97 99 99 2010 ordering theory OT 96.31% 16.33 14.05% 90.25% 14.80 22.11% 95.79% 14.10 25.79% 90.59% 12.39 34.79%
Adaptive Diagnostic &
Remedial Teaching System
based on the KSAT,
Matrix Unit of High School Math,
for example
Abstract
How to detect students' learning process of the blind spot, and be corrected, to assist them in to new heights, is the key point of education . In this study, based on" the 99 Syllabus", we build the expert knowledge structure of the "matrix" unit of high school , and construct the corresponding items bank for each concept , has been able to load computerized adaptive testing requirements. Based expert knowledge structure and the corresponding exam as a blueprint to build a computerized adaptive optional question test and diagnostic systems. The interface of this system is simple and easy to operate, and applicable to a computer or mobile vehicle , can reduce manpower and time burden. The saving item rate relating to the optional question the computerized adaptive test diagnostic system performance assessment results: pretest, the average item numbers are 16.33 under the prediction accuracy of 96.31% , a savings of approximately 14.05% items; and the average applied to
approximately 25.79% of the questions; average number of questions when the prediction accuracy of 90.59%, 12.39 title, a savings of approximately 34.79 % of the questions. That regardless of the pretest or posttest the post therapy confirm student questions structure adaptive topics to accept the whole test can save questions than the students. After we analyzed the score changes between each group, viewed the score changes of different level between each group, compared the saving item rate of each adaptive topics, measured pretest and posttest differences within each group, surveyed pretest and posttest differences of different level inside each group, and observed the change of computerized tests topics before and after, we could make sure this adaptive testing system can diagnose the learning status of high school students in the Matrix unit, to identify students' individual blind spots, supplemented by targeted remedial instruction structure, adaptive remedial instruction, and confirmed the remedial teaching effectiveness of computerized adaptive testing system is better than the traditional group remedial teaching style to enhance the effectiveness of high school teaching can expect.
Keywords: Computer adaptive testing, expert knowledge structure, Matrix, Saving item rate, Remedial teaching
………. 01 ……… 01 ……… 05 ……… 06 ……… 06 ……… 10 ……… 13 CAT ……… 15 ……… 19 ……… 20 ……… 21 ………. 23 ……… 23 ……… 23 ……… 25 ……… 25 ……… 26 ……… 27 ……… 30 ……… 32 ……… 34 ……… 37 ………. 38 ……… 38 ……… 41 ……… 48 ………. 71
2.1.1. ……….. 08 2.2.1. j k ……… 10 2.3.1. ……….. 13 3.4.1. ……….. 26 3.6.1. ……….. 27 4.1.1. ………… 39 4.4.1. A B ……… 50 4.4.2. A B ……… 51 4.4.3. A B ……… 51 4.4.4. B C ……… 52 4.4.5. B C ……… 52 4.4.6. B C ……… 53 4.4.7. A B ……… 53 4.4.8. A B ……… 54 4.4.9. A B ……… 54 4.4.10. A B ……… 55 4.4.11. A B ……… 55 4.4.12. A B ……… 56 4.4.13. B C ……… 56 4.4.14 B C ……… 57 4.4.15. B C ……… 57 4.4.16. B C ……… 58 4.4.17. B C ……… 58 4.4.18. B C ……… 59 4.4.19. ……… 59 4.4.20. ………... 60 4.4.21. ………... 61 4.4.22. A ……….. 62 4.4.23. B ……….. 62 4.4.24. C ……….. 63 4.4.25. ………... 63 4.4.26. A ……….. 64 4.4.27. A ……….. 64
4.4.32. ………... 66
4.4.33. A ………... 68
4.4.34. B ………... 69
2.1.1. ……….. 08 2.1.2. ……….. 09 2.4.1. Chang et al, 1998 ………… 17 2.4.2. ……….. 18 2.9.1. ……….. 22 3.1.1. ……….. 24 3.5.1. 1 ……… 26 3.5.2. 2 ……… 27 3.6.1. - A ……… 28 3.6.2. - A ……… 28 3.6.3. - B ……… 29 3.6.4. - C ……… 29 3.7.1. ……….. 30 3.7.2. ……….. 31 3.7.3. Camtasia Studio……… 31 3.7.4. Camtasia Studio……… 32 3.8.1. A ………. 33 3.8.2. B ………. 33 3.8.3. C ………. 34 3.9.1. - A ……… 35 3.9.2. - A ……… 35 3.9.3. - B ……… 36 3.9.4. - C ……… 36 4.1.1. ……….. 40 4.2.1. ……… 41 4.2.2. ……….. 42 4.2.3. ……….. 43 4.2.4. ….. 43 4.2.5. ………….. 43 4.2.6. ……….. 44 4.2.7. ……….. 45 4.2.8. ……….. 46 4.2.9. 1 ………... 46 4.2.10. 2 ………... 47 4.3.1. 1 ………... 48
adaptive
testing 2006
Testing Service F. Lord 2009
computerized adaptive testing CAT
testing KSAT knowledge structure 2006 KSAT 2007 KSAT 2010 2008 2007 2007 2006 2005 2005 2006 2003 2004 2005
2012
60 2004
97 1 24
1993 2000 Bower 1972 Shavelson 1972 2005 2006
Larkin, McDermott, Simon & Simon 1980
Chi , Glaser & Rees 1982
Hall 1988 Rewey, Dansereau, Skaggs, Hall & Pitre 1989 Skaggs 1988
concept scaffold 1997 2006
2006 2.1.1. 2.1.1. 1 2 3 4 5 6 2.1.1.
