Determinants Te...

In a triangle ABC, with usual notations, if $$\begin{vmatrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{vmatrix}=0,$$ then $$4sin^2A+24sin^2B+36sin^2C$$ is equal to 

If adj $$A=\begin{bmatrix}20 & -20 \\ 10 & 10 \end{bmatrix}$$ , then $$|A|=$$..... 

Sum of the real roots of the equation $$\begin{vmatrix} 1 & 4 & 20\\ 1 & -2 & 5 \\ 1 & 2x & 5{x}^{2} \end{vmatrix}=0$$ is

Let $$\omega =-\dfrac {1}{2}+i\dfrac {\sqrt {3}}{2}$$. Then the value of the determinant.
$$\begin{vmatrix} 1 & 1 & 1 \\ 1 & -1-{ \omega  }^{ 2 } & { \omega  }^{ 2 } \\ 1 & { \omega  }^{ 2 } & { \omega  }^{ 2 } \end{vmatrix}$$ is

The determinant $$\begin{bmatrix} b_{1}+c_{1} & c_{1}+a_{1} & a_{1}+b_{1} \\ b_{2}+c_{2} & c_{2}+a_{2} & a_{2}+b_{2} \\ b_{3}+c_{3} & c_{3}+a_{3} & a_{3}+b_{3}\end{bmatrix}=$$_____

Find the values of $$x$$ if, $$\left| \begin{matrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5x^{ 2 } \end{matrix} \right| =0$$

29 If $$z=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ where 0, I are 2x2 null and identity matrix then det $$\left( \left[ z \right]  \right) $$ is  _______________.

The cofactor of the element $$4$$ in the determinant $$\begin{vmatrix} 1 & 3 & 5 & 1\\ 2 & 3 & 4 & 2\\ 8 & 0 & 1 & 1\\ 0 & 2 & 1 & 1\end{vmatrix}$$ is?

If $$\Delta =\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{vmatrix}$$ for $$x \neq 0, y \neq 0$$, then $$\Delta $$ is 

The number of distinct real roots of the equation,$$\left| \begin{matrix} cosx & sinx & sinx \\ sinx & cosx & sinx \\ sinx & sinx & cosx \end{matrix} \right| =0$$In t interval $$\left[ -\dfrac { \pi  }{ 4 } \dfrac { \pi  }{ 4 }  \right]$$ is/are: