- 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

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- 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

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- 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