2005 2007
2.1.2.
2.1.2.
ordering theory OT Airasian & Bart 1973 Item Relationship Structure analysis IRS Takaya 1991 Diagnosys Appleby et al., 1997
(OT)
j k j k slip guess j k ( 0, 1) jk P Xj Xk j k threshold0.02 0.04 Airasian & Bart 1973 jk j k k j X X j k j k 2.2.1. j k j k 1 k X 0 k X 1 j X ( 1, 1) j k P X X P X( j 1,Xk 0) P X( j 1) 0 j X ( 0, 1) j k P X X P X( j 0,Xk 0) P X( j 0)
( j 1) P X j P X( j 0) j P X( k 1) k P X( k 0) k P X( j 1,Xk 1) j k P X( j 1,Xk 0) j k P X( j 0,Xk 1) j k P X( j 0,Xk 0) j k
(IRS)
Takeya 1991; 1999 1 ( 0, 1) ( 0) ( 1) j k jk j k P X X r P X P X ( j 0, k 1) P X X P X( j 0) P X( k 1) r r 0.5 r r jk j k k j X X j k 2006 X j Xk j k X X k j X X j k OT IRSDiagnosys
Diagnosys 2004 Diagnosys f X( j 1,Xk 0) f X( j 0,Xk 1) j k X X j k Appleby, Samuel Treasure-Jones 1997 f X( j 1,Xk 1) f X( j 0,Xk 0) ( j 1, k 0) ( j 0, k 1) f X X f X X Xj Xk j k
OT
IRS
Diagnosys
2003 2004 2004 Kuo, Liu, Sheu,
Pai, Ko,Yang & Lin 2004 Wu, Kuo & Yang 2012 OT IRS Diagnosys
IRS
Diagnosys
OT 2006 2.3.1. 2.3.1. 1. A 2×2 A2 A A A3 A A A 1 1 A 1 1 1 1 A 1 1 1 1 2 A b a b a 0 (1) 1 1 A 1 1 1 1 A 1 1 1 1 1 1 1 1 1 1 A 1 1 1 1 1 1 1 1 1 A 2 1 1 1 1 1 1 1 1 1
(2) 1 1 2 A 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 1 1 1 0.5 0.5~0.7 0.7 1/4 0.3~0.6 0.6~0.8 0.8 1/4 0.3 0.3~0.6 0.6~0.8 (1) (2) 0.3 0.3~0.5 0.5 IRT a= , b= , c= ( ) ( ) ( ) ( ) (2006)
Wainer, 1990 2004 Item Response
Theory IRT Wainer 2000
knowledge structure Appleby, Samuels & Treasure-Jones, 1997 Brown & Burton, 1978 Chang, Liu & Chen, 1998 VanLehn, 1988 Wenger, 1987
1998 Chang, Liu & Chen M1 M2 M3 M4 2.4.1. 1 1a 2 2 2a 2b 2c M1 M2 M3 1b M4 1c M1 M4 2007 1. 2. 1. 2.
2.4.1. Chang et al, 1998 2003 2004 2005 (I)(II)(III) Chang OT 2.4.2. A B C B C A A B C 1 1a 1b 1c 2a 2b 2c M1 M3 M4 M2
D E F G H I A B C 2.4.2. Chang OT A I 9 A B C D E F G H I 8 A B C B C D E F G H I Chang
2002
2007
computer-based adaptive remedial instruction CBRI
2004
2008
97 1 24 99
99 1. 2. 3. Caley-Hamilton 4. 5. 6. 7. ± 0 1 2 9 1 3 3 28 2 3 3 2011
2.9.1.
99
OT
99
60
3.1.1.
3.1.1.
A B C
A B C
99
20
99 102
78 101
3.4.1.
Cronbach's Alpha
.807 19
PHP Personal Home Page 3.5.1.
3.5.1. 1
3.4.3.
3.5.2. 2
3.6.1.
3.6.1.
A 125 B 121
3.6.1. - A
3.6.3. - B
Camtasia Studio
A B
C
5 10
3.7.2.
3.7.4. Camtasia Studio
Camtasia Studio
A B 246
computer-based adaptive remedial instruction CBRI C 298
teacher-based adaptive remedial instruction TBRI
246
3.8.1. A
3.8.3. C
A
3.9.1. - A
99 99 20 4.1.1. 99 4.1.1. 4.1.1.
4.1.1. 1 2 3 4 5 6 7 -8 9 10 11 12 13 14 15 16 17 18 19 20
OT 4.1.1.
4.1.1.
