| { |
| "schema_version": "1", |
| "exam_id": "ceec-115-gsat-mathematics-a", |
| "title": "115學年度學科能力測驗試題：數學A考科", |
| "subject": "數學A", |
| "language": "zh-TW", |
| "description": "考試時間：100分鐘\n\n作答方式：\n\n- 選擇（填）題用 2B 鉛筆在「答題卷」上作答；更正時以橡皮擦擦拭，切勿使用修正帶（液）。\n- 除題目另有規定外，非選擇題用筆尖較粗之黑色墨水的筆在「答題卷」上作答；更正時，可以使用修正帶（液）。\n- 考生須依上述規定劃記或作答，若未依規定而導致答案難以辨識或評閱時，恐將影響成績。\n- 答題卷每人一張，不得要求增補。\n- 選填題考生必須依各題的格式填答，且每一個列號只能在一個格子劃記。\n\n※試題中參考的附圖均為示意圖，試題後附有參考公式及數值。", |
| "total_points": 100.0, |
| "sections": [ |
| { |
| "id": "s1", |
| "title": "第壹部分、選擇（填）題：一、單選題（占30分）", |
| "instructions": [ |
| { |
| "type": "text", |
| "markdown": "說明：第1題至第6題，每題5分。\n\n單選題：每題有 $n$ 個選項，其中只有一個是正確或最適當的選項。各題答對者，得該題的分數；答錯、未作答或劃記多於一個選項者，該題以零分計算。" |
| } |
| ], |
| "questions": [ |
| { |
| "id": "q1", |
| "number": "1.", |
| "type": "single_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "財神廟舉辦抽發財金活動：參加者抽兩次籤，每次抽籤出現「吉」、「祥」的機率皆為 $\\frac{1}{3}$。如果兩次都抽得「吉」，獲得獎金180元；如果兩次都抽得「祥」，獲得獎金90元；其餘情況則無獎金。試問參加者可獲獎金的期望值為何？" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "20元" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "30元" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "45元" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "60元" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "90元" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q2", |
| "number": "2.", |
| "type": "single_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "對任一實數 $a$，令 $[a]$ 代表滿足 $[a]\\leq a<[a]+1$ 的整數，例如：$[3]=3$，$[3.1]=3$，$[-3.1]=-4$。關於函數 $f(x)=[\\sqrt{99-x}]+[\\sqrt{99+x}]$，其中 $-99\\leq x\\leq 99$；試選出正確的選項。" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$f(-20)\\leq f(0)<f(1)$" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$f(-20)<f(1)\\leq f(0)$" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$f(1)<f(-20)\\leq f(0)$" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$f(0)<f(-20)\\leq f(1)$" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$f(0)\\leq f(1)<f(-20)$" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q3", |
| "number": "3.", |
| "type": "single_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "設 $f(x)=a^x$，其中 $a$ 為正實數。已知 $c_1,c_2,c_3$ 是公差為 $\\frac{10}{3}$ 的等差數列，且 $f(c_1),f(c_2),f(c_3)$ 是公比為4的等比數列。則等比數列 $f(10),f(8),f(6)$ 的公比為何？" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$2^{-\\frac{6}{5}}$" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$2^{-\\frac{3}{5}}$" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$2^{\\frac{3}{5}}$" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$2^{\\frac{6}{5}}$" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$2^{\\frac{5}{3}}$" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q4", |
| "number": "4.", |
| "type": "single_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "某網遊有16種材料，其中6種為基本材料，10種為進階材料。任選3種不同材料可以合成出草藥、食物、藥水中的1類道具，其合成規則如下：若3種材料均為基本材料，則合成結果必為同一種草藥；若3種材料中2種為基本材料、1種為進階材料，則合成結果會根據不同的進階材料得到不同種的食物，但不會受到基本材料不同而改變；其他的組合都會合成出不同種的藥水。試問此網遊總共可合成出多少種道具？" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "256" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "370" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "401" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "455" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "560" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q5", |
