Add AST math
Browse files- AST_mathematics.jsonl +57 -0
- AST_mathematics_dev.jsonl +5 -0
- AST_mathematics_test.jsonl +52 -0
- README.md +82 -79
AST_mathematics.jsonl
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{"id": "AST mathematics - 109補 - 1", "question": "考慮兩個函數 $f(x)=\\left\\{\\begin{array}{cc}1+x, & x \\leq 1 \\\\ 1, & x>1\\end{array} 、 g(x)=\\left\\{\\begin{array}{cc}1, & x \\leq 1 \\\\ 3-x, & x>1\\end{array}\\right.\\right.$ 關於函數的極限, 試選出正確的選項。", "correct_choices": ["$\\lim _{x \\rightarrow 1} f(x)$ 不存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 不存在、 $\\lim _{x \\rightarrow 1}(f(x)+g(x))$ 存在"], "incorrect_choices": ["$\\lim _{x \\rightarrow 1} f(x)$ 存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 存在、 $\\lim _{x \\rightarrow 1}(f(x)+g(x))$ 存在", "$\\lim _{x \\rightarrow 1} f(x)$ 存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 不存在、 $\\lim _{x \\rightarrow 1}(f(x)+g(x))$ 不存在", "$\\lim _{x \\rightarrow 1} f(x)$ 不存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 存在、 $\\lim _{x \\rightarrow 1}(f(x)+g(x))$ 不存在", "$\\lim _{x \\rightarrow 1} f(x)$ 不存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 不存在 $、 \\lim _{x \\rightarrow 1}(f(x)+g(x))$ 不存在"], "metadata": {"timestamp": "2024-01-09T01:16:58.113416", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 2", "question": "某質點在數線上移動, 已知其位置坐標為 $s(t)=\\int_{0}^{t}\\left(-x^{2}+6 x\\right) d x$, 其中 $t$ 表時間且 $0 \\leq t \\leq 10$ 。若此質點的速度在時段 $0 \\leq t<a$ 遞增, 且在時段 $a<t \\leq 10$ 遞減, 試選出正確的 $a$ 值。", "correct_choices": ["3"], "incorrect_choices": ["4", "5", "6", "7"], "metadata": {"timestamp": "2024-01-09T01:16:58.113446", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 3", "question": "在坐標平面上, 其 $x$ 坐標與 $y$ 坐標都是整數的點稱為 $「$ 格子點 $」$ 試問滿足方程式 $\\log _{2}(x-1)=\\log _{4}\\left(25-y^{2}\\right)$ 的格子點 $(x, y)$ 共有幾個 ?", "correct_choices": ["5 個"], "incorrect_choices": ["4 個", "6 個", "8 個", "12 個"], "metadata": {"timestamp": "2024-01-09T01:16:58.113450", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 4", "question": "設二階方陣 $M$ 為在坐標平面上定義的線性變換, $O$ 為原點。已知 $M$ 可將不共線的三點 $O 、 A 、 B$ 映射至不共線的三點 $O 、 A^{\\prime} 、 B^{\\prime}$, 試選出正確的選項。", "correct_choices": ["$M$ 為可逆矩陣", "若 $M$ 將點 $C$ 映射至點 $C^{\\prime}$ 且 $\\overrightarrow{O C}=2 \\overrightarrow{O A}+3 \\overrightarrow{O B}$, 則 $\\overrightarrow{O C^{\\prime}}=2 \\overrightarrow{O A^{\\prime}}+3 \\overrightarrow{O B^{\\prime}}$", "$\\triangle O A^{\\prime} B^{\\prime}$ 的面積 $=\\triangle O A B$ 的面積 $\\times|\\operatorname{det}(M)|$"], "incorrect_choices": ["$\\angle A O B=\\angle A^{\\prime} O B^{\\prime}$", "$\\overline{O A}: \\overline{O B}=\\overline{O A^{\\prime}}: \\overline{O B^{\\prime}}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113454", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 5", "question": "下列選項中, 試選出與 $\\cos \\frac{\\pi}{7}+i \\sin \\frac{\\pi}{7}$ 相乘之後會得到實數的選項。( 註 : $\\left.i=\\sqrt{-1}\\right)$", "correct_choices": ["$\\cos \\frac{\\pi}{7}-i \\sin \\frac{\\pi}{7}$", "$-\\sin \\frac{5 \\pi}{14}+i \\cos \\frac{5 \\pi}{14}$"], "incorrect_choices": ["$\\cos \\frac{\\pi}{7}+i \\sin \\frac{\\pi}{7}$", "$\\sin \\frac{\\pi}{7}+i \\cos \\frac{\\pi}{7}$", "$\\sin \\frac{\\pi}{7}-i \\cos \\frac{\\pi}{7}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113458", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 6", "question": "持續投擲一枚公正股子, 在過程中若出現連續兩次點數的和為 7 時, 就停止投擲 。例如：若前兩次投擲分別出現點數 1 、 4 , 點數和不等於 7 , 所以繼續投擲; 若第三次投出點數 3 , 因為第二次與第三次點數和為 7 , 所以此時即停止投擲。關於此機率事件, 試選出正確的選項。", "correct_choices": ["在第一次投擲的點數為 6 的情況下，總共投擲兩次就停的機率為 $\\frac{1}{6}$", "總共投擲兩次就停止的機率為 $\\frac{1}{6}$"], "incorrect_choices": ["在第一次投擲的點數為 5 的情況下，總共投擲三次恰好停止的機率為 $\\frac{1}{6}$", "總共投擲三次恰好停止的機率大於 $\\frac{1}{6}$", "至少投擲三次才停止的機率為 $\\frac{1}{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113461", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 7", "question": "關於非常數的實係數多項式函數 $f(x)$, 試選出正確的選項。", "correct_choices": ["若 $f(1) f(2)<0$, 則存在 $c \\in(1,2)$ 滿足 $f(c)=0$", "若 $\\left(\\int_{0}^{1} f(x) d x \\right)\\left(\\int_{0}^{2} f(x) d x\right)<0$, 則存在 $c \\in(1,2)$ 滿足 $\\int_{0}^{c} f(x) d x=0$"], "incorrect_choices": ["若 $f(1) f(2)>0$, 則對任意的 $c \\in(1,2), f(c) \neq 0$ 均成立", "若 $f(1) f(2) f(3)<0$, 則存在 $c \\in(1,3)$ 滿足 $f(c)=0$", "若 $\\int_{1}^{2} f(x) d x=0$, 則 $f(1) f(2)<0$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113464", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 8", "question": "設 $a, b, c$ 為三實數, 且 $a>b>c$ 。已知 $2^{a}, 2^{b}, 2^{c}$ 三數依序成等差數列。試選出正確的選項。", "correct_choices": ["$2^{a+100}, 2^{b+100}, 2^{c+100}$ 三數依序成等差數列", "$a<b+1$", "$b \\geq \\frac{a+c}{2}$"], "incorrect_choices": ["$a, b, c$ 三數依序成等比數列", "$4^{a}, 4^{b}, 4^{c}$ 三數依序成等差數列"], "metadata": {"timestamp": "2024-01-09T01:16:58.113468", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 1", "question": "已知 $45^{\\circ}<\\theta<50^{\\circ}$, 且設 $a=1-\\cos ^{2} \\theta 、 b=\\frac{1}{\\cos \\theta}-\\cos \\theta 、 c=\\frac{\\tan \\theta}{\\tan ^{2} \\theta+1}$ 。關於 $a, b, c$ 三個數值的大小, 試選出正確的選項。", "correct_choices": ["$c<a<b$"], "incorrect_choices": ["$a<b<c$", "$a<c<b$", "$b<a<c$", "$b<c<a$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113684", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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。", "correct_choices": ["$E_{2}=E_{3}>E_{1}$"], "incorrect_choices": ["$E_{1}>E_{2}>E_{3}$", "$E_{1}=E_{2}>E_{3}$", "$E_{1}=E_{3}>E_{2}$", "$E_{3}>E_{2}>E_{1}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113688", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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： $\\log 2 \\approx 0.3010 ， \\log 3 \\approx 0.4771$ )", "correct_choices": ["70 小時"], "incorrect_choices": ["63 小時", "77 小時", "84 小時", "91 小時"], "metadata": {"timestamp": "2024-01-09T01:16:58.113691", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 4", "question": "在坐標平面上, 設 $O$ 為原點, 且 $A 、 B$ 為異於 $O$ 的相異兩點。令 $C_{1}, C_{2}, C_{3}$ 為平面上三個點, 且滿足 $\\overrightarrow{O C}_{n}=\\overrightarrow{O A}+n \\overrightarrow{O B}, n=1,2,3$, 試選出正確的選項。", "correct_choices": ["$\\overrightarrow{O C}_{1} \\cdot \\overrightarrow{O B}<\\overrightarrow{O C}_{2} \\cdot \\overrightarrow{O B}<\\overrightarrow{O C}_{3} \\cdot \\overrightarrow{O B}$", "$C_{1}, C_{2}, C_{3}$ 在同一直線上"], "incorrect_choices": ["$\\overrightarrow{O C}_{1} \\neq \\overrightarrow{0}$", "$\\overline{O C_{1}}<\\overline{O C_{2}}<\\overline{O C_{3}}$", "$\\overrightarrow{O C}_{1} \\cdot \\overrightarrow{O A}<\\overrightarrow{O C}_{2} \\cdot \\overrightarrow{O A}<\\overrightarrow{O C}_{3} \\cdot \\overrightarrow{O A}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113695", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 5", "question": "對一實數 $a$, 以 $[a]$ 表示不大於 $a$ 的最大整數, 例如 : $[1.2]=[\\sqrt{2}]=1,[-1.2]=-2$ 。考慮無理數 $\\theta=\\sqrt{10001}$, 試選出正確的選項。", "correct_choices": ["$a-1<[a] \\leq a$ 對任意實數 $a$ 均成立", "數列 $e_{n}=n\\left[\\frac{-\\theta}{n}\\right]$ 發散, $n$ 為正整數"], "incorrect_choices": ["數列 $b_{n}=\\frac{[n \\theta]}{n}$ 發散, $n$ 為正整數", "數列 $c_{n}=\\frac{[-n \\theta]}{n}$ 發散, $n$ 為正整數", "數列 $d_{n}=n\\left[\\frac{\\theta}{n}\\right]$ 發散, $n$ 為正整數"], "metadata": {"timestamp": "2024-01-09T01:16:58.113698", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 6", "question": "設 $F(x)$ 、 $f(x)$ 皆為實係數多項式函數。已知 $F^{\\prime}(x)=f(x)$, 試選出正確的選項。", "correct_choices": ["若 $a \\geq 0$, 則 $F(a)-F(0)=\\int_{0}^{a} f(t) d t$", "若 $F(x)$ 除以 $x$ 的商式為 $Q(x)$, 則 $Q(0)=f(0)$"], "incorrect_choices": ["若 $f(x)$ 可被 $x+1$ 整除, 則 $F(x)-F(0)$ 可被 $(x+1)^{2}$ 整除", "若對所有實數 $x, F(x) \\geq \\frac{x^{2}}{2}$ 都成立, 則對所有實數 $x, f(x) \\geq x$ 也都成立", "若對所有 $x>0 ， f(x) \\geq x$ 都成立, 則對所有 $x>0, F(x) \\geq \\frac{x^{2}}{2}$ 也都成立"], "metadata": {"timestamp": "2024-01-09T01:16:58.113701", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 7", "question": "在複數平面上, 設 $O$ 為原點, 且 $A 、 B$ 分別表示坐標為複數 $z 、 z+1$ 的點。已知點 $A$ 、點 $B$ 都在以 $O$ 為圓心的單位圓上, 試選出正確的選項。", "correct_choices": ["直線 $A B$ 與實數軸平行", "$z^{3}=1$", "坐標為 $1+\\frac{1}{z}$ 的點也在同一單位圓上"], "incorrect_choices": ["$\\triangle O A B$ 為直角三角形", "點 $A$ 在第二象限"], "metadata": {"timestamp": "2024-01-09T01:16:58.113705", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 8", "question": "設二階實係數方陣 $A$ 代表坐標平面的一個鏡射變換且滿足 $A^{3}=\\left[\\begin{array}{cc}0 & -1 \\\\ -1 & 0\\end{array}\\right]$; 另設二階實係數方陣 $B$ 代表坐標平面的一個（以原點為中心的）旋轉變換且滿足 $B^{3}=\\left[\\begin{array}{cc}-1 & 0 \\\\ 0 & -1\\end{array}\\right]$, 試選出正確的選項。", "correct_choices": ["$B$ 恰有三種可能", "$B A B A=\\left[\\begin{array}{ll}1 & 0 \\\\ 0 & 1\\end{array}\\right]$"], "incorrect_choices": ["$A$ 恰有三種可能", "$A B=B A$", "二階方陣 $A B$ 代表坐標平面的一個旋轉變換"], "metadata": {"timestamp": "2024-01-09T01:16:58.113708", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 1", "question": "坐標平面上, 一質點由點 $(-3,-2)$ 出發, 沿著向量 $(a, 1)$ 的方向移動 5 單位長之後剛好抵達 $x$ 軸, 其中 $a$ 為正實數。