If $x, y \in R$ and $\left|\begin{array}{lll}\left(a^x+a^{-x}\right)^2 & \left(a^x-a^{-x}\right)^2 & 1 \\ \left(b^x+b^{-x}\right)^2 & \left(b^x-b^{-x}\right)^2 & 1 \\ \left(c^x+c^{-x}\right)^2 & \left(c^x-c^{-x}\right)^2 & 1\end{array}\right| = 2y+6$,then $y=$

The value of $\left| {\begin{array}{*{20}{c}} {{(b + c)}^2} & {{a^2}} & {{a^2}} \\ {{b^2}} & {{(a + c)}^2} & {{b^2}} \\ {{c^2}} & {{c^2}} & {{(a + b)}^2} \end{array}} \right|$ is equal to

Using the property of determinants and without expanding,prove that:
$\left|\begin{array}{lll}1 & bc & a(b+c) \\ 1 & ca & b(c+a) \\ 1 & ab & c(a+b)\end{array}\right|=0$

If ${f_n}(x)$,${g_n}(x)$,${h_n}(x)$ for $n = 1, 2, 3$ are polynomials in $x$ such that ${f_n}(a) = {g_n}(a) = {h_n}(a)$ for $n = 1, 2, 3$,then the determinant $F(x) = \left| \begin{matrix} {f_1}(x) & {f_2}(x) & {f_3}(x) \\ {g_1}(x) & {g_2}(x) & {g_3}(x) \\ {h_1}(x) & {h_2}(x) & {h_3}(x) \end{matrix} \right|$ at $x = a$ is equal to:

Using properties of determinants,prove that:
$\left|\begin{array}{lll}x & x^{2} & 1+p x^{3} \\ y & y^{2} & 1+p y^{3} \\ z & z^{2} & 1+p z^{3}\end{array}\right|=(1+p x y z)(x-y)(y-z)(z-x),$ where $p$ is any scalar.

If $a, b, c$ are all different from zero and $\left| \begin{array}{ccc} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{array} \right| = 0$,then the value of $a^{-1} + b^{-1} + c^{-1}$ is