Sujoy Kumar Datta

You are given a right circular cone of base radius 4 units and height 10 units. You are asked to place a cylinder inside the cone such that one of the flat surfaces rests on the base of the cone. What is the largest possible total surface area (in square units) of the cylinder?

Two circles of radii 1 unit each intersect in such a manner that the circumference of each circle passes through the centre of the other. What is the area, in square units, of the intersecting region?

There are two different cloth cutting methods. In the first method, n circular pieces are cut from a square cloth piece of side a in the following steps: the square of side a is divided into n smaller squares, not necessarily of the same size, then a circle of the maximum possible area is cut from those smaller squares. In the second method, only one circle of maximum possible area is cut from the square of side a and the process ends. The cloth pieces remaining after cutting the circles are scrapped in both methods. The ratio of the total area of cloth scrapped in the former to that in the latter is