\(\quad \int \frac { 3 - x } { x \left( x ^ { 2 } + 1 \right) } \mathrm { d } x\)
\(\quad \int \frac { \sinh 2 x } { \sqrt { \cosh ^ { 4 } x - 9 \cosh ^ { 2 } x } } \mathrm {~d} x\)
The matrix \(\mathbf { M }\) is defined by
$$\mathbf { M } = \left( \begin{array} { c c c }
12 & 30 & 8
18 & 25 & 20
19 & 50 & 16
\end{array} \right)$$
Given that \(\operatorname { det } \mathbf { M } = - 1040\), give a geometrical interpretation of the solution to the following equation.
$$\left( \begin{array} { c c c }
12 & 30 & 8
18 & 25 & 20
19 & 50 & 16
\end{array} \right) \left( \begin{array} { l }
x
y
z
\end{array} \right) = \left( \begin{array} { l }
2668
3402
4581
\end{array} \right)$$
Three hotels \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) each have different types of room available to book: single, double and family rooms. For each type of room, the price per night is the same in each of the three hotels.
The table below gives, for each hotel, details of the number of each type of room and the total revenue per night when the hotel is full.