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Miscellaneous

Misc 1

Misc. 2 Important Deleted for CBSE Board 2023 Exams

Misc 3

Misc 4

Misc 5

Misc 6 Important

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11 Important Deleted for CBSE Board 2023 Exams

Misc 12 Important Deleted for CBSE Board 2023 Exams

Misc. 13 Deleted for CBSE Board 2023 Exams You are here

Misc 14 Deleted for CBSE Board 2023 Exams

Misc. 15 Important Deleted for CBSE Board 2023 Exams

Misc. 16 Important

Misc 17 (MCQ) Important Deleted for CBSE Board 2023 Exams

Misc 18 (MCQ)

Misc 19 (MCQ) Important

Matrices and Determinants - Formula Sheet and Summary Important

Chapter 4 Class 12 Determinants

Serial order wise

Last updated at May 29, 2018 by Teachoo

Misc. 13 Using properties of determinants, prove that: 3a a+b a+c b+a 3b b+c c+a c+b 3c = 3 ( a + b + c) (ab + bc + ac) Taking L.H. S 3a a+b a+c b+a 3b b+c c+a c+b 3c Applying C1 C1 + C2 + C3 = 3a a+b a+c a+b a+c b+a 3b b+c 3b b+c c+a c+b 3c c+b 3c = + + a+b a+c + + 3b b+c + + c+b 3c Taking (a + b + c) common from C1 = ( + + ) 1 a+b a+c 1 3b b+c 1 c+b 3c Applying R1 R1 R2 = (a+b+c) a+b 3b a+c ( b+c) 1 3b b+c 1 c+b 3c = (a+b+c) a 2b a+c+b c 1 3b b+c 1 c+b 3c = (a+b+c) 0 a 2b a+b 1 3b b+c 1 c+b 3c Applying R2 R2 R3 =(a+b+c) 0 a 2b a+b 3b b+c 1 c+b 3c 3c =(a+b+c) 0 a 2b a+b 2b+c b 2c 1 c+b 3c Expanding determinant along C1 = (a + b + c ) 0 2b+c b 2c c+b 3c 0 a 2b a+b c+b 3c +1 a 2b a+b 2b+c b 2c = (a + b + c ) b+2c a+2b (2b+c)( a+b) = (a + b + c ) b+2c a+2b (2b+c)( a+b) = (a + b + c ) ab+2b2+2ca+4ab (2b( a+b) c( a+b) = (a + b + c ) ab+2b2+2ca+4ab ( 2ba+2b2+ac bc) = (a + b + c) ab+2b2+2ca+4ab ( 2ba+2b2+ac bc) = (a + b + c ) ab+2b2+2ca+4ab+2ba 2b2+ca cb = (a + b + c ) ab+2ba+2ca+ +4 +2 2 2 2 = (a + b + c ) (3ab + 3ac + 3bc + 0) = (a + b + c ) (3ab + 3ac + 3bc) = (a + b + c ). 3 (ab + ac + bc) = 3 (a + b + c ) (ab + ac + bc) = R.H.S Hence Proved