Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above.

The ratio of the area of the regular convex to the area of the original equilateral triangle is

Computers are ubiquitous. They are used to improve efficiency in almost all fields from agriculture to space exploration. Artificial intelligence (AI) is currently a hot topic AI enables computers to learn, given enough training data. For humans, sitting in front of a computer for long hours can lead to health issues.

Which of the following can be deducted from the above passage?

(i) Nowadays, computers are present in almost all places.

(ii) Computers cannot be used for solving problems in engineering

(iii) For humans, there are both positive and negative effects of using computers.

(iv) Artificial intelligence can be done without data.

Consider a square sheet of side 1 unit. In the first step, it is cut along the main diagonal to get two triangles. In the next step, one of the cut triangles is revolved about its short edge to form a solid cone. The volume of the resulting cone, in

p and q are positive integers and $${p \over q} + {q \over p} = 3$$, then $${{{p^2}} \over {{q^2}}} + {{{q^2}} \over {{p^2}}}$$ =