Prosser 1974 10
50 Reckase 1981
computerized adaptive testing 2009
4.2.1.
4.2.2.
4.2.4.
4.2.5.
4.2.6.
4.2.7.
4.2.7.
4.2.8.
4.2.9. 4.2.10.
Camtasia Studio A
B C
4.3.1. 4.3.2.
4.3.3. Camtasia Studio
A B
B C
A B
A B
4.4.1. A B III df F * 79.804 1 79.804 .879 .349 21964.882 242 90.764 23520.459 245 a. R = .066 ( R = .055) A B A B A B A B 4.579 4.4.2. A B 95% A 95.687a .869 93.975 97.399 B 91.108a .884 89.367 92.849 a. = 71.50
4.4.3. A B (I) (J) (I-J) a 95% a 4.579* 1.264 .000 2.089 7.069 *. .05 a. B C B C 4.4.4. B C 4.4.4. B C III df F * 240.412 1 240.412 .678 .411 158553.725 447 354.706 170113.725 450 a. R = .068 R = .062 B C B C
10.161 4.4.5. B C 95% B 90.982a 1.712 87.619 94.346 C 80.822a 1.036 78.785 82.858 a. = 65.45 4.4.6. B C (I) (J) (I-J) a 95% a B C 10.161* 2.001 .000 6.229 14.093 *. .05 a.
A B A B 4.4.7. A B 4.4.7. A B III df F * 99.908 1 99.908 1.143 .287 10491.542 120 87.430 11286.742 123 a. R = .070 R = .047 A B 4.4.8. A B 95% A 95.619a 1.443 92.763 98.475 B 92.820a 1.474 89.902 95.738 a. = 85.73
4.4.9. A B (I) (J) (I-J) a 95% a 2.799 2.384 .243 -1.921 7.519 *. .05 a. A B A B 4.4.10. A B 4.4.10. A B III df F * .590 1 .590 .006 .937 11206.137 118 94.967 12070.893 121 a. R = .072 R = .048
A B 4.4.11. A B 95% A 95.322a 1.254 92.838 97.806 B 89.817a 1.276 87.291 92.343 a. = 64.56 4.4.12. A B (I) (J) (I-J) a 95% a 5.505* 1.820 .003 1.901 9.109 *. .05 a. B C B C 4.4.13. B C
4.4.13. B C III df F * 811.307 1 811.307 2.764 .098 65458.191 223 293.534 74365.956 226 a. R = .120 R = .108 B C 4.4.14 B C 95% B 91.899a 2.202 87.559 96.239 C 83.447a 1.335 80.816 86.078 a. = 78.99
4.4.15. B C (I) (J) (I-J) a 95% a B C 8.452* 2.575 .001 3.376 13.527 *. .05 a. B C B C 4.4.16. B C 4.4.16. B C III df F * 166.616 1 166.616 .423 .516 86572.955 220 393.513 93594.839 223 a. R = .075 R = .062
B C 4.4.17. B C 95% B 89.939a 2.558 84.898 94.979 C 78.205a 1.547 75.157 81.254 a. = 51.74 4.4.18. B C (I) (J) (I-J) a 95% a B C 11.733* 2.989 .000 5.842 17.624 *. .05 a. 4.4.19. A B B C
4.4.19. I J I-J A B 2.799 5.505 B C 8.452 11.733 A B 2.799 A 5-fold cross-validation 4.4.20. 4.4.21.
4.4.20. 0.001 0.006 0.011 0.016 0.021 0.9972 0.9967 0.9936 0.9925 0.9847 18.2374 18.2181 17.9045 17.7832 17.5497 0.026 0.031 0.036 0.041 0.046 0.9631 0.9663 0.9313 0.9333 0.9212 16.9355 17.0336 15.9742 15.8697 15.5471 0.051 0.056 0.061 0.066 0.071 0.9078 0.9060 0.9025 0.8761 0.8735 15.08 14.9742 14.8039 13.9716 13.8774 0.076 0.081 0.086 0.091 0.096 0.8628 0.8477 0.8348 0.8249 0.8101 13.4981 12.9832 12.6116 12.1471 11.7600 4.4.21. 0.001 0.006 0.011 0.016 0.021 0.996604 0.996333 0.990696 0.987504 0.986553 17.8929 17.98323 16.95484 16.8671 16.40903 0.026 0.031 0.036 0.041 0.046 0.980985 0.981664 0.967199 0.957895 0.943905 15.49935 15.40258 14.45419 14.09806 13.51484 0.051 0.056 0.061 0.066 0.071 0.929508 0.917487 0.905874 0.870357 0.872122
0.076 0.081 0.086 0.091 0.096 0.857317 0.844618 0.826486 0.81365 0.798234 10.88129 10.44258 9.994839 9.464516 9.12129 96.31% 16.33 14.05% 90.25% 14.80 22.11% 95.79% 14.10 25.79% 90.59% 12.39 34.79% A 4.4.22. A ( ) 95%
B 4.4.23. B ( ) 95% – -20.169 24.756 2.123 -24.367 -15.971 .000 C 4.4.24. C ( ) 95% – -20.641 20.343 1.178 -22.960 -18.322 .000 4.4.25.
