AI 狂想曲：10亿+高质量题库引爆“数据燃料”革命

{"id": "f2e4e9c0c97367cb4600b6ae", "title": "设函数$f$在闭区间$[a,b]$上连续，并且$\\int_a^b fdx = 0$。如果对于任意$x\\in [a,b]$，都有$f\\geq 0$，证明：对所有$x \\in [a, b]$有$f=0$。", "option": null, "answer": "利用反证法结合连续函数的介值定理和积分性质进行证明。", "parse": "假设存在$x_0\\in[a,b]$使得$f(x_0)>0$。根据连续函数的局部保号性，在$x_0$附近存在一个小区间$(x_0-\\delta,x_0+\\delta)$（适当缩小以保证仍在$[a,b]$内），在这个小区间内$f>0$。那么$\\int_a^b fdx\\geq\\int_{x_0-\\delta}^{x_0+\\delta}fdx>0$，这与已知条件$\\int_a^b fdx = 0$矛盾。所以对所有$x \\in [a, b]$有$f=0$。", "qtype": "证明题", "subject": "数学/数学分析", "grade": "大学", "has_img": false, "image_analysis": {"img0": {"image_present": false}}, "classification": {"major_category": "Science", "sub_category": "Mathematics", "subject": "Math. Anal."}, "correctness": {"is_correct": true}, "difficulty": {"knowledge_point": "Definite integral, Properties of continuous functions, Intermediate Value Theorem, Proof by contradiction", "knowledge_level": "high school", "educational_stage": "university"}, "quality": {"logical_coherence": 4, "educational_value": 4, "clarity": 4}, "question_metadata": {"question_type": "Short Answer", "language": "Chinese", "ai_generation_likelihood": 4}}
{"id": "dcb705347f19e67c20de8a97", "title": "已知某金属离子$M^{n+}$与氨水反应形成配合物$[M(NH_3)_6]^{n+}$。若该配合物的磁矩为零，且金属离子的电子排布为$d^6$，试确定该金属离子在周期表中的位置，并说明理由。", "option": null, "answer": "铁", "parse": "根据题目信息，金属离子的电子排布为$d^6$，并且形成的配合物磁矩为零，这意味着所有未成对电子都已配对。对于$d^6$电子排布，在八面体场中只有当形成低自旋态时，即所有电子都进入$t_{2g}$轨道并完全配对时，磁矩才会为零。因此，该金属离子应处于第VIII族，常见$d^6$电子排布且能形成低自旋态的金属离子是铁。铁在周期表中的位置为第四周期，第VIII族。", "qtype": "简答题", "subject": "化学/化学", "grade": "大学", "has_img": false, "image_analysis": {"img0": {"image_present": false}}, "classification": {"major_category": "Science", "sub_category": "Chemistry", "subject": "Inorg. Chem."}, "correctness": {"is_correct": true}, "difficulty": {"knowledge_point": "Coordination chemistry, Crystal field theory, Electronic configuration, Magnetic moment, Periodic table", "knowledge_level": "high school", "educational_stage": "university"}, "quality": {"logical_coherence": 4, "educational_value": 4, "clarity": 4}, "question_metadata": {"question_type": "Short Answer", "language": "Chinese", "ai_generation_likelihood": 4}}
{"id": "6aef2e4e9002c2467cb7daeb", "title": "设$D$是由$x=0, y=0, x+y=2$所围成的三角形闭区域，计算二重积分\n$$\nI = \\iint_D (x^2 + y^2) dxdy.\n$$", "option": null, "answer": "$\\frac{8}{3}$", "parse": "首先确定积分区域$D$为由$x=0, y=0, x+y=2$所围成的三角形。选择先对$y$后对$x$积分，则$x$的范围是从$0$到$2$，对于固定的$x$，$y$的范围是从$0$到$2-x$。