- 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 \(\displaystyle{ \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+…+\frac{1}{\sqrt{10000}} }\). - 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 \(\displaystyle{\lim_{n\to \infty} P_n}\)
- 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.
- 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 \(\displaystyle{|A_k|-|B_k|=0 \textrm{ or } 1}\).
- 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 .
- 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 \(\displaystyle{(a_j,a_{j+1})}\)
- 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. - 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?