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College Mathematics Corner

Jensen’s Inequality (NBHM 2017 problem 5.2)

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 …

Contraction of a function – advanced Cheenta seminar

It is almost like deflating a balloon. But the effect is exponential.  Today (29th January 2018, Monday), we have a …

Increasing function and continuity (TIFR 2015 problem 7)

Question: Let \(f\) and \(g\) be two functions from \(0,1\) to \(0,1\) with \(f\) strictly increasing. Which of the following statements …

Groups without commuting elements (TIFR 2015 problem 4)

Question: Let \(S\) be the collection of isomorphism classes of groups \(G\) such that every element of G commutes with …

Groups with no subgroups

Claim: If G has no subgroups H /= (e), G, then G must be cyclic of prime order. Proof:  One …

Trace of product of matrix (TIFR 2015 problem 3)

Question: Let \(A\) be a \(10\times 10\) matrix with complex entries such that all its eigenvalues are non-negative real numbers, …

Image of continuous function (TIFR 2015 problem 2)

Question: Let \(f: \mathbb{R} \to \mathbb{R} \) be a continuous function. Which of the following can not be the image of …

Invertible matrix with sum of each row 1 (TIFR 2015 problem 1)

Question: Let \(A\) be an invertible \(10 \times 10\) matrix with real entries such that the sum of each row is …

Homomorphisms from \(S_n\) (TIFR 2014 problem 23)

Question: There exists an onto group homomorphism A. from \(S_5\) to \(S_4\) B. from \(S_4\) to \(S_2\) C. from \(S_5\) …

Number of maps (TIFR 2014 problem 30)

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 …

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