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Grade 12Algebra

Prove using properties of dterminants that
| a sin x cos x |
| -sin x -a 1 |
| cos x 1 a |
is independent of x.

Profile image of Kartik Ohri
7 Years agoGrade 12
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1 Answer

Profile image of Mrunal Sonawane
7 Years ago
                                             |    a         sin x        cos x |
                                              | -sin x       -a              1    |   
                                              | cos x         1             a    |
 
 
=  a(-a×a – 1×1) – [ sin x (a×sin x –  cos x ) ] +cosx ( -sin x + a×cos x)
=  a(-a^2 – 1) – sin x (-asin x – cos x) + cosx (-sin x+a cos x)
= -a^2-a+ a sin^2 x + sinx . cosx –  sinx . cos x + a cos^2x
=-a^3-a+a(sin^2 x+cos^2 x)
= -a^3 -a +a(1)...............as sin^2\theta +cos^2\theta=1
hence
= -a^3 -a +a
=-a^3
hence -a^3 is independent of x

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