| "number": "5.", |
| "type": "single_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "已知實數三階方陣 $A$ 滿足 $A\\begin{bmatrix}1\\\\1\\\\0\\end{bmatrix}=\\begin{bmatrix}0\\\\-1\\\\1\\end{bmatrix}$，$A\\begin{bmatrix}0\\\\-1\\\\1\\end{bmatrix}=\\begin{bmatrix}1\\\\1\\\\0\\end{bmatrix}$，$A\\begin{bmatrix}1\\\\0\\\\-1\\end{bmatrix}=\\begin{bmatrix}0\\\\0\\\\0\\end{bmatrix}$。試問有多少個行向量 $\\vec v=\\begin{bmatrix}v_1\\\\v_2\\\\v_3\\end{bmatrix}$ 滿足 $A\\vec v=\\begin{bmatrix}1\\\\0\\\\1\\end{bmatrix}$ 且 $\\vec v$ 垂直於行向量 $\\begin{bmatrix}0\\\\1\\\\0\\end{bmatrix}$？" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "1個" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "2個" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "3個" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "0個" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "無窮多個" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q6", |
| "number": "6.", |
| "type": "single_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "坐標平面上有 $A(2,-2),B(-1,2)$ 兩點，試問直線 $y=-6$ 上有多少個點 $C$ 使得 $\\triangle ABC$ 為等腰三角形？" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "1" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "2" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "3" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "4" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "5" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| } |
| ] |
| }, |
| { |
| "id": "s2", |
| "title": "第壹部分、選擇（填）題：二、多選題（占30分）", |
| "instructions": [ |
| { |
| "type": "text", |
| "markdown": "說明：第7題至第12題，每題5分。\n\n多選題：每題有 $n$ 個選項，其中至少有一個是正確的選項。各題之選項獨立判定，所有選項均答對者，得該題全部的分數；答錯 $k$ 個選項者，得該題 $\\frac{n-2k}{n}$ 的分數；但得分低於零分或所有選項均未作答者，該題以零分計算。" |
| } |
| ], |
| "questions": [ |
| { |
| "id": "q7", |
| "number": "7.", |
| "type": "multiple_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "坐標平面上同時滿足 $\\begin{cases}2x-y-3>0\\\\x+2y+1<0\\end{cases}$ 的點 $P(x,y)$ 可能位在下列哪些選項？" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "第一象限" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "第二象限" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "第三象限" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "第四象限" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$x$軸" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q8", |
| "number": "8.", |
| "type": "multiple_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "已知 $A=\\begin{bmatrix}2&1\\\\1&0\\end{bmatrix}$，且對所有正整數 $n\\geq 2$，令 $A^n=\\begin{bmatrix}a_n&b_n\\\\c_n&d_n\\end{bmatrix}$。試選出正確的選項。" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$b_2<c_2$" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$A^2=2A+\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix}$" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$c_{n+2}=c_{n+1}+2c_n$" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$\\begin{bmatrix}a_n&b_n\\\\c_n&d_n\\end{bmatrix}\\begin{bmatrix}0\\\\1\\end{bmatrix}=\\begin{bmatrix}b_{n+1}\\\\d_{n+1}\\end{bmatrix}$" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$d_{2n}-a_{2n}=(d_n)^2-(a_n)^2$" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q9", |
