Question

If $P,Q$ and $R$ are angles of $∆PQR$ then the value of $\left|\begin{array}{ccc}-1& \cos R& \cos Q\\ \cos R& -1& \cos P\\ \cos Q& \cos P& -1\end{array}\right|$

• A. $-1$
• B. $0$
• C. $\frac{1}{2}$
• D. $1$

Answer 1

The correct option is B

$0$

Step 1: Given information

If $P,Q$ and $R$ are angles of $∆PQR$

$\mathrm{Let},∆=\left|\begin{array}{ccc}-1& \cos R& \cos Q\\ \cos R& -1& \cos P\\ \cos Q& \cos P& -1\end{array}\right|$

Step 2: Multiply the column $\left(C_1\right)$ first by $p$

$∆=\frac{1}{p}\left|\begin{array}{ccc}-p& \cos R& \cos Q\\ p\cos R& -1& \cos P\\ p\cos Q& \cos P& -1\end{array}\right|$

Apply the operation $C_1\to C_1+q{C}_2+r{C}_3$

$$\begin{gathered} ∆=\frac{1}{p}\left|\begin{array}{ccc}-p+q\cos R+r\cos Q& \cos R& \cos Q\\ p\cos R-q+r\cos P& -1& \cos P\\ p\cos Q+q\cos P-r& \cos P& -1\end{array}\right| \\ ∆=\frac{1}{p}\left|\begin{array}{ccc}0& \cos R& \cos Q\\ 0& -1& \cos P\\ 0& \cos P& -1\end{array}\right| \end{gathered}$$

According to the determinant rule:

If all the elements of a row or column are zeros, then the value of the determinant is zero

$\therefore ∆=0$, where $∆$is a symbol of the determinant.

Hence, option ‘B’ is correct.