The ordered pair $(a, b)$, for which the system of linear equations $3 x-2 y+z=b$ ; $5 x-8 y+9 z=3$ ; $2 x+y+a z=-1$ has no solution, is
$\left(3, \frac{1}{3}\right)$
$\left(-3, \frac{1}{3}\right)$
$\left(-3,-\frac{1}{3}\right)$
$\left(3,-\frac{1}{3}\right)$
The value of $\left| {\begin{array}{*{20}{c}}
1&x&y\\
2&{\sin x + 2x}&{\sin y + 2y}\\
3&{\cos x + 3x}&{\cos y + 3y}
\end{array}} \right|$ is
If $\omega $ be a complex cube root of unity, then $\left| {\,\begin{array}{*{20}{c}}1&\omega &{ - {\omega ^2}/2}\\1&1&1\\1&{ - 1}&0\end{array}\,} \right| = $
The values of the determinant $\left| {\,\begin{array}{*{20}{c}}1&{\cos (\alpha - \beta )}&{\cos \alpha }\\{\cos (\alpha - \beta )}&1&{\cos \beta }\\{\cos \alpha }&{\cos \beta }&1\end{array}\,} \right|$ is
If the system of linear equations $2 x + y - z =7$ ; $x-3 y+2 z=1$ ; $x +4 y +\delta z = k$, where $\delta, k \in R$ has infinitely many solutions, then $\delta+ k$ is equal to