4.4.25. – A -18.456 B -25.521 C -20.641 A 4.4.26. A ( ) 95% – -4.333 5.708 .719 -5.771 -2.896 .000 A 4.4.27. A ( ) 95%
B 4.4.28. B ( ) 95% – -12.836 14.424 1.847 -16.530 -9.142 .000 B 4.4.29. B ( ) 95% – -38.417 17.151 2.214 -42.847 -33.986 .000 C 4.4.30. C ( ) 95%
C 4.4.31. C ( ) 95% – -26.384 26.713 2.086 -30.503 -22.265 .000 4.4.32. 20 30 4.4.32. – A -4.333 -32.806 B -12.836 -38.417
A 4.4.33. 19 /19 100 1 4 6 8 9 11 7 24 98 12 50 94 17 /19 90 100 B 4.4.34. 15 /19 3 69 99 7 26 99 10 56 93 12 66 96 13 50 82 14 55 94 15 34 90 16 52 90 17 46 95 18 53 96 4 /19 2 5 6 11 C 4.4.35. 14 /19 2 63 96 7 24 95 9 67 90 12 58 90 13 52 93 14 57 94 15 32 87 16 47 92 17 51 89 5 /19 1 4 5 6 8 A B C +
4.4.33. A 1 2 1 96% 100% 2 87% 99% 3 69% 96% 4 96% 100% 5 93% 97% 6 95% 100% 7 24% 98% 8 94% 100% 9 92% 100% 10 91% 99% 11 89% 100% 12 50% 94% 13 81% 99% 14 81% 99% 15 79% 98% 16 81% 99% 17 78% 97% 18 53% 75% 19 46% 71%
4.4.34. B 1 2 1 81% 93% 2 96% 91% 3 69% 99% 4 72% 94% 5 97% 95% 6 95% 66% 7 26% 99% 8 87% 89% 9 82% 93% 10 56% 93% 11 93% 87% 12 66% 96% 13 50% 82% 14 55% 94% 15 34% 90% 16 52% 90% 17 46% 95% 18 53% 96% 19 57% 81%
4.4.35. C 1 2 1 92% 84% 2 63% 96% 3 72% 97% 4 95% 85% 5 97% 93% 6 95% 51% 7 24% 95% 8 84% 81% 9 67% 90% 10 83% 87% 11 80% 88% 12 58% 90% 13 52% 93% 14 57% 94% 15 32% 87% 16 47% 92% 17 51% 89% 18 56% 69% 19 52% 74%
99 99 OT Prosser 1974 10 96.31% 16.33 14.05% 90.25% 14.80 22.11% 95.79% 14.10 25.79% 90.59% 12.39 34.79%
KSAT
A
B C
2002 1997 42 1-16 2008 2009 IRT 2004 2008 TWELF 2008 97 5 17 2006 14(2) 1-16 (2008) 2007 8(2) 191-197 2012 DINA
GCCCE2012 2012 5 28 - 6 1 2004 2000 2011 ( ) 198 http://www.ceec.edu.tw/CeecMag/Articles/198/198-04.htm 2008 SCORM 2004 16(2) 31-58 2010 97 99 http://www.edu.tw/high-school/content.aspx?site_content_sn=8403 (2010) 1993 NSC82-0301-H017-001 2005 — 2003 (I)
NSC91-2520-S-142-001-2005 (III) (2005) 2011 TWELF 2011 2011 11 4-5 2007 2006 2006 12 15 -16 2005 -2007 (2008 2006 14(2) 17-35
Airasian, P.W. & Bart, W.M. (1973). Ordering Theory: A new and useful measurementmodel. Journal of Educational Technology, 5, 56-60.
Appleby, J., Samules, P. & Treasure-Jones, T. (1997). Diagnosys a knowledge-based diagnostic test of basic mathematical skills. Computers
Education, 28(2), 113-131.
Bower, G.H. (1972). Mental imagery and associative learning. In L. Gregg (Ed.), Cognition in learning and memory, 51-88.
Brown, J. S. & Burton, R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2:155-192.
Chang, K. E., Liu, S. H. & Chen, S. W. (1998). A testing system for diagnosing misconceptions in DC electric circuits. Computers & Education, 31, 195-210.
Chi, M.T.H., Glaser, R. & Rees, E. (1982). Expertise in problem solving. In R.J. Sternberg (Ed.), Advances in psychology of human intelligence, 1, 7-75. Hillsdale, NJ: Erlbaum.
Hall, R. (1988). Knowledge maps and the presentation of related information domains. Unpublished doctoral dissertation, Texas Christian University, Fort Worth, TX.
Hambleton, R. K. & Swaminathan, H. (1985). Item response theory: Principles and applications. Boston, MA: Kluwer-Nijhoff.
Kuo, B-C., Liu, H-C., Sheu, T-W., Pai, C-H., Ko, L-W., Yang, J-M. & Lin, W-C. (2004). A Computerized Adaptive Testing System Based on
Larkin, J., McDermott, J., Simon, D. P., & Simon, H. A. (1980). Expert and novice performance in solving physics problems. Science, 208, 1335-1342.