| "number": "9.", |
| "type": "multiple_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "$T$ 分數為評量成績的一種方式，其計算方式如下：設全班平均成績為 $\\mu$ 且標準差為 $\\sigma$。若某生原始成績為 $S$，則他該科之 $T$ 分數為 $T=50+10\\left(\\frac{S-\\mu}{\\sigma}\\right)$。已知某班期末數學和英文兩科的平均成績皆為60，數學成績的標準差為12，英文成績的標準差為8。試選出正確的選項。" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "若甲生英文的原始成績為52，則其 $T$ 分數為40" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "各生數學的 $T$ 分數不會超過其原始成績" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "若乙生兩科的原始成績平均比丙生兩科的原始成績平均高，則乙生兩科的 $T$ 分數平均比丙生兩科的 $T$ 分數平均高" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "若該班級兩科的及格標準均為 $T$ 分數大於或等於40，則數學及格的原始成績比英文及格的原始成績低" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "該班原始成績數學對英文的迴歸直線（即最適直線）之斜率與該班 $T$ 分數數學對英文的迴歸直線之斜率相同" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q10", |
| "number": "10.", |
| "type": "multiple_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "已知四邊形 $ABCD$ 中，$\\overline{AB}$ 平行 $\\overline{DC}$，$\\overline{AC}$ 與 $\\overline{BD}$ 交於 $E$。若 $\\overrightarrow{AB}=(2,-6)$，$\\overrightarrow{AD}=(1,5)$ 且 $\\triangle ABE$ 面積為3。試選出正確的選項。" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$\\cos\\angle BAD=\\frac{-7\\sqrt{65}}{65}$" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$\\triangle ABD$ 面積為9" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$\\overrightarrow{AE}=(\\frac{3}{2},\\frac{1}{2})$" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "四邊形 $ABCD$ 面積為 $\\frac{65}{3}$" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$\\overline{BC}<\\frac{8}{3}$" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q11", |
| "number": "11.", |
| "type": "multiple_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "令 $\\Gamma$ 為坐標平面上 $y=\\cos(\\frac{\\pi}{2}x)$ 的圖形。對任一實數 $m\\neq 0$，以 $L_m$ 表示直線 $y=mx+1$。試選出正確的選項。" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$m>0$ 時，$L_m$ 和 $\\Gamma$ 交點的 $x$ 坐標皆為負" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "若 $(a,b)$ 為 $L_m$ 和 $\\Gamma$ 的交點，則 $(-a,b)$ 為 $L_{-m}$ 和 $\\Gamma$ 的交點" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "可以找到一實數 $m\\neq 0$ 使得 $L_m$ 和 $\\Gamma$ 交於點 $(\\frac{20}{3},\\frac{1}{2})$" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "若 $L_m$ 與 $\\Gamma$ 有一交點在直線 $y=-1$ 上，則 $\\frac{1}{m}$ 是奇數" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "若 $L_m$ 與 $\\Gamma$ 有一交點在 $x$ 軸上，則 $L_m$ 與 $\\Gamma$ 有偶數個交點" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q12", |
| "number": "12.", |
| "type": "multiple_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "令 $f(x)$、$g(x)$ 為實係數三次多項式且 $f(x)$ 的首項係數為1，已知 $f(x)-g(x)=2x^3+2x$。令 $\\Gamma_1$ 和 $\\Gamma_2$ 分別為 $f(x)$ 和 $g(x)$ 在坐標平面上的函數圖形，其對稱中心分別為 $(a_1,b_1),(a_2,b_2)$。試選出正確的選項。" |
| } |
| ], |
