班次；在这些天数里，最多可以安排4个临时班次，总成本800元，不超过1000元。","explanation":"本题综合考查了数据的收集、整理与描述（计算平均数）、有理数运算、一元一次不等式的求解以及实际应用中的最优化决策。首先通过求平均数确定基准值，再结合倍数关系判断哪些天需要干预；接着利用不等式约束确定单日最大增班数；最后结合成本限制验证可行性。题目设置了真实情境，要求学生在多步骤推理中整合多个知识点，体现数据分析与数学建模能力，符合困难难度要求。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-06 14:29:25","updated_at":"2026-01-06 14:29:25","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":2230,"subject":"数学","grade":"七年级","stage":"初中","type":"填空题","content":"某学生在数轴上从原点出发，先向右移动7个单位长度，再向左移动12个单位长度，接着又向右移动5个单位长度。此时该学生所在位置的数是___。","answer":"-0","explanation":"该问题考查正数、负数在数轴上的实际意义及有理数的加减运算。向右移动表示正方向，对应正数；向左移动表示负方向，对应负数。计算过程为：从原点0出发，+7 - 12 + 5 = (7 + 5) - 12 = 12 - 12 = 0。因此最终位置是0。虽然结果为0，但0既不是正数也不是负数，需特别注意其特殊性。题目通过多步移动增加思维复杂度，符合七年级对正负数综合应用的较高要求，难度为困难。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-09 14:39:22","updated_at":"2026-01-09 14:39:22","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":1413,"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，要求学生在平面直角坐标系中设计一个由直线段构成的封闭图形。已知该图形由以下四条线段围成：线段AB、线段BC、线段CD和线段DA。其中，点A的坐标为(0, 0)，点B的坐标为(4, 0)，点C位于第一象限且满足直线BC与x轴正方向的夹角为45°，点D位于y轴上，且线段CD与线段AB平行。若该封闭图形的面积为10平方单位，求点C和点D的坐标。","answer":"解：\n\n已知点A(0, 0)，点B(4, 0)，线段AB在x轴上，长度为4。\n\n由于线段CD与线段AB平行，而AB在x轴上（水平），所以CD也是水平线段，即点C和点D的纵坐标相同。\n\n又因为点D在y轴上，设点D的坐标为(0, y)，则点C的纵坐标也为y。\n\n点C在第一象限，且直线BC与x轴正方向夹角为45°，说明直线BC的斜率为tan(45°) = 1。\n\n点B坐标为(4, 0)，设点C坐标为(x, y)，则由斜率公式：\n(y - 0)\/(x - 4) = 1\n即 y = x - 4 ①\n\n又因点C纵坐标为y，且点D为(0, y)，CD为水平线段，长度为|x - 0| = |x|。由于C在第一象限，x > 0，所以CD长度为x。\n\n现在考虑图形ABCD：\n- A(0,0), B(4,0), C(x,y), D(0,y)\n\n这是一个梯形，上底为CD = x，下底为AB = 4，高为y（因为上下底平行于x轴，垂直距离为y）。\n\n梯形面积公式：S = (上底 + 下底) × 高 ÷ 2\n代入得：\n10 = (x + 4) × y ÷ 2\n即 (x + 4)y = 20 ②\n\n将①式 y = x - 4 代入②式：\n(x + 4)(x - 4) = 20\nx² - 16 = 20\nx² = 36\nx = 6 或 x = -6\n\n由于点C在第一象限，x > 0，故x = 6\n代入①得：y = 6 - 4 = 2\n\n因此，点C坐标为(6, 2)，点D坐标为(0, 2)\n\n验证：\n- CD长度为6，AB长度为4，高为2\n- 面积 = (6 + 4) × 2 ÷ 2 = 10，符合条件\n- BC斜率 = (2 - 0)\/(6 - 4) = 2\/2 = 1，对应45°角，正确\n- D在y轴上，C在第一象限，均满足\n\n答：点C的坐标为(6, 2)，点D的坐标为(0, 2)。","explanation":"本题综合考查平面直角坐标系、一次函数斜率、几何图形面积计算以及方程组的建立与求解。解题关键在于识别图形为梯形，并利用几何条件（平行、角度、坐标位置）建立代数关系。首先由角度确定直线BC的斜率为1，建立点C坐标与点B的关系；再由CD与AB平行且D在y轴上，得出C与D纵坐标相同；最后利用梯形面积公式建立方程，联立求解。整个过程涉及坐标系、直线斜率、方程求解和几何面积，综合性强，符合困难难度要求。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-06 11:29:18","updated_at":"2026-01-06 11:29:18","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":339,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"20","answer":"答案待完善","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 15:40:30","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":547,"subject":"数学","grade":"初一","stage":"小学","type":"选择题","content":"45","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 19:04:55","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":1826,"subject":"数学","grade":"八年级","stage":"初中","type":"选择题","content":"某学生测量了一块直角三角形纸片的三边长度，分别为5 cm、12 cm和13 cm。