NIMCET 2015 — Mathematics PYQ

Given $A+B+C=\pi$:

• $\sin (A+B+C)=\sin (\pi )=0$

• Since $A+B=\pi -C$, then $\cos (A+B)=\cos (\pi -C)=-\cos C$

2. Substitute the simplified values into the determinant

Let the determinant be $\Delta$:

3. Identify the Matrix Property

Observe the structure of the matrix $M$ corresponding to this determinant:

Notice that the diagonal elements are all $0$, and for any element $a_{ij}$, we have $a_{ij}=-a_{ji}$.

For example:

• $a_{12}=\sin B$ and $a_{21}=-\sin B$

• $a_{13}=\cos C$ and $a_{31}=-\cos C$

• $a_{23}=\tan A$ and $a_{32}=-\tan A$

This is a Skew-Symmetric Matrix.

4. Apply the Property of Skew-Symmetric Determinants

The determinant of a skew-symmetric matrix of odd order (in this case, $3\times 3$) is always zero.

Alternatively, by direct expansion along the first row:

Correct Option: (a)