Prosser, F. (1974). Item banking. In G. Lippey (Ed.), Computer-assisted test construction, 29-66. Englewood Cliffs, NJ: Educational Technology. Reckase, M. D. (1981). Tailored testing, measurement problems and latent trait
theory. Paper presented at the annual meeting of the National Council for Measurement in Education, Los Angeles.
Rewey, K. L., Dansereau, D. F., Skaggs, L. P., Hall, R. H. & Pitre, U. (1989). Effects of scripted cooperation and knowledge maps on the processing of technical material. Journal of Educational Psychology, 81(4), 604-609. Shavelson, R. J. (1972). Some aspects of the correspondence between conten
structure and cognitive structure in physics instruction. Journal of Educational Psychology, 63, 225-234.
Shavelson, R. J., & Stanton, G. C. (1975). Construct validation: methodology and application to three measures of cognitive structure. Journal of Educational Measurement, 12(2), 67-85.
Skaggs, L. P. (1988). The effects of knowledge maps and pictures on the acquisition of scientific information. Unpublished doctoral dissertation, Texas Christian University, Fort Worth, TX.
Solso, R. L. (1995). Cognitive Psychology. (4th ed.). Boston: Allyn, & Bacon. Takeya, M. (1991). New Test Theory: Structure Analysis Methods for
Educational Information. Tokyo: Waseda University Press. (in Japanese). Takeya, M. (1999). Structure Analysis Methods for Instruction. Takushoku
physics concepts. Journal of Educational Psychology, 70(6), 971-978. VanLehn. K.(1988). Student Modeling. In M.C. Polson and J.J Richardson.
Foundations of Intelligent Tutoring Systems. Hillsdale, NJ: Lawrence Erlbaum Publishers.
Wainer, ). Computerized adaptive testing: A primer. Hillsdale, NJ: Lawrence Erlbaum Associates.
Wainer, H. (2000). Computerized adaptive testing: a primer (2nd ed). Hillsdale, NJ: Lawrence Erlbaum Publishers.
Wenger, E. (1987). Artificial Intelligence and Tutoring Systems:Computational and cognitive approaches to the communication of knowledge. Los Altos, California: Morgan Kaufmann Publishers.
Wu, H.-M., Kuo, B.-C. & Yang, J.-M. (2012). Evaluating Knowledge Structure-based Adaptive Testing Algorithms and System Development. Educational Technology & Society, 15(2), 7388.
(19 100 ) 1. A 2×2 A2 A A A3 A A A 1 1 A 1 1 1 1 A 1 1 1 1 2 A b a b a 2. 3 2 1 1 A 1 5 6 3 B A B d c b a d c b a 3. ? (1) 0 2 2 0 (2) 1 1 2 2 (3) 2 1 2 2 (4) 2 1 1 1 4. 6 2 3 0 3 2 0 3 2 z y x z y x z y x c b a 1 0 0 0 1 0 0 0 1 m n c b a m ,n 1 2 3 4 5 6 7
