The lower corner of a leaf in a book is folded over so as to reach the inner edge of the page. Show that the fraction of the width folded over when the area of the folded part is minimum is `2//3.`

If a leaf of a book is folded so that one corner moves along an opposite side,then prove that the line of crease will always touch parabola.

A note book is made up of sheets folded in the middle and stapled. Each sheet form two leaves i.e., four pages. On removing some papers of the first half and second half of the book, Joe found the number of the leaves, in the first case as odd and in the second case as even. If the sum of the numbers of the pages on the last leaf of the book is 63, then what could be the maximum possible sum of the numbers on the pages of leaves that were left in the book.

A particle A of mass 2m is held on a smooth horizontal table and is attached to one end of an inelastic string which runs over a smooth light pulley at the edge of the table. At the other end of the string there hangs another particle B of mass m, the distance from A to the pulley is l. The particle A is then projected towards the pulley with velocity u. (a) Find the time before the string becomes taut, and shown that after the string becomes taut, the initial velocity of A and B is 4u//3 . (b) Find the common velocity when A reaches the pulley (assume that B has not yet reached the ground).

The length and breadth of a rectangular sheet of a paper are 50 cm and 30 cm, respectively. A square of side 5 cm is cut and removed form the four corners of the sheet, and the rest of the pater is folded to form a cuboid (without the top face). Find the edge of the cube whose volume is one-fourth the volume of the cuboid so formed.

The figure shows a long coaxial cable in which a current I flows down through the inner cylinder of radius a and the same current flows back up through the outer cylinder of radius 2a. The cylinders are insulated from each other and the current is uniformly distributed over the area of the cross-section in each cylinder. The strength of the magnetic field at a distance 3a//2 from the axis of the cable is

A circular metallic sheet is divided into two parts in such away that each part can be folded in to a cone.If the ratio of their curved surface areas is 1:2, then the ratio of their volumes is