Show that the equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has no real solution.

The largest interval for which $${x^{12}} - {x^9} + {x^4} - x + 1 > 0$$ is

$$mn$$ squares of equal size are arranged to from a rectangle of dimension $$m$$ by $$n$$, where $$m$$ and $$n$$ are natural numbers. Two squares will be called ' neighbours ' if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares.Show that this is possible only if all the numbers used are equal.

Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built at