by Aritra Bhattacharya

Question: Let \(f\) and \(g\) be two functions from \([0,1]\) to \([0,1]\) with \(f\) strictly increasing. Which of the following statements is always correct? A. If \(g\) is continuous, then \(fog\) is continuous B. If \(f\) is continuous, then \(fog\) is continuous...
by Aritra Bhattacharya

Question: Let \(S\) be the collection of isomorphism classes of groups \(G\) such that every element of G commutes with only the identity element and itself. Then what is \(|S|\)? Discussion: Given any \(g\in G\), it commutes with few obvious elements:...
by Aritra Bhattacharya

Question: The number of irreducible polynomials of the form \(x^2+ax+b\) , with \(a,b\) in the field \(\mathbb{F}_7\) of 7 elements is: A. 7 B. 21 C. 35 D. 49 Discussion: First, what is the number of polynomials of the form \(x^2+ax+b\) in \(\mathbb{F}_7\) ? \(a\) has...
by Aritra Bhattacharya

Question: Let \(f:\mathbb{R}^2 \to \mathbb{R} \) be a continuous map such that \(f(x)=0 \) for only finitely many values of \(x\). Which of the following is true? A. either \(f(x) \le 0 \)for all \(x\) or \(f(x) \ge 0 \) for all \(x\). B. the map \(f\) is onto C. the...
by Aritra Bhattacharya

Question: Let \(C\) denote the cube \([-1,1]^3\subset \mathbb{R}^3 \). How many rotations are there in \(\mathbb{R}^3\) which take \(C\) to itself? Discussion: Let us label the six faces of the cube by \(F_1,F_2,…,F_6\). Let \(G\) be the set consisting of all...