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प्रश्न
Prove that the determinant `|(x,sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1, x)|` is independent of θ.
उत्तर
Let, Δ = `[(x,sintheta,costheta),(-sintheta,-x,1),(costheta,1,x)]`
= x(-x2 - 1) - sin θ (-xsin θ - cos θ) + cos θ(-sin θ + xcosθ)
= -x(x2 + 1) + xsin2θ + sinθcosθ - sinθcosθ + xcos2θ
= -x(x2 + 1) + x (sin2θ + cos2θ)
= -x(x2 + 1) + x = -x[x2 + 1 - 1]
= -x3
which is independent of θ.
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