\n\n\n$$\nI = \\int_0^2 dx \\int_0^{2-x} (x^2 + y^2) dy\n$$\n\n$$\n\\int_0^{2-x} (x^2 + y^2) dy = [x^2y + \\frac{1}{3}y^3]_0^{2-x} = x^2(2-x) + \\frac{1}{3}(2-x)^3\n$$\n\n$$\nI = \\int_0^2 [x^2(2-x) + \\frac{1}{3}(2-x)^3] dx\n$$\n将上式展开并逐项积分可得结果为$\\frac{8}{3}$。", "qtype": "计算题", "subject": "数学/高等数学", "grade": "大学", "has_img": false, "image_analysis": {"img0": {"image_present": false}}, "classification": {"major_category": "Science", "sub_category": "Mathematics", "subject": "Math. Anal."}, "correctness": {"is_correct": true}, "difficulty": {"knowledge_point": "double integral, iterated integral, region of integration, polynomial integration", "knowledge_level": "high school", "educational_stage": "university"}, "quality": {"logical_coherence": 4, "educational_value": 4, "clarity": 4}, "question_metadata": {"question_type": "Calculation", "language": "Chinese", "ai_generation_likelihood": 4}}
{"id": "6aef2e4e97f67cb12d7b4b45", "title": "求解以下初值问题：\n$$ y'' - 5y' + 6y = e^{2t}, \\quad y(0) = 1, \\quad y'(0) = 0 $$", "option": null, "answer": "$y = 3e^{2t} - 2e^{3t} + te^{2t}$", "parse": "\n首先求解对应的齐次方程 $y'' - 5y' + 6y = 0$ 的通解。特征方程为 $r^2 - 5r + 6 = 0$，解得 $r_1 = 2$ 和 $r_2 = 3$。因此齐次方程的通解为 $y_h = C_1e^{2t} + C_2e^{3t}$。\n接下来使用待定系数法求特解。由于非齐次项为 $e^{2t}$，设特解形式为 $y_p = Ate^{2t}$。代入原方程得到 $A = 1$，因此特解为 $y_p = te^{2t}$。\n方程的通解为 $y = y_h + y_p = C_1e^{2t} + C_2e^{3t} + te^{2t}$。\n利用初始条件 $y(0) = 1$ 和 $y'(0) = 0$ 确定常数 $C_1$ 和 $C_2$。由 $y(0) = 1$ 得到 $C_1 + C_2 = 1$；由 $y'(0) = 0$ 得到 $2C_1 + 3C_2 + 1 = 0$。解此线性方程组得到 $C_1 = 3$ 和 $C_2 = -2$。\n因此最终解为 $y = 3e^{2t} - 2e^{3t} + te^{2t}$。", "qtype": "计算题", "subject": "数学/常微分方程", "grade": "大学", "has_img": false, "image_analysis": {"img0": {"image_present": false}}, "classification": {"major_category": "Science", "sub_category": "Mathematics", "subject": "Ord. Diff. Eq."}, "correctness": {"is_correct": true}, "difficulty": {"knowledge_point": "Second-order linear ordinary differential equation, Homogeneous equation, Characteristic equation, Method of undetermined coefficients, Initial value problem, Linear system of equations", "knowledge_level": "high school", "educational_stage": "university"}, "quality": {"logical_coherence": 5, "educational_value": 4.5, "clarity": 5}, "question_metadata": {"question_type": "Calculation", "language": "Chinese", "ai_generation_likelihood": 4}}
{"id": "b3b42b6ae67cf2e4e9b3c866", "title": "设$R=\\mathbb{Z}[x]$，理想$I=(3,x^2+x+1)$。求商环$R/I$中元素的个数。", "option": null, "answer": "9", "parse": "首先注意到$\\mathbb{Z}[x]/(3,x^2+x+1)$可以看作先对$x^2+x+1$取模，再对3取模的结果。