by Aritra Bhattacharya

Question: Let \(n\in\mathbb{N}\), \(n\ge 2\). Let \(x_1,x_2,…,x_n\in(0,\pi)\). Set \(x=\frac{x_1+x_2+…+x_n}{n}\). Which of the following are true? A) \(\prod_{k=1}^{n} sinx_k \ge sin^nx \) B) \(\prod_{k=1}^{n} sinx_k \le sin^nx \) C) Neither (A) or (B) is...
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: Let \(A\) be a \(10\times 10\) matrix with complex entries such that all its eigenvalues are non-negative real numbers, and at least one eigenvalue is positive. Which of the following statements is always false? A. There exists a matrix B such that...
by Aritra Bhattacharya

Question: Let \(f: \mathbb{R} \to \mathbb{R} \) be a continuous function. Which of the following can not be the image of \((0,1]\) under \(f\)? A. {0} B. \((0,1)\) C. \([0,1)\) D. \([0,1]\) Discussion: If f is the constant function constantly mapping to 0, which is...