fog(x) = f {g(x)}

          =  f(x - 7)

          =  {(x - 7) + 7}

          =  x

$\therefore$ fog (7) = 7

We know that domain and range of the principal value branch of function  sin^-1 is defined as:

$$\begin{gathered} {\sin }^{-1}:\left[-1, 1\right] \to \left[-\frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right] \\ \therefore \sin \left[\frac{\mathrm{\pi }}{3} - {\sin }^{-1}\left(\frac{-1}{2}\right)\right] \\ =\sin \left[\frac{\mathrm{\pi }}{3} - \left( -\frac{\mathrm{\pi }}{6}\right)\right] \\ = \sin \left[\frac{\mathrm{\pi }}{3} +\frac{\mathrm{\pi }}{6}\right] \\ = \sin \left[\frac{\mathrm{\pi }}{2} \right] \\ = 1 \end{gathered}$$

$$\begin{gathered} 2\left[\begin{array}{cc}1& 3\\ 0& \mathrm{x}\end{array}\right] + \left[\begin{array}{cc}\mathrm{y}& 0\\ 1& 2\end{array}\right] = \left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right] \\ \left[\begin{array}{cc}2& 6\\ 0& 2\mathrm{x}\end{array}\right] + \left[\begin{array}{cc}\mathrm{y}& 0\\ 1& 2\end{array}\right] = \left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right] \\ \left[\begin{array}{cc}2 + \mathrm{y}& 6 + 0\\ 0 + 1& 2\mathrm{x} + 2\end{array}\right] = \left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right] \\ \left[\begin{array}{cc}2 + \mathrm{y}& 6\\ 1& 2(\mathrm{x} + 1)\end{array}\right] = \left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right] \end{gathered}$$

On comparing the corresponding elements of the matrices on both sides, We get:

2 + y = 5  $\Rightarrow$ y = 3

2 ( x + 1 ) = 8  $\Rightarrow$ x = 3

$$\begin{gathered} \left[\begin{array}{cc}\mathrm{a} + \mathrm{ib}& \mathrm{c} + \mathrm{id}\\ -\mathrm{c} + \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right] = (\mathrm{a} + \mathrm{ib}) (\mathrm{a} - \mathrm{ib}) - (\mathrm{c} + \mathrm{id}) (-\mathrm{c} + \mathrm{id}) \\ = ( {\mathrm{a}}^2 - {\mathrm{i}}^2{\mathrm{b}}^2) - (-{\mathrm{c}}^2 + {\mathrm{i}}^2{\mathrm{d}}^2) \\ = ( {\mathrm{a}}^2 + {\mathrm{b}}^2) - (-{\mathrm{c}}^2 - {\mathrm{d}}^2) .........(\because {\mathrm{i}}^2 = -1) \\ = {\mathrm{a}}^2 + {\mathrm{b}}^2 +{\mathrm{c}}^2 + {\mathrm{d}}^2) \end{gathered}$$

$\left|\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right|$

Cofactor of a₁₂ = ( -1 )¹⁺² [ 6(-7) - 4(1) ]

                      = (-1) [ -42 - 4 ]

                      = 46