Question

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⁡⁡θ

This question was addressed to me in my homework.

My question is based upon Properties of Determinants in section Determinants of Mathematics – Class 12

Answer 1

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 along  C1, we get

0-1(cos^2⁡⁡θ+sinθ)=sin⁡θ-cos^2⁡⁡θ.