Here, you will find all the questions of ISI Entrance Paper 2005 from the Indian Statistical Institute's B.Math Entrance. You will also get the solutions soon to all the previous year's problems.
Problem 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 \(\displaystyle{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}}}\).
Problem 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 \(\displaystyle{\lim_{n\to \infty} P_n}\)
Problem 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.
Problem 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 \(\displaystyle{|A_k|-|B_k|=0 \textrm{ or } 1}\).
Problem 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.
Problem 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 \(\displaystyle{(a_j,a_{j+1})}\)
Problem 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.
Problem 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 \( -1 \)?
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