79182 The roots of the determinant \(\left|\begin{array}{lll}a & a & x \\ m & m & m\end{array}\right|=0\) are

79183 If \(\left|\begin{array}{lll}\mathbf{a}+\mathbf{b} & \mathbf{b}+\mathbf{c} & \mathbf{c}+\mathbf{a} \\ \mathbf{b}+\mathbf{c} & \mathbf{c}+\mathbf{a} & \mathbf{a}+\mathbf{b} \\ \mathbf{c}+\mathbf{a} & \mathbf{a}+\mathbf{b} & \mathbf{b}+\mathbf{c}\end{array}\right|=\mathbf{k}\left|\begin{array}{ccc}\mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{b} & \mathbf{c} & \mathbf{a} \\ \mathbf{c} & \mathbf{a} & \mathbf{b}\end{array}\right|\), then \(\mathbf{k} \quad\) is equal to

79184 Suppose \(\Delta=\begin{array}{lll}\mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2}\end{array}\) and
\(\left|\begin{array}{lll}
\mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3} \end{array}\right|\)
\(\Delta^{\prime}=\left|\begin{array}{lll}
\mathbf{a}_{1}+\mathbf{p} b_{1} & \mathbf{b}_{1}+q c_{1} & \mathbf{c}_{1}+\mathbf{r} \mathbf{a}_{1} \\ \mathbf{a}_{2}+\mathbf{p} \mathbf{b}_{2} & \mathbf{b}_{2}+\mathbf{q} \mathbf{c}_{2} & \mathbf{c}_{2}+\mathbf{r} \mathbf{a}_{2} \\ \mathbf{a}_{3}+\mathbf{p} \mathbf{b}_{3} & \mathbf{b}_{3}+\mathbf{q} \mathbf{c}_{3} & \mathbf{c}_{3}+\mathbf{r} \mathbf{a}_{3} \end{array}\right| \text {, then }\)

79185 If \(P, Q\) and \(R\) are angles of \(\triangle P Q R\), then the value
of \(\left|\begin{array}{ccc}-1 & \cos R & \cos Q \\ \cos R & -1 & \cos P \\ \cos Q & \cos P & -1\end{array}\right|\) is equal to

79182 The roots of the determinant \(\left|\begin{array}{lll}a & a & x \\ m & m & m\end{array}\right|=0\) are

79183 If \(\left|\begin{array}{lll}\mathbf{a}+\mathbf{b} & \mathbf{b}+\mathbf{c} & \mathbf{c}+\mathbf{a} \\ \mathbf{b}+\mathbf{c} & \mathbf{c}+\mathbf{a} & \mathbf{a}+\mathbf{b} \\ \mathbf{c}+\mathbf{a} & \mathbf{a}+\mathbf{b} & \mathbf{b}+\mathbf{c}\end{array}\right|=\mathbf{k}\left|\begin{array}{ccc}\mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{b} & \mathbf{c} & \mathbf{a} \\ \mathbf{c} & \mathbf{a} & \mathbf{b}\end{array}\right|\), then \(\mathbf{k} \quad\) is equal to

79184 Suppose \(\Delta=\begin{array}{lll}\mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2}\end{array}\) and
\(\left|\begin{array}{lll}
\mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3} \end{array}\right|\)
\(\Delta^{\prime}=\left|\begin{array}{lll}
\mathbf{a}_{1}+\mathbf{p} b_{1} & \mathbf{b}_{1}+q c_{1} & \mathbf{c}_{1}+\mathbf{r} \mathbf{a}_{1} \\ \mathbf{a}_{2}+\mathbf{p} \mathbf{b}_{2} & \mathbf{b}_{2}+\mathbf{q} \mathbf{c}_{2} & \mathbf{c}_{2}+\mathbf{r} \mathbf{a}_{2} \\ \mathbf{a}_{3}+\mathbf{p} \mathbf{b}_{3} & \mathbf{b}_{3}+\mathbf{q} \mathbf{c}_{3} & \mathbf{c}_{3}+\mathbf{r} \mathbf{a}_{3} \end{array}\right| \text {, then }\)

79185 If \(P, Q\) and \(R\) are angles of \(\triangle P Q R\), then the value
of \(\left|\begin{array}{ccc}-1 & \cos R & \cos Q \\ \cos R & -1 & \cos P \\ \cos Q & \cos P & -1\end{array}\right|\) is equal to