5. A B A 1 2 3 4 B 3 2 4 1 A 2B x y z u 2x u y (1) 14 (2) 11 (3) 8 (4) 6 6. A 3 1 2 5 B 1 5 0 3 2 4 4AB (1) 4 0 1 0 4 0 (2) 500 24 4 40 68 0 (3) 24 0 80 68 52 0 (4) 24 52 16 68 0 80 7. ABCD A 3 1 A A m n n m 8. x y z 6 2x y 3z 9 x 3y z 4 xyz 9. 1 5 4 3 2 y x y x A 1 5 4 3 2 1 A A (1) 0 0 0 4 1 5 4 3 2 1 0 0 1 2 3 D C B A 8 9 10 11 12 13
(3) 0 4 0 1 1 5 4 3 2 1 (4) 1 4 0 1 1 5 4 3 2 1 10. 1 0 1 1 A A20 d c b a d c b a 11. A [ aij ]3×3 aij i2 3j (1) 80 (2) 96 (3) 102 (4) 122 12. 7 8 9 9 (1) 8 7 9 9 (2) 9 9 8 7 (3) 9 8 9 7 (4) 7 9 8 9 13. 1 2 2 2 2 b y a x (1) a (2) 2 b (3) ab (4) 2 b a 14. L y x d c b a d c b a 15. x 3 3x 2y 2 ax by c a b c 14 15 16 17 18 19 20 21
A d c b a d c b a 17. P(5 8) x y 3 P' m ,n n m 18. A cos50° sin50° sin50° cos50° B cos70° sin70° sin70° cos70° AB (1) 1 2 3 2 3 2 1 2 (2) 1 2 3 2 3 2 1 2 (3) 1 2 3 2 3 2 1 2 (4) 1 2 3 2 3 2 1 2 19. 2 1 3 0 0 1 1 0 d c b a 1 0 2 1 d c b a 22 23 24 25 26
1. A 2×2 A2 A A A3 A A A 1 1 A 1 1 1 1 A 1 1 1 1 2 A b a b a 0 (1) 1 1 A 1 1 1 1 A 1 1 1 1 1 1 1 1 1 1 A 1 1 1 1 1 1 1 1 1 A 2 1 1 1 1 1 1 1 1 1 2 1 0 2 2 0 0 1 1 0 (2) 1 1 2 A 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 1 1 1 2. 3 2 1 1 A 1 5 6 3 B A B d c b a d c b a 22 4 7 7 4 1 5 6 3 3 2 1 1 1 2 3
3. ? (1) 0 2 2 0 (2) 1 1 2 2 (3) 2 1 2 2 (4) 2 1 1 1 (1) 4. 6 2 3 0 3 2 0 3 2 z y x z y x z y x c b a 1 0 0 0 1 0 0 0 1 m n c b a m ,n 5 51 m n 6 2 1 3 0 3 1 2 0 3 2 1 6 7 5 0 0 9 5 0 0 3 2 1 6 7 5 0 0 9 5 0 0 15 10 5 5 1 R 6 2 0 0 0 9 5 0 0 3 0 5 3 1 0 0 0 9 5 0 0 3 0 5 2 1 3 R 27 0 5 0 9 0 0 5 5 27 0 1 0 5 9 0 0 1 5 6 7
c b a, , ,3 5 27 , 5 9 5 51 c b a 5. A B A 1 2 3 4 B 3 2 4 1 A 2B x y z u 2x u y (1) 14 (2) 11 (3) 8 (4) 6 (1) A 1 2 3 4 B 3 2 4 1 A 2B 1 2 3 4 6 4 8 2 7 6 11 6 x y z u 2x u y 14 6 6 14 6. A 3 1 2 5 B 1 5 0 3 2 4 4AB (1) 4 0 1 0 4 0 (2) 500 24 4 40 68 0 (3) 24 0 80 68 52 0 (4) 24 52 16 68 0 80 (4) 4AB 4 3 1 2 5 1 5 0 3 2 4 4 6 13 4 17 0 20 24 52 16 68 0 80 7. ABCD A 3 1 A A m n n m 34 7 2 1 0 27 7 9 2 3 1 0 27 6 9 3 0 1 1 0 1 1 3 1 3 1 0 3 1 3 1 3 1 3 1 0 20 7 2 1 81 20 27 7 9 2 3 1 81 21 27 6 9 3 0 D C B A 8 9 10 11
9. x y z 6 2x y 3z 9 x 3y z 4 xyz 6 x y z 6 1 2x y 3z 9 2 x 3y z 4 3 1×( 2) 2 1×( 1) 3 x y z 1 1 3y z 3 4 2y 2z 2 5 4× 2 3 5 x y z 6 1 3y z 3 4 4 3 z 4 6 z 3 y 2 x 1 9. 1 5 4 3 2 y x y x A 1 5 4 3 2 1 A A (1) 0 0 0 4 1 5 4 3 2 1 (2) 0 4 0 0 1 5 4 3 2 1 (3) 0 4 0 1 1 5 4 3 2 1 (4) 1 4 0 1 1 5 4 3 2 1 3 2 1 12 13
10. 1 0 1 1 A A20 d c b a d c b a 18 B I A 0 0 1 0 1 0 0 1 1 0 1 1 B2 B3 0 A20 I B 20 20 19 1 1 20 20 0 I
C
I BC