由于$x^2+x+1$在$\\mathbb{Z}_3[x]$中是不可约多项式（没有根），因此$\\mathbb{Z}_3[x]/(x^2+x+1)$是一个有9个元素的有限域，即每个元素都可以表示为$a+bx+(x^2+x+1)$的形式，其中$a,b\\in\\mathbb{Z}_3$，共有$3\\times3=9$种组合。", "qtype": "计算题", "subject": "数学/抽象代数", "grade": "大学", "has_img": false, "image_analysis": {"img0": {"image_present": false}}, "classification": {"major_category": "Science", "sub_category": "Mathematics", "subject": "Adv. Algebra"}, "correctness": {"is_correct": true}, "difficulty": {"knowledge_point": "Ideal, Quotient Ring, Finite Field, Irreducible Polynomial, Modular Arithmetic", "knowledge_level": "high school", "educational_stage": "university"}, "quality": {"logical_coherence": 4, "educational_value": 4, "clarity": 4}, "question_metadata": {"question_type": "Calculation", "language": "Chinese", "ai_generation_likelihood": 4}}

{
    "id": "9c0c9330b6ae736f2e4e7cb46",
    "title": "Дано $$P(-5,0)$$, точка $$Q$$ лежит на окружности $$(x-5)^{2}+y^{2}=36$$, $$M$$ — середина отрезка $$PQ$$. $$($$Ⅰ$$)$$ Найдите уравнение траектории $$C$$ точки $$M$$. $$($$Ⅱ$$)$$ Прямая $$l$$, проходящая через точку $$P$$, пересекает траекторию $$C$$ в двух точках $$A$$ и $$B$$ ($$A$$ и $$B$$ не совпадают$$)$$. $$①$$ Если $$|AB|=4$$, найдите уравнение прямой $$l$$. $$②$$ Найдите значение $$ \\overrightarrow{PA}⋅ \\overrightarrow{PB}$$.",
    "option": null,
    "answer": "(Ⅰ) Используя формулу координат середины отрезка и тот факт, что точка Q лежит на заданной окружности, найти уравнение траектории точки M. (Ⅱ) ① Записать уравнение прямой, проходящей через точку P, решить систему уравнений прямой и траектории C. Используя теорему Виета и формулу длины хорды, найти уравнение прямой l. ② Вычислить скалярное произведение векторов, используя их координаты и теорему Виета, чтобы найти его значение.",
    "parse": "$$($$Ⅰ$$)$$ Решение: Пусть $$M(x,y)$$, тогда точка, симметричная $$P(-5,0)$$ относительно $$M$$, есть $$Q(2x+5,2y)$$. $$∵$$ Точка $$Q$$ лежит на окружности $$(x-5)^{2}+y^{2}=36$$, $$∴(2x+5-5)^{2}+(2y)^{2}=36$$, то есть $$x^{2}+y^{2}=9$$. Следовательно, уравнение траектории $$C$$ есть $$x^{2}+y^{2}=9$$.\n\n$$($$Ⅱ$$)$$\n①$$ Решение: Пусть $$A(x_{1},y_{1})$$ , $$B(x_{2},y_{2})$$. По условию, угловой коэффициент прямой $$l$$ существует, обозначим его через $$k$$, тогда уравнение прямой $$l$$ есть $$y=k(x+5)$$. Из системы уравнений $$ \\begin{cases} y=k(x+5) \\\\ x^{2}+y^{2}=9\\end{cases}$$, получаем $$(1+k^{2})x^{2}+10k^{2}x+25k^{2}-9=0$$. Из $$ \\triangle =(10k^{2})^{2}-4(1+k^{2})(25k^{2}-9) > 0$$, получаем $$- \\dfrac {3}{4} < k < \\dfrac {3}{4}$$ $$∴x_{1}+x_{2}=- \\dfrac {10k^{2}}{1+k^{2}},x_{1}x_{2}= \\dfrac {25k^{2}-9}{1+k^{2}}$$. $$∵|AB|=4$$, $$∴ \\sqrt {1+k^{2}}|x_{1}-x_{2}|=4$$, $$∴ \\sqrt {1+k^{2}}\\cdot \\sqrt {(x_{1}+x_{2})^{2}-4x_{1}x_{2}}=4$$, $$∴ \\sqrt {1+k^{2}}\\cdot \\sqrt {(- \\dfrac {10k^{2}}{1+k^{2}})^{2}- \\dfrac {4(25k^{2}-9)}{1+k^{2}}}=4$$. Решая, находим $$k=± \\dfrac {1}{2}$$. $$∴$$ Уравнение прямой $$l$$ есть $$y=± \\dfrac {1}{2}(x+5)$$, то есть $$x+2y+5=0$$ или $$x-2y+5=0$$.