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# College Mathematics Corner

### Checking for Uniform continuity(TIFR 2014 problem 4)

Question: Let $$f:0,\infty)\to \mathbb{R}$$ be defined by $$f(x)=x^{2/3}logx$$  for $$x>0$$ $$f(x)=0$$ if $$x=0$$ Then A. f is discontinuous …

### Function bounds from derivative limits (TIFR 2014 problem 3)

Question: Let $$f:\mathbb{R}\to \mathbb{R}$$ be a differentiable function such that $$\lim_{x\to \infty} f'(x)=1$$ then A. f is bounded B. f …

### Continuous Bounded Function (TIFR 2014 problem 2)

Question: (MCQ) Let $$f:\mathbb{R}\to \mathbb{R}$$ be a continuous bounded function. Then A. f has to be uniformly continuous B. there …

### Negation (TIFR 2014 problem 1)

Question: (MCQ) Let A,B,C be three subsets of $$\mathbb{R}$$ The negation of the following statement: For every $$\epsilon >1$$, there …

### Convergence of alternating series (TIFR 2013 problem 40)

Question: True/False? The series $$1-\frac{1}{\sqrt2}+\frac{1}{\sqrt3}-\frac{1}{\sqrt4}+…$$ is divergent. Hint: Recall the alternating series test (or the Leibniz test) Discussion: Let $$a_n=\frac{1}{\sqrt{n}}$$. …

### Rank and Trace of Idempotent matrix (TIFR 2013 problem 39)

Question: True/False? If $$A$$ is a complex nxn matrix with $$A^2=A$$, then rank$$A$$=trace$$A$$. Hint: What are the eigenvalues of $$A$$? …

Question: True/False? Let $$V$$ be the vector space of polynomials with real coefficients in variable $$t$$ of degree $$\le … ### Checking irreducibility over \(\mathbb{R}$$ (TIFR 2013 problem 37)

Question: True/False? The polynomial $$x^3+3x-2\pi$$ is irreducible over $$\mathbb{R}$$ Hint: When is a odd degree polynomial irreducible over $$\mathbb{R}$$? …

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