Q1 of 127 Page 141

Prove that the determinant is independent of θ.

Let Δ =

Expanding the above determinant along R1 i.e. Row 1


Δ = x (-x2 – 1) – sinθ (-x × sinθ – cosθ × 1) + cosθ (-sinθ × 1 + x cosθ)


Δ = -x3 – x + xsin2θ + sinθ × cos θ – sinθ × cosθ + x cos2θ


Δ = -x3 – x + x (sin2θ + cos2θ)


Since, sin2θ + cos2θ = 1


∴ Δ = -x3 – x + x


Hence Δ is independent of θ.


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