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Question

Prove that the determinant ∣ ∣xsinθcosθsinθx1cosθ1x∣ ∣ is independent of θ.

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Solution

∣ ∣xsinθcosθsinθx1cosθ1x∣ ∣

=x(x21)sinθ(xsinθcosθ)+cosθ(sinθ+xcosθ)

=x3x+xsin2θ+sinθcosθsinθcosθ+xcos2θ

=x3x+x(sin2θ+cos2θ)

=x3x+x

=x3 ( Which is Independent of θ )

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