Determinants Te...

The value of the determinant $$\displaystyle \left | \begin{matrix}
1 &\omega ^{3}  &\omega ^{5} \\
 \omega ^{3}&1  &\omega ^{4} \\
 \omega ^{5}&\omega ^{4}  &1
\end{matrix} \right |$$ , where $$\omega$$ is an imaginary cube root of unity,is

Let $$\displaystyle A=\left [ \begin{matrix}1 &0  &0 \\ 2 &1  &0 \\ 3 &2  &1 \end{matrix} \right ]$$ and $$\displaystyle U_{1}, U_{2}, U_{3}$$ be column
matrices satisfying $$\displaystyle AU_{1}=\left [ \begin{matrix}1\\ 0\\ 0\end{matrix} \right ], AU_{2}=\left [ \begin{matrix}2\\ 3\\ 0\end{matrix} \right ], AU_{3}=\left [ \begin{matrix}2\\ 3\\ 1\end{matrix} \right ]$$. If U is
$$\displaystyle 3\times 3$$ matrix whose columns are  $$\displaystyle U_{1}, U_{2}, U_{3}$$, then $$\displaystyle \left | U \right |=$$

If the lines $$L_{1}:\lambda ^{2}x-y-1=0$$ $$L_{2}:x-\lambda ^{2}y+1=0$$ $$L_{3}:x+y-\lambda ^{2}=0$$ pass through the same point the value(s) of $$\lambda$$ equals

If A is a square matrix such that  $$\displaystyle \left | \begin{matrix}4 &0  &0 \\0 &4  &0 \\0 &0  &4\end{matrix} \right |$$=

If $$\displaystyle \omega$$ is an imaginary cube root of unity,then the value of
$$\left | \begin{matrix}
a  &b\omega ^{2}  & a\omega \\
 b\omega & c &b\omega ^{2} \\
 c\omega ^{2}&a\omega   &c
\end{matrix} \right |$$,is ?

If $$\displaystyle a\neq b\neq c,$$ are value of x which satisfies the equation $$\displaystyle \left | \begin{matrix}0 &x-a  &x-b \\ x+a &0  &x-c \\ x+b &x+c  &0 \end{matrix} \right |=0$$ is given by

If $$A\displaystyle= \left | \begin{matrix}a &b  &c \\ x &y  &z \\ p &q  &r \end{matrix} \right |$$ and $$B=\left | \begin{matrix}q &-b  &y \\  -p&a  &-x \\  r&-c  &z \end{matrix} \right |$$, then 

Number of values of $$a$$ for which the lines $$2x+y-1=0, ax+3y-3=0, 3x+2y-2=0$$ are concurrent is

The value of $$\displaystyle \left | \begin{matrix}
11 & 12 &13 \\
 12&13  &14 \\
 13&14  &15
\end{matrix} \right |$$,is

Let $$\begin{vmatrix} x & 2 & x \\ { x }^{ 2 } & x & 6 \\ x & x & 6 \end{vmatrix}=A{ x }^{ 4 }+B{ x }^{ 3 }+C{ x }^{ 2 }+Dx+E$$. Then the value of $$5A+4B+3C+2D+E$$ is equal to