試問 $a$ 值等於下列哪一個選項？", "correct_choices": ["$\\frac{\\sqrt{21}}{2}$"], "incorrect_choices": ["$\\frac{\\sqrt{13}}{2}$", "2", "$\\sqrt{5}$", "$2 \\sqrt{6}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113935", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 2", "question": "放射性物質的半衰期 $T$ 定義為 $\\ulcorner$ 每經過時間 $T$, 該物質的質量會衰退成原來的一半」。鉛製容器中有 $A 、 B$ 兩種放射性物質, 其半衰期分別為 $T_{A} 、 T_{B}$ 。開始記錄時這兩種物質的質量相等, 112 天後測量發現物質 $B$ 的質量為物質 $A$ 的質量的四分之一。根據上述, 試問 $T_{A} 、 T_{B}$ 滿足下列哪一個關係式?", "correct_choices": ["$2+\\frac{112}{T_{A}}=\\frac{112}{T_{B}}$"], "incorrect_choices": ["$-2+\\frac{112}{T_{A}}=\\frac{112}{T_{B}}$", "$-2+\\log _{2} \\frac{112}{T_{A}}=\\log _{2} \\frac{112}{T_{B}}$", "$2+\\log _{2} \\frac{112}{T_{A}}=\\log _{2} \\frac{112}{T_{B}}$", "$2 \\log _{2} \\frac{112}{T_{A}}=\\log _{2} \\frac{112}{T_{B}}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113939", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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?", "correct_choices": ["$\\int_{0}^{3} \\sqrt{4+x^{2}} d x$"], "incorrect_choices": ["$\\int_{0}^{3} \\sqrt{1+x^{2}} d x$", "$\\int_{0}^{3} \\sqrt{1+9 x^{2}} d x$", "$\\int_{0}^{3} \\sqrt{4+9 x^{2}} d x$", "$\\int_{0}^{3} \\sqrt{4 x^{2}+9} d x$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113942", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 4", "question": "設 $a, b$ 為實數。已知四個數 $-3,-1,4,7$ 皆滿足 $x$ 的不等式 $|x-a| \\leq b$, 試選出正確的選項。", "correct_choices": ["$\\sqrt{10}$ 也滿足 $x$ 的不等式 $|x-a| \\leq b$", "$3,1,-4,-7$ 滿足 $x$ 的不等式 $|x+a| \\leq b$"], "incorrect_choices": ["$-\\frac{3}{2},-\\frac{1}{2}, 2, \\frac{7}{2}$ 滿足 $x$ 的不等式 $|x-a| \\leq \\frac{b}{2}$", "$b$ 可能等於 4", "$a, b$ 可能相等"], "metadata": {"timestamp": "2024-01-09T01:16:58.113946", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 5", "question": "考慮實係數多項式 $f(x)=x^{4}-4 x^{3}-2 x^{2}+a x+b$ 。已知方程式 $f(x)=0$ 有虛根 $1+2 i$\n（其中 $i=\\sqrt{-1}$ ), 試選出正確的選項。", "correct_choices": ["$1-2 i$ 也是 $f(x)=0$ 的根", "$f^{\\prime}(2.1)<0$"], "incorrect_choices": ["$a, b$ 皆為正數", "函數 $y=f(x)$ 在 $x=1$ 有局部極小值", "$y=f(x)$ 圖形反曲點的 $x$ 坐標皆大於 0"], "metadata": {"timestamp": "2024-01-09T01:16:58.113949", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 6", "question": "設 $a, b, c, d, r, s, t$ 皆為實數, 已知坐標空間中三個非零向量 $\\vec{u}=(a, b, 0) 、 \\vec{v}=(c, d, 0)$ 及 $\\vec{w}=(r, s, t)$ 滿足內積 $\\vec{w} \\cdot \\vec{u}=\\vec{w} \\cdot \\vec{v}=0$ 。考慮三階方陣 $A=\\left[\\begin{array}{lll}a & b & 0 \\\\ c & d & 0 \\\\ r & s & t\\end{array}\\right]$, 試選出正確的選項。", "correct_choices": ["若 $\\vec{u} \\cdot \\vec{v}=0$, 則行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$", "若對任意三個實數 $e, f, g$, 向量 $(e, f, g)$ 都可以表示成 $\\vec{u}, \\vec{v}, \\vec{w}$ 的線性組合, 則行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$", "若行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$, 則 $A$ 的行列式不等於 0"], "incorrect_choices": ["若 $t \\neq 0$, 則行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$", "若存在一個向量 $\\overrightarrow{w^{\\prime}}$ 滿足 $\\overrightarrow{w^{\\prime}} \\cdot \\vec{u}=\\overrightarrow{w^{\\prime}} \\cdot \\vec{v}=0$ 且外積 $\\overrightarrow{w^{\\prime}} \\times \\vec{w} \\neq \\overrightarrow{0}$, 則行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113952", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 1", "question": "某公司尾牙舉辦「紅包大放送」活動。每位員工擲兩枚均匀銅板一次, 若出現兩個反面可得獎金 400 元; 若出現一正一反可得獎金 800 元; 若出現兩個正面可得獎金 800 元並且獲得再擲一次的機會，其獲得獎金規則與前述相同，但不再有繼續投擲銅板的機會（也就是說每位員工最多有兩次擲銅板的機會）。試問每位參加活動的員工可獲得獎金的期望值為何?", "correct_choices": ["875 元"], "incorrect_choices": ["850 元", "900 元", "925 元", "950 元"], "metadata": {"timestamp": "2024-01-09T01:16:58.114165", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 2", "question": "設 $n$ 為正整數。第 $n$ 個費馬數 (Fermat Number) 定義為 $F_{n}=2^{\\left(2^{n}\\right)}+1$, 例如 $F_{1}=2^{\\left(2^{1}\\right)}+1=2^{2}+1=5, F_{2}=2^{\\left(2^{2}\\right)}+1=2^{4}+1=17$ 。試問 $\\frac{F_{13}}{F_{12}}$ 的整數部分以十進位表示時,其位數最接近下列哪一個選項 ? $(\\log 2 \\approx 0.3010)$", "correct_choices": ["1200"], "incorrect_choices": ["120", "240", "600", "900"], "metadata": {"timestamp": "2024-01-09T01:16:58.114168", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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 ? $\\left(\\cot 14^{\\circ} \\approx 4.01, \\cot 18^{\\circ} 30^{\\prime} \\approx 2.99\\right)$", "correct_choices": ["31 公尺"], "incorrect_choices": ["27 公尺", "29 公尺", "33 公尺", "35 公尺"], "metadata": {"timestamp": "2024-01-09T01:16:58.114171", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 4", "question": "設 $\\Gamma$ 為坐標平面上通過 $(7,0)$ 與 $\\left(0, \\frac{7}{2}\\right)$ 兩點的圓。試選出正確的選項。", "correct_choices": ["當 $\\Gamma$ 的半徑達到最小可能值時, $\\Gamma$ 通過原點", "若 $\\Gamma$ 的圓心在第三象限, 則 $\\Gamma$ 的半徑大於 8"], "incorrect_choices": ["$\\Gamma$ 的半徑大於或等於 5", "$\\Gamma$ 與直線 $x+2 y=6$ 有交點", "$\\Gamma$ 的圓心不可能在第四象限"], "metadata": {"timestamp": "2024-01-09T01:16:58.114174", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 5", "question": "袋中有 2 顆紅球、 3 顆白球與 1 顆藍球, 其大小皆相同。今將袋中的球逐次取出,每次隨機取出一顆, 取後不放回, 直到所有球被取出為止。試選出正確的選項。", "correct_choices": ["「取出的第一顆為紅球」的機率等於 「取出的第二顆為紅球」的機率", "「取出的前三顆皆為白球」的機率小於 $\\ulcorner$ 取出的前三顆球顏色皆相異」的機率"], "incorrect_choices": ["「取出的第一顆為紅球」與 $\\ulcorner$ 取出的第二顆為紅球」兩者為獨立事件", "「取出的第一顆為紅球」與 $\\ulcorner$ 取出的第二顆為白球或藍球」兩者為互斥事件", "「取出的第一、二顆皆為紅球 $\\lrcorner$ 的機率等於 $\\ulcorner$ 取出的第一、二顆皆為白球 $\\lrcorner$的機率"], "metadata": {"timestamp": "2024-01-09T01:16:58.114177", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 6", "question": "設 $\\left\\langle a_{n}\\right\\rangle 、\\left\\langle b_{n}\\right\\rangle$ 為兩實數數列, 且對所有的正整數 $n, a_{n}<b_{n}{ }^{2}<a_{n+1}$ 均成立。若已知 $\\lim _{n \\rightarrow \\infty} a_{n}=4$, 試選出正確的選項。", "correct_choices": ["對所有的正整數 $n, b_{n}{ }^{2}<b_{n+1}{ }^{2}$ 均成立", "$\\lim _{n \\rightarrow \\infty} b_{n}^{2}=4$"], "incorrect_choices": ["對所有的正整數 $n, a_{n}>3$ 均成立", "存在正整數 $n$, 使得 $a_{n+1}>4$", "$\\lim _{n \\rightarrow \\infty} b_{n}=2$ 或 $\\lim _{n \\rightarrow \\infty} b_{n}=-2$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114180", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 8", "question": "坐標平面上以原點 $O$ 為圓心的單位圓上三相異點 $A 、 B 、 C$ 滿足 $2 \\overrightarrow{O A}+3 \\overrightarrow{O B}+4 \\overrightarrow{O C}=\\overrightarrow{0}$, 其中 $A$ 點的坐標為 $(1,0)$ 。試選出正確的選項。", "correct_choices": ["向量 $2 \\overrightarrow{O A}+3 \\overrightarrow{O B}$ 的長度為 4", "$3 \\sin \\angle A O B=4 \\sin \\angle A O C$"], "incorrect_choices": ["內積 $\\overrightarrow{O A} \\cdot \\overrightarrow{O B}<0$", "$\\angle B O C 、 \\angle A O C 、 \\angle A O B$ 中, 以 $\\angle B O C$ 的度數為最小", "$\\overline{A B}>\\frac{3}{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114183", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 1", "question": "設 $A$ 為 $3 \\times 3$ 矩陣, 且對任意實數 $a, b, c, A\\left[\\begin{array}{l}a \\\\ b \\\\ c\\end{array}\\right]=\\left[\\begin{array}{l}b \\\\ c \\\\ a\\end{array}\\right]$ 均成立。試問矩陣 $A^{2}\\left[\\begin{array}{c}1 \\\\ 0 \\\\ -1\\end{array}\\right]$ 為何?", "correct_choices": ["$\\left[\\begin{array}{c}-1 \\\\ 1 \\\\ 0\\end{array}\\right]$"], "incorrect_choices": ["$\\left[\\begin{array}{l}0 \\\\ 1 \\\\ 1\\end{array}\\right]$", "$\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 1\\end{array}\\right]$", "$\\left[\\begin{array}{c}0 \\\\ 1 \\\\ -1\\end{array}\\right]$", "$\\left[\\begin{array}{c}-1 \\\\ 0 \\\\ 1\\end{array}\\right]$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114397", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 2", "question": "坐標平面上, 考慮 $A(2,3)$ 與 $B(-1,3)$ 兩點, 並設 $O$ 為原點。令 $E$ 為滿足 $\\overrightarrow{O P}=a \\overrightarrow{O A}+b \\overrightarrow{O B}$ 的所有點 $P$ 所形成的區域, 其 中 $-1 \\leq a \\leq 1,0 \\leq b \\leq 4$ 。考慮函數 $f(x)=x^{2}+5$, 試問當限定 $x$ 為區域 $E$ 中的點 $P(x, y)$ 的横坐標時, $f(x)$ 的最大值為何 ?", "correct_choices": ["41"], "incorrect_choices": ["5", "9", "30", "54"], "metadata": {"timestamp": "2024-01-09T01:16:58.114401", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 3", "question": "某零售商店販賣 $\\ulcorner$ 熊大 」與 $\\ulcorner$ 皮卡丘 $\\lrcorner$ 兩種玩偶, 其進貨來源有 $A, B, C$ 三家廠商。