# Example 30 - Chapter 4 Class 12 Determinants (Term 1)

Last updated at May 29, 2018 by

Last updated at May 29, 2018 by

Transcript

Example 30 If a, b, c are positive and unequal, show that value of the determinant Δ = abcbcacab is negative Δ = abcbcacab Applying C1→ C1 + C2 + C3 = 𝐚+𝐛+𝐜bc𝐚+𝐛+𝐜ca𝐚+𝐛+𝐜ab Taking common a + b + c from C1 = (𝐚+𝐛+𝐜) 1bc1ca1ab Applying R2 → R2 – R1 = (a+b+c) 1bc𝟏−𝟏c−ba−c1ab = (a+b+c) 1bc𝟎c−ba−c1ab Applying R3 → R2 – R1 = (a+b+c) 1bc0c−ba−c𝟏−𝟏a−bb−c = (a+b+c) 1bc0c−ba−c𝟎a−bb−c Expanding determinant along C1 = (a + b + c ) 1 c−ba−cb−cb−c−0 bc𝑎−𝑏b−c+0 bcc−ba−c = (a + b + c ) 1 𝑐−𝑎 𝑏−𝑐− 𝑎−𝑏 𝑎−𝑐−0+0 = (a + b + c ) 𝑏−𝑐 − 𝑏−𝑐−(𝑎−𝑏)(𝑎−𝑐) = (a + b + c ) − 𝑏2+𝑐2−2𝑏𝑐−(𝑎2−𝑎𝑐−𝑏𝑎+𝑏𝑐) = (a + b + c ) −a2−b2 −𝑐2+𝑎𝑏+𝑏𝑐+𝑐𝑎 = – (a + b + c ) a2+b2+𝑐2−𝑎𝑏−𝑏𝑐−𝑐𝑎 Multiplying & Dividing by 2 = – 1 × 22 (a + b + c ) a2+b2+𝑐2−𝑎𝑏−𝑏𝑐−𝑐𝑎 = −12 (a + b + c ) 2a2+2b2+2𝑐2−2𝑎𝑏−2𝑏𝑐−2𝑐𝑎 = −12 (a + b + c ) 𝑎2+𝑎2+𝑏2+𝑏2+𝑐2+𝑐2−2𝑎𝑏−2𝑏𝑐−2𝑐𝑎 = −12 (a + b + c ) 𝒂𝟐+𝒄𝟐−𝟐𝒄𝒂+𝑎2+𝑏2−2𝑎𝑏+𝑏2+𝑐2−2𝑏𝑐 = −12 (a + b + c ) 𝒂−𝒄𝟐+ 𝑎−𝑏2+ 𝑐−𝑎2 Now 𝑎−𝑐2+ 𝑎−𝑏2+ 𝑐−𝑎2 > 0 & a + b + c > 0 ∴ ∆ = −12(a + b + c ) 𝑎−𝑐2+ 𝑎−𝑏2+ 𝑐−𝑎2 < 0 Hence ∆ is negative Hence Shown

Examples

Example 1

Example 2

Example 3

Example 4

Example 5 Important

Example 6 Deleted for CBSE Board 2022 Exams

Example 7 Deleted for CBSE Board 2022 Exams

Example 8 Deleted for CBSE Board 2022 Exams

Example 9 Important Deleted for CBSE Board 2022 Exams

Example 10 Important Deleted for CBSE Board 2022 Exams

Example 11 Deleted for CBSE Board 2022 Exams

Example 12 Deleted for CBSE Board 2022 Exams

Example 13 Deleted for CBSE Board 2022 Exams

Example 14 Important Deleted for CBSE Board 2022 Exams

Example 15 Important Deleted for CBSE Board 2022 Exams

Example 16 Important Deleted for CBSE Board 2022 Exams

Example 17

Example 18 Important

Example 19

Example 20

Example 21

Example 22 Important

Example 23

Example 24 Important

Example 25

Example 26 Important

Example 27

Example 28 Important

Example 29

Example 30 Deleted for CBSE Board 2022 Exams You are here

Example 31 Important Deleted for CBSE Board 2022 Exams

Example 32 Important Deleted for CBSE Board 2022 Exams

Example 33 Important

Example 34 Important Deleted for CBSE Board 2022 Exams

Chapter 4 Class 12 Determinants (Term 1)

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.