Question

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 1

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