已知此零售商店從每家廠商進貨的玩偶總數相同, 且三家廠商製作的每一種玩偶外觀也一樣, 而從 $A, B, C$ 這三家廠商進貨的玩偶中, $\\ulcorner$ 皮卡丘 $\\lrcorner$ 所占的比例分別為 $\\frac{1}{4} 、 \\frac{2}{5} 、 \\frac{1}{2}$ 。阿德從這家零售商店隨機挑選一隻 $\\ulcorner$ 皮卡丘」送給小安作為生日禮物, 試問此「皮卡丘」出自 $C$ 廠商的機率為何?", "correct_choices": ["$\\frac{10}{23}$"], "incorrect_choices": ["$\\frac{1}{3}$", "$\\frac{2}{5}$", "$\\frac{10}{19}$", "$\\frac{5}{9}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114404", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 4", "question": "設 $f(x)=-x^{2}+499$, 且\n$A=\\int_{0}^{10} f(x) d x 、 B=\\sum_{n=0}^{9} f(n) 、 C=\\sum_{n=1}^{10} f(n) 、 D=\\sum_{n=0}^{9} \\frac{f(n)+f(n+1)}{2}$\n試選出正確的選項。", "correct_choices": ["$A$ 表示在坐標平面上函數 $y=-x^{2}+499$ 的圖形與直線 $y=0 、 x=0 、 x=10$ 所圍成的有界區域的面積", "$C<D$"], "incorrect_choices": ["$B<C$", "$B<A$", "$A<D$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114406", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 5", "question": "坐標平面上, 已知直線 $L$ 與函數 $y=\\log _{2} x$ 的圖形有兩個交點 $P(a, b), Q(c, d)$,且 $\\overline{P Q}$ 的中點在 $x$ 軸上。試選出正確的選項。", "correct_choices": ["$L$ 的斜率大於 0", "$a c=1$", "$L$ 的 $x$ 截距大於 1"], "incorrect_choices": ["$b d=-1$", "$L$ 的 $y$ 截距大於 -1"], "metadata": {"timestamp": "2024-01-09T01:16:58.114410", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 6", "question": "坐標空間中, 有 $\\vec{a} 、 \\vec{b} 、 \\vec{c} 、 \\vec{d}$ 四個向量, 滿足外積 $\\vec{a} \\times \\vec{b}=\\vec{c}, \\vec{a} \\times \\vec{c}=\\vec{d}$, 且 $\\vec{a} 、 \\vec{b} 、 \\vec{c}$ 的向量長度均為 4 。設向量 $\\vec{a}$ 與 $\\vec{b}$ 的夾角為 $\\theta$ (其中 $0 \\leq \\theta \\leq \\pi$ ), 試選出正確的選項。", "correct_choices": ["$\\vec{a} 、 \\vec{b} 、 \\vec{c}$ 所張出的平行六面體的體積為 16", "$\\vec{a} 、 \\vec{c} 、 \\vec{d}$ 兩兩互相垂直"], "incorrect_choices": ["$\\cos \\theta=\\frac{1}{4}$", "$\\vec{d}$ 的長度等於 4", "$\\vec{b}$ 與 $\\vec{d}$ 的夾角等於 $\\theta$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114416", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 7", "question": "設 $O$ 為複數平面上的原點, 並令點 $A, B$ 分別代表複數 $z_{1}, z_{2}$, 且滿足 $\\left|z_{1}\\right|=2,\\left|z_{2}\\right|=3$, $\\left|z_{2}-z_{1}\\right|=\\sqrt{5}$ 。若 $\\frac{z_{2}}{z_{1}}=a+b i$, 其中 $a, b$ 為實數, $i=\\sqrt{-1}$ 。試選出正確的選項。", "correct_choices": ["$\\cos \\angle A O B=\\frac{2}{3}$", "$a>0$"], "incorrect_choices": ["$\\left|z_{2}+z_{1}\\right|=\\sqrt{23}$", "$b>0$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114419", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 8", "question": "設 $f(x)$ 為一定義在非零實數上的實數值函數。已知極限 $\\lim _{x \\rightarrow 0} f(x) \\frac{|x|}{x}$ 存在, 試選出正確的選項。", "correct_choices": ["$\\lim _{x \\rightarrow 0}\\left(\\frac{x}{|x|}\\right)^{2}$ 存在", "$\\lim _{x \\rightarrow 0} f(x) \\frac{x}{|x|}$ 存在", "$\\lim _{x \\rightarrow 0} f(x)^{2}$ 存在"], "incorrect_choices": ["$\\lim _{x \\rightarrow 0}(f(x)+1) \\frac{x}{|x|}$ 存在", "$\\lim _{x \\rightarrow 0} f(x)$ 存在"], "metadata": {"timestamp": "2024-01-09T01:16:58.114422", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 1", "question": "假設 2 階方陣 $\\left[\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right]$ 所代表的線性變換將坐標平面上三點 $O(0,0), A(1,0), B(0,1)$ 分別映射到 $O(0,0), A^{\\prime}(3, \\sqrt{3}), B^{\\prime}(-\\sqrt{3}, 3)$ ，並將與原點距離為 1 的點 $C(x, y)$ 映射到點 $C^{\\prime}\\left(x^{\\prime}, y^{\\prime}\\right)$ 。試選出正確的選項。", "correct_choices": ["$\\overrightarrow{O C}$ 和 $\\overrightarrow{O C^{\\prime}}$ 的夾角為 $60^{\\circ}$"], "incorrect_choices": ["行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right|=6$", "$\\overline{O C^{\\prime}}=2 \\sqrt{3}$", "有可能 $y=y^{\\prime}$", "若 $x<y$ 則 $x^{\\prime}<y^{\\prime}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114724", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 2", "question": "設 $c$ 為實數使得三元一次方程組 $\\left\\{\\begin{array}{c}x-y+z=0 \\\\ 2 x+c y+3 z=1 \\\\ 3 x-3 y+c z=0\\end{array}\\right.$ 無解。試選出 $c$ 之值。", "correct_choices": ["-2"], "incorrect_choices": ["-3", "0", "2", "3"], "metadata": {"timestamp": "2024-01-09T01:16:58.114728", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 3", "question": "坐標空間中 $O$ 為原點, 點 $P$ 在第一卦限且 $\\overline{O P}=1$ 。已知直線 $O P$ 與 $x$ 軸有一夾角為 $45^{\\circ}$, 且 $P$ 點到 $y$ 軸的距離為 $\\frac{\\sqrt{6}}{3}$ 。試選出點 $P$ 的 $z$ 坐標。", "correct_choices": ["$\\frac{\\sqrt{6}}{6}$"], "incorrect_choices": ["$\\frac{1}{2}$", "$\\frac{\\sqrt{2}}{4}$", "$\\frac{\\sqrt{3}}{3}$", "$\\frac{\\sqrt{3}}{6}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114731", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 4", "question": "設多項式 $f(x)=x^{3}+2 x^{2}-2 x+k 、 g(x)=x^{2}+a x+1$, 其 中 $k, a$ 為實數。已知 $g(x)$ 整除 $f(x)$, 且方程式 $g(x)=0$ 有虛根���試選出為方程式 $f(x)=0$ 的根之選項。", "correct_choices": ["-3", "$\\frac{1+\\sqrt{-3}}{2}$"], "incorrect_choices": ["0", "1", "$\\frac{3+\\sqrt{-5}}{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114734", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 5", "question": "坐標平面上有一圖形 $\\Gamma$, 其方程式為 $(x-1)^{2}+(y-1)^{2}=101$ 。試選出正確的選項。", "correct_choices": ["$\\Gamma$ 與 $x$ 軸負向、 $y$ 軸負向分別交於 $(-9,0)$ 、 $(0,-9)$", "$\\Gamma$ 上的點與原點距離的最大值為 $\\sqrt{2}+\\sqrt{101}$", "$\\Gamma$ 經旋轉線性變換後, 其圖形仍可用一個不含 $x y$ 項的二元二次方程式表示"], "incorrect_choices": ["$\\Gamma$ 上 $x$ 坐標最大的點是點 $(11,0)$", "$\\Gamma$ 在第三象限的點之極坐標可用 $[9, \\theta]$ 表示, 其中 $\\pi<\\theta<\\frac{3}{2} \\pi$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114738", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 8", "question": "假設兩數列 $\\left\\langle a_{n}\\right\\rangle 、\\left\\langle b_{n}\\right\\rangle$, 對所有正整數 $n$ 都滿足 $b_{n}+\\frac{4 n-1}{n}<a_{n}<3 b_{n}$ 。已知 $\\lim _{n \\rightarrow \\infty} a_{n}=6$,試選出正確的選項。", "correct_choices": ["$b_{n}>\\frac{4 n-1}{2 n}$", "$a_{10000}>5.9$"], "incorrect_choices": ["$b_{n}<6-\\frac{4 n-1}{n}$", "數列 $\\left\\langle b_{n}\\right\\rangle$ 有可能發散", "$a_{10000}<6.1$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114741", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 1", "question": "設 $F(x)$ 為一實係數多項式且 $F^{\\prime}(x)=f(x)$ 。已知 $f^{\\prime}(x)>x^{2}+1.1$ 對所有的實數 $x$ 均成立, 試選出正確的選項。", "correct_choices": ["$f(f(x))$ 為遞增函數"], "incorrect_choices": ["$f^{\\prime}(x)$ 為遞增函數", "$f(x)$ 為遞增函數", "$F(x)$ 為遞增函數", "$[f(x)]^{2}$ 為遞增函數"], "metadata": {"timestamp": "2024-01-09T01:16:58.114932", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 2", "question": "研究團隊採用某快篩試劑的檢驗, 以了解保護區內生物因環境汗染而導致體內毒素累積超過標準的比率。此試劑檢驗結果只有紅色、黃色兩種。依據過去的經驗得知：若體內毒素累積超過標準, 經此試劑檢驗後, 有 $75 \\%$ 顯示為紅色; 若體內毒素累積未超過標準, 經此試劑檢驗後, 有 $95 \\%$ 顯示為黃色。已知此保護區的某類生物經試劑檢驗後, 有 $7.8 \\%$ 的結果顯示為紅色。假設此類生物實際體內毒素累積超過標準的比率為 $p \\%$, 試選出正確的選項。", "correct_choices": ["$3 \\leq p<5$"], "incorrect_choices": ["$1 \\leq p<3$", "$5 \\leq p<7$", "$7 \\leq p<9$", "$9 \\leq p<11$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114935", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 3", "question": "試求極限 $\\lim _{n \\rightarrow \\infty} \\frac{10^{10}}{n^{10}}\\left[1^{9}+2^{9}+3^{9}+\\cdots+(2 n)^{9}\\right]$ 的值 。", "correct_choices": ["$10^{9} \\times 2^{10}$"], "incorrect_choices": ["$10^{9}$", "$10^{9} \\times\\left(2^{10}-1\\right)$", "$2^{9} \\times\\left(10^{10}-1\\right)$", "$2^{9} \\times 10^{10}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114938", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 4", "question": "某電子公司有數百名員工, 其用餐方式分為自備、外食兩種。經長期調查發現：若當日用餐為自備的員工, 則隔天會有 $10 \\%$ 轉為外食; 若當日用餐為外食的員工, 則隔天會有 $20 \\%$ 轉為自備。\n假設 $x_{0} 、 y_{0}$ 分別代表該公司今日用餐自備人數與外食人數占員工總人數的比例,其中 $x_{0} 、 y_{0}$ 皆為正數, 且 $x_{n} 、 y_{n}$ 分別代表經過 $n$ 日後用餐自備人數與外食人數占員工總人數的比例。在該公司員工不變動的情形下, 試選出正確的選項。", "correct_choices": ["$\\left[\\begin{array}{l}x_{n+1} \\\\ y_{n+1}\\end{array}\\right]=\\left[\\begin{array}{ll}0.9 & 0.2 \\\\ 0.1 & 0.8\\end{array}\\right]\\left[\\begin{array}{l}x_{n} \\\\ y_{n}\\end{array}\\right]$", "若 $\\frac{x_{0}}{y_{0}}=\\frac{2}{1}$, 則 $\\frac{x_{n}}{y_{n}}=\\frac{2}{1}$ 對任意正整數 $n$ 均成立"], "incorrect_choices": ["$y_{1}=0.9 y_{0}+0.2 x_{0}$", "若 $y_{0}>x_{0}$, 則 $y_{1}>x_{1}$", "若 $x_{0}>y_{0}$, 則 $x_{0}>x_{1}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114941", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 5", "question": "假設 $f(x)$ 為五次實係數多項式, 且 $f(x)$ 除以 $x^{n}-1$ 的餘式為 $r_{n}(x), n$ 是正整數。試選出正確的選項。", "correct_choices": ["$r_{1}(x)=f(1)$", "$r_{4}(x)$ 除以 $x^{2}-1$ 所得的餘式等於 $r_{2}(x)$"], "incorrect_choices": ["$r_{2}(x)$ 是一次實係數多項式", "$r_{5}(x)=r_{6}(x)$", "若 $f(-x)=-f(x)$, 則 $r_{3}(-x)=-r_{3}(x)$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114944", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 6", "question": "一個標有 1 至 12 號格子的 12 格翟翟樂遊戲, 每回遊戲以投擲一枚均匀銅板四次來決定要戳哪些格子。