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2008학년도 3월 고3 전국연합학력평가 문제지

제 2 교시
‘나’형
성명

수 리 영 역
수험번호
3. lim  의    →∞   
① 
 ② 

3
       
값은? [2점]
 ④ 

1
⑤ 

◦ 먼저 수험생이 선택한 응시 유형의 문제지인지 확인하시오. ◦ 문제지에 성명과 수험 번호를 정확히 기입하시오. ◦ 답안지에 수험 번호, 응시 유형 및 답을 표기할 때는 반드시 ‘수험생이 지켜야 할 일’에 따라 표기하시오. ◦ 단답형 답의 숫자에 0 이 포함된 경우, 0 을 OMR 답안지에 반드시 표기해야 합니다. ◦ 문항에 따라 배점이 다르니, 각 물음의 끝에 표시된 배점을 참고하시오. 배점은 2점, 3점 또는 4점입니다. ◦ 계산은 문제지의 여백을 활용하시오. ◦ 이 문제지에는 아래의 예와 같이 국립국어연구원의 ‘한글 맞 춤법’에 의한 사이시옷 표기법을 사용하였습니다. (예) 최솟값(종전 표기: 최소값)
 ×  1.  의


③ 

값은? [2점] ③  ④  ⑤ 

① 

② 

2.

두 행렬  

    ,       
③ 

 에 대하여      를 만 



4.

  log    일 때,      의 값은? [3점]   ②     ⑤   ③ 

족시키는 행렬  의 모든 성분의 합은? [2점] ①  ②  ④  ⑤ 

 ①   ④   

1

8

2

수 리 영 역
    

‘나’형

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 5. 연립방정식       가        이외의 해를 갖도록 7. 이차정사각행렬  의 역행렬이    일 때,    의 역행렬     은    이다. 두 실수    의 합    의 값은? (단,  는 하는 모든 실수  의 값의 합은? [3점] 단위행렬이다.) [3점] ①  ②  ③  ④  ⑤  ①  ②  ③  ④  ⑤ 

6.

→∞

   자연수  에 대하여  의 정수부분을  이라 할 때,    의 값은? [3점]  lim       ②  

8.

이차정사각행렬    에 대하여 에서 옳은 것을 모두

고른 것은? (단,  는 영행렬,  는 단위행렬이다.) [3점] < 보 기 >

① 

  ③  

④ 

 ⑤ 

ㄱ.  ≠  이고    이면    이다. ㄴ.         이면      이다. ㄷ.   의 역행렬이 존재하면   의 역행렬도 존재한다. ① ㄱ ④ ㄴ, ㄷ ② ㄴ ⑤ ㄱ, ㄴ, ㄷ ③ ㄱ, ㄷ

2

8

‘나’형

수 리 영 역
 ③    ④    ⑤  

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━  11. 수열  을 다음과 같이 정의한다. 9. 등비수열 에서 첫째항부터 제 항까지의 합이  이고     ,      ,             ⋯       에서 옳은 것을 모두 고른 것은? [4점] 곱이  일 때,          의 값은? [3점]     
 ①    ②  

3

< 보 기 > ㄱ.     ㄴ.     (  는 자연수)이면      이다. ㄷ.       ( 는 자연수)이면      이다. ① ㄱ ④ ㄴ, ㄷ ② ㄴ ⑤ ㄱ, ㄴ, ㄷ ③ ㄱ, ㄷ

10.

수열 은 다음과 같이  으로 나누어 떨어지지 않는 자연수

12.

자연수  을  개 이어 붙여 만든 자연수를  이라 하자.

