Misc 9 - Chapter 4 Class 12 Determinants

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Misc 9 Evaluate + + + Let = + + + Applying R1 R1 + R2 + R3 = + + + + + + + + + + + = 2x+2y 2x+2y 2x+2y y x+y x x+y x y = ( + ) ( + ) ( + ) y x+y x x+y x y Taking common 2(x + y), from R1 = ( + ) 1 1 1 y x+y x x+y x y Applying C2 C2 C1 = 2(x+y) 1 1 y x+y x x+y x x y y = 2(x+y) 1 1 y x x x+y y x Applying C3 C3 C1 = 2(x+y) 1 0 y x x y x+y y y x y = 2(x+y) 1 0 y x x y x+y y x Expanding determinant along R1 = 2(x+y) 1 0 + +0 + = 2(x+y) 1 0+0 = 2(x+y) (1( x2 ( y) (x y)) ) = 2(x+y) ( x2 + y (x y)) = 2(x+y) ( x2 + xy y2) = 2(x+y) ( x2 + y2 xy) = 2(x3+y3) Hence , = 2( + )