規則如下：\n（一）第一次投擲銅板, 若是正面, 則翟 1 號格子; 若是反面, 則翟 3 號格子。\n（二）第二、三、四次投擲銅板, 若是正面, 則所翟格子的號碼為前一次所翟格子的號碼加 1 ; 若是反面, 則所翟格子的號碼為前一次所羽格子的號碼加 3 , 依此類推。\n例如：投鄭銅板四次的結果依序為 $\\ulcorner$ 正、反、反、正」, 則會㪬編號分別為 1 、 4、7、8 號的四個格子。\n假設 $p_{m}$ 代表在每回遊戲中 $m$ 號格子被翟到的機率, 試選出正確的選項。", "correct_choices": ["$p_{2}=\\frac{1}{4}$", "$\\quad p_{4}=\\frac{1}{2} p_{1}+\\frac{1}{2} p_{3}$", "$p_{8}>p_{10}$"], "incorrect_choices": ["$p_{3}=\\frac{1}{2}$", "在 4 號格子被翟到的條件下, 3 號格子被翟到的機率為 $\\frac{1}{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114947", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 8", "question": "已知 $z_{1} 、 z_{2} 、 z_{3} 、 z_{4}$ 為四個相異複數, 且其在複數平面上所對應的點, 依序可連成一個平行四邊形, 試問下列哪些選項必為實數?", "correct_choices": ["$z_{1}-z_{2}+z_{3}-z_{4}$", "$\\frac{z_{1}-z_{2}}{z_{3}-z_{4}}$"], "incorrect_choices": ["$\\left(z_{1}-z_{3}\\right)\\left(z_{2}-z_{4}\\right)$", "$z_{1}+z_{2}+z_{3}+z_{4}$", "$\\left(\\frac{z_{2}-z_{4}}{z_{1}-z_{3}}\\right)^{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114950", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 1", "question": "從所有二位正整數中隨機選取一個數, 設 $p$ 是其十位數字小於個位數字的機率。關於 $p$ 值的範圍, 試選出正確的選項。", "correct_choices": ["$0.33 \\leq p<0.44$"], "incorrect_choices": ["$0.22 \\leq p<0.33$", "$0.44 \\leq p<0.55$", "$0.55 \\leq p<0.66$", "$0.66 \\leq p<0.77$"], "metadata": {"timestamp": "2024-01-09T01:16:58.115023", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 2", "question": "設 $a=\\sqrt[3]{10}$ 。關於 $a^{5}$ 的範圍, 試選出正確的選項。", "correct_choices": ["$45 \\leq a^{5}<50$"], "incorrect_choices": ["$25 \\leq a^{5}<30$", "$30 \\leq a^{5}<35$", "$35 \\leq a^{5}<40$", "$40 \\leq a^{5}<45$"], "metadata": {"timestamp": "2024-01-09T01:16:58.115027", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 3", "question": "試問在 $0 \\leq x \\leq 2 \\pi$ 的範圍 中, $y=3 \\sin x$ 的函數圖形與 $y=2 \\sin 2 x$ 的函數圖形有幾個交點?", "correct_choices": ["5 個交點"], "incorrect_choices": ["2 個交點", "3 個交點", "4 個交點", "6 個交點"], "metadata": {"timestamp": "2024-01-09T01:16:58.115030", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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 ?", "correct_choices": ["-5"], "incorrect_choices": ["-3", "0", "3", "5"], "metadata": {"timestamp": "2024-01-09T01:16:58.115033", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 5", "question": "設 $\\vec{u}$ 與 $\\vec{v}$ 為兩非零向量, 夾角為 $120^{\\circ}$ 。若 $\\vec{u}$ 與 $\\vec{u}+\\vec{v}$ 垂直, 試選出正確的選項。", "correct_choices": ["$\\vec{v}$ 與 $\\vec{u}+\\vec{v}$ 的夾角為 $30^{\\circ}$", "$\\vec{u}$ 與 $\\vec{u}-\\vec{v}$ 的夾角為銳角"], "incorrect_choices": ["$\\vec{u}$ 的長度是 $\\vec{v}$ 的長度的 2 倍", "$\\vec{v}$ 與 $\\vec{u}-\\vec{v}$ 的夾角為銳角", "$\\vec{u}+\\vec{v}$ 的長度大於 $\\vec{u}-\\vec{v}$ 的長度"], "metadata": {"timestamp": "2024-01-09T01:16:58.115036", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 6", "question": "已知複數 $z$ 滿足 $z^{n}+z^{-n}+2=0$, 其 中 $n$ 為正整數。將 $z$ 用極式表示為 $r(\\cos \\theta+i \\sin \\theta)$,且 $r>0$ 。試選出正確的選項。", "correct_choices": ["$r=1$", "$\\theta$ 可能是 $\\frac{3 \\pi}{7}$"], "incorrect_choices": ["$n$ 不能是偶數", "對給定的 $n$, 恰有 $2 n$ 個不同的複數 $z$ 滿足題設", "$\\theta$ 可能是 $\\frac{4 \\pi}{7}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.115039", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 7", "question": "設實係數三次多項式 $f(x)$ 的首項係數為正。已知 $y=f(x)$ 的圖形和直線 $y=g(x)$ 在 $x=1$ 相切, 且兩圖形只有一個交點。試選出正確的選項。", "correct_choices": ["=0$", "$f^{\\prime}(1)=g^{\\prime}(1)$", "$f^{\\prime \\prime}(1)=0$"], "incorrect_choices": ["存在實數 $a \\neq 1$ 使得 $f^{\\prime}(a)=g^{\\prime}(a)$", "存在實數 $a \\neq 1$ 使得 $f^{\\prime \\prime}(a)=g^{\\prime \\prime}(a)$"], "metadata": {"timestamp": "2024-01-09T01:16:58.115042", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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。", "correct_choices": ["$E_{2}=E_{3}>E_{1}$"], "incorrect_choices": ["$E_{1}>E_{2}>E_{3}$", "$E_{1}=E_{2}>E_{3}$", "$E_{1}=E_{3}>E_{2}$", "$E_{3}>E_{2}>E_{1}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113688", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 5", "question": "坐標平面上有一圖形 $\\Gamma$, 其方程式為 $(x-1)^{2}+(y-1)^{2}=101$ 。試選出正確的選項。", "correct_choices": ["$\\Gamma$ 與 $x$ 軸負向、 $y$ 軸負向分別交於 $(-9,0)$ 、 $(0,-9)$", "$\\Gamma$ 上的點與原點距離的最大值為 $\\sqrt{2}+\\sqrt{101}$", "$\\Gamma$ 經旋轉線性變換後, 其圖形仍可用一個不含 $x y$ 項的二元二次方程式表示"], "incorrect_choices": ["$\\Gamma$ 上 $x$ 坐標最大的點是點 $(11,0)$", "$\\Gamma$ 在第三象限的點之極坐標可用 $[9, \\theta]$ 表示, 其中 $\\pi<\\theta<\\frac{3}{2} \\pi$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114738", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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?", "correct_choices": ["$\\int_{0}^{3} \\sqrt{4+x^{2}} d x$"], "incorrect_choices": ["$\\int_{0}^{3} \\sqrt{1+x^{2}} d x$", "$\\int_{0}^{3} \\sqrt{1+9 x^{2}} d x$", "$\\int_{0}^{3} \\sqrt{4+9 x^{2}} d x$", "$\\int_{0}^{3} \\sqrt{4 x^{2}+9} d x$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113942", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 7", "question": "設 $O$ 為複數平面上的原點, 並令點 $A, B$ 分別代表複數 $z_{1}, z_{2}$, 且滿足 $\\left|z_{1}\\right|=2,\\left|z_{2}\\right|=3$, $\\left|z_{2}-z_{1}\\right|=\\sqrt{5}$ 。若 $\\frac{z_{2}}{z_{1}}=a+b i$, 其中 $a, b$ 為實數, $i=\\sqrt{-1}$ 。試選出正確的選項。", "correct_choices": ["$\\cos \\angle A O B=\\frac{2}{3}$", "$a>0$"], "incorrect_choices": ["$\\left|z_{2}+z_{1}\\right|=\\sqrt{23}$", "$b>0$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114419", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 4", "question": "設二階方陣 $M$ 為在坐標平面上定義的線性變換, $O$ 為原點。已知 $M$ 可將不共線的三點 $O 、 A 、 B$ 映射至不共線的三點 $O 、 A^{\\prime} 、 B^{\\prime}$, 試選出正確的選項。", "correct_choices": ["$M$ 為可逆矩陣", "若 $M$ 將點 $C$ 映射至點 $C^{\\prime}$ 且 $\\overrightarrow{O C}=2 \\overrightarrow{O A}+3 \\overrightarrow{O B}$, 則 $\\overrightarrow{O C^{\\prime}}=2 \\overrightarrow{O A^{\\prime}}+3 \\overrightarrow{O B^{\\prime}}$", "$\\triangle O A^{\\prime} B^{\\prime}$ 的面積 $=\\triangle O A B$ 的面積 $\\times|\\operatorname{det}(M)|$"], "incorrect_choices": ["$\\angle A O B=\\angle A^{\\prime} O B^{\\prime}$", "$\\overline{O A}: \\overline{O B}=\\overline{O A^{\\prime}}: \\overline{O B^{\\prime}}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113454", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 1", "question": "考慮兩個函數 $f(x)=\\left\\{\\begin{array}{cc}1+x, & x \\leq 1 \\\\ 1, & x>1\\end{array} 、 g(x)=\\left\\{\\begin{array}{cc}1, & x \\leq 1 \\\\ 3-x, & x>1\\end{array}\\right.\\right.$ 關於函數的極限, 試選出正確的選項。", "correct_choices": ["$\\lim _{x \\rightarrow 1} f(x)$ 不存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 不存在、 $\\lim _{x \\rightarrow 1}(f(x)+g(x))$ 存在"], "incorrect_choices": ["$\\lim _{x \\rightarrow 1} f(x)$ 存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 存在、 $\\lim _{x \\rightarrow 1}(f(x)+g(x))$ 存在", "$\\lim _{x \\rightarrow 1} f(x)$ 存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 不存在、 $\\lim _{x \\rightarrow 1}(f(x)+g(x))$ 不存在", "$\\lim _{x \\rightarrow 1} f(x)$ 不存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 存在、 $\\lim _{x \\rightarrow 1}(f(x)+g(x))$ 不存在", "$\\lim _{x \\rightarrow 1} f(x)$ 不存在 、$ \\lim _{x \\rightarrow 1} g(x)$ 不存在 $、 \\lim _{x \\rightarrow 1}(f(x)+g(x))$ 不存在"], "metadata": {"timestamp": "2024-01-09T01:16:58.113416", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 2", "question": "某質點在數線上移動, 已知其位置坐標為 $s(t)=\\int_{0}^{t}\\left(-x^{2}+6 x\\right) d x$, 其中 $t$ 表時間且 $0 \\leq t \\leq 10$ 。若此質點的速度在時段 $0 \\leq t<a$ 遞增, 且在時段 $a<t \\leq 10$ 遞減, 試選出正確的 $a$ 值。", "correct_choices": ["3"], "incorrect_choices": ["4", "5", "6", "7"], "metadata": {"timestamp": "2024-01-09T01:16:58.113446", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 3", "question": "在坐標平面上, 其 $x$ 坐標與 $y$ 坐標都是整數的點稱為 $「$ 格子點 $」$ 試問滿足方程式 $\\log _{2}(x-1)=\\log _{4}\\left(25-y^{2}\\right)$ 的格子點 $(x, y)$ 共有幾個 ?", "correct_choices": ["5 個"], "incorrect_choices": ["4 個", "6 個", "8 個", "12 個"], "metadata": {"timestamp": "2024-01-09T01:16:58.113450", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 5", "question": "下列選項中, 試選出與 $\\cos \\frac{\\pi}{7}+i \\sin \\frac{\\pi}{7}$ 相乘之後會得到實數的選項。