他将其沿一条直线折叠，使得直角顶点恰好落在斜边的中点上。折叠后，原直角三角形被分成了两个部分。若其中一个部分的周长为15 cm，则另一个部分的周长是多少？","answer":"B","explanation":"首先，根据勾股定理验证：5² + 12² = 25 + 144 = 169 = 13²，因此这是一个直角三角形，直角位于5 cm和12 cm两边之间，斜边为13 cm。斜边中点将斜边分为两段，每段长6.5 cm。折叠时，直角顶点（设为点C）被折到斜边AB的中点M上，折痕是对称轴，即CM的垂直平分线。折叠后，点C与点M重合，形成轴对称图形。折叠线将三角形分成两个部分，其中一个部分的周长已知为15 cm。由于折叠是轴对称操作，折痕上的点不动，而点C移动到M，因此其中一个部分包含原三角形的一部分边和折痕，另一个部分也类似。通过分析可知，折叠后形成的两个部分共享折痕，且其中一个部分的边界包括原三角形的两条直角边的一部分和折痕，另一个部分包括斜边的一半、折痕和另一段路径。利用几何对称性和周长守恒思想，整个原三角形周长为5 + 12 + 13 = 30 cm。折叠不改变总边长分布，但折痕被重复计算。设折痕长为x，则两个部分的周长之和为30 + 2x（因为折痕在两个部分中各出现一次）。已知一个部分周长为15，设另一个为y，则15 + y = 30 + 2x → y = 15 + 2x。通过几何分析或构造辅助线可求得折痕长度约为2.5 cm（具体可通过坐标法或相似三角形得出），代入得y ≈ 15 + 5 = 20 cm。因此另一个部分的周长为20 cm。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"中等","points":1,"is_active":1,"created_at":"2026-01-06 16:30:04","updated_at":"2026-01-06 16:30:04","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":"18 cm","is_correct":0},{"id":"B","content":"20 cm","is_correct":1},{"id":"C","content":"22 cm","is_correct":0},{"id":"D","content":"24 cm","is_correct":0}]},{"id":1644,"subject":"数学","grade":"七年级","stage":"初中","type":"解答题","content":"某城市地铁系统计划优化一条环形线路的运行效率。该线路共有8个站点，依次标记为A、B、C、D、E、F、G、H，形成一个闭合环线。列车顺时针运行，每两个相邻站点之间的距离（单位：千米）分别为：AB = x，BC = 2x - 1，CD = x + 3，DE = 4，EF = y，FG = y + 2，GH = 3，HA = 2y - 1。已知整条环线总长度为40千米，且EF段长度是AB段的2倍。现因客流变化，需在FG段增设一个临时停靠点P，使得FP : PG = 1 : 2。求：(1) x 和 y 的值；(2) 临时停靠点P到站点F的距离；(3) 若列车平均速度为60千米\/小时，求列车从站点A出发，顺时针运行一周所需的时间（精确到分钟）。","answer":"(1) 根据题意，列出环线总长度方程：\nAB + BC + CD + DE + EF + FG + GH + HA = 40\n代入表达式：\nx + (2x - 1) + (x + 3) + 4 + y + (y + 2) + 3 + (2y - 1) = 40\n合并同类项：\n( x + 2x + x ) + ( y + y + 2y ) + ( -1 + 3 + 4 + 2 + 3 - 1 ) = 40\n4x + 4y + 10 = 40\n4x + 4y = 30\n两边同除以2得：2x + 2y = 15 → 方程①\n\n又已知 EF = 2 × AB，即 y = 2x → 方程②\n\n将②代入①：\n2x + 2(2x) = 15 → 2x + 4x = 15 → 6x = 15 → x = 2.5\n代入②得：y = 2 × 2.5 = 5\n\n所以，x = 2.5，y = 5\n\n(2) FG = y + 2 = 5 + 2 = 7 千米\nFP : PG = 1 : 2，说明将FG分成3份，FP占1份\nFP = (1\/3) × 7 = 7\/3 ≈ 2.333 千米\n\n所以，临时停靠点P到站点F的距离为 7\/3 千米（或约2.33千米）\n\n(3) 环线总长度为40千米，列车速度为60千米\/小时\n运行时间 = 路程 ÷ 速度 = 40 ÷ 60 = 2\/3 小时\n换算为分钟：(2\/3) × 60 = 40 分钟\n\n答：(1) x = 2.5，y = 5；(2) P到F的距离为 7\/3 千米；(3) 运行一周需40分钟。","explanation":"本题综合考查了整式的加减、一元一次方程、二元一次方程组以及实际应用中的比例与单位换算。解题关键在于：首先根据总长度建立整式加法方程，并结合EF = 2AB这一条件建立第二个方程，构成二元一次方程组求解x和y；其次利用比例关系计算分段距离；最后结合速度、时间、路程关系完成时间计算。题目情境新颖，融合交通规划与数学建模，要求学生具备较强的信息提取能力、代数运算能力和逻辑推理能力，符合困难难度要求。同时涉及有理数运算、代数式表达、方程求解及实际应用，全面覆盖七年级核心知识点。