The roots of the equation $\left| {\,\begin{array}{*{20}{c}}0&x&{16}\\x&5&7\\0&9&x\end{array}\,} \right| = 0$ are
$0,\,\,12,\,\,12$
$0, 12, -12$
$0, 12, 16$
$0, 9, 16$
The system of equations $\lambda x + y + z = 0,$ $ - x + \lambda y + z = 0,$ $ - x - y + \lambda z = 0$, will have a non zero solution if real values of $\lambda $ are given by
Let $A_1, A_2, A_3$ be the three A.P. with the same common difference $d$ and having their first terms as $A , A +1, A +2$, respectively. Let $a , b , c$ be the $7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}$ terms of $A_1, A_2, A_3$, respectively such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$ If $a=29$, then the sum of first $20$ terms of an $AP$ whose first term is $c - a - b$ and common difference is $\frac{ d }{12}$, is equal to $........$.
If $A, B, C$ are the angles of triangle then the value of determinant $\left| {\begin{array}{*{20}{c}}
{\sin \,2A}&{\sin \,C}&{\sin \,B} \\
{\sin \,C}&{\sin \,2B}&{\sin A} \\
{\sin \,B}&{\sin \,A}&{\sin \,2C}
\end{array}} \right|$ is