0 0 1 0 20 1 0 0 1 1 0 20 1 11. A [ aij ]3×3 aij i2 3j (1) 80 (2) 96 (3) 102 (4) 122 (2) j = 1 3 i = 1 3 ( i2 3j ) j = 1 3 ( i = 1 3 i2 i = 1 3 3j ) j = 1 3 ( 12 22 32 ) 3×3j j = 1 3 14 9 j = 1 3 j 3×14 9 ( 1 2 3 ) 96 A 1 3 1 6 1 9 4 3 4 6 4 9 9 3 9 6 9 9 4 7 10 7 10 13 12 15 18 ( 4 7 18 ) 96 12. 7 8 9 9 (1) 8 7 9 9 (2) 9 9 8 7 (3) 9 8 9 7 (4) 7 9 8 9 (4) 14 15 16 17 1813. 1 2 2 2 2 b y a x (1) a (2) 2 b (3) ab (4) 2 b a (3) 1 2 2 2 2 b y a x 1 2 2 y x y x y x b a 1 0 0 1 b a 1 14. L y x d c b a d c b a 2 x y 1 tan45 0 1 1 0 90 cos 90 sin 90 sin 90 cos 15. x 3 3x 2y 2 ax by c a b c 5 x x x 3 0 3 x 3x x x 3 1 2 1 b a b a 1 1 b a 19 20 21
16. A (2 0) (1 1) (4 2) (1 2) A d c b a d c b a 3 A 2 4 0 2 A 2 1 1 1 A 2 2 1 4 1 0 1 2 2 0 1 1 2 1 1 0 1 2 1 A 2 2 2 4 2 1 2 0 1 1 2 1 2 2 1 4 1 0 1 2 2 2 1 4 1 A 1 1 1 2 18. P(5 8) x y 3 P' m ,n n m 37 8 29 8 5 1 0 3 1 y x 18. A cos50° sin50° sin50° cos50° B cos70° sin70° sin70° cos70° AB (1) 1 2 3 2 3 2 1 2 (2) 1 2 3 2 3 2 1 2 (3) 1 2 3 2 3 2 1 2 (4) 1 2 3 2 3 2 1 2 (3) AB cos120° sin120° sin120° cos120° 1 2 3 2 3 2 1 2 22 23 24 25
19. 2 1 3 0 0 1 1 0 d c b a 1 0 2 1 d c b a 4 2 1 3 0 3 0 2 1 12 R 1 0 2 1 2 3 1 R 1 0 0 1 1 2 2R R 1 0 2 1 3 1 0 0 1 0 1 1 0 2 1 3 0 1 0 0 1 2 1 3 0 0 1 1 0 3 0 0 1 1 0 2 1 26
(19 100 ) 1. A 1 2 3 7 A 5 1 4 9 A d b c a A 5 1 1 2 d c b a 2. 0 2 8 2 2 3 3 5 z x z y x z y x c b a 1 0 0 0 1 0 0 0 1 c b a 3. 3 2 1 1 A 1 5 6 3 B X A B X d b c a d c b a 4. 1 5 4 3 7 3 y x y x A 1 5 4 3 7 3 A A (1) 0 0 0 4 1 5 4 3 7 3 (2) 0 4 0 0 1 5 4 3 7 3 (3) 1 4 0 1 1 5 4 3 7 3 (4) 1 4 3 7 3 1 2 3 5 4
5. A [aij]3×3 aij 1 i j 2 i j 3 i j A (1) 10 (2) 18 (3) 16 (4) 15 (5) 14 6. 2 2 12 y + 3 3 yx z z x y x z y 2 4 13 2 3 2 x + y + z = 7. A 3 1 2 1 2 5 3 4 1 B 2 0 5 1 2 2 AB (1) 7 5 22 8 24 6 (2) 5 7 8 22 6 24 (3) 10 5 6 1 8 2 (4) 1 0 0 0 1 0 0 0 1 8. P ABCDE A 1 2 1 P A 4 B m n n m 9. 2x 3y z 6 2x 2y 3z 7 x y 2z 0 z y x 10. (A) 0 0 0 0 (B) 3 3 3 3 (C) 2 1 1 5 (D) 0 2 2 0
(1)(A)(D) (2)(B)(C)(D) (3)(A)(B)(C) (4)(A)(C)(D)
A B C D E 6 7 8 11 12 n n b a n a 13 14 9 10
12. 2 1 1 0 (1) 2 1 1 0 (2) 2 1 1 0 (3) 2 1 0 1 (4) 1 2 1 0 13. y 2 3x 2y 2 14. 2 1 3 0 0 1 1 0 d b c a 1 0 2 1 abcd 15. A ( 1 , 1 ) ( 1 , 2 ) A ( 0 , 3 ) ( 3 , 0 ) A = d b c a d c b a 16. A ( 1 , 2 ) x y 2 n m , m n 17. A cos40° sin40° sin40° cos40° B cos80° sin80° sin80° cos80° AB (1) 1 2 3 2 3 2 1 2 (2) 1 2 3 2 3 2 1 2 (3) 1 2 3 2 3 2 1 2 (4) 1 2 3 2 3 2 1 2 (5) 1 2 3 2 3 2 1 2 18. x2 y2 16 1 9 16 2 2 y x 1 x x 2 C E C m n m 16 17 x y 18 22 19 20 21 23
19. L y x d b c a d c b a 24 25
(19 100 ) 1. A 1 2 3 7 A 5 1 4 9 A d b c a A 5 1 1 2 d c b a 1 1 2 3 7 A 5 1 4 9 A 5 1 1 2 4 3 9 7 A 5 1 1 2 4 3 9 7 5 1 1 2 d b c a 1 0 0 1 4 3 9 7 d b c a 1 1 7 3 9 4 1 0 0 1 4 3 9 7 1 0 0 1 1 d b c a 7 3 9 4 d c b a, , , 4, 3, 9,7 a b c d 1 2. 0 2 8 2 2 3 3 5 z x z y x z y x c b a 1 0 0 0 1 0 0 0 1 c b a 4 8 1 1 2 2 3 3 5 1 2 3