( 註 : $\\left.i=\\sqrt{-1}\\right)$", "correct_choices": ["$\\cos \\frac{\\pi}{7}-i \\sin \\frac{\\pi}{7}$", "$-\\sin \\frac{5 \\pi}{14}+i \\cos \\frac{5 \\pi}{14}$"], "incorrect_choices": ["$\\cos \\frac{\\pi}{7}+i \\sin \\frac{\\pi}{7}$", "$\\sin \\frac{\\pi}{7}+i \\cos \\frac{\\pi}{7}$", "$\\sin \\frac{\\pi}{7}-i \\cos \\frac{\\pi}{7}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113458", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 6", "question": "持續投擲一枚公正股子, 在過程中若出現連續兩次點數的和為 7 時, 就停止投擲 。例如：若前兩次投擲分別出現點數 1 、 4 , 點數和不等於 7 , 所以繼續投擲; 若第三次投出點數 3 , 因為第二次與第三次點數和為 7 , 所以此時即停止投擲。關於此機率事件, 試選出正確的選項。", "correct_choices": ["在第一次投擲的點數為 6 的情況下，總共投擲兩次就停的機率為 $\\frac{1}{6}$", "總共投擲兩次就停止的機率為 $\\frac{1}{6}$"], "incorrect_choices": ["在第一次投擲的點數為 5 的情況下，總共投擲三次恰好停止的機率為 $\\frac{1}{6}$", "總共投擲三次恰好停止的機率大於 $\\frac{1}{6}$", "至少投擲三次才停止的機率為 $\\frac{1}{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113461", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 7", "question": "關於非常數的實係數多項式函數 $f(x)$, 試選出正確的選項。", "correct_choices": ["若 $f(1) f(2)<0$, 則存在 $c \\in(1,2)$ 滿足 $f(c)=0$", "若 $\\left(\\int_{0}^{1} f(x) d x \\right)\\left(\\int_{0}^{2} f(x) d x\right)<0$, 則存在 $c \\in(1,2)$ 滿足 $\\int_{0}^{c} f(x) d x=0$"], "incorrect_choices": ["若 $f(1) f(2)>0$, 則對任意的 $c \\in(1,2), f(c) \neq 0$ 均成立", "若 $f(1) f(2) f(3)<0$, 則存在 $c \\in(1,3)$ 滿足 $f(c)=0$", "若 $\\int_{1}^{2} f(x) d x=0$, 則 $f(1) f(2)<0$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113464", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109補 - 8", "question": "設 $a, b, c$ 為三實數, 且 $a>b>c$ 。已知 $2^{a}, 2^{b}, 2^{c}$ 三數依序成等差數列。試選出正確的選項。", "correct_choices": ["$2^{a+100}, 2^{b+100}, 2^{c+100}$ 三數依序成等差數列", "$a<b+1$", "$b \\geq \\frac{a+c}{2}$"], "incorrect_choices": ["$a, b, c$ 三數依序成等比數列", "$4^{a}, 4^{b}, 4^{c}$ 三數依序成等差數列"], "metadata": {"timestamp": "2024-01-09T01:16:58.113468", "source": "AST mathematics - 109補"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 1", "question": "已知 $45^{\\circ}<\\theta<50^{\\circ}$, 且設 $a=1-\\cos ^{2} \\theta 、 b=\\frac{1}{\\cos \\theta}-\\cos \\theta 、 c=\\frac{\\tan \\theta}{\\tan ^{2} \\theta+1}$ 。關於 $a, b, c$ 三個數值的大小, 試選出正確的選項。", "correct_choices": ["$c<a<b$"], "incorrect_choices": ["$a<b<c$", "$a<c<b$", "$b<a<c$", "$b<c<a$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113684", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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： $\\log 2 \\approx 0.3010 ， \\log 3 \\approx 0.4771$ )", "correct_choices": ["70 小時"], "incorrect_choices": ["63 小時", "77 小時", "84 小時", "91 小時"], "metadata": {"timestamp": "2024-01-09T01:16:58.113691", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 4", "question": "在坐標平面上, 設 $O$ 為原點, 且 $A 、 B$ 為異於 $O$ 的相異兩點。令 $C_{1}, C_{2}, C_{3}$ 為平面上三個點, 且滿足 $\\overrightarrow{O C}_{n}=\\overrightarrow{O A}+n \\overrightarrow{O B}, n=1,2,3$, 試選出正確的選項。", "correct_choices": ["$\\overrightarrow{O C}_{1} \\cdot \\overrightarrow{O B}<\\overrightarrow{O C}_{2} \\cdot \\overrightarrow{O B}<\\overrightarrow{O C}_{3} \\cdot \\overrightarrow{O B}$", "$C_{1}, C_{2}, C_{3}$ 在同一直線上"], "incorrect_choices": ["$\\overrightarrow{O C}_{1} \\neq \\overrightarrow{0}$", "$\\overline{O C_{1}}<\\overline{O C_{2}}<\\overline{O C_{3}}$", "$\\overrightarrow{O C}_{1} \\cdot \\overrightarrow{O A}<\\overrightarrow{O C}_{2} \\cdot \\overrightarrow{O A}<\\overrightarrow{O C}_{3} \\cdot \\overrightarrow{O A}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113695", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 5", "question": "對一實數 $a$, 以 $[a]$ 表示不大於 $a$ 的最大整數, 例如 : $[1.2]=[\\sqrt{2}]=1,[-1.2]=-2$ 。考慮無理數 $\\theta=\\sqrt{10001}$, 試選出正確的選項。", "correct_choices": ["$a-1<[a] \\leq a$ 對任意實數 $a$ 均成立", "數列 $e_{n}=n\\left[\\frac{-\\theta}{n}\\right]$ 發散, $n$ 為正整數"], "incorrect_choices": ["數列 $b_{n}=\\frac{[n \\theta]}{n}$ 發散, $n$ 為正整數", "數列 $c_{n}=\\frac{[-n \\theta]}{n}$ 發散, $n$ 為正整數", "數列 $d_{n}=n\\left[\\frac{\\theta}{n}\\right]$ 發散, $n$ 為正整數"], "metadata": {"timestamp": "2024-01-09T01:16:58.113698", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 6", "question": "設 $F(x)$ 、 $f(x)$ 皆為實係數多項式函數。已知 $F^{\\prime}(x)=f(x)$, 試選出正確的選項。", "correct_choices": ["若 $a \\geq 0$, 則 $F(a)-F(0)=\\int_{0}^{a} f(t) d t$", "若 $F(x)$ 除以 $x$ 的商式為 $Q(x)$, 則 $Q(0)=f(0)$"], "incorrect_choices": ["若 $f(x)$ 可被 $x+1$ 整除, 則 $F(x)-F(0)$ 可被 $(x+1)^{2}$ 整除", "若對所有實數 $x, F(x) \\geq \\frac{x^{2}}{2}$ 都成立, 則對所有實數 $x, f(x) \\geq x$ 也都成立", "若對所有 $x>0 ， f(x) \\geq x$ 都成立, 則對所有 $x>0, F(x) \\geq \\frac{x^{2}}{2}$ 也都成立"], "metadata": {"timestamp": "2024-01-09T01:16:58.113701", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 7", "question": "在複數平面上, 設 $O$ 為原點, 且 $A 、 B$ 分別表示坐標為複數 $z 、 z+1$ 的點。已知點 $A$ 、點 $B$ 都在以 $O$ 為圓心的單位圓上, 試選出正確的選項。", "correct_choices": ["直線 $A B$ 與實數軸平行", "$z^{3}=1$", "坐標為 $1+\\frac{1}{z}$ 的點也在同一單位圓上"], "incorrect_choices": ["$\\triangle O A B$ 為直角三角形", "點 $A$ 在第二象限"], "metadata": {"timestamp": "2024-01-09T01:16:58.113705", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 109 - 8", "question": "設二階實係數方陣 $A$ 代表坐標平面的一個鏡射變換且滿足 $A^{3}=\\left[\\begin{array}{cc}0 & -1 \\\\ -1 & 0\\end{array}\\right]$; 另設二階實係數方陣 $B$ 代表坐標平面的一個（以原點為中心的）旋轉變換且滿足 $B^{3}=\\left[\\begin{array}{cc}-1 & 0 \\\\ 0 & -1\\end{array}\\right]$, 試選出正確的選項。", "correct_choices": ["$B$ 恰有三種可能", "$B A B A=\\left[\\begin{array}{ll}1 & 0 \\\\ 0 & 1\\end{array}\\right]$"], "incorrect_choices": ["$A$ 恰有三種可能", "$A B=B A$", "二階方陣 $A B$ 代表坐標平面的一個旋轉變換"], "metadata": {"timestamp": "2024-01-09T01:16:58.113708", "source": "AST mathematics - 109"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 1", "question": "坐標平面上, 一質點由點 $(-3,-2)$ 出發, 沿著向量 $(a, 1)$ 的方向移動 5 單位長之後剛好抵達 $x$ 軸, 其中 $a$ 為正實數。試問 $a$ 值等於下列哪一個選項？", "correct_choices": ["$\\frac{\\sqrt{21}}{2}$"], "incorrect_choices": ["$\\frac{\\sqrt{13}}{2}$", "2", "$\\sqrt{5}$", "$2 \\sqrt{6}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113935", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 2", "question": "放射性物質的半衰期 $T$ 定義為 $\\ulcorner$ 每經過時間 $T$, 該物質的質量會衰退成原來的一半」。鉛製容器中有 $A 、 B$ 兩種放射性物質, 其半衰期分別為 $T_{A} 、 T_{B}$ 。開始記錄時這兩種物質的質量相等, 112 天後測量發現物質 $B$ 的質量為物質 $A$ 的質量的四分之一。根據上述, 試問 $T_{A} 、 T_{B}$ 滿足下列哪一個關係式?", "correct_choices": ["$2+\\frac{112}{T_{A}}=\\frac{112}{T_{B}}$"], "incorrect_choices": ["$-2+\\frac{112}{T_{A}}=\\frac{112}{T_{B}}$", "$-2+\\log _{2} \\frac{112}{T_{A}}=\\log _{2} \\frac{112}{T_{B}}$", "$2+\\log _{2} \\frac{112}{T_{A}}=\\log _{2} \\frac{112}{T_{B}}$", "$2 \\log _{2} \\frac{112}{T_{A}}=\\log _{2} \\frac{112}{T_{B}}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113939", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 4", "question": "設 $a, b$ 為實數。已知四個數 $-3,-1,4,7$ 皆滿足 $x$ 的不等式 $|x-a| \\leq b$, 試選出正確的選項。", "correct_choices": ["$\\sqrt{10}$ 也滿足 $x$ 的不等式 $|x-a| \\leq b$", "$3,1,-4,-7$ 滿足 $x$ 的不等式 $|x+a| \\leq b$"], "incorrect_choices": ["$-\\frac{3}{2},-\\frac{1}{2}, 2, \\frac{7}{2}$ 滿足 $x$ 的不等式 $|x-a| \\leq \\frac{b}{2}$", "$b$ 可能等於 4", "$a, b$ 可能相等"], "metadata": {"timestamp": "2024-01-09T01:16:58.113946", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 5", "question": "考慮實係數多項式 $f(x)=x^{4}-4 x^{3}-2 x^{2}+a x+b$ 。已知方程式 $f(x)=0$ 有虛根 $1+2 i$\n（其中 $i=\\sqrt{-1}$ ), 試選出正確的選項。", "correct_choices": ["$1-2 i$ 也是 $f(x)=0$ 的根", "$f^{\\prime}(2.1)<0$"], "incorrect_choices": ["$a, b$ 皆為正數", "函數 $y=f(x)$ 在 $x=1$ 有局部極小值", "$y=f(x)$ 圖形反曲點的 $x$ 坐標皆大於 0"], "metadata": {"timestamp": "2024-01-09T01:16:58.113949", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 112 - 6", "question": "設 $a, b, c, d, r, s, t$ 皆為實數, 已知坐標空間中三個非零向量 $\\vec{u}=(a, b, 0) 、 \\vec{v}=(c, d, 0)$ 及 $\\vec{w}=(r, s, t)$ 滿足內積 $\\vec{w} \\cdot \\vec{u}=\\vec{w} \\cdot \\vec{v}=0$ 。考慮三階方陣 $A=\\left[\\begin{array}{lll}a & b & 0 \\\\ c & d & 0 \\\\ r & s & t\\end{array}\\right]$, 試選出正確的選項。", "correct_choices": ["若 $\\vec{u} \\cdot \\vec{v}=0$, 則行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$", "若對任意三個實數 $e, f, g$, 向量 $(e, f, g)$ 都可以表示成 $\\vec{u}, \\vec{v}, \\vec{w}$ 的線性組合, 則行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$", "若行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$, 則 $A$ 的行列式不等於 0"], "incorrect_choices": ["若 $t \\neq 0$, 則行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$", "若存在一個向量 $\\overrightarrow{w^{\\prime}}$ 滿足 $\\overrightarrow{w^{\\prime}} \\cdot \\vec{u}=\\overrightarrow{w^{\\prime}} \\cdot \\vec{v}=0$ 且外積 $\\overrightarrow{w^{\\prime}} \\times \\vec{w} \\neq \\overrightarrow{0}$, 則行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right| \\neq 0$"], "metadata": {"timestamp": "2024-01-09T01:16:58.113952", "source": "AST mathematics - 112"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 1", "question": "某公司尾牙舉辦「紅包大放送」活動。