# Misc 4 - Chapter 4 Class 12 Determinants

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Misc 4 If a, b and c are real numbers, and ∆ = b+cc+aa+bc+aa+bb+ca+bb+cc+a = 0 , Show that either a + b + c = 0 or a = b = c Solving ∆ ∆ = b+cc+aa+bc+aa+bb+ca+bb+cc+a Applying R1→ R1 + R2 + R3 = b+c+c+a+a+b𝑐+𝑎+𝑎+𝑏+𝑏+𝑐a+b+b+c+c+a𝑐+𝑎a+bb+c𝑎+𝑏b+cc+a = 𝟐(𝐚+𝐛+𝐜)𝟐(𝐚+𝐛+𝐜)𝟐(𝐚+𝐛+𝐜)𝑐+𝑎a+bb+c𝑎+𝑏b+cc+a Taking Common 2(𝑎+𝑏+𝑐) From R1 = 𝟐(𝐚+𝐛+𝐜) 111𝑐+𝑎a+bb+c𝑎+𝑏b+cc+a Applying C2→ C2 – C1 = 2(a+b+c) 1𝟏−𝟏1𝑐+𝑎a+b−c−ab+c𝑎+𝑏b+c−a−bc+a = 2(a+b+c) 1𝟎1𝑐+𝑎a−cb+c𝑎+𝑏c−ac+a Applying C3→ C3 – C1 = 2(a+b+c) 10𝟏−𝟏𝑐+𝑎b−cb+c−c−a𝑎+𝑏c−ac+a−a−b = 2(a+b+c) 10𝟎𝑐+𝑎b−cb−a𝑎+𝑏c−ac−b Expanding determinant along R1 = 2(a+b+c) 1 𝑏−𝑐𝑏−𝑎𝑐−𝑎𝑐−𝑏−0 𝑐+𝑎𝑏−𝑎𝑎+𝑏𝑐−𝑏+0 𝑐+𝑎𝑏−𝑐𝑎+𝑏𝑐−𝑎 = 2(a+b+c) 1 𝑏−𝑐𝑏−𝑎𝑐−𝑎𝑐−𝑏−0+0 = 2(a+b+c) ((c – b) (b – c) – (b – a) (c – a)) = 2(a+b+c) ( – (c – b) (c – b) – (bc – ab – ac + a2)) = 2(a+b+c) ( – (c – b)2 – (bc – ab – ac + a2)) = 2(a+b+c) ( – (c2 + b2 – 2cd) – bc + ab + ac – a2) = 2(a+b+c) ( – c2 – b2 + 2cb – bc + ab + ac – a2) = 2(a+b+c) ( – (a2 + b2 + c2) + ab + bc + ac) ∴ ∆ = 2(a+b+c) ( – (a2 + b2 + c2) + ab + bc + ac) Given ∆ = 0 2(a+b+c) ( – (a2 + b2 + c2) + ab + bc + ac) = 0 So, either (a + b + c) = 0 or a = b = c Hence proved

Miscellaneous

Misc 1

Misc. 2 Important Deleted for CBSE Board 2021 Exams only

Misc 3

Misc 4 You are here

Misc 5

Misc 6 Important

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11 Important Deleted for CBSE Board 2021 Exams only

Misc 12 Important Deleted for CBSE Board 2021 Exams only

Misc. 13 Deleted for CBSE Board 2021 Exams only

Misc 14 Deleted for CBSE Board 2021 Exams only

Misc. 15 Important Deleted for CBSE Board 2021 Exams only

Misc. 16 Important

Misc 17 Important Deleted for CBSE Board 2021 Exams only

Misc 18

Misc 19 Important

Matrices and Determinants - Formula Sheet and Summary

Chapter 4 Class 12 Determinants

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.