Determinants Te...

If $$\begin{vmatrix} 1+x & 2 & 3 \\ 1 & 2+x & 3 \\ 1 & 2 & 3+x \end{vmatrix}=0$$ then $$x=$$

If $$A = {\left[ {\begin{array}{*{20}{c}}a\\b\\c\end{array}\begin{array}{*{20}{c}}p\\q\\r\end{array}} \right]_{3 \times 2}}$$ then determinant $$\left( {A{A^T}} \right)$$ is equal to

If $${t_{1,}}{t_2}\,$$ and $${t_3}$$ distinct. and the points $$\left( {{t_1}.2a{t_1} + a{t_1}^3} \right).\left( {{t_2}.2a{t_2} + a{t_2}^3} \right),\left( {{t_3}.2a{t_3} + a{t_3}^3} \right)$$ are collinear, then $${t_1} + {t_2} + {t_3} = $$

If $$\Delta  = \left| {\begin{array}{*{20}{c}}1&1&1\\1&{1 + x}&1\\1&1&{1 + y}\end{array}} \right|$$ for $$x \ne 0,\,y \ne 0$$ then $$\Delta $$ is

If $$\alpha, \beta, \gamma, $$ are the roots of $$x^3+ax^2+b=0$$, then the value of $$\left| \begin{matrix} \alpha  & { \beta  } & \gamma  \\ \beta  & \gamma  & \alpha  \\ \gamma  & \alpha  & \beta  \end{matrix} \right| $$ is

Find the value of the determinant $$\begin{vmatrix} 1 & 0 & 0 \\ 2 & \cos { x }  & \sin { x }  \\ 3 & \sin { x }  & \cos { x }  \end{vmatrix}$$.

The value of $$x$$ for which the matrix $$A=\begin{bmatrix} x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1 \end{bmatrix}$$
Is non-singular

If the points $$(k, 2-2k)$$, $$(1-k, 2k)$$ and $$(-k-4, 6-2k)$$ be collinear, the number of possible values of $$k$$ are 

If $$f(x)=\begin{bmatrix} \cos { x }  & -\sin { x }  & 0 \\ \sin { x }  & \cos { x }  & 0 \\ 0 & 0 & 1 \end{bmatrix}$$, then $$f(x+y)$$ is equal to:

If $$A=\quad \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{bmatrix}, B=(adj\quad A)$$ and $$C=5A$$, then $$\cfrac { \left| adj\quad B \right|  }{ \left| C \right|  } $$ is equal to