每位員工擲兩枚均匀銅板一次, 若出現兩個反面可得獎金 400 元; 若出現一正一反可得獎金 800 元; 若出現兩個正面可得獎金 800 元並且獲得再擲一次的機會，其獲得獎金規則與前述相同，但不再有繼續投擲銅板的機會（也就是說每位員工最多有兩次擲銅板的機會）。試問每位參加活動的員工可獲得獎金的期望值為何?", "correct_choices": ["875 元"], "incorrect_choices": ["850 元", "900 元", "925 元", "950 元"], "metadata": {"timestamp": "2024-01-09T01:16:58.114165", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 2", "question": "設 $n$ 為正整數。第 $n$ 個費馬數 (Fermat Number) 定義為 $F_{n}=2^{\\left(2^{n}\\right)}+1$, 例如 $F_{1}=2^{\\left(2^{1}\\right)}+1=2^{2}+1=5, F_{2}=2^{\\left(2^{2}\\right)}+1=2^{4}+1=17$ 。試問 $\\frac{F_{13}}{F_{12}}$ 的整數部分以十進位表示時,其位數最接近下列哪一個選項 ? $(\\log 2 \\approx 0.3010)$", "correct_choices": ["1200"], "incorrect_choices": ["120", "240", "600", "900"], "metadata": {"timestamp": "2024-01-09T01:16:58.114168", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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 ? $\\left(\\cot 14^{\\circ} \\approx 4.01, \\cot 18^{\\circ} 30^{\\prime} \\approx 2.99\\right)$", "correct_choices": ["31 公尺"], "incorrect_choices": ["27 公尺", "29 公尺", "33 公尺", "35 公尺"], "metadata": {"timestamp": "2024-01-09T01:16:58.114171", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 4", "question": "設 $\\Gamma$ 為坐標平面上通過 $(7,0)$ 與 $\\left(0, \\frac{7}{2}\\right)$ 兩點的圓。試選出正確的選項。", "correct_choices": ["當 $\\Gamma$ 的半徑達到最小可能值時, $\\Gamma$ 通過原點", "若 $\\Gamma$ 的圓心在第三象限, 則 $\\Gamma$ 的半徑大於 8"], "incorrect_choices": ["$\\Gamma$ 的半徑大於或等於 5", "$\\Gamma$ 與直線 $x+2 y=6$ 有交點", "$\\Gamma$ 的圓心不可能在第四象限"], "metadata": {"timestamp": "2024-01-09T01:16:58.114174", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 5", "question": "袋中有 2 顆紅球、 3 顆白球與 1 顆藍球, 其大小皆相同。今將袋中的球逐次取出,每次隨機取出一顆, 取後不放回, 直到所有球被取出為止。試選出正確的選項。", "correct_choices": ["「取出的第一顆為紅球」的機率等於 「取出的第二顆為紅球」的機率", "「取出的前三顆皆為白球」的機率小於 $\\ulcorner$ 取出的前三顆球顏色皆相異」的機率"], "incorrect_choices": ["「取出的第一顆為紅球」與 $\\ulcorner$ 取出的第二顆為紅球」兩者為獨立事件", "「取出的第一顆為紅球」與 $\\ulcorner$ 取出的第二顆為白球或藍球」兩者為互斥事件", "「取出的第一、二顆皆為紅球 $\\lrcorner$ 的機率等於 $\\ulcorner$ 取出的第一、二顆皆為白球 $\\lrcorner$的機率"], "metadata": {"timestamp": "2024-01-09T01:16:58.114177", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 6", "question": "設 $\\left\\langle a_{n}\\right\\rangle 、\\left\\langle b_{n}\\right\\rangle$ 為兩實數數列, 且對所有的正整數 $n, a_{n}<b_{n}{ }^{2}<a_{n+1}$ 均成立。若已知 $\\lim _{n \\rightarrow \\infty} a_{n}=4$, 試選出正確的選項。", "correct_choices": ["對所有的正整數 $n, b_{n}{ }^{2}<b_{n+1}{ }^{2}$ 均成立", "$\\lim _{n \\rightarrow \\infty} b_{n}^{2}=4$"], "incorrect_choices": ["對所有的正整數 $n, a_{n}>3$ 均成立", "存在正整數 $n$, 使得 $a_{n+1}>4$", "$\\lim _{n \\rightarrow \\infty} b_{n}=2$ 或 $\\lim _{n \\rightarrow \\infty} b_{n}=-2$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114180", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 108 - 8", "question": "坐標平面上以原點 $O$ 為圓心的單位圓上三相異點 $A 、 B 、 C$ 滿足 $2 \\overrightarrow{O A}+3 \\overrightarrow{O B}+4 \\overrightarrow{O C}=\\overrightarrow{0}$, 其中 $A$ 點的坐標為 $(1,0)$ 。試選出正確的選項。", "correct_choices": ["向量 $2 \\overrightarrow{O A}+3 \\overrightarrow{O B}$ 的長度為 4", "$3 \\sin \\angle A O B=4 \\sin \\angle A O C$"], "incorrect_choices": ["內積 $\\overrightarrow{O A} \\cdot \\overrightarrow{O B}<0$", "$\\angle B O C 、 \\angle A O C 、 \\angle A O B$ 中, 以 $\\angle B O C$ 的度數為最小", "$\\overline{A B}>\\frac{3}{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114183", "source": "AST mathematics - 108"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 1", "question": "設 $A$ 為 $3 \\times 3$ 矩陣, 且對任意實數 $a, b, c, A\\left[\\begin{array}{l}a \\\\ b \\\\ c\\end{array}\\right]=\\left[\\begin{array}{l}b \\\\ c \\\\ a\\end{array}\\right]$ 均成立。試問矩陣 $A^{2}\\left[\\begin{array}{c}1 \\\\ 0 \\\\ -1\\end{array}\\right]$ 為何?", "correct_choices": ["$\\left[\\begin{array}{c}-1 \\\\ 1 \\\\ 0\\end{array}\\right]$"], "incorrect_choices": ["$\\left[\\begin{array}{l}0 \\\\ 1 \\\\ 1\\end{array}\\right]$", "$\\left[\\begin{array}{l}1 \\\\ 0 \\\\ 1\\end{array}\\right]$", "$\\left[\\begin{array}{c}0 \\\\ 1 \\\\ -1\\end{array}\\right]$", "$\\left[\\begin{array}{c}-1 \\\\ 0 \\\\ 1\\end{array}\\right]$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114397", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 2", "question": "坐標平面上, 考慮 $A(2,3)$ 與 $B(-1,3)$ 兩點, 並設 $O$ 為原點。令 $E$ 為滿足 $\\overrightarrow{O P}=a \\overrightarrow{O A}+b \\overrightarrow{O B}$ 的所有點 $P$ 所形成的區域, 其 中 $-1 \\leq a \\leq 1,0 \\leq b \\leq 4$ 。考慮函數 $f(x)=x^{2}+5$, 試問當限定 $x$ 為區域 $E$ 中的點 $P(x, y)$ 的横坐標時, $f(x)$ 的最大值為何 ?", "correct_choices": ["41"], "incorrect_choices": ["5", "9", "30", "54"], "metadata": {"timestamp": "2024-01-09T01:16:58.114401", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 3", "question": "某零售商店販賣 $\\ulcorner$ 熊大 」與 $\\ulcorner$ 皮卡丘 $\\lrcorner$ 兩種玩偶, 其進貨來源有 $A, B, C$ 三家廠商。已知此零售商店從每家廠商進貨的玩偶總數相同, 且三家廠商製作的每一種玩偶外觀也一樣, 而從 $A, B, C$ 這三家廠商進貨的玩偶中, $\\ulcorner$ 皮卡丘 $\\lrcorner$ 所占的比例分別為 $\\frac{1}{4} 、 \\frac{2}{5} 、 \\frac{1}{2}$ 。阿德從這家零售商店隨機挑選一隻 $\\ulcorner$ 皮卡丘」送給小安作為生日禮物, 試問此「皮卡丘」出自 $C$ 廠商的機率為何?", "correct_choices": ["$\\frac{10}{23}$"], "incorrect_choices": ["$\\frac{1}{3}$", "$\\frac{2}{5}$", "$\\frac{10}{19}$", "$\\frac{5}{9}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114404", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 4", "question": "設 $f(x)=-x^{2}+499$, 且\n$A=\\int_{0}^{10} f(x) d x 、 B=\\sum_{n=0}^{9} f(n) 、 C=\\sum_{n=1}^{10} f(n) 、 D=\\sum_{n=0}^{9} \\frac{f(n)+f(n+1)}{2}$\n試選出正確的選項。", "correct_choices": ["$A$ 表示在坐標平面上函數 $y=-x^{2}+499$ 的圖形與直線 $y=0 、 x=0 、 x=10$ 所圍成的有界區域的面積", "$C<D$"], "incorrect_choices": ["$B<C$", "$B<A$", "$A<D$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114406", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 5", "question": "坐標平面上, 已知直線 $L$ 與函數 $y=\\log _{2} x$ 的圖形有兩個交點 $P(a, b), Q(c, d)$,且 $\\overline{P Q}$ 的中點在 $x$ 軸上。試選出正確的選項。", "correct_choices": ["$L$ 的斜率大於 0", "$a c=1$", "$L$ 的 $x$ 截距大於 1"], "incorrect_choices": ["$b d=-1$", "$L$ 的 $y$ 截距大於 -1"], "metadata": {"timestamp": "2024-01-09T01:16:58.114410", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 6", "question": "坐標空間中, 有 $\\vec{a} 、 \\vec{b} 、 \\vec{c} 、 \\vec{d}$ 四個向量, 滿足外積 $\\vec{a} \\times \\vec{b}=\\vec{c}, \\vec{a} \\times \\vec{c}=\\vec{d}$, 且 $\\vec{a} 、 \\vec{b} 、 \\vec{c}$ 的向量長度均為 4 。設向量 $\\vec{a}$ 與 $\\vec{b}$ 的夾角為 $\\theta$ (其中 $0 \\leq \\theta \\leq \\pi$ ), 試選出正確的選項。", "correct_choices": ["$\\vec{a} 、 \\vec{b} 、 \\vec{c}$ 所張出的平行六面體的體積為 16", "$\\vec{a} 、 \\vec{c} 、 \\vec{d}$ 兩兩互相垂直"], "incorrect_choices": ["$\\cos \\theta=\\frac{1}{4}$", "$\\vec{d}$ 的長度等於 4", "$\\vec{b}$ 與 $\\vec{d}$ 的夾角等於 $\\theta$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114416", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 107 - 8", "question": "設 $f(x)$ 為一定義在非零實數上的實數值函數。已知極限 $\\lim _{x \\rightarrow 0} f(x) \\frac{|x|}{x}$ 存在, 試選出正確的選項。", "correct_choices": ["$\\lim _{x \\rightarrow 0}\\left(\\frac{x}{|x|}\\right)^{2}$ 存在", "$\\lim _{x \\rightarrow 0} f(x) \\frac{x}{|x|}$ 存在", "$\\lim _{x \\rightarrow 0} f(x)^{2}$ 存在"], "incorrect_choices": ["$\\lim _{x \\rightarrow 0}(f(x)+1) \\frac{x}{|x|}$ 存在", "$\\lim _{x \\rightarrow 0} f(x)$ 存在"], "metadata": {"timestamp": "2024-01-09T01:16:58.114422", "source": "AST mathematics - 107"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 1", "question": "假設 2 階方陣 $\\left[\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right]$ 所代表的線性變換將坐標平面上三點 $O(0,0), A(1,0), B(0,1)$ 分別映射到 $O(0,0), A^{\\prime}(3, \\sqrt{3}), B^{\\prime}(-\\sqrt{3}, 3)$ ，並將與原點距離為 1 的點 $C(x, y)$ 映射到點 $C^{\\prime}\\left(x^{\\prime}, y^{\\prime}\\right)$ 。試選出正確的選項。", "correct_choices": ["$\\overrightarrow{O C}$ 和 $\\overrightarrow{O C^{\\prime}}$ 的夾角為 $60^{\\circ}$"], "incorrect_choices": ["行列式 $\\left|\\begin{array}{ll}a & b \\\\ c & d\\end{array}\\right|=6$", "$\\overline{O C^{\\prime}}=2 \\sqrt{3}$", "有可能 $y=y^{\\prime}$", "若 $x<y$ 則 $x^{\\prime}<y^{\\prime}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114724", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 2", "question": "設 $c$ 為實數使得三元一次方程組 $\\left\\{\\begin{array}{c}x-y+z=0 \\\\ 2 x+c y+3 z=1 \\\\ 3 x-3 y+c z=0\\end{array}\\right.$ 無解。試選出 $c$ 之值。", "correct_choices": ["-2"], "incorrect_choices": ["-3", "0", "2", "3"], "metadata": {"timestamp": "2024-01-09T01:16:58.114728", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 3", "question": "坐標空間中 $O$ 為原點, 點 $P$ 在第一卦限且 $\\overline{O P}=1$ 。