| "points": 5.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$\\Gamma_1$ 和 $\\Gamma_2$ 恰交於三點" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$a_1+a_2$ 可唯一確定" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$b_1+b_2$ 可唯一確定" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "若 $a_1=a_2$，則 $b_1=b_2$" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "若 $b_1=b_2$，則 $a_1=a_2$" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| } |
| ] |
| }, |
| { |
| "id": "s3", |
| "title": "第壹部分、選擇（填）題：三、選填題（占25分）", |
| "instructions": [ |
| { |
| "type": "text", |
| "markdown": "說明：第13題至第17題，每題5分。\n\n選填題每題有 $n$ 個空格，須全部答對才給分，答錯不倒扣。" |
| } |
| ], |
| "questions": [ |
| { |
| "id": "q13", |
| "number": "13.", |
| "type": "fill_in_blank", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "某高中聘用的全體教師 $\\frac{1}{4}$ 只有學士學位，$\\frac{3}{4}$ 有碩士學位。只有學士學位的教師中有 $\\frac{1}{5}$ 通過英聽檢定，有碩士學位的教師中有 $\\frac{3}{5}$ 通過英聽檢定。已知每位教師被抽到的機會相等，若隨機抽選一位通過英聽檢定的教師，則該教師有碩士學位的條件機率為 $\\dfrac{\\boxed{\\text{13-1}}}{\\boxed{\\text{13-2}}\\boxed{\\text{13-3}}}$。（化為最簡分數）" |
| } |
| ], |
| "points": 5.0, |
| "options": null, |
| "blanks": [ |
| { |
| "id": "b1", |
| "label": "13-1" |
| }, |
| { |
| "id": "b2", |
| "label": "13-2" |
| }, |
| { |
| "id": "b3", |
| "label": "13-3" |
| } |
| ], |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q14", |
| "number": "14.", |
| "type": "fill_in_blank", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "坐標平面上，向量 $(a,b)$ 與直線 $y=bx-1$ 垂直，則 $a+b$ 的最大可能值為 $\\dfrac{\\boxed{\\text{14-1}}}{\\boxed{\\text{14-2}}}$。（化為最簡分數）" |
| } |
| ], |
| "points": 5.0, |
| "options": null, |
| "blanks": [ |
| { |
| "id": "b1", |
| "label": "14-1" |
| }, |
| { |
| "id": "b2", |
| "label": "14-2" |
| } |
| ], |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q15", |
| "number": "15.", |
| "type": "fill_in_blank", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "已知三正數 $a,b,c$ 成一等差數列，其中 $a<b<c$，且坐標平面上三點 $(a,\\log 3a)$、$(b,\\log 4b)$、$(c,\\log 6c)$ 在同一直線上，則 $\\frac{b}{a}$ 之值為 $\\dfrac{\\boxed{\\text{15-1}}}{\\boxed{\\text{15-2}}}$。（化為最簡分數）" |
| } |
| ], |
| "points": 5.0, |
| "options": null, |
| "blanks": [ |
| { |
| "id": "b1", |
| "label": "15-1" |
| }, |
| { |
| "id": "b2", |
| "label": "15-2" |
| } |
| ], |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q16", |
| "number": "16.", |
| "type": "fill_in_blank", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "坐標平面上，已知二次函數圖形 $\\Gamma:y=f(x)$ 的頂點 $P$ 在直線 $y=1+2x$ 上，且交 $x$ 軸於點 $A(-\\frac{1}{2},0),B(\\frac{1}{2},0)$。將 $\\Gamma$ 平移使得平移後圖形的頂點 $Q$ 仍在直線 $y=1+2x$ 上，且亦通過點 $B(\\frac{1}{2},0)$，此時 $P$、$Q$ 為兩相異點，則 $\\overline{PQ}=\\dfrac{\\boxed{\\text{16-1}}\\sqrt{\\boxed{\\text{16-2}}}}{\\boxed{\\text{16-3}}}$。（化為最簡根式）" |
| } |
| ], |
| "points": 5.0, |
| "options": null, |
| "blanks": [ |
| { |
| "id": "b1", |
| "label": "16-1" |
| }, |
| { |
| "id": "b2", |
| "label": "16-2" |
| }, |
| { |
| "id": "b3", |
| "label": "16-3" |
| } |
| ], |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q17", |
| "number": "17.", |
