Which of the following values of $\alpha$ satisfy the equation

$\left|\begin{array}{lll}(1+\alpha)^2 & (1+2 \alpha)^2 & (1+3 \alpha)^2 \\ (2+\alpha)^2 & (2+2 \alpha)^2 & (2+3 \alpha)^2 \\ (3+\alpha)^2 & (3+2 \alpha)^2 & (3+3 \alpha)^2\end{array}\right|=-648 \alpha$ ?

$(A)$ $-4$ $(B)$ $9$ $(C)$ $-9$ $(D)$ $4$

If $A$, $B$ and $C$ are the angles of a triangle then the value of the determinant

$\left| {\begin{array}{*{20}{c}}
  { – 1 + \cos B}&{\cos C + \cos B}&{\cos B} \\ 
  {\cos C + \cos A}&{ – 1 + \cos A}&{\cos A} \\ 
  { – 1 + \cos B}&{ – 1 + \cos A}&{ – 1} 
\end{array}} \right|$

Let $a, b, c, d$ be in arithmetic progression with common difference $\lambda$. If

$\left|\begin{array}{lll} x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c \end{array}\right|=2$

then value of $\lambda^{2}$ is equal to $…..$

Let $\alpha $, $\beta$ $\gamma$, $\delta$ are distinct imaginary roots of

$z^5=1$ then value of $\left| {\begin{array}{*{20}{c}}
  {{e^\alpha }}&{{e^{2\alpha }}}&{{e^{3\alpha  + 1}}}&{ – {e^{ – \delta }}} \\ 
  {{e^\beta }}&{{e^{2\beta }}}&{{e^{3\beta  + 1}}}&{ – {e^{ – \delta }}} \\ 
  {{e^\gamma }}&{{e^{2\gamma }}}&{{e^{3\gamma  + 1}}}&{ – {e^{ – \delta }}} 
\end{array}} \right|$

If $a,b,c$ are unequal what is the condition that the value of the following determinant is zero $\Delta = \left| {\,\begin{array}{*{20}{c}}a&{{a^2}}&{{a^3} + 1}\\b&{{b^2}}&{{b^3} + 1}\\c&{{c^2}}&{{c^3} + 1}\end{array}\,} \right|$