1.

Evaluate \(\begin{vmatrix}cos⁡θ&-cos⁡θ&1\\sin^2⁡θ&cos^2⁡θ&1\\sin⁡θ&-sin⁡θ&1\end{vmatrix}\).(a) sin⁡θ+cos^2⁡θ(b) -sin⁡θ-cos^2⁡⁡θ(c) -sin⁡θ+cos^2⁡⁡θ(d) sin⁡θ-cos^2⁡⁡θThis question was addressed to me in my homework.My question is based upon Properties of Determinants in section Determinants of Mathematics – Class 12

Answer»

The correct choice is (d) sin⁡θ-cos^2⁡⁡θ

The explanation is: Δ=\(\begin{vmatrix}cos⁡θ&-cos⁡θ&1\\sin^2⁡θ&cos^2⁡θ&1\\sin⁡θ&-sin⁡θ&1\end{vmatrix}\)

APPLYING C1→C1+C2

Δ=\(\begin{vmatrix}cos⁡θ-cos⁡θ&-cos⁡θ&1\\sin^2⁡θ+cos^2⁡θ&cos^2⁡θ&1\\sinθ-sin⁡θ&-sin⁡θ&1\end{vmatrix}\)=\(\begin{vmatrix}0&-cos⁡θ&1\\1&cos^2⁡θ&1\\0&-sin⁡θ&1\end{vmatrix}\)

Expanding alongC1, we get

0-1(cos^2⁡⁡θ+sinθ)=sin⁡θ-cos^2⁡⁡θ.


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