{"id": "CAP_mathematics_dev-102-23", "question": "判斷下列各式的值，何者最小？", "A": "$8 - (- \\frac{13}{29})^{20}$", "B": "$8 - (- \\frac{13}{29})^{17}$", "C": "$8 - (- \\frac{13}{29})^{11}$", "D": "$8 - (- \\frac{13}{29})^{14}$", "E": null, "F": null, "answer": "D", "explanation": "1. $8 - (- \\frac{13}{29})^{17}$, $8 - (- \\frac{13}{29})^{11}$ > 8\n2. $8 - (- \\frac{13}{29})^{20}$, $8 - (- \\frac{13}{29})^{14}$ < 8\n3. 又 13/29 <1 -> $(-13/29)^14$ > $(-13/29)^20$ 所以 D < A\n", "metadata": {"timestamp": "2023-10-29 01:58:12.626759", "source": "CAP_mathematics-102", "explanation_source": ""}, "human_evaluation": {"quality": "", "comments": ""}}
{"id": "CAP_mathematics_dev-112-1", "question": "$(-3)^3$ 之值為何？", "A": "-27", "B": "9", "C": "27", "D": "-9", "E": null, "F": null, "answer": "A", "explanation": "(-3) x (-3) x (-3) = 9 x (-3) = -27，故(A)正確。", "metadata": {"timestamp": "2023-10-29 01:58:12.628968", "source": "CAP_mathematics-112", "explanation_source": ""}, "human_evaluation": {"quality": "", "comments": ""}}
{"id": "CAP_mathematics_dev-111-22", "question": "已知坐標平面上有二次函數 $y=-(x+6)^2+5$ 的圖形，函數圖形與$x$ 軸相交 於 $(a,0)$ 、 $(b,0)$ 兩點，其中 $a<b$。今將此函數圖形往上平移，平移後函數形與$x$ 軸相交於 $(c,0)$ 、 $(d,0)$ 兩點，其中 $c<d$，判斷下列敘述何者正確？", "A": "$(a+b)=(c+d)$，$(b-a)>(d-c)$", "B": "$(a+b)<(c+d)$，$(b-a)<(d-c)$", "C": "$(a+b)=(c+d)$，$(b-a)<(d-c)$", "D": "$(a+b)<(c+d)$，$(b-a)>(d-c)$", "E": null, "F": null, "answer": "C", "explanation": "圖形平移前後皆對稱 $x=-6$, 也就是 $\\left\\{\\begin{array}{l}(a+b) \\div 2=-6 \\\\ (c+d) \\div 2=-6\\end{array} \\Rightarrow a+b=c+d\\right.$原圖形凹向下, 往上平移後圖形與 $x$ 軸兩交點距離變大, 即 $(d-c)>(b-a)$, 故選 $(C)$", "metadata": {"timestamp": "2023-10-29 01:58:12.628946", "source": "CAP_mathematics-111", "explanation_source": "https://chu246.blogspot.com/2022/05/111_22.html"}, "human_evaluation": {"quality": "", "comments": ""}}
{"id": "CAP_mathematics_dev-109-16", "question": "中秋節時阿柚製作的廣式月餅、蛋黃酥、鳳梨酥的數量比為 2:1:3 ,其中只有製作廣式月餅和蛋黃酥時使用鹹蛋黃。若阿柚製作每個廣式月餅時使用 2 顆鹹蛋黃，製作每個蛋黃酥時使用 1 顆鹹蛋黃，且總共使用 120 顆鹹蛋黃，則他製作了幾個鳳梨酥？", "A": "72", "B": "60", "C": "120", "D": "45", "E": null, "F": null, "answer": "A", "explanation": "已知廣式月餅、蛋黃酥、鳳梨酥的數量比為 $2: 1: 3$,\n可設廣式月餅、蛋黃酤、鳳梨酥的數量分別為 $2r$、$r$、$3r$, r 不等於 0 ,所以共用了 $2 \\times 2 r+1 \\times r=4 r+r=5 r$ 顆战蛋黃,\n已知總共使用 120 顆閨蛋黃, 可列出 $5 r=120 \\rightarrow r=24$,因此鳳梨酥一共製作了 $3 \\times 24=72$ 個。", "metadata": {"timestamp": "2023-10-29 01:58:12.628434", "source": "CAP_mathematics-109", "explanation_source": "https://www.wincenter.com.tw/activity/wincenter/exam/2020exam/doc/109wincenter-math.pdf"}, "human_evaluation": {"quality": "", "comments": ""}}
{"id": "CAP_mathematics_dev-105-6", "question": "多項式$77x^{2} −13x−30$可因式分解成$(7x+a)(bx+c)$，其中a、b、c均為 整數，求 a + b + c 之值為何?", "A": "22", "B": "0", "C": "12", "D": "10", "E": null, "F": null, "answer": "C", "explanation": "1. $77x^2-13x-30 =(7x+a)(bx+c) =(7x-5)(11x+6)$  (乘法分配律反推: -55+42=-13)\n$\\Rightarrow  a=-5, b=11, c=6$\n$a+b+c=-5+11+6=12$，故(C)正確。\n", "metadata": {"timestamp": "2023-10-29 01:58:12.627386", "source": "CAP_mathematics-105", "explanation_source": "https://yamol.tw/item-6.+%E5%A4%9A%E9%A0%85%E5%BC%8F+77x2+%E2%88%92+13x+%E2%88%92+30+%E5%8F%AF%E5%9B%A0%E5%BC%8F%E5%88%86%E8%A7%A3%E6%88%90+%28+7x+%2B+a+%29..-1291673.htm"}, "human_evaluation": {"quality": "", "comments": ""}}