by Aritra Bhattacharya

Question: Let \(A\) be an invertible \(10 \times 10\) matrix with real entries such that the sum of each row is 1. Then A. The sum of the entries of each row of the inverse of A is 1 B. The sum of the entries of each column of the inverse of A is 1 C. The trace of the... by Aritra Bhattacharya

TIFR 2014 Problem 23 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D.,... by Aritra Bhattacharya

Question: How many maps \(\phi: \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}\) are there satisfying \(\phi(ab)=\phi(a)+\phi(b)\) , for all \(a,b\in \mathbb{N} \cup \{0\}\) ? Discussion: Take \(n\in \mathbb{N} \cup \{0\} \). By the given equation \(\phi(n\times... by Aritra Bhattacharya

Question: Let \(f: X\to Y \) be a continuous map between metric spaces. Then \(f(X)\) is a complete subset of \(Y\) if A. X is compact B. Y is compact C. X is complete D. Y is complete Discussion: Let \(X\) be compact. Then \(f(X)\) is compact. (continuous image of... by Aritra Bhattacharya

Question: Let \(X\) be a topological space such that every function \(f: X \to \mathbb{R}\) is continuous. Then A. \(X\) has the discrete topology. B. \(X\) has the indiscrete topology. C. \(X\) is compact. D. \(X\) is not connected. Discussion: We know that if \(Y\)...