# Recent questions tagged tifrmaths2013

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True/False Question : The inequality $\sqrt{n+1}-\sqrt{n}< \frac{1}{\sqrt{n}}$ is false for all $n$ such that $101\leq n\leq 2000.$
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True/False Question : $\underset{n\rightarrow \infty }{lim}\left ( n+1 \right )^{1/3}-n^{1/3}=\infty$.
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True/False Question : There exists a bijection between $\mathbb{R}^{2}$and the open interval $\left ( 0,1 \right ).$
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True/False Question : Let $S$ be the set of all sequence $\left \{ a_{1},a_{2},\dots,a_{n},\dots \right \}$ where each entry $a_{i}$ is either $0$ or $1$. Then $S$ is countable.
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True/False Question : Let $\left \{ a_{n} \right \}$ be any non-constant sequence in $\mathbb{R}$ such that $a_{n+1}=\frac{a_{n}+a_{n+2}}{2}$ for all $n\geq 1$. Then $\left \{ a_{n} \right \}$ is unbounded.
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True/False Question : The function $f:\mathbb{Z}\rightarrow \mathbb{R}$ defined by $f\left ( n \right )=n^{3}-3n$ in injective.
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True/False Question : The polynomial $x^{3}+3x-2\pi$ is irreducible over $\mathbb{R}.$
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True/False Question : Let $V$ be the vector space consisting of polynomials with real coefficients in variable $t$ of degree $\leq 9$. Let $D:V\rightarrow V$be the linear operator defined by $D\left ( f \right ):=\frac{df}{dt}.$ Then $0$ is an eigenvalue of $D.$
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True/False Question : If $A$ is a complex $n\times n$ matrix with $A^{2}=A$, then rank $A$ = trace $A$.
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True/False Question : The series $1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\cdots$ is divergent.
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True/False Question : Every homeomorphism of the $2$-sphere to itself has a fixed point.
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True/False Question : The intervals $\left [ 0,1\right )$ and $\left (0,1\right )$ are homeomorphic.
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True/False Question : Let $X$ be a complete metric space such that distance between any two points is less than $1$. Then $X$ is compact.
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True/False Question : There exists a continuous surjective function from $S^{1}$ onto $\mathbb{R}$.
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True/False Question : There exists a complete metric on the open interval $\left ( 0,1 \right )$ inducing the usual topology.
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True/False Question : There exists a continuous surjective map from the complex plane onto the non-zero reals.
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True/False Question : If every differentiable function on a subset $X\subset \mathbb{R}^{n}$(i.e., restriction of a differentiable function on a neighbourhood of $X$) is bounded, then $X$ is compact.
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True/False Question : Let $f:X\rightarrow Y$ be a continuous map between metric spaces. If $f$ is a bijection, then its inverse is also continuous.
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True/False Question : Let $f$ be a function on the closed interval $\left [ 0,1 \right ]$defined by $f\left ( x \right )=x$ if $x$ is rational $f\left ( x \right )=x^{2}$ if $x$ is irrational Then $f$ is continuous at $0$ and $1$.
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True/False Question : There exists an infinite subset $S\subset \mathbb{R}^{3}$such that any three vectors in $S$ are linearly independent.
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True/False Question : Consider the sequences $x_{n}=\sum_{j=1}^{n}\frac{1}{j}$ $y_{n}=\sum_{j=1}^{n}\frac{1}{j^{2}}$ Then $\left \{ x_{n} \right \}$ is Cauchy but $\left \{ y_{n} \right \}$ is not.
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True/False Question : $\underset{x\rightarrow 0}{lim}\frac{sin\left ( x^{2} \right )}{x^{2}}sin\left ( \frac{1}{x} \right )=1$.
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True/False Question : Let $f:\left [ a,b \right ]\rightarrow \left [ c,d \right ]$ and $g:\left [ c,d \right ]\rightarrow \mathbb{R}$ be Riemann integrable functions defined on the closed intervals $\left [ a,b \right ]$ and $\left [ c,d \right ]$ respectively. Then the composite $g\circ f$ is also Riemann integrable.
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True/False Question : Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined by $f\left ( x \right )=sin\:x ^{3}$. Then $f$ is continuous but not uniformly continuous.
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True/False Question : Let $x_{1}\in\left ( 0,1 \right )$ be a real number between $0$ and $1$. For $n> 1$, define $x_{n+1}=x_{n}-x_{n}^{n+1}.$ Then $\underset{n\rightarrow \infty }{lim}x_{n}$ exists.
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True/False Question : Suppose $\left \{ a_{i} \right \}$ is a sequence in $\mathbb{R}$ such that $\sum \left | a_{i} \right |\left | x_{i} \right |< \infty$ whenever $\sum \left | x_{i} \right |< \infty$. Then $\left \{ a_{i} \right \}$ is a bounded sequence.
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True/False Question : The integral $\int_{0}^{\infty }e^{-x^{5}}dx$ is convergent.
True/False Question : Let $P\left ( x \right )=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots +\frac{x^{n}}{n!}$ where $n$ is a large positive integer. Then $\underset{x\rightarrow \infty }{lim}\:\frac{e^{x}}{p\left ( x \right )}=1.$
True/False Question : Every differentiable function $f:\left ( 0,1 \right )\rightarrow \left [ 0,1 \right ]$ is uniformly continuous.
True/False Question : Consider the function $f\left ( x \right )=ax+b$ with $a,b\in \mathbb{R}$. Then the iteration $x_{n+1}=f\left ( x_{n} \right ); \: \:\:\:\:\:n\geq 0$ for a given $x_{0}$ converges to $b/\left ( 1-a \right )$ whenever $0<a<1.$