\n\n$$②$$ Решение: $$ \\overrightarrow{PA}=(x_1+5, y_1)$$ и $$ \\overrightarrow{PB}=(x_2+5, y_2)$$. \n$$ \\overrightarrow{PA} \\cdot \\overrightarrow{PB} = (x_1+5)(x_2+5) + y_1y_2 $$. \nТак как $$y_1=k(x_1+5)$$ и $$y_2=k(x_2+5)$$, то $$y_1y_2 = k^2(x_1+5)(x_2+5)$$. \n$$ \\overrightarrow{PA} \\cdot \\overrightarrow{PB} = (1+k^2)(x_1+5)(x_2+5) = (1+k^2)(x_1x_2 + 5(x_1+x_2) + 25) $$. \nПодставляя выражения из теоремы Виета: \n$$ = (1+k^2)\\left( \\frac{25k^2-9}{1+k^2} + 5\\left(-\\frac{10k^2}{1+k^2}\\right) + 25 \\right) $$ \n$$ = (1+k^2) \\frac{25k^2-9 - 50k^2 + 25(1+k^2)}{1+k^2} $$ \n$$ = 25k^2-9 - 50k^2 + 25 + 25k^2 = 16 $$.\n$$∴$$ Значение $$ \\overrightarrow{PA}\\cdot \\overrightarrow{PB}$$ равно $$16$$. (Это значение является степенью точки P относительно окружности C и не зависит от наклона секущей $$l$$).",
    "qtype": "Задача с развернутым ответом",
    "subject": "Математика/Аналитическая геометрия",
    "grade": "Университет",
    "has_img": false,
    "image_analysis": {
        "img0": {
            "image_present": false
        }
    },
    "classification": {
        "major_category": "Science",
        "sub_category": "Mathematics",
        "subject": "Anal. Geom."
    },
    "correctness": {
        "is_correct": true
    },
    "difficulty": {
        "knowledge_point": "Уравнение геометрического места точек, Уравнение окружности, Уравнение прямой, Теорема Виета, Формула длины хорды, Скалярное произведение векторов, Степень точки относительно окружности",
        "knowledge_level": "high school",
        "educational_stage": "high school"
    },
    "quality": {
        "logical_coherence": 5,
        "educational_value": 4,
        "clarity": 4
    },
    "question_metadata": {
        "question_type": "Short Answer",
        "language": "Russian",
        "ai_generation_likelihood": 4
    }
}

{
    "id": "dcb705347f19e67c20de8a97",
    "title": "已知某金属离子$M^{n+}$与氨水反应形成配合物$[M(NH_3)_6]^{n+}$。若该配合物的磁矩为零，且金属离子的电子排布为$d^6$，试确定该金属离子在周期表中的位置，并说明理由。",
    "option": null,
    "answer": "铁",
    "parse": "根据题目信息，金属离子的电子排布为$d^6$，并且形成的配合物磁矩为零，这意味着所有未成对电子都已配对。对于$d^6$电子排布，在八面体场中只有当形成低自旋态时，即所有电子都进入$t_{2g}$轨道并完全配对时，磁矩才会为零。因此，该金属离子应处于第VIII族，常见$d^6$电子排布且能形成低自旋态的金属离子是铁。铁在周期表中的位置为第四周期，第VIII族。",
    "qtype": "简答题",
    "subject": "化学/化学",
    "grade": "大学",
    "has_img": false,
    "image_analysis": {
        "img0": {
            "image_present": false
        }
    },
    "classification": {
        "major_category": "Science",
        "sub_category": "Chemistry",
        "subject": "Inorg. Chem."
