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  1. Chapter 4 Class 12 Determinants
  2. Serial order wise

Transcript

Misc 9 Evaluate |โ– 8(๐‘ฅ&๐‘ฆ&๐‘ฅ+๐‘ฆ@๐‘ฆ&๐‘ฅ+๐‘ฆ&๐‘ฅ@๐‘ฅ+๐‘ฆ&๐‘ฅ&๐‘ฆ)| Let โˆ† = |โ– 8(๐‘ฅ&๐‘ฆ&๐‘ฅ+๐‘ฆ@๐‘ฆ&๐‘ฅ+๐‘ฆ&๐‘ฅ@๐‘ฅ+๐‘ฆ&๐‘ฅ&๐‘ฆ)| Applying R1โ†’ R1 + R2 + R3 = |โ– 8(๐‘ฅ+๐‘ฆ+๐‘ฅ+๐‘ฆ&๐‘ฆ+๐‘ฅ+๐‘ฆ+๐‘ฅ&๐‘ฅ+๐‘ฆ+๐‘ฅ+๐‘ฆ@๐‘ฆ&๐‘ฅ+๐‘ฆ&๐‘ฅ@๐‘ฅ+๐‘ฆ&๐‘ฅ&๐‘ฆ)| = |โ– 8(2x+2y&2x+2y&2x+2y@y&x+y&x@x+y&x&y)| = |โ– 8(๐Ÿ(๐ฑ+๐ฒ)&๐Ÿ(๐ฑ+๐ฒ)&๐Ÿ(๐ฑ+๐ฒ)@y&x+y&x@x+y&x&y)| Taking common 2(x + y), from R1 = ๐Ÿ(๐ฑ+๐ฒ) |โ– 8(1&1&1@y&x+y&x@x+y&x&y)| Applying C2โ†’ C2 โ€“ C1 = 2(x+y) |โ– 8(1&๐Ÿโˆ’๐Ÿ&1@y&x+yโˆ’๐‘ฆ&x@x+y&xโˆ’xโˆ’y&y)| = 2(x+y) |โ– 8(1&๐ŸŽ&1@y&x&x@x+y&โˆ’y&y)| Applying C3 โ†’C3 โ€“ C1 = 2(x+y) |โ– 8(1&0&๐Ÿโˆ’๐Ÿ@y&x&xโˆ’y@x+y&โˆ’y&yโˆ’(x+y))| = 2(x+y) |โ– 8(1&0&๐ŸŽ@y&x&xโˆ’y@x+y&โˆ’y&โˆ’x)| Expanding determinant along R1 = 2(x+y) (1|โ– 8(๐‘ฅ&๐‘ฅโˆ’๐‘ฆ@โˆ’๐‘ฆ&โˆ’๐‘ฅ)|โˆ’0|โ– 8(๐‘ฆ&๐‘ฅโˆ’๐‘ฆ@๐‘ฅ+๐‘ฆ&โˆ’๐‘ฅ)|+0|โ– 8(๐‘ฆ&๐‘ฅ@๐‘ฅ+๐‘ฆ&โˆ’๐‘ฆ)|) = 2(x+y) (1|โ– 8(๐‘ฅ&๐‘ฅโˆ’๐‘ฆ@โˆ’๐‘ฆ&โˆ’๐‘ฅ)|โˆ’0+0) = 2(x+y) (1( โ€“ x2 โ€“ ( โ€“y) (x โ€“ y)) ) = 2(x+y) ( โ€“ x2 + y (x โ€“ y)) = 2(x+y) ( โ€“ x2 + xy โ€“ y2) = โ€“ 2(x+y) ( x2 + y2 โ€“ xy) = โˆ’ 2(x3+y3) Hence , โˆ† = โ€“ 2(๐ฑ๐Ÿ‘+๐ฒ๐Ÿ‘) (Using a3 + b3 = (a + b) (a2 + b2 โ€“ ab))

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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.