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Question

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

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Solution

∣ ∣xsinθcosθsinθx1cosθ1x∣ ∣

Expanding along the first row, we get
Δ=x(x21)sinθ(xsinθcosθ)+cosθ(sinθ+xcosθ)
=x3x+xsin2θ+sinθcosθsinθcosθ+xcos2θ
=x3x+x(sin2θ+cos2θ)
Δ=x3 (sin2θ+cos2θ=1)
Hence, the given determinant is independent of θ


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