| "type": "fill_in_blank", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "直角 $\\triangle ABC$ 中，$\\angle CAB$ 為直角，$\\overline{AB}$ 邊上一點 $D$，滿足 $\\angle BCD=2\\angle ACD$，且 $\\overline{BC}=2\\overline{BD}$。若 $\\overrightarrow{AD}=k\\overrightarrow{AB}$，則 $k=\\dfrac{\\boxed{\\text{17-1}}}{\\boxed{\\text{17-2}}\\boxed{\\text{17-3}}}$。（化為最簡分數）" |
| } |
| ], |
| "points": 5.0, |
| "options": null, |
| "blanks": [ |
| { |
| "id": "b1", |
| "label": "17-1" |
| }, |
| { |
| "id": "b2", |
| "label": "17-2" |
| }, |
| { |
| "id": "b3", |
| "label": "17-3" |
| } |
| ], |
| "matching": null, |
| "children": [] |
| } |
| ] |
| }, |
| { |
| "id": "s4", |
| "title": "第貳部分、混合題或非選擇題（占15分）", |
| "instructions": [ |
| { |
| "type": "text", |
| "markdown": "說明：本部分共有1題組，單選題每題3分，非選擇題配分標於題末。限在答題卷標示題號的作答區內作答。\n\n選擇（填）題與「非選擇題作圖部分」使用2B鉛筆作答，更正時以橡皮擦擦拭，切勿使用修正帶（液）。非選擇題請由左而右橫式書寫，作答時必須寫出計算過程或理由，否則將酌予扣分。" |
| } |
| ], |
| "questions": [ |
| { |
| "id": "q18", |
| "number": "18-20題為題組", |
| "type": "composite", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "坐標空間中有一平行六面體 $PQRS-ABCD$，如圖所示。已知 $\\overrightarrow{AB}\\times\\overrightarrow{AD}=(-5,5,5)$、$\\overrightarrow{AD}\\times\\overrightarrow{AP}=(-2,0,-4)$、$\\overrightarrow{AP}\\times\\overrightarrow{AB}=(6,-10,-8)$，$\\overline{AP}=6$。試回答下列問題。" |
| }, |
| { |
| "type": "image", |
| "asset": "assets/q18-fig1.png", |
| "alt": "平行六面體PQRS-ABCD的透視線框示意圖。底面頂點A、B、C、D中，D在左前方、C在右前方、B在右後方，A為左後方的隱藏頂點；上方對應頂點依序為P、Q、R、S。外框與可見稜以實線表示，從A連到D、B、P的隱藏稜以虛線表示。", |
| "caption": null |
| } |
| ], |
| "points": 15.0, |
| "options": null, |
| "blanks": null, |
| "matching": null, |
| "children": [ |
| { |
| "id": "q18.a", |
| "number": "18.", |
| "type": "single_choice", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "試問平行四邊形 $ABCD$ 的面積為何？（單選題，3分）" |
| } |
| ], |
| "points": 3.0, |
| "options": [ |
| { |
| "id": "1", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$2\\sqrt{5}$" |
| } |
| ] |
| }, |
| { |
| "id": "2", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$5\\sqrt{2}$" |
| } |
| ] |
| }, |
| { |
| "id": "3", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$5\\sqrt{3}$" |
| } |
| ] |
| }, |
| { |
| "id": "4", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$6\\sqrt{3}$" |
| } |
| ] |
| }, |
| { |
| "id": "5", |
| "content": [ |
| { |
| "type": "text", |
| "markdown": "$10\\sqrt{2}$" |
| } |
| ] |
| } |
| ], |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q18.b", |
| "number": "19.", |
| "type": "short_answer", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "設 $B$ 點坐標為 $(1,2,0)$，試求平面 $ABCD$ 的平面方程式。（非選擇題，4分）" |
| } |
| ], |
| "points": 4.0, |
| "options": null, |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| }, |
| { |
| "id": "q18.c", |
| "number": "20.", |
| "type": "essay", |
| "prompt": [ |
| { |
| "type": "text", |
| "markdown": "試求平行六面體的體積，並求平行六面體上（含邊界）距點 $A$ 的最長距離。（非選擇題，8分）" |
| } |
| ], |
| "points": 8.0, |
| "options": null, |
| "blanks": null, |
| "matching": null, |
| "children": [] |
| } |
| ] |
| } |
| ] |
| } |
| ], |
| "metadata": { |
| "exam_year": "115學年度", |
| "exam_name": "學科能力測驗", |
| "duration": "100分鐘", |
| "source_exam": "input/01-matha-exam.pdf", |
| "source_answer_key": "input/02-matha-key.pdf" |
| } |
| } |
|
|