를 작은 수부터 차례로 나열한 것이다.        ⋯ 이때 ① 


예를 들어    ,    ⋯  (  자리의 수)이다. log  의 지표와 가수를 각각     이라 할 때, 에

  의 값은? [3점]




서 옳은 것을 모두 고른 것은? [4점] ④  ⑤  < 보 기 > ㄱ.    ㄴ.     인 자연수  은  뿐이다. ㄷ.      는 자연수)이면           이다. ① ㄱ ④ ㄱ, ㄴ ② ㄴ ⑤ ㄴ, ㄷ ③ ㄷ

② 

③ 

3

8

4

수 리 영 역
 
① ②

‘나’형

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━      14. 실수   에 대하여 행렬         의 역행렬이 존재 13. 자연수  에 대하여         ⋯   로 정의한다.     하지 않을 때, 좌표평면 위에서 점     가 나타내는 그래프의 다음은  이상인 모든 자연수  에 대하여 등식 개형은? [4점]       ⋯         이 성립함을 증명한 것이다. (1)    일 때, 좌변  우변  진 등식은 성립한다. (2)    일 때, 주어진 등식이 성립한다고 가정하면
      ⋯        

(가)

이므로 주어

양변에  를 더하면
      ⋯   

③ (나) 이므로



그런데      

(다)

      ⋯          

그러므로      일 때도 성립한다. 따라서  이상인 모든 자연수  에 대하여 주어진 등식 은 성립한다. 위 증명에서 (가), (나), (다)에 알맞은 것은? [3점] (가) ① ② ③
  



(나)
                        

(다)
              

 ④    ⑤  

4

8

‘나’형
P 
    로 정의하자.

수 리 영 역
지는 부분이 정육각형이 되도록 분)를 그린다.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━  15. 좌표평면 위의 점 P   에 대하여 행렬  P 를 17. 반지름의 길이가  인 원이 있다. 그림과 같이 이 원에 내접하는 두 정삼각형이 겹쳐지는 부분이 정육각형이 되도록 모양의 도형   (어두운 부분)을 그린다. 또,   의 정육각형 에 내접하는 원을 그리고, 이 원에 내접하는 두 정삼각형이 겹쳐

5

임의의 두 점 P Q 에 대하여 에서 옳은 것을 모두 고 른 것은? (단,  는 단위행렬이다.) [4점] < 보 기 > ㄱ.  P  Q   R 인 점 R 가 존재한다. ㄴ.  P  Q   Q  P ㄷ. 점 P 가 원점이 아니면  P  S   인 점 S 가 존재한다. ① ㄱ ④ ㄴ, ㄷ ② ㄴ ⑤ ㄱ, ㄴ, ㄷ ③ ㄱ, ㄴ

모양의 도형   (어두운 부

이와 같은 방법으로 모양의 도형       ⋯    을 그릴 때, 도형   의 넓이는? [4점]

  ①     ④  

  ②     ⑤  

  ③  

16.

어느 연구소의 보고서에 따르면 앞으로 LPG 경차 사용이 늘

어나 자동차 휘발유 소비량이 감소할 것이라고 한다.  년 A 지역의 연간 자동차 휘발유 소비량은  톤이고, 매년 이 지역의 연간 자동차 휘발유 소비량은 전년도에 비하여 일정한 비율로  감소하여  년에는   톤이 된다고 한다.  년 이후에도  이와 같은 비율로 계속 감소한다고 할 때, A 지역에서  년부 터  년까지  년 동안 사용되는 자동차 휘발유 소비량의 총   합은? (단,    로 계산한다.) [4점]
 ①  톤   ④  톤   ②  톤   ⑤  톤   ③  톤 

5

8

6
18.

수 리 영 역
구하시오. [3점] 등비수열  에 대하여    ,     일 때,   의 값

‘나’형

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 21. 두 정수   가      를 만족할 때,      의 최솟값을 단답형(18~25)

을 구하시오. [3점]

19.

이차정사각행렬           

가
                  

22.

 부터  까지 자연수의 합을  이라 할 때,



  log





  의

을 만족시킬 때, 행렬     의 모든 성분의 합을 구하시오. [3점]

  
 

값을 구하시오. (단,    는 보다 크지 않은 최대의 정수이다.) [4점]

20.

두 등차수열    에 대하여
     ,

23.