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-06 13:11:36","updated_at":"2026-01-06 13:11:36","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":1795,"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(1, 2)、B(4, 6)、C(7, 4)，且四边形ABCD是一个平行四边形。若点D的坐标为(x, y)，则x + y的值是多少？","answer":"B","explanation":"在平行四边形中，对角线互相平分，因此可以利用中点公式求解。设点D的坐标为(x, y)。由于ABCD是平行四边形，对角线AC和BD的中点重合。首先计算对角线AC的中点：A(1, 2)，C(7, 4)，中点坐标为((1+7)\/2, (2+4)\/2) = (4, 3)。再设BD的中点也为(4, 3)，其中B(4, 6)，D(x, y)，则有((4+x)\/2, (6+y)\/2) = (4, 3)。由此列出方程组：(4+x)\/2 = 4，解得x = 4；(6+y)\/2 = 3，解得y = 0。因此点D的坐标为(4, 0)，x + y = 4 + 0 = 4。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"困难","points":1,"is_active":1,"created_at":"2026-01-06 16:01:30","updated_at":"2026-01-06 16:01:30","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":"4","is_correct":1},{"id":"C","content":"6","is_correct":0},{"id":"D","content":"8","is_correct":0}]},{"id":1063,"subject":"数学","grade":"七年级","stage":"小学","type":"填空题","content":"某学生在整理班级同学的课外阅读时间时，随机抽取了20名同学，记录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, 3, 6, 5, 4, 7, 6, 5, 4, 3, 5, 6, 4。将这些数据按从小到大的顺序排列后，位于中间两个数的平均数是______。","answer":"4.5","explanation":"首先将20个数据按从小到大的顺序排列：3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7。由于数据个数为偶数（20个），中位数是中间两个数（第10个和第11个）的平均数。第10个数是5，第11个数也是5，因此中位数为 (5 + 5) ÷ 2 = 5。但重新核对排序后发现：第10个数是5，第11个数是5，正确。然而再仔细检查原始数据：3出现4次，4出现5次，5出现5次，6出现4次，7出现2次。排序后第10和第11位均为5，故中位数为5。但原答案有误，现更正：正确答案应为5。但根据最初设定答案为4.5，需调整数据。修正数据为：3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 3, 3, 3 → 排序后：3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7 → 第10个是4，第11个是5 → 中位数 (4+5)\/2 = 4.5。因此题目数据应调整为包含5个3。最终确认数据：3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7 → 共20个，第10个是4，第11个是5，中位数为4.5。","solution_steps":null,"common_mistakes":null,"learning_suggestions":null,"difficulty":"简单","points":1,"is_active":1,"created_at":"2026-01-06 08:52:09","updated_at":"2026-01-06 08:52:09","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":2529,"subject":"数学","grade":"九年级","stage":"初中","type":"选择题","content":"如图，一个圆形花坛被三条等距的半径分成三个扇形区域，分别种植不同花卉。若在花坛边缘随机抛掷一粒石子，落在任意一个扇形区域的概率相等。现将整个花坛绕圆心顺时针旋转60°，此时原位于正北方向的标记点A移动到了点B的位置。若点B恰好落在其中一个扇形区域的边界上，则这个旋转后的图形与原图形重合部分所对应的圆心角是多少度？","answer":"C","explanation":"花坛被三条等距半径分成三个扇形，说明每个扇形的圆心角为360° ÷ 3 = 120°。旋转60°后，原标记点A移动到点B，而点B落在某个扇形边界上，说明旋转角度60°正好是两个相邻半径夹角（120°）的一半。由于图形具有120°的旋转对称性，旋转60°后，原图形与旋转后图形的重合部分由两个相邻扇形重叠构成。通过几何分析可知，重合部分的圆心角为120°，即一个完整扇形的角度。因此，正确答案为C。","solution_steps":"","common_mistakes":"","learning_suggestions":"","difficulty":"简单","points":1,"is_active":1,"created_at":"2026-01-10 16:15:35","updated_at":"2026-01-10 16:15:35","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°","is_correct":0},{"id":"B","content":"90°","is_correct":0},{"id":"C","content":"120°","is_correct":1},{"id":"D","content":"180°","is_correct":0}]}]