# Ex 4.2, 8 - Chapter 4 Class 12 Determinants

Last updated at Jan. 22, 2020 by Teachoo

Last updated at Jan. 22, 2020 by Teachoo

Transcript

Ex 4.2, 8 By using properties of determinants, show that: (i) |โ 8(1&๐&๐2@1&๐&๐2@1&๐&๐2)| = (a - b) (b - c)(c โ a) Solving L.H.S |โ 8(1&๐&๐2@1&๐&๐2@1&๐&๐2)| Applying R1 โ R1 โ R2 = |โ 8(๐โ๐&๐โ๐&๐^2โ๐^2@1&๐&๐2@1&๐&๐2 ) | = |โ 8(๐&(๐โ๐)&(๐โ๐)(๐+๐)@1&๐&๐2@1&๐&๐2 ) | = |โ 8(0(๐โ๐)&(๐โ๐)&(๐โ๐)(a+b)@1&b&b2@1&c&c2 ) | Taking Common (a โ b) from R1 = (๐โ๐) |โ 8(0&1&a+b@1&b&b2@1&c&c2 ) | Applying R2 โ R2 โ R3 = (aโb) |โ 8(0&1&a+b@๐โ๐&bโc&b2โc2@1&c&c2 ) | = (a โ b) |โ 8(0&1&a+๐@๐&bโc&(bโc)(b+c)@1&c&c2 ) | Taking common (b โ c) from R2 = (a โ b) (b โ c) |โ 8(0&1&a+b@0&1&b+c@1&c&c2 ) | Expanding Determinant along C1 = (a โ b) (b โ c) ( 0|โ 8(1&๐+๐@๐&๐2)|โ0|โ 8(1&๐+๐@๐&๐2)|+1|โ 8(1&๐+๐@1&๐+๐)|) = (a โ b) (b โ c) ( 0โ0+1|โ 8(1&๐+๐@1&๐+๐)|) = (a โ b) (b โ c) (1(b + c) โ 1(a + b) ) = (a โ b) (b โ c) (b + c โ a โ b) = (a โ b) (b โ c)(c โ a) = R.H.S Hence Proved Ex 4.2, 8 By using properties of determinants, show that: (ii) |โ 8(1&1&1@a&b&c@a3&b3&c3)| = (a โ b) (b โ c) (c โ a) (a + b + c) Solving L.H.S |โ 8(1&1&1@a&b&c@a3&b3&c3)| Applying C1 โ C1 โ C2 = |โ 8(๐โ๐&1&1@aโb&b&c@๐๐ โ๐๐&b3 &c3)| = |โ 8(๐&1&1@aโb&b&c@(๐ โ๐)(๐๐+๐๐+๐๐) &b3&c3)| = |โ 8(0&1&1@๐โ๐&b&c@(๐ โ๐)(a2+b2+ab) &b3&c3)| Taking Common (a โ b) from C1 = (a โ b) |โ 8(0&1&1@1&b&c@(a2+b2+ab)&b3&c3)| Applying C2 โ C2 โ C3 = (a โ b) |โ 8(0&๐โ๐&1@1&bโc&c@(a2+b2+ab)&b3โc3&c3)| (x3 โ y3 = (x โ y)(x2 + y2 +xy)) = (a โ b) |โ 8(0&๐&1@1&bโc&c@(a2+b2+ab)&(bโc)(b2+c2+bc)&c3)| Taking Common (b โ c) from C2 = (a โ b) (b โ c) |โ 8(0&0&1@1&1&c@a2+b2+ab&b2+c2+bc&c3)| Expanding determinant along R1 = (a โ b) (b โ c) (0|โ 8(1&๐@๐2+๐2+๐๐&๐3)|โ0|โ 8(1&1@๐2+๐2+๐๐&๐3)|+1|โ 8(1&1@๐2+๐2+๐๐&๐2+๐2+๐๐)|) = (a โ b) (b โ c) (0โ0+1|โ 8(1&1@๐2+๐2+๐๐&๐2+๐2+๐๐)|) = (a โ b) (b โ c) (1((b2 + c2 + bc) โ (a2 + b2 + ab)) = (a โ b) (b โ c) (b2 + c2 + bc โ a2 โ b2 โ ab) = (a โ b) (b โ c) (c2 โ a2 + bc โ ab) = (a โ b) (b โ c) ((c โ a) (c + a) + b (c โ a)) = (a โ b) (b โ c) ((c โ a) (c + a + b)) = (a โ b) (b โ c) ((c โ a) (a + b + c)) = R.H.S Hence Proved

Ex 4.2

Ex 4.2, 1
Deleted for CBSE Board 2022 Exams

Ex 4.2, 2 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 3 Deleted for CBSE Board 2022 Exams

Ex 4.2, 4 Deleted for CBSE Board 2022 Exams

Ex 4.2, 5 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 6 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 7 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 8 Important Deleted for CBSE Board 2022 Exams You are here

Ex 4.2, 9 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 10 Deleted for CBSE Board 2022 Exams

Ex 4.2, 11 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 12 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 13 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 14 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 15 Important Deleted for CBSE Board 2022 Exams

Ex 4.2, 16 Important Deleted for CBSE Board 2022 Exams

Chapter 4 Class 12 Determinants (Term 1)

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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.