Misc 1 - Chapter 4 Class 12 Determinants

Transcript

Misc 1 Prove that the determinant x sin cos sin x 1 cos 1 x is independent of . Let = x sin cos sin x 1 cos 1 x = x 1 1 sin sin 1 cos + cos sin cos 1 = x ( x2 1) sin ( xsin cos ) + cos ( sin + x cos ) = x3 x + x. sin 2 + sin cos sin cos + x cos2 = x3 x + x sin2 + x+ cos2 + sin cos sin cos = x3 x + x (sin2 + cos2 ) + 0 = x3 x + x (1) = x3 Hence = x3 Which has no term Thus, the determinant is independent of Hence Proved