| {"id": "AST mathematics - 109 - 2", "question": "有 $A, B$ 兩個箱子, 其中 $A$ 箱有 6 顆白球與 4 顆紅球, $B$ 箱有 8 顆白球與 2 顆監球。現有三種抽獎方式（各箱中每顆球被抽取的機率相同）:\n(一) 先在 $A$ 箱中抽取一球, 若抽中紅球則停止, 若抽到白球則再從 $B$ 箱中抽取一球;\n(二) 先在 $B$ 箱中抽取一球, 若抽中藍球則停止, 若抽到白球則再從 $A$ 箱中抽取一球;\n(三) 同時分別在 $A, B$ 箱中各抽取一球。\n給獎方式為: 在紅、藍這兩種色球當中, 若只抽到紅球得 50 元獎金; 若只抽到藍球得 100 元獎金; 若兩種色球都抽到, 則仍只得 100 元獎金; 若都沒抽到, 則無獎金。將上列 (一)、(二)、(三) 這 3 種抽獎方式所得獎金的期望值分別記為 $E_{1} 、 E_{2} 、 E_{3}$, 試選出正確的選項。", "A": "$E_{3}>E_{2}>E_{1}$", "B": "$E_{1}=E_{3}>E_{2}$", "C": "$E_{1}=E_{2}>E_{3}$", "D": "$E_{1}>E_{2}>E_{3}$", "E": "$E_{2}=E_{3}>E_{1}$", "F": null, "answer": "E", "explanation": "", "metadata": {"timestamp": "2024-01-09T01:16:58.113688", "source": "AST mathematics - 109", "explanation_source": ""}, "human_evaluation": {"quality": "", "comments": ""}} | |
| {"id": "AST mathematics - 112 - 3", "question": "試問極限\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{3}{n^{2}}\\left(\\sqrt{4 n^{2}+9 \\times 1^{2}}+\\sqrt{4 n^{2}+9 \\times 2^{2}}+\\cdots+\\sqrt{4 n^{2}+9 \\times(n-1)^{2}}\\right)\n$$\n的值可用下列哪一個定積分表示?", "A": "$\\int_{0}^{3} \\sqrt{4 x^{2}+9} d x$", "B": "$\\int_{0}^{3} \\sqrt{1+x^{2}} d x$", "C": "$\\int_{0}^{3} \\sqrt{4+9 x^{2}} d x$", "D": "$\\int_{0}^{3} \\sqrt{4+x^{2}} d x$", "E": "$\\int_{0}^{3} \\sqrt{1+9 x^{2}} d x$", "F": null, "answer": "D", "explanation": "", "metadata": {"timestamp": "2024-01-09T01:16:58.113942", "source": "AST mathematics - 112", "explanation_source": ""}, "human_evaluation": {"quality": "", "comments": ""}} | |
| {"id": "AST mathematics - 106 - 4", "question": "已知一實係數三次多項式 $f(x)$ 在 $x=1$ 有極大值 3 , 且圖形 $y=f(x)$ 在 $(4, f(4))$ 之切線方程式為 $y-f(4)+5(x-4)=0$, 試問 $\\int_{1}^{4} f^{\\prime \\prime}(x) d x$ 之值為下列哪一選項 ?", "A": "5", "B": "0", "C": "-3", "D": "-5", "E": "3", "F": null, "answer": "D", "explanation": "", "metadata": {"timestamp": "2024-01-09T01:16:58.115033", "source": "AST mathematics - 106", "explanation_source": ""}, "human_evaluation": {"quality": "", "comments": ""}} | |
| {"id": "AST mathematics - 109 - 3", "question": "根據實驗統計, 某種細菌繁殖, 其數量平均每 3.5 小時會擴增為 2.4 倍。假設實驗室的試管一開始有此種細菌 1000 隻, 根據指數函數模型, 試問大約在多少小時後此種細菌的數量會到達 $4 \\times 10^{10}$ 隻左右？（註： $\\log 2 \\approx 0.3010 , \\log 3 \\approx 0.4771$ ) \\\\ \n", "A": "84 小時", "B": "63 小時", "C": "91 小時", "D": "70 小時", "E": "77 小時", "F": null, "answer": "D", "explanation": "", "metadata": {"timestamp": "2024-01-09T01:16:58.113691", "source": "AST mathematics - 109", "explanation_source": ""}, "human_evaluation": {"quality": "", "comments": ""}} | |
| {"id": "AST mathematics - 108 - 3", "question": "在一座尖塔的正南方地面某點 $A$, 測得塔頂的仰角為 $14^{\\circ}$; 又在此尖塔正東方地面某點 $B$, 測得塔頂的仰角為 $18^{\\circ} 30^{\\prime}$, 且 $A 、 B$ 兩點距離為 65 公尺。已知當在線段 $\\overline{A B}$ 上移動時, 在 $C$ 點測得塔頂的仰角為最大, 則 $C$ 點到塔底的距離最接近下列哪一個選項 ? $\\left(\\cot 14^{\\circ} \\approx 4.01, \\cot 18^{\\circ} 30^{\\prime} \\approx 2.99\\right)$", "A": "29 公尺", "B": "31 公尺", "C": "33 公尺", "D": "35 公尺", "E": "27 公尺", "F": null, "answer": "B", "explanation": "", "metadata": {"timestamp": "2024-01-09T01:16:58.114171", "source": "AST mathematics - 108", "explanation_source": ""}, "human_evaluation": {"quality": "", "comments": ""}} | |