 는 세 자리의 자연수이고,  는  보다 큰 세 자리의 자연

      
  





수이다. log  의 가수가 log  의 가수의  배일 때, 자연수  의 값을 구하시오. [4점]

일 때,    의 값을 구하시오. [3점]

6

8

‘나’형

수 리 영 역



━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 24. 표는 두 등산화 A B 를 각각 한 5지 선다형 (단위 : kg) 켤레씩 만드는 데 필요한 가죽과 고 A B 무의 양을 나타낸 것이다.     가죽 가죽  kg, 고무  kg을 모두 사 26. 무한등비수열  sin  이 수렴하도록 하는  이하의  용하여 등산화 A 를  켤레, 등산화   고무 자연수  의 개수는? [3점] B 를  켤레 만든다고 할 때, 다음은 ①  ②  ③  ④  ⑤   와  의 값을 구하는 식을 행렬로 나타낸 것이다.            

7

 

 

이때 두 상수   에 대하여    의 값을 구하시오. [4점]

25.

그림과 같이 반지름의 길이가  인 원을  개의 부채꼴로 나

27.

실수  에 대하여          ,          일 때,   을

누었더니 부채꼴의 넓이가 작은 것부터 차례로 등차수열을 이루 었다. 가장 큰 부채꼴의 넓이가 가장 작은 부채꼴의 넓이의  배 일 때, 가장 큰 부채꼴의 넓이는  이다. 이때  의 값을 구하 시오. [4점]

  를 이용하여 나타낸 것은? [3점]  ①     ②     ③    ④    ⑤  

7

8

8

수 리 영 역
30.
< 보 기 > 음 규칙에 따라 수를 나열한 것이다.

‘나’형

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 28. 수열 의 첫째항부터 제 항까지의 합을  이라 하면 단답형

      이 성립한다. 에서 옳은 것을 모두 고른 것
은? (단,  는  이 아닌 상수이다.) [4점] 그림은 직사각형 모양을 이루고 있는  ×   개의 칸에 다

 ㄱ.    

ㄴ. 수열 은 등비수열이다.
 ㄷ.    이면 수열 은 수렴한다. 

(가) 제  행에는    ⋯  을 차례로 나열하고, 각 행의 첫 칸에는 모두  을 나열한다. (나) 그림에 있는  ×  개의 칸으로 이루어진 임의의 직사각형   에서 등식        가 성립하도록 한다.     예를 들면 에서        가 성립한다.  

① ㄱ ④ ㄱ, ㄷ

② ㄴ ⑤ ㄱ, ㄴ, ㄷ

③ ㄱ, ㄴ

제 행  제 행  제 행  제 행  제 행 

    

    

    

    

    

⋯ ⋯ ⋯ ⋯ ⋯

  




이때 제  행(어두운 부분)에 나열된  개의 수의 합을 구하 시오. [4점]

29.

넓이가     ⋯ 인 등비수열을 이루는 정사각형들을

그림과 같이 왼쪽부터 차례로 배열하고, 각 정사각형의 내부에 정사각형과 한 변을 공유하는 정삼각형을 그린다.

정삼각형의 외부와 정사각형의 내부의 공통부분(어두운 부분) 의 넓이를 왼쪽부터 차례로    ⋯ 이라 할 때,
→∞

lim  의 값은? [4점] 
  ②        ⑤    ③   

    ⋯   

 ①        ④  

※ 확인 사항 문제지와 답안지의 해당란을 정확히 기입(표기)했는지 확인하시오. 8 8

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Sun Case

...Computervision’s decision? Does Sun have a better product? Can Sun be regarded as a reliable longterm vendor? If I understand well the case there is many obstacles for Sun to be chose by Computervision such as: * The fact that the company is quite new and Sun don’t have the reputation of Apolo * There is also the problem that Apalo has the industry standard and the good amount of employees when Sun has only “40 person” as said in the phone call from the president. * And the sun product is seen as incomplete. So if we stay on those elements, Vinod Khosla don’t really has chance to change the decision of Computervision. But if we look more in detail, there is also argument in favor of Sun: * The advantage of Sun is that they have an open product which can be modified by the user, this is the reason why in the case the software developers are attracted by their product. It also mean that the product will be a part of the improvement of computer technology and may grow with it while the product of Apolo for example is nonstandard so it is not flexible as the Sun machine. * The price of the Apolo product is also higher so for the long run benefit, the Sun computer may be more interesting because more competitive. * The Sun machine is already adopted by university which mean that the product is viable and well appreciate * Moreover, in the phone call of the president, it is mentioned that the number of employees of sun doesn’t play in their favor...

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