ISI Entrance Paper BMath 2011 – from Indian Statistical Institute’s Entrance

Also see: ISI and CMI Entrance Course at Cheenta

- Given and . Also . What can you say about a? Justify your answer.
- Given two cubes R and S with integer sides of lengths r and s units respectively . If the difference between volumes of the two cubes is equal to the difference in their surface areas , then prove that r=s.
- For prove that

Solution - Let be real numbers. Consider a function given by . Show that f(x) will attain minimum value at
- Consider a sequence denoted by F_n of non-square numbers . and so on . Now , if . Then prove that m is the integer closest to
- Let and let g be a function defined as for every integer , a straight line joining (k,f(k)) and (k+1,f(k+1)) . Find the area between the graphs of f and g.
- If are not necessarily distinct real numbers such that for all i, then show that we can choose three of them such that they are the lengths of the sides of a triangle.
- In a triangle ABC , we have a point O on BC . Now show that there exists a line l such that l||AO and l divides the triangle ABC into two halves of equal area.

solution to ques no 2 and 8 of isi bmath 2011

Solutions???