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# I.S.I. B.Math Subjective Paper 2005

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?
May 6, 2014