Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

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 \mathbf{ a,x\in\mathbb{R}} and \mathbf{x\geq 0,a\geq 0} . Also \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 \mathbf{n\in\mathbb{N}} prove that \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 \mathbf{t_1 < t_2 < t_3 < \cdots < t_{99}} be real numbers. Consider a function \mathbf{f: \mathbb{R} to \mathbb{R}} given by \mathbf{f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|} . Show that f(x) will attain minimum value at \mathbf{x=t_{50}}
  5. Consider a sequence denoted by F_n of non-square numbers . \mathbf{F_1=2,F_2=3,F_3=5} and so on . Now , if \mathbf{m^2\leq F_n<(m+1)^2} . Then prove that m is the integer closest to \mathbf{\sqrt{n}}
  6. Let \mathbf{f(x)=e^{-x} for all x\geq 0} and let g be a function defined as for every integer \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 \mathbf{a_1, a_2, \cdots, a_7} are not necessarily distinct real numbers such that \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.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com