已知直線 $O P$ 與 $x$ 軸有一夾角為 $45^{\\circ}$, 且 $P$ 點到 $y$ 軸的距離為 $\\frac{\\sqrt{6}}{3}$ 。試選出點 $P$ 的 $z$ 坐標。", "correct_choices": ["$\\frac{\\sqrt{6}}{6}$"], "incorrect_choices": ["$\\frac{1}{2}$", "$\\frac{\\sqrt{2}}{4}$", "$\\frac{\\sqrt{3}}{3}$", "$\\frac{\\sqrt{3}}{6}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114731", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 4", "question": "設多項式 $f(x)=x^{3}+2 x^{2}-2 x+k 、 g(x)=x^{2}+a x+1$, 其 中 $k, a$ 為實數。已知 $g(x)$ 整除 $f(x)$, 且方程式 $g(x)=0$ 有虛根。試選出為方程式 $f(x)=0$ 的根之選項。", "correct_choices": ["-3", "$\\frac{1+\\sqrt{-3}}{2}$"], "incorrect_choices": ["0", "1", "$\\frac{3+\\sqrt{-5}}{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114734", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 111 - 8", "question": "假設兩數列 $\\left\\langle a_{n}\\right\\rangle 、\\left\\langle b_{n}\\right\\rangle$, 對所有正整數 $n$ 都滿足 $b_{n}+\\frac{4 n-1}{n}<a_{n}<3 b_{n}$ 。已知 $\\lim _{n \\rightarrow \\infty} a_{n}=6$,試選出正確的選項。", "correct_choices": ["$b_{n}>\\frac{4 n-1}{2 n}$", "$a_{10000}>5.9$"], "incorrect_choices": ["$b_{n}<6-\\frac{4 n-1}{n}$", "數列 $\\left\\langle b_{n}\\right\\rangle$ 有可能發散", "$a_{10000}<6.1$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114741", "source": "AST mathematics - 111"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 1", "question": "設 $F(x)$ 為一實係數多項式且 $F^{\\prime}(x)=f(x)$ 。已知 $f^{\\prime}(x)>x^{2}+1.1$ 對所有的實數 $x$ 均成立, 試選出正確的選項。", "correct_choices": ["$f(f(x))$ 為遞增函數"], "incorrect_choices": ["$f^{\\prime}(x)$ 為遞增函數", "$f(x)$ 為遞增函數", "$F(x)$ 為遞增函數", "$[f(x)]^{2}$ 為遞增函數"], "metadata": {"timestamp": "2024-01-09T01:16:58.114932", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 2", "question": "研究團隊採用某快篩試劑的檢驗, 以了解保護區內生物因環境汗染而導致體內毒素累積超過標準的比率。此試劑檢驗結果只有紅色、黃色兩種。依據過去的經驗得知：若體內毒素累積超過標準, 經此試劑檢驗後, 有 $75 \\%$ 顯示為紅色; 若體內毒素累積未超過標準, 經此試劑檢驗後, 有 $95 \\%$ 顯示為黃色。已知此保護區的某類生物經試劑檢驗後, 有 $7.8 \\%$ 的結果顯示為紅色。假設此類生物實際體內毒素累積超過標準的比率為 $p \\%$, 試選出正確的選項。", "correct_choices": ["$3 \\leq p<5$"], "incorrect_choices": ["$1 \\leq p<3$", "$5 \\leq p<7$", "$7 \\leq p<9$", "$9 \\leq p<11$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114935", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 3", "question": "試求極限 $\\lim _{n \\rightarrow \\infty} \\frac{10^{10}}{n^{10}}\\left[1^{9}+2^{9}+3^{9}+\\cdots+(2 n)^{9}\\right]$ 的值 。", "correct_choices": ["$10^{9} \\times 2^{10}$"], "incorrect_choices": ["$10^{9}$", "$10^{9} \\times\\left(2^{10}-1\\right)$", "$2^{9} \\times\\left(10^{10}-1\\right)$", "$2^{9} \\times 10^{10}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114938", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 4", "question": "某電子公司有數百名員工, 其用餐方式分為自備、外食兩種。經長期調查發現：若當日用餐為自備的員工, 則隔天會有 $10 \\%$ 轉為外食; 若當日用餐為外食的員工, 則隔天會有 $20 \\%$ 轉為自備。\n假設 $x_{0} 、 y_{0}$ 分別代表該公司今日用餐自備人數與外食人數占員工總人數的比例,其中 $x_{0} 、 y_{0}$ 皆為正數, 且 $x_{n} 、 y_{n}$ 分別代表經過 $n$ 日後用餐自���人數與外食人數占員工總人數的比例。在該公司員工不變動的情形下, 試選出正確的選項。", "correct_choices": ["$\\left[\\begin{array}{l}x_{n+1} \\\\ y_{n+1}\\end{array}\\right]=\\left[\\begin{array}{ll}0.9 & 0.2 \\\\ 0.1 & 0.8\\end{array}\\right]\\left[\\begin{array}{l}x_{n} \\\\ y_{n}\\end{array}\\right]$", "若 $\\frac{x_{0}}{y_{0}}=\\frac{2}{1}$, 則 $\\frac{x_{n}}{y_{n}}=\\frac{2}{1}$ 對任意正整數 $n$ 均成立"], "incorrect_choices": ["$y_{1}=0.9 y_{0}+0.2 x_{0}$", "若 $y_{0}>x_{0}$, 則 $y_{1}>x_{1}$", "若 $x_{0}>y_{0}$, 則 $x_{0}>x_{1}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114941", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 5", "question": "假設 $f(x)$ 為五次實係數多項式, 且 $f(x)$ 除以 $x^{n}-1$ 的餘式為 $r_{n}(x), n$ 是正整數。試選出正確的選項。", "correct_choices": ["$r_{1}(x)=f(1)$", "$r_{4}(x)$ 除以 $x^{2}-1$ 所得的餘式等於 $r_{2}(x)$"], "incorrect_choices": ["$r_{2}(x)$ 是一次實係數多項式", "$r_{5}(x)=r_{6}(x)$", "若 $f(-x)=-f(x)$, 則 $r_{3}(-x)=-r_{3}(x)$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114944", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 6", "question": "一個標有 1 至 12 號格子的 12 格翟翟樂遊戲, 每回遊戲以投擲一枚均匀銅板四次來決定要戳哪些格子。規則如下：\n（一）第一次投擲銅板, 若是正面, 則翟 1 號格子; 若是反面, 則翟 3 號格子。\n（二）第二、三、四次投擲銅板, 若是正面, 則所翟格子的號碼為前一次所翟格子的號碼加 1 ; 若是反面, 則所翟格子的號碼為前一次所羽格子的號碼加 3 , 依此類推。\n例如：投鄭銅板四次的結果依序為 $\\ulcorner$ 正、反、反、正」, 則會㪬編號分別為 1 、 4、7、8 號的四個格子。\n假設 $p_{m}$ 代表在每回遊戲中 $m$ 號格子被翟到的機率, 試選出正確的選項。", "correct_choices": ["$p_{2}=\\frac{1}{4}$", "$\\quad p_{4}=\\frac{1}{2} p_{1}+\\frac{1}{2} p_{3}$", "$p_{8}>p_{10}$"], "incorrect_choices": ["$p_{3}=\\frac{1}{2}$", "在 4 號格子被翟到的條件下, 3 號格子被翟到的機率為 $\\frac{1}{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114947", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 110 - 8", "question": "已知 $z_{1} 、 z_{2} 、 z_{3} 、 z_{4}$ 為四個相異複數, 且其在複數平面上所對應的點, 依序可連成一個平行四邊形, 試問下列哪些選項必為實數?", "correct_choices": ["$z_{1}-z_{2}+z_{3}-z_{4}$", "$\\frac{z_{1}-z_{2}}{z_{3}-z_{4}}$"], "incorrect_choices": ["$\\left(z_{1}-z_{3}\\right)\\left(z_{2}-z_{4}\\right)$", "$z_{1}+z_{2}+z_{3}+z_{4}$", "$\\left(\\frac{z_{2}-z_{4}}{z_{1}-z_{3}}\\right)^{2}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.114950", "source": "AST mathematics - 110"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 1", "question": "從所有二位正整數中隨機選取一個數, 設 $p$ 是其十位數字小於個位數字的機率。關於 $p$ 值的範圍, 試選出正確的選項。", "correct_choices": ["$0.33 \\leq p<0.44$"], "incorrect_choices": ["$0.22 \\leq p<0.33$", "$0.44 \\leq p<0.55$", "$0.55 \\leq p<0.66$", "$0.66 \\leq p<0.77$"], "metadata": {"timestamp": "2024-01-09T01:16:58.115023", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 2", "question": "設 $a=\\sqrt[3]{10}$ 。關於 $a^{5}$ 的範圍, 試選出正確的選項。", "correct_choices": ["$45 \\leq a^{5}<50$"], "incorrect_choices": ["$25 \\leq a^{5}<30$", "$30 \\leq a^{5}<35$", "$35 \\leq a^{5}<40$", "$40 \\leq a^{5}<45$"], "metadata": {"timestamp": "2024-01-09T01:16:58.115027", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 3", "question": "試問在 $0 \\leq x \\leq 2 \\pi$ 的範圍 中, $y=3 \\sin x$ 的函數圖形與 $y=2 \\sin 2 x$ 的函數圖形有幾個交點?", "correct_choices": ["5 個交點"], "incorrect_choices": ["2 個交點", "3 個交點", "4 個交點", "6 個交點"], "metadata": {"timestamp": "2024-01-09T01:16:58.115030", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"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 ?", "correct_choices": ["-5"], "incorrect_choices": ["-3", "0", "3", "5"], "metadata": {"timestamp": "2024-01-09T01:16:58.115033", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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{"id": "AST mathematics - 106 - 5", "question": "設 $\\vec{u}$ 與 $\\vec{v}$ 為兩非零向量, ���角為 $120^{\\circ}$ 。若 $\\vec{u}$ 與 $\\vec{u}+\\vec{v}$ 垂直, 試選出正確的選項。", "correct_choices": ["$\\vec{v}$ 與 $\\vec{u}+\\vec{v}$ 的夾角為 $30^{\\circ}$", "$\\vec{u}$ 與 $\\vec{u}-\\vec{v}$ 的夾角為銳角"], "incorrect_choices": ["$\\vec{u}$ 的長度是 $\\vec{v}$ 的長度的 2 倍", "$\\vec{v}$ 與 $\\vec{u}-\\vec{v}$ 的夾角為銳角", "$\\vec{u}+\\vec{v}$ 的長度大於 $\\vec{u}-\\vec{v}$ 的長度"], "metadata": {"timestamp": "2024-01-09T01:16:58.115036", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
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| 51 |
+
{"id": "AST mathematics - 106 - 6", "question": "已知複數 $z$ 滿足 $z^{n}+z^{-n}+2=0$, 其 中 $n$ 為正整數。將 $z$ 用極式表示為 $r(\\cos \\theta+i \\sin \\theta)$,且 $r>0$ 。試選出正確的選項。", "correct_choices": ["$r=1$", "$\\theta$ 可能是 $\\frac{3 \\pi}{7}$"], "incorrect_choices": ["$n$ 不能是偶數", "對給定的 $n$, 恰有 $2 n$ 個不同的複數 $z$ 滿足題設", "$\\theta$ 可能是 $\\frac{4 \\pi}{7}$"], "metadata": {"timestamp": "2024-01-09T01:16:58.115039", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
|
| 52 |
+
{"id": "AST mathematics - 106 - 7", "question": "設實係數三次多項式 $f(x)$ 的首項係數為正。已知 $y=f(x)$ 的圖形和直線 $y=g(x)$ 在 $x=1$ 相切, 且兩圖形只有一個交點。試選出正確的選項。", "correct_choices": ["=0$", "$f^{\\prime}(1)=g^{\\prime}(1)$", "$f^{\\prime \\prime}(1)=0$"], "incorrect_choices": ["存在實數 $a \\neq 1$ 使得 $f^{\\prime}(a)=g^{\\prime}(a)$", "存在實數 $a \\neq 1$ 使得 $f^{\\prime \\prime}(a)=g^{\\prime \\prime}(a)$"], "metadata": {"timestamp": "2024-01-09T01:16:58.115042", "source": "AST mathematics - 106"}, "human_evaluation": {"quality": "", "comments": ""}}
|
README.md
CHANGED
|
@@ -13,7 +13,13 @@ configs:
|
|
| 13 |
- split: test
|
| 14 |
path: "AST_chinese_test.jsonl"
|
| 15 |
- split: dev
|
| 16 |
-
path: "AST_chinese_dev.jsonl"
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 17 |
- config_name: AST_biology
|
| 18 |
data_files:
|
| 19 |
- split: test
|
|
@@ -327,111 +333,108 @@ Users should be made aware of the risks, biases and limitations of the dataset.
