Questions from previous years' IIT Entrance Examination (Indian Institute of Technology)
1. mn squares of equal size are arranged to form a rectangle of dimension m by n; where m and n are natural numbers. Two squares will be called neighbours, if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares. Show that this is possible only if all the numbers used are equal.
2. Use mathematical induction to prove that
3. Prove by mathematical induction that
4. A student is allowed to select at most 'n' books from a collection of (2n+1) books. If the total number of ways in which he can select atleast one book is 63, find the value of n.
5. A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to be included in the committee? In how many of these committees
(a) the women are in majority?
(b) the men are in majority?
6. In how how many ways can a pack of 52 cards be
(a) divided equally among 4 players in order?
(b) formed into 4 groups of 13 cards each?
(c) divided into 4 sets, three of them having 17 cards each and the fourth just one card?
Character is who you are when no one is looking.