The corners and mid-points of the sides of a triangle are named using the distinct letters P, Q, R, S, T and U, but not necessarily in the same order. Consider the following statements :

$$\bullet$$ The line joining P and R is parallel to the line joining Q and S.

$$\bullet$$ P is placed on the side opposite to the corner T.

$$\bullet$$ S and U cannot be placed on the same side.

Which one of the following statements is correct based on the above information?

A plot of land must be divided between four families. They want their individual plots to be similar in shape, not necessarily equal in area. The land has equally spaced poles, marked as dots in the below figure. Two ropes, R1 and R2, are already present and cannot be moved.

What is the least number of additional straight ropes needed to create the desired plots? A single rope can pass through three poles that are aligned in a straight line.

If $${\left( {x - {1 \over 2}} \right)^2} - {\left( {x - {3 \over 2}} \right)^2} = x + 2$$, then the value of x is :

If $$\theta$$ is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then cos $$\theta$$ = _______