ISI Entrance Paper BMath 2011 - Subjective

ISI Entrance Paper BMath 2011 - from Indian Statistical Institute's Entrance

Also see: ISI and CMI Entrance Course at Cheenta

1. Given $latex \mathbf{ a,x\in\mathbb{R}}$ and $latex \mathbf{x\geq 0,a\geq 0}$ . Also $latex \mathbf{sin(\sqrt{x+a})=sin(\sqrt{x})}$ . What can you say about a? Justify your answer.
2. 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.
3. For $latex \mathbf{n\in\mathbb{N}}$ prove that $latex \mathbf{\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}}$
   Solution
4. Let $latex \mathbf{t_1 < t_2 < t_3 < \cdots < t_{99}}$ be real numbers. Consider a function $latex \mathbf{f: \mathbb{R} to \mathbb{R}}$ given by $latex \mathbf{f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|}$ . Show that f(x) will attain minimum value at $latex \mathbf{x=t_{50}}$
5. Consider a sequence denoted by F_n of non-square numbers . $latex \mathbf{F_1=2,F_2=3,F_3=5}$ and so on . Now , if $latex \mathbf{m^2\leq F_n<(m+1)^2}$ . Then prove that m is the integer closest to $latex \mathbf{\sqrt{n}}$
6. Let $latex \mathbf{f(x)=e^{-x} for all x\geq 0}$ and let g be a function defined as for every integer $latex \mathbf{k \ge 0}$, a straight line joining (k,f(k)) and (k+1,f(k+1)) . Find the area between the graphs of f and g.
7. If $latex \mathbf{a_1, a_2, \cdots, a_7}$ are not necessarily distinct real numbers such that $latex \mathbf{1 < a_i < 13}$ for all i, then show that we can choose three of them such that they are the lengths of the sides of a triangle.
8. 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.