2 3 3 5 8 1 1 2 0 2 0 1 2 7 3 0 8 5 1 0 0 2 0 1 22 22 0 0 8 5 1 0 0 2 0 1 1 1 0 0 8 5 1 0 0 2 0 1 22 1 3 R 1 1 0 0 3 0 1 0 2 0 0 1 c b a, , 2,3, 1 3. 3 2 1 1 A 1 5 6 3 B X A B X d b c a d c b a 8 2 3 5 2 3 2 1 1 1 5 6 3 A B X 4. 1 5 4 3 7 3 y x y x A 1 5 4 3 7 3 A A (1) 0 0 0 4 1 5 4 3 7 3 (2) 0 4 0 0 1 5 4 3 7 3 (3) 1 4 0 1 1 5 4 3 7 3 5 4
1 5 4 7 27 13 (4) 5. A [aij]3×3 aij 1 i j 2 i j 3 i j A (1) 10 (2) 18 (3) 16 (4) 15 (5) 14 (2) a11 a22 a33 1 a21 a32 a31 2 a12 a13 a23 3 1 1 1 2 2 2 3 3 3 18 6. 2 2 1 2 y + 3 x z 3 y x z y x z y 2 4 13 2 3 2 x + y + z = 6 3x + 4 = 2y + z 2 + 3z = 3x + 2y 5y 4x + 2z 3x 2y z = 4 3x + 2y 3z = 2 4x 5y + 2z = 0 4y 2z = 2 23y 18z = 8 y = 2 z = 3 x = 1 7. A 3 1 2 1 2 5 3 4 1 B 2 0 5 1 2 2 AB (1) 7 5 22 8 24 6 (2) 5 7 8 22 6 24 (3) 10 5 6 1 8 2 (4) 1 0 0 0 1 0 0 0 1 (1) AB 3 1 2 1 2 5 3 4 1 2 0 5 1 2 2 7 5 22 8 24 6 6 7 8
8. P ABCDE A 1 2 1 P A 4 B m n n m 17 8 3 0 2 1 0 8 1 4 1 0 0 8 1 4 1 0 0 8 3 0 2 1 0 0 4 2 0 1 0 2 1 0 0 2 1 2 1 0 2 1 0 0 0 2 1 0 2 1 0 0 0 2 1 0 2 1 2 1 0 0 2 1 0 16 1 8 3 0 2 1 16 4 8 1 4 1 0 16 4 8 1 4 1 0 16 1 8 3 0 2 1 16 6 0 4 2 0 9. 2x 3y z 6 2x 2y 3z 7 x y 2z 0 z y x 3 2x 3y z 6 2x 2y 3z 7 x y 2z 0 x y 2z 0 2x 2y 3z 7 2x 3y z 6 2) 2) A B C D E 11 9 10
x y 2z 0 y 5z 6 z 1 z 1 y 5 6 y 1 x 1 2 0 x 1 ( x , y , z ) ( 1 , 1 , 1 ) 10. (A) 0 0 0 0 (B) 3 3 3 3 (C) 2 1 1 5 (D) 0 2 2 0
(1)(A)(D) (2)(B)(C)(D) (3)(A)(B)(C) (4)(A)(C)(D) (1) 11. n a b a 0 n n n a b a n a 0 1 0 0 0 b B 0 0 0 0 0 0 0 2 2 b B n n b a a a b a 0 0 0 0 0 0 n b I a 0 0 0 2 C a I C n an I B n n n n 2 1 1 2 2 2 2 2 a I B Cn n n 0 0 0 1 0 0 1 1 b a n a n n 0 0 0 0 0 n a 1 b a a n n n n n n a b a n a 0 1 12 n n n a b a n a 13 14 15
12. 2 1 1 0 (1) 2 1 1 0 (2) 2 1 1 0 (3) 2 1 0 1 (4) 1 2 1 0 (1) 13. y 2 3x 2y 2 2 3x y y x y x y x 2 2 0 0 1 y y x x 2 y y x x 2 1 2 2 3x y 2 2 1 2 3 x y 3x y 2 14. 2 1 3 0 0 1 1 0 d b c a 1 0 2 1 abcd 0 2 1 3 0 3 0 2 1 12 R 1 0 2 1 2 3 1 R 1 0 0 1 1 2 2R R 1 0 2 1 3 1 0 0 1 0 1 1 0 2 1 3 0 1 0 0 1 2 1 3 0 0 1 1 0 3 0 0 1 1 0 2 1 16 17 x y 18 19
d c b a A 0 3 3 0 2 1 1 1 d c b a 1 1 2 1 1 1 0 3 3 0 2 1 1 1 2 1 1 1 d c b a 1 2 1 1 3 1 3 1 3 1 3 2 0 3 3 0 d c b a 16. A ( 1 , 2 ) x y 2 n m , m n 6 1 2 0 1 1 2 3 2 17. A cos40° sin40° sin40° cos40° B cos80° sin80° sin80° cos80° AB (1) 1 2 3 2 3 2 1 2 (2) 1 2 3 2 3 2 1 2 (3) 1 2 3 2 3 2 1 2 (4) 1 2 3 2 3 2 1 2 (5) 1 2 3 2 3 2 1 2 (3) AB cos(40°+80°) sin(40°+80°) sin(40°+80°) cos(40°+80°) cos120° sin120° sin120° cos120° 1 3 22 21
18. x2 y2 16 1 9 16 2 2 y x 1 x x 2 C E E C n m n m 7 1 4 4 2 2 y x 1 3 4 2 2 y x y x y x 3 4 0 0 1 3 4 3 4 0 0 1 E C 19. L y x d b c a d c b a 2 x y 1 tan 45 0 1 1 0 90 cos 90 sin 90 sin 90 cos 23 24 25
