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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 is always correct? A. If $$g$$ is continuous, then $$fog$$ is continuous B. If  $$f$$ is continuous, then $$fog$$ is continuous...

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 only the identity element and itself. Then what is $$|S|$$? Discussion: Given any $$g\in G$$, it commutes with few obvious elements:...

Number of irreducible polynomials (TIFR 2014 problem 26)

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...

An application of intermediate value theorem (TIFR 2014 problem 22)

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...

Symmetries of Cube (TIFR 2014 problem 20)

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...