The value of the determinant $\left| {\begin{array}{*{20}{c}}1&1&1\\{\cos (nx)}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin (nx)}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}} \right|$ is independent of:

If $1, \omega, \omega^2$ are the cube roots of unity,then $\Delta = \begin{vmatrix} 1 & \omega^n & \omega^{2n} \\ \omega^n & \omega^{2n} & 1 \\ \omega^{2n} & 1 & \omega^n \end{vmatrix}$ is equal to

If $\left| \begin{matrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matrix} \right| = (a + b + c)(x + a + b + c)^2$,$x \ne 0$ and $a + b + c \ne 0$,then $x$ is equal to

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:

If the determinant $\left| \begin{array}{ccc} a+p & 1+x & u+f \\ b+q & m+y & v+g \\ c+r & n+z & w+h \end{array} \right|$ splits into exactly $K$ determinants of order $3$,each element of which contains only one term,then the value of $K$ is:

For non-zero,real $a, b$ and $c$,if $\left| \begin{array}{ccc} \frac{a^2+b^2}{c} & c & c \\ a & \frac{b^2+c^2}{a} & a \\ b & b & \frac{c^2+a^2}{b} \end{array} \right| = \alpha abc$,then the value of $\alpha$ is