Ex 4.2, 12 - Show |1 x x2 x2 1 x x x2 1| = (1 - x3)2 - Ex 4.2

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  1. Class 12
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Ex 4.2, 12 By using properties of determinants, show that: 1 x x2 x2 1 x x x2 1 = (1 x3)2 Taking L.H.S 1 x x2 x2 1 x x x2 1 Applying R1 R1 + R2 + R3 = + + + + + + x2 1 x x x2 1 Taking (1 + x + x2) Common from 1st row = ( + + ) 1 1 1 x2 1 x x x2 1 Applying C1 C1 C2 = (1+x+x2) 1 1 x2 1 1 x x x2 x2 1 = (1+x+x2) 0 1 1 x2 1 1 x x(1 x) x2 1 = (1+x+x2) 0 1 1 (x 1)( ) 1 x x ( ) x2 1 Taking (x 1) common from C1 = (x 1) (1+x+x2) 0 1 1 (x+1) 1 x x x2 1 Applying C2 C2 C3 = (x 1) (1+x+x2) 0 1 x+1 1 x x x x2 1 1 = (x 1) (1+x+x2) 0 1 x+1 ( ) x x x 1 ( ) 1 Taking (x 1) common from 2nd Column = (x 1) (1+x+x2) (x 1) 0 0 1 x+1 1 x x +1 1 Expanding Determinant along R1 = (x 1)2 (1+x+x2) 0 1 +1 1 0 +1 x 1 +1 +1 1 x +1 = (x 1)2 (1+x+x2) 0 0+1 +1 1 x +1 = (x 1)2 (1+x+x2) 0 0+1( +1 2 ) = (x 1)2 (1+x+x2) +1 2 = (x 1)2 (1 + x + x2) ((x2 + 1 + 2x) x) = (x 1)2 (1 + x + x2) (1 + x + x2) = (x 1)2 (1 + x + x2)2 = ((x 1) (1 + x + x2))2 = ( (1 x) (1 + x + x2))2 = ((1 x) (1 + x + x2))2 = (13 x3)2 = (1 x3)2 = R.H.S Hence proved

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.