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Evaluate \(\begin{vmatrix}sin⁡ \,y&0&sin⁡ \,y\\cos⁡ \,y&1&cos⁡ \,x\\sin⁡ \,y&1&sin ⁡\,y \end{vmatrix}\)(a) sin⁡ y (cos⁡ y-cos⁡ x)(b) sin⁡ x (cos⁡ y-cos⁡ x)(c) sin ⁡x (cos⁡ x-cos⁡ y)(d) sin ⁡y (cos⁡ 2y-cos⁡ x)The question was posed to me in semester exam.Question is from Determinant in portion Determinants of Mathematics – Class 12

Answer» CORRECT choice is (a) sin⁡ y (COS⁡ y-cos⁡ X)

For explanation: Δ=\(\begin{vmatrix}sin⁡ \,y&0&sin⁡ \,y\\cos⁡ \,y&1&cos⁡ \,x\\sin⁡ \,y&1&sin⁡ \,y \end{vmatrix}\)

Δ=sin⁡ y \(\begin{vmatrix}1&cos⁡ \,x\\1&sin \,⁡y \end{vmatrix}\)-0\(\begin{vmatrix}cos ⁡\,y&cos⁡ \,x \\sin⁡ \,y&sin \,y \end{vmatrix}\)+sin⁡ y \(\begin{vmatrix}cos ⁡\,y&1\\sin \,⁡y&1\end{vmatrix}\)

Δ=sin ⁡y (sin⁡ y-cos⁡ x)-0+sin ⁡y (cos⁡ y-sin ⁡y)

Δ=sin^2⁡y-sin ⁡ycos⁡ x+sin⁡ ycos ⁡y-sin^2⁡y=sin ⁡y (cos⁡ y-cos⁡ x)


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