    },
    "correctness": {
        "is_correct": true
    },
    "difficulty": {
        "knowledge_point": "Coordination chemistry, Crystal field theory, Electronic configuration, Magnetic moment, Periodic table",
        "knowledge_level": "high school",
        "educational_stage": "university"
    },
    "quality": {
        "logical_coherence": 4,
        "educational_value": 4,
        "clarity": 4
    },
    "question_metadata": {
        "question_type": "Short Answer",
        "language": "Chinese",
        "ai_generation_likelihood": 4
    }
}

{
    "id": "6807164f991513c1ae30f45a",
    "title": "二氧化碳（CO_{2}$）的来源是以下哪一过程？",
    "option": [
        "A. 半水煤气经脱硫变换后的气体",
        "B. 石灰窑燃烧产生的废气",
        "C. 工业废气直接收集",
        "D. 天然气直接燃烧"
    ],
    "answer": "A",
    "parse": "$侯氏联合制碱法通过将半水煤气进行脱硫和变换处理，获得含CO_{2}的变换气作为原料，这一过程既满足制碱需求，又解决了合成氨原料气脱碳的问题。B选项是索尔维法的CO_{2}$$来源，C选项未明确具体工业废气来源且不符合工艺描述，D选项天然气燃烧生成的CO_{2}$未提及脱硫等预处理步骤，均不符合题意。",
    "qtype": "单选题",
    "subject": "化学/工业化学",
    "grade": "大学",
    "has_img": false,
    "image_analysis": {
        "img0": {
            "image_present": false
        }
    },
    "classification": {
        "major_category": "Engineering",
        "sub_category": "Chemical Engineering and Technology",
        "subject": "Mass Trans. & Sep. Process in Chem. Eng."
    },
    "correctness": {
        "is_correct": false,
        "error_type": "Wrong Answer"
    },
    "difficulty": {
        "knowledge_point": "Industrial Chemistry, CO2 sources, Haber-Bosch process, Solvay process",
        "knowledge_level": "high school",
        "educational_stage": "university"
    },
    "quality": {
        "logical_coherence": 3,
        "educational_value": 4,
        "clarity": 2
    },
    "question_metadata": {
        "question_type": "Single Choice",
        "language": "Chinese",
        "ai_generation_likelihood": 4
    }
}

{
    "id": "67ff4799a366742fe5edf95b",
    "title": "在泊松方程的九点差分格式中!!误差项 $R_{i,k}^*$ 的阶数为：",
    "option": [
        "A. $O(h^2)$",
        "B. $O(h^4)$",
        "C. $O(h^6)$",
        "D. $O(h^8)$"
    ],
    "answer": "C",
    "parse": "根据九点差分格式的推导，误差项 $R_{i,k}^*$ 的估计式中包含 $h^6$ 的项，因此其阶数为 $O(h^6)$。",
    "qtype": "单选题",
    "subject": "数学/数值分析",
    "grade": "大学",
    "has_img": false,
    "image_analysis": {
        "img0": {
            "image_present": false
        }
    },
    "classification": {
        "major_category": "Science",
        "sub_category": "Mathematics",
        "subject": "Num. Anal."
    },
    "correctness": {
        "is_correct": true
    },
    "difficulty": {
        "knowledge_point": "Poisson equation, nine-point difference scheme, error term, order of accuracy, numerical analysis",
        "knowledge_level": "university",
        "educational_stage": "university"
    },
    "quality": {
        "logical_coherence": 4,
        "educational_value": 3,
        "clarity": 4
    },
    "question_metadata": {
        "question_type": "Single Choice",
        "language": "Chinese",
        "ai_generation_likelihood": 4
    }
}