壹、前言
數學甲的題型有選擇(填)題、混合題型與非選擇題,其中非選擇題主要評量考生是否能夠清楚表達推理論證過程,答題時應將推理或解題過程說明清楚,且得到正確答案,方可得到滿分。若過程中列式正確,但計算錯誤,則酌給部分分數。如果只有答案對,但觀念錯誤,或過程不合理,則無法得到分數。以下提供非選擇題參考答案、評分原則,以及考生作答中無法得到滿分的情形。
貳、各題滿分參考答案及評分原則
第 13 題
一、滿分參考答案
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)
與
所截邊長相等,設此邊長為
,由相似形可得
,所以
。
所截矩形另一邊長為
,所以矩形面積為
。
二、評分原則
能否依據題意所給條件,以相似形求得矩形邊長,並證得矩形面積。
三、正確解題步驟
根據題幹條件,利用所學數學策略求出矩形邊長,例如相似形的性質,以證得矩形面積。
四、常見的錯誤概念或解法
(一) 矩形長寬計算錯誤,如:將矩形寬算成
。
(二) 算出矩形長寬但未算矩形面積。
(三) 僅由題目告知之面積湊出答案,未寫出證明過程,如:
。
第 14 題
一、滿分參考答案
1. 由
,與過
的水平面所截矩形面積為
, 再乘上高
,得黎曼和
。
2. 體積的定積分式為
。
利用反導函數得積分值為
。
二、評分原則
能否由切片方法寫下估計積木體積的黎曼和,並能利用定積分表示積木體積,進而求得積木體積值。
三、正確解題步驟
利用前一題得出的矩形面積,以切片方法寫下估計積木體積的黎曼和,並利用定積分表示積木體積,進而求得積木體積值。
四、常見的錯誤概念或解法
(一) 黎曼和形式錯誤,如:黎曼和寫成
或![](data:image/png;base64,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)
(二) 未寫出正確的定積分式,如:
,或
。
(三) 定積分式正確但反導函數錯誤,如:![](data:image/png;base64,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)
(四) 計算錯誤,如:
。
第 15 題
一、滿分參考答案
、
、
所形成的三角形邊長分別為
、
、
。
由餘弦定理得
,
微分得
。
二、評分原則
能否正確操作餘弦定理並得出
,並以微分除法律求其導函數。
三、正確解題步驟
根據題幹條件得出三角形的三邊長,利用餘弦定理得出
,並以微分除法律求其導函數。
四、常見的錯誤概念或解法
(一) 餘弦公式錯誤或僅寫出餘弦公式而未代入向量
、
之長度,如:
,或
。
(二) 向量長度與向量內積混淆,如:
。
(三) 計算錯誤,如:
或
。
第 16 題
一、滿分參考答案
法一:微分
由上題的
可得:
當
時,
,故此時
為遞減;
時
,此時
為遞增。
當
時,
,
有最小值,此時夾角
為最大。
法二:配方法
由上題的
,其中
的分子為常數,將分母配方得
,當
時,分母遞增,此時
為遞減;
時,分母遞減,此時
為遞增。
當
時,分母有最大值,
有最小值。此時夾角
為最大。
二、評分原則
以一階微分
的正負來說明
在哪個區域遞增、遞減,以說明
為多少時,夾角
為最大。
三、正確解題步驟
利用
的正負情形來說明
在哪個區域遞增、遞減,並指出
為多少時,夾角
為最大。
四、常見的錯誤概念或解法
(一) 不清楚導函數的正負情形與函數遞增、遞減的關係,如:寫出
時
,但誤以為
遞增。
(二) 以配方法解題,誤以為極值會出現在
或
。
第 17 題
一、滿分參考答案
因為 5 在 4.96 附近且易於計算,所以是最合適的估計參考點。在 5 附近的一次估計為
,由
與
,可得
。
二、評分原則
會寫出一次估計形式,並找到合理的估計參考點說明當
時,
約為多少。
三、正確解題步驟
根據題幹條件找出合理估計點,並藉此寫出
的一次估計形式,進而得出
的估計值。
四、常見的錯誤概念或解法
(一) 僅以符號寫出一次估計形式,如:
。
(二) 計算錯誤。
參、結語
數學科非選擇題的解法通常不只一種,且有些解法並不屬於高中課程範圍,在此提供屬於高中課程,且多數考生可能採用的解法以供各界參考。不管採取哪種解法,均需於答題卷上清楚表達推理或解題過程,且得到正確答案,方可得到滿分。如果只有答案對,但觀念錯誤,或過程不合理,則無法得到分數。本文說明正確的解題概念與步驟,以及得部分分數與無法得分的可能情形,主要用意在於提供老師教學或學生平常練習時的參考。