\(\quad\) Factorise \(\left| \begin{array} { c c c } 2 a + b + x & x + b & x ^ { 2 } + b ^ { 2 } 0 & a & - a ^ { 2 } a + b & b & b ^ { 2 } \end{array} \right|\) as fully as possible.
8
The matrix \(\mathbf { M }\) is defined by
$$\mathbf { M } = \left[ \begin{array} { c c c }
13 + x & x + 3 & x ^ { 2 } + 9
0 & 5 & - 25
8 & 3 & 9
\end{array} \right]$$
Under the transformation represented by \(\mathbf { M }\), a solid of volume \(0.625 \mathrm {~m} ^ { 3 }\) becomes a solid of volume \(300 \mathrm {~m} ^ { 3 }\)
Use your answer to part (a) to find the possible values of \(x\).
Use \(\mathbf { C }\) to show that \(\cos \frac { \pi } { 12 }\) can be written in the form \(\frac { \sqrt { \sqrt { m } + n } } { 2 }\), where \(m\) and \(n\) are integers.