|
|
| 327 |
#### GSAT
|
| 328 |
##### ChatGPT
|
| 329 |
Total: 217 / 534 (0.4064)
|
| 330 |
-
|
| 331 |
| Subject | Accuracy | correct / total |
|
| 332 |
|:------------- | -------- |:--------------- |
|
| 333 |
-
| chinese | 0.
|
| 334 |
-
| mathematics | 0.
|
| 335 |
-
| physics | 0.
|
| 336 |
-
| chemistry | 0.
|
| 337 |
-
| biology | 0.
|
| 338 |
-
| earth science | 0.
|
| 339 |
-
| geography | 0.
|
| 340 |
-
| history | 0.
|
| 341 |
-
| civics | 0.
|
| 342 |
|
| 343 |
##### GPT-4-turbo
|
| 344 |
-
Total:
|
| 345 |
-
|
| 346 |
| Subject | Accuracy | correct / total |
|
| 347 |
|:------------- | -------- |:--------------- |
|
| 348 |
-
| chinese | 0.
|
| 349 |
-
| mathematics | 0.
|
| 350 |
-
| physics | 0.
|
| 351 |
-
| chemistry | 0.
|
| 352 |
-
| biology | 0.
|
| 353 |
-
| earth science | 0.
|
| 354 |
-
| geography | 0.
|
| 355 |
-
| history | 0.
|
| 356 |
-
| civics | 0.
|
| 357 |
-
|
| 358 |
-
##### Claude-
|
| 359 |
-
Total:
|
| 360 |
| Subject | Accuracy | correct / total |
|
| 361 |
|:------------- | -------- |:--------------- |
|
| 362 |
-
| chinese | 0.
|
| 363 |
-
| mathematics | 0.
|
| 364 |
-
| physics | 0.
|
| 365 |
-
| chemistry | 0.
|
| 366 |
-
| biology | 0.
|
| 367 |
-
| earth science | 0.
|
| 368 |
-
| geography | 0.
|
| 369 |
-
| history | 0.
|
| 370 |
-
| civics | 0.
|
| 371 |
|
| 372 |
##### Claude-2
|
| 373 |
-
Total:
|
| 374 |
| Subject | Accuracy | correct / total |
|
| 375 |
|:------------- | -------- |:--------------- |
|
| 376 |
-
| chinese | 0.
|
| 377 |
-
| mathematics | 0.
|
| 378 |
-
| physics | 0.
|
| 379 |
-
| chemistry | 0.
|
| 380 |
-
| biology | 0.
|
| 381 |
-
| earth science | 0.
|
| 382 |
-
| geography | 0.
|
| 383 |
-
| history | 0.
|
| 384 |
-
| civics | 0.
|
| 385 |
|
| 386 |
#### AST
|
| 387 |
##### ChatGPT
|
| 388 |
Total: 197 / 492 (0.4004)
|
| 389 |
-
|
| 390 |
| Subject | Accuracy | correct / total |
|
| 391 |
-
|
| 392 |
-
| chinese | 0.
|
| 393 |
-
| physics | 0.
|
| 394 |
-
| chemistry | 0.
|
| 395 |
-
| biology | 0.
|
| 396 |
-
| geography | 0.
|
| 397 |
-
| history | 0.
|
| 398 |
-
| civics | 0.
|
| 399 |
|
| 400 |
##### GPT-4-turbo
|
| 401 |
-
Total:
|
| 402 |
| Subject | Accuracy | correct / total |
|
| 403 |
|:--------- | -------- |:--------------- |
|
| 404 |
-
| chinese | 0.
|
| 405 |
-
| physics | 0.
|
| 406 |
-
| chemistry | 0.
|
| 407 |
-
| biology | 0.
|
| 408 |
-
| geography | 0.
|
| 409 |
-
| history | 0.
|
| 410 |
-
| civics | 0.
|
| 411 |
-
|
| 412 |
-
##### Claude-
|
| 413 |
-
Total:
|
| 414 |
| Subject | Accuracy | correct / total |
|
| 415 |
|:--------- | -------- |:--------------- |
|
| 416 |
-
| chinese | 0.
|
| 417 |
-
| physics | 0.
|
| 418 |
-
| chemistry | 0.
|
| 419 |
-
| biology | 0.
|
| 420 |
-
| geography | 0.
|
| 421 |
-
| history | 0.
|
| 422 |
-
| civics | 0.
|
| 423 |
|
| 424 |
##### Claude-2
|
| 425 |
-
Total:
|
| 426 |
| Subject | Accuracy | correct / total |
|
| 427 |
|:--------- | -------- |:--------------- |
|
| 428 |
-
| chinese | 0.
|
| 429 |
-
| physics | 0.
|
| 430 |
-
| chemistry | 0.
|
| 431 |
-
| biology | 0.
|
| 432 |
-
| geography | 0.
|
| 433 |
-
| history | 0.
|
| 434 |
-
| civics | 0.
|
| 435 |
|
| 436 |
#### CAP
|
| 437 |
##### ChatGPT
|
|
|
|
| 13 |
- split: test
|
| 14 |
path: "AST_chinese_test.jsonl"
|
| 15 |
- split: dev
|
| 16 |
+
path: "AST_chinese_dev.jsonl"
|
| 17 |
+
- config_name: AST_mathematics
|
| 18 |
+
data_files:
|
| 19 |
+
- split: test
|
| 20 |
+
path: "AST_mathematics_test.jsonl"
|
| 21 |
+
- split: dev
|
| 22 |
+
path: "AST_mathematics_dev.jsonl"
|
| 23 |
- config_name: AST_biology
|
| 24 |
data_files:
|
| 25 |
- split: test
|
|
|
|
| 333 |
#### GSAT
|
| 334 |
##### ChatGPT
|
| 335 |
Total: 217 / 534 (0.4064)
|
|
|
|
| 336 |
| Subject | Accuracy | correct / total |
|
| 337 |
|:------------- | -------- |:--------------- |
|
| 338 |
+
| chinese | 0.3280 | 41 / 125 |
|
| 339 |
+
| mathematics | 0.1569 | 8 / 51 |
|
| 340 |
+
| physics | 0.3529 | 12 / 34 |
|
| 341 |
+
| chemistry | 0.2683 | 11 / 41 |
|
| 342 |
+
| biology | 0.3774 | 20 / 53 |
|
| 343 |
+
| earth science | 0.2571 | 9 / 35 |
|
| 344 |
+
| geography | 0.5455 | 24 / 44 |
|
| 345 |
+
| history | 0.6049 | 49 / 81 |
|
| 346 |
+
| civics | 0.6143 | 43 / 70 |
|
| 347 |
|
| 348 |
##### GPT-4-turbo
|
| 349 |
+
Total: 347 / 534 (0.6498)
|
|
|
|
| 350 |
| Subject | Accuracy | correct / total |
|
| 351 |
|:------------- | -------- |:--------------- |
|
| 352 |
+
| chinese | 0.6240 | 78 / 125 |
|
| 353 |
+
| mathematics | 0.1176 | 6 / 51 |
|
| 354 |
+
| physics | 0.5882 | 20 / 34 |
|
| 355 |
+
| chemistry | 0.4634 | 19 / 41 |
|
| 356 |
+
| biology | 0.7170 | 38 / 53 |
|
| 357 |
+
| earth science | 0.7143 | 25 / 35 |
|
| 358 |
+
| geography | 0.8864 | 39 / 44 |
|
| 359 |
+
| history | 0.8519 | 69 / 81 |
|
| 360 |
+
| civics | 0.7571 | 53 / 70 |
|
| 361 |
+
|
| 362 |
+
##### Claude-instant-1
|
| 363 |
+
Total: 254 / 534 (0.4757)
|
| 364 |
| Subject | Accuracy | correct / total |
|
| 365 |
|:------------- | -------- |:--------------- |
|
| 366 |
+
| chinese | 0.4320 | 54 / 125 |
|
| 367 |
+
| mathematics | 0.1373 | 7 / 51 |
|
| 368 |
+
| physics | 0.4706 | 16 / 34 |
|
| 369 |
+
| chemistry | 0.2195 | 9 / 41 |
|
| 370 |
+
| biology | 0.4906 | 26 / 53 |
|
| 371 |
+
| earth science | 0.3429 | 12 / 35 |
|
| 372 |
+
| geography | 0.6818 | 30 / 44 |
|
| 373 |
+
| history | 0.7160 | 58 / 81 |
|
| 374 |
+
| civics | 0.6000 | 42 / 70 |
|
| 375 |
|
| 376 |
##### Claude-2
|
| 377 |
+
Total: 260 / 534 (0.4869)
|
| 378 |
| Subject | Accuracy | correct / total |
|
| 379 |
|:------------- | -------- |:--------------- |
|
| 380 |
+
| chinese | 0.4800 | 60 / 125 |
|
| 381 |
+
| mathematics | 0.1176 | 6 / 51 |
|
| 382 |
+
| physics | 0.5000 | 17 / 34 |
|
| 383 |
+
| chemistry | 0.3659 | 15 / 41 |
|
| 384 |
+
| biology | 0.4717 | 25 / 53 |
|
| 385 |
+
| earth science | 0.3714 | 13 / 35 |
|
| 386 |
+
| geography | 0.6818 | 30 / 44 |
|
| 387 |
+
| history | 0.6914 | 56 / 81 |
|
| 388 |
+
| civics | 0.5429 | 38 / 70 |
|
| 389 |
|
| 390 |
#### AST
|
| 391 |
##### ChatGPT
|
| 392 |
Total: 197 / 492 (0.4004)
|
|
|
|
| 393 |
| Subject | Accuracy | correct / total |
|
| 394 |
+
|:--------- | -------- |:--------------- |
|
| 395 |
+
| chinese | 0.3544 | 56 / 158 |
|
| 396 |
+
| physics | 0.2549 | 13 / 51 |
|
| 397 |
+
| chemistry | 0.1765 | 9 / 51 |
|
| 398 |
+
| biology | 0.3810 | 24 / 63 |
|
| 399 |
+
| geography | 0.5094 | 27 / 53 |
|
| 400 |
+
| history | 0.7241 | 42 / 58 |
|
| 401 |
+
| civics | 0.4483 | 26 / 58 |
|
| 402 |
|
| 403 |
##### GPT-4-turbo
|
| 404 |
+
Total: 307 / 492 (0.6240)
|
| 405 |
| Subject | Accuracy | correct / total |
|
| 406 |
|:--------- | -------- |:--------------- |
|
| 407 |
+
| chinese | 0.6582 | 104 / 158 |
|
| 408 |
+
| physics | 0.3333 | 17 / 51 |
|
| 409 |
+
| chemistry | 0.2745 | 14 / 51 |
|
| 410 |
+
| biology | 0.7143 | 45 / 63 |
|
| 411 |
+
| geography | 0.6981 | 37 / 53 |
|
| 412 |
+
| history | 0.8966 | 52 / 58 |
|
| 413 |
+
| civics | 0.6552 | 38 / 58 |
|
| 414 |
+
|
| 415 |
+
##### Claude-instant-1
|
| 416 |
+
Total: 228 / 492 (0.4634)
|
| 417 |
| Subject | Accuracy | correct / total |
|
| 418 |
|:--------- | -------- |:--------------- |
|
| 419 |
+
| chinese | 0.4810 | 76 / 158 |
|
| 420 |
+
| physics | 0.2549 | 13 / 51 |
|
| 421 |
+
| chemistry | 0.1373 | 7 / 51 |
|
| 422 |
+
| biology | 0.3492 | 22 / 63 |
|
| 423 |
+
| geography | 0.6038 | 32 / 53 |
|
| 424 |
+
| history | 0.6724 | 39 / 58 |
|
| 425 |
+
| civics | 0.6724 | 39 / 58 |
|
| 426 |
|
| 427 |
##### Claude-2
|
| 428 |
+
Total: 213 / 492 (0.4329)
|
| 429 |
| Subject | Accuracy | correct / total |
|
| 430 |
|:--------- | -------- |:--------------- |
|
| 431 |
+
| chinese | 0.4304 | 68 / 158 |
|
| 432 |
+
| physics | 0.2157 | 11 / 51 |
|
| 433 |
+
| chemistry | 0.1765 | 9 / 51 |
|
| 434 |
+
| biology | 0.3651 | 23 / 63 |
|
| 435 |
+
| geography | 0.6604 | 35 / 53 |
|
| 436 |
+
| history | 0.6379 | 37 / 58 |
|
| 437 |
+
| civics | 0.5172 | 30 / 58 |
|
| 438 |
|
| 439 |
#### CAP
|
| 440 |
##### ChatGPT
|