Differentiate between:
Physical region and Political region.
Physical region: A physical region is characterized by natural elements like mountains, rivers, and climate. These regions are defined by the landscape's features, such as the Sahara Desert or the Amazon Rainforest.
Political region: A political region is defined by human-made boundaries, such as countries, states, or provinces. These regions are determined by political decisions and borders, like the United States or the European Union.
The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.
Solve the following L.P.P. by graphical method:
Maximize:
\[ z = 10x + 25y. \] Subject to: \[ 0 \leq x \leq 3, \quad 0 \leq y \leq 3, \quad x + y \leq 5. \]