In the given figure, PQRSTV is a regular hexagon with each side of length 5 cm. A circle is drawn with its centre at V such that it passes through P. What is the area (in cm² ) of the shaded region? (The diagram is representative) 

An ant walks in a straight line on a plane leaving behind a trace of its movement. The initial position of the ant is at point P facing East.

The ant first turns 72$$^\circ$$ anticlockwise at P, and then does the following two steps in sequence exactly FIVE times before halting.

1. moves forward for 10 cm.

2. turns 144$$^\circ$$ clockwise

The pattern made by the trade left behind by the ant is

$$x:y:z = {1 \over 2}:{1 \over 3}:{1 \over 4}$$

What is the value of $${{x + z - y} \over y}$$ = ?

Both the numerator and the denominator of $${3 \over 4}$$ are increased by a positive $${15 \over 17}$$ integer, x, and those of _____ are decreased by the same integer. This operation results in the same value for both the fractions. What is the value of x?