79182 The roots of the determinant \(\left|\begin{array}{lll}a & a & x \\ m & m & m\end{array}\right|=0\) are

79183 If \(\left|\begin{array}{lll}\mathbf{a}+\mathbf{b} & \mathbf{b}+\mathbf{c} & \mathbf{c}+\mathbf{a} \\ \mathbf{b}+\mathbf{c} & \mathbf{c}+\mathbf{a} & \mathbf{a}+\mathbf{b} \\ \mathbf{c}+\mathbf{a} & \mathbf{a}+\mathbf{b} & \mathbf{b}+\mathbf{c}\end{array}\right|=\mathbf{k}\left|\begin{array}{ccc}\mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{b} & \mathbf{c} & \mathbf{a} \\ \mathbf{c} & \mathbf{a} & \mathbf{b}\end{array}\right|\), then \(\mathbf{k} \quad\) is equal to

79184 Suppose \(\Delta=\begin{array}{lll}\mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2}\end{array}\) and
\(\left|\begin{array}{lll}
\mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3} \end{array}\right|\)
\(\Delta^{\prime}=\left|\begin{array}{lll}
\mathbf{a}_{1}+\mathbf{p} b_{1} & \mathbf{b}_{1}+q c_{1} & \mathbf{c}_{1}+\mathbf{r} \mathbf{a}_{1} \\ \mathbf{a}_{2}+\mathbf{p} \mathbf{b}_{2} & \mathbf{b}_{2}+\mathbf{q} \mathbf{c}_{2} & \mathbf{c}_{2}+\mathbf{r} \mathbf{a}_{2} \\ \mathbf{a}_{3}+\mathbf{p} \mathbf{b}_{3} & \mathbf{b}_{3}+\mathbf{q} \mathbf{c}_{3} & \mathbf{c}_{3}+\mathbf{r} \mathbf{a}_{3} \end{array}\right| \text {, then }\)

79185 If \(P, Q\) and \(R\) are angles of \(\triangle P Q R\), then the value
of \(\left|\begin{array}{ccc}-1 & \cos R & \cos Q \\ \cos R & -1 & \cos P \\ \cos Q & \cos P & -1\end{array}\right|\) is equal to

79182 The roots of the determinant \(\left|\begin{array}{lll}a & a & x \\ m & m & m\end{array}\right|=0\) are

79183 If \(\left|\begin{array}{lll}\mathbf{a}+\mathbf{b} & \mathbf{b}+\mathbf{c} & \mathbf{c}+\mathbf{a} \\ \mathbf{b}+\mathbf{c} & \mathbf{c}+\mathbf{a} & \mathbf{a}+\mathbf{b} \\ \mathbf{c}+\mathbf{a} & \mathbf{a}+\mathbf{b} & \mathbf{b}+\mathbf{c}\end{array}\right|=\mathbf{k}\left|\begin{array}{ccc}\mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{b} & \mathbf{c} & \mathbf{a} \\ \mathbf{c} & \mathbf{a} & \mathbf{b}\end{array}\right|\), then \(\mathbf{k} \quad\) is equal to

79184 Suppose \(\Delta=\begin{array}{lll}\mathbf{a}_{2} & \mathbf{b}_{2} & \mathbf{c}_{2}\end{array}\) and
\(\left|\begin{array}{lll}
\mathbf{a}_{3} & \mathbf{b}_{3} & \mathbf{c}_{3} \end{array}\right|\)
\(\Delta^{\prime}=\left|\begin{array}{lll}
\mathbf{a}_{1}+\mathbf{p} b_{1} & \mathbf{b}_{1}+q c_{1} & \mathbf{c}_{1}+\mathbf{r} \mathbf{a}_{1} \\ \mathbf{a}_{2}+\mathbf{p} \mathbf{b}_{2} & \mathbf{b}_{2}+\mathbf{q} \mathbf{c}_{2} & \mathbf{c}_{2}+\mathbf{r} \mathbf{a}_{2} \\ \mathbf{a}_{3}+\mathbf{p} \mathbf{b}_{3} & \mathbf{b}_{3}+\mathbf{q} \mathbf{c}_{3} & \mathbf{c}_{3}+\mathbf{r} \mathbf{a}_{3} \end{array}\right| \text {, then }\)

79185 If \(P, Q\) and \(R\) are angles of \(\triangle P Q R\), then the value
of \(\left|\begin{array}{ccc}-1 & \cos R & \cos Q \\ \cos R & -1 & \cos P \\ \cos Q & \cos P & -1\end{array}\right|\) is equal to