$\left| {\,\begin{array}{*{20}{c}}{11}&{12}&{13}\\{12}&{13}&{14}\\{13}&{14}&{15}\end{array}\,} \right| = $
$1$
$0$
$-1$
$67$
(b)Apply ${C_3} \to {C_3} – {C_2}$ and ${C_2} \to {C_2} – {C_1}$.
Find equation of line joining $(3,1)$ and $(9,3)$ using determinants
If $a, b, c$ are non-zero real numbers and if the system of equations $(a – 1 )x = y + z,$ $(b – 1 )y = z + x ,$ $(c – 1 )z= x + y,$ has a non-trivial solution, then $ab + bc + ca$ equals
Let $\theta \in\left(0, \frac{\pi}{2}\right)$. If the system of linear equations
$\left(1+\cos ^{2} \theta\right) x+\sin ^{2} \theta y+4 \sin 3 \theta z=0$
$\cos ^{2} \theta x+\left(1+\sin ^{2} \theta\right) y+4 \sin 3 \theta z=0$
$\cos ^{2} \theta x+\sin ^{2} \theta y+(1+4 \sin 3 \theta) z=0$
has a non-trivial solution, then the value of $\theta$ is :
If $a,b,c$ be positive and not all equal, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ is
If $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = k(a + b + c)({a^2} + {b^2} + {c^2}$ $ – bc – ca – ab)$, then $k =$
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