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Last updated at May 29, 2018 by Teachoo

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

Example 6 Not in Syllabus - CBSE Exams 2021

Example 7 Not in Syllabus - CBSE Exams 2021

Example 8 Not in Syllabus - CBSE Exams 2021

Example 9 Important Not in Syllabus - CBSE Exams 2021

Example 10 Important Not in Syllabus - CBSE Exams 2021

Example 11 Not in Syllabus - CBSE Exams 2021

Example 12 Not in Syllabus - CBSE Exams 2021

Example 13 Not in Syllabus - CBSE Exams 2021

Example 14 Important Not in Syllabus - CBSE Exams 2021

Example 15 Important Not in Syllabus - CBSE Exams 2021

Example 16 Important Not in Syllabus - CBSE Exams 2021

Example 17

Example 18 Important

Example 19

Example 20

Example 21

Example 22

Example 23

Example 24 Important

Example 25

Example 26 Important

Example 27 Not in Syllabus - CBSE Exams 2021

Example 28 Important Not in Syllabus - CBSE Exams 2021

Example 29 Not in Syllabus - CBSE Exams 2021

Example 30 You are here

Example 31 Important Not in Syllabus - CBSE Exams 2021

Example 32 Important Not in Syllabus - CBSE Exams 2021

Example 33 Important Not in Syllabus - CBSE Exams 2021

Example 34 Important Not in Syllabus - CBSE Exams 2021

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 9 years. He provides courses for Maths and Science at Teachoo.