Geometry - General Aptitude (GA) Previous Year Questions

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \(r\) cm as shown in the figure. The side of the dodecagon is \(d\) cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side \(r\) cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.

Consider two rectangular sheets, Sheet M and Sheet N of dimensions 6 cm x 4 cm each. Folding operation 1: The sheet is folded into half by joining the short edges of the current shape. Folding operation 2: The sheet is folded into half by joining the long edges of the current shape. Folding operation 1 is carried out on Sheet M three times. Folding operation 2 is carried out on Sheet N three times. The ratio of perimeters of the final folded shape of Sheet N to the final folded shape of Sheet M is ______.

A sphere of radius r cm is packed in a box of cubical shape.
What should be the minimum volume (in cm³) of the box that can enclose the sphere?

As shown in the figure, circle \(C_1\) with center \(O_1\) and radius \(r_1\) touches the square \(VWXY\) at points \(P\) and \(Q\) while circle \(C_2\) with center \(O_2\) and radius \(r_2\) touches the square \(VWXY\) at points \(R\) and \(S\). The two circles touch each other at \(T\).
Given \(r_1 = 1\) cm and \(\overline{VY} = \overline{VW} = 4\) cm, \(r_2 = \)_____ cm.