ISI Entrance Paper BMath 2005 – from Indian Statistical Institute’s Entrance

Also see: ISI and CMI Entrance Course at Cheenta

  1. For any k \in\mathbb{Z}^+ , prove that:-
    \displaystyle{ 2(\sqrt{k+1}-\sqrt{k})<\frac{1}{\sqrt{k}}<2(\sqrt{k}-\sqrt{k-1}) }
    Also compute integral part of \displaystyle{ \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}} }.
  2. Let \displaystyle{a_1=1 } and \displaystyle{a_n=n(a_{n-1}+1)} for all \displaystyle{n\ge 2} . Define : \displaystyle{P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)} Compute \displaystyle{\lim_{n\to \infty} P_n}
  3. Let ABCD be a quadrilateral such that the sum of a pair of opposite sides equals the sum of other pair of opposite sides (AB+CD=AD+BC). Prove that the circles inscribed in triangles ABC and ACD are tangent to each other.
  4. For a set S we denote its cardinality by |S|. Let \displaystyle{e_1,e_2,\ldots,e_k } be non-negative integers. Let \displaystyle{A_k} (respectively \displaystyle{B_k}) be the set of all k-tuples \displaystyle{(f_1,f_2,\ldots,f_k)} of integers such that \displaystyle{0\leq f_i\leq e_i} for all i and \displaystyle{\sum_{i=1}^k f_i } is even (respectively odd). Show that \displaystyle{|A_k|-|B_k|=0 \textrm{ or } 1}.
  5. Find the point in the closed unit disc \displaystyle{D={ (x,y) | x^2+y^2\le 1 }} at which the function f(x,y)=x+y attains its maximum .
  6. Let a_0=0<a_1<a_2<... \displaystyle{\int_{a_j}^{a_{j+1}} p(t),dt = 0 \forall 0 \le j\le n-1} Show that , for \displaystyle{0\le j\le n-1} , the polynomial p(t) has exactly one root in the interval \displaystyle{(a_j,a_{j+1})}
  7. Let M be a point in the triangle ABC such that \displaystyle{\text{area}(ABM)=2 \cdot \text{area}(ACM)}
    Show that the locus of all such points is a straight line.
  8. In how many ways can one fill an n*n matrix with +1 and -1 so that the product of the entries in each row and each column equals -1?