{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"id": "AST mathematics - 112 - 2", "question": "放射性物質的半衰期 $T$ 定義為「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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"id": "AST mathematics - 107 - 3", "question": "某零售商店販賣「熊大」與「皮卡丘」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,「皮卡丘」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「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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}
{"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": ""}}