# Recent questions tagged tifrmaths2019

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True/False Question : There exists a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f\left ( \mathbb{Q} \right )\subseteq \mathbb{R}-\mathbb{Q}$ and $f\left ( \mathbb{R-Q} \right )\subseteq \mathbb{Q}.$
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True/False Question : If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
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True/False Question : Let $X\subseteq \mathbb{Q}^{2}$. Suppose each continuous function $f:X\rightarrow \mathbb{R}^{2}$ is bounded. Then $X$ is necessarily finite.
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True/False Question : If $A$ is a $2\times2$ complex matrix that is invertible and diagonalizable, and such that $A$ and $A^{2}$ have the same characteristic polynomial, then $A$ is the $2\times2$ identity matrix.
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True/False Question : Suppose $A,B,C$ are $3\times3$ real matrices with Rank $A =2$, Rank $B=1$, Rank $C=2$. Then Rank $(ABC)=1$.
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True/False Question : For any $n\geq 2$, there exists an $n\times n$ real matrix $A$ such that the set $\left \{ A^{p} \mid p\geq 1 \right \}$ spans the $\mathbb{R}$-vector space $M_{n}\left ( \mathbb{R} \right )$.
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True/False Question : The matrices $\begin{pmatrix} 0 & i & 0\\ 0& 0& 1\\ 0& 0 & 0 \end{pmatrix} and \begin{pmatrix} 0 & 0 & 0\\ -i& 0& 0\\ 0& 1 & 0 \end{pmatrix}$ are similar.
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True/False Question : Consider the set $A\subset M_{3}\left ( \mathbb{R} \right )$ of $3\times 3$ real matrices with characteristic polynomial. $x^{3}-3x^{2}+2x-1$. Then $A$ is a compact subset of $M_{3}\left ( \mathbb{R} \right )\cong \mathbb{R}^{9}$.
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True/False Question : There exists an injective ring homomorphism from the product ring $\mathbb{R}\times \mathbb{R}$ into $C\left ( \mathbb{R} \right )$, where $C\left ( \mathbb{R} \right )$ denotes the ring of all continuous functions $\mathbb{R}\rightarrow \mathbb{R}$ under pointwise addition and multiplication.
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True/False Question : $\mathbb{R}$ and $\mathbb{R}\oplus \mathbb{R}$ are isomorphic as vector spaces over $\mathbb{Q}$.
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True/False Question : If $0$ is a limit point of a set $A\subseteq \left ( 0,\infty \right )$, then the set of all $x\in\left ( 0,\infty \right )$ that can be expressed as a sum of (not necessarily distinct) elements of $A$ is dense in $\left ( 0,\infty \right )$.
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True/False Question : The only idempotents in the ring $\mathbb{Z}_{51} \left ( i.e.,\mathbb{Z}/51\mathbb{Z} \right )$ are $0$ and $1$. (An idempotent is an element $x$ such that $x^{2}=x$).
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True/False Question : Let $A$ be a commutative ring with $1$, and let $a,b,c\in A$. Suppose there exist $x,y,z\in A$ such that $ax+by+cz=1.$ Then there exist ${x}',{y}',{z}'\in A$ such that $a^{50}{x}'+b^{20}{y}'+c^{15}{z}'=1$.
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True/False Question : The ring $\mathbb{R}\left [ x \right ]/\left ( x^{5} +x-3\right )$ is an integral domain.
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True/False Question : Given any group $G$ of order $12$, and any $n$ that divides $12$, there exists a subgroup $H$ of $G$ of order $n$.
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True/False Question : Let $H,N$ be subgroups of a finite group $G$, with $N$ a normal subgroup of $G$. If the orders of $G/N$ and $H$ are relatively prime, then $H$ is necessarily contained in $N$.
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True/False Question : If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is cyclic.
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True/False Question : Each solution of the differential equation ${y}''+e^{x}y=0$ remains bounded as $x\rightarrow \infty$.
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True/False Question : There exists a uniformly continuous function $f:\left ( 0,\infty \right )\rightarrow \left ( 0,\infty \right )$ such that $\sum_{n=1 }^{\infty }\frac{1}{f\left ( n \right )}$ converges.
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True/False Question : Let $v:\mathbb{R}\rightarrow \mathbb{R}^{2}$ be $C^{\infty }$ (i.e., has derivatives of all orders). Then there exists $t_{0}\in \left ( 0,1 \right )$ such that $v\left ( 1 \right )-v\left ( 0 \right )$ is a scalar multiple of $\frac{\mathrm{dv} }{\mathrm{dt} }\mid _{t=t_{0}}$.
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The following sum of numbers (expressed in decimal notation) $1+11+111+\cdots +\underset{n}{\underbrace{11\dots1}}$ is equal to $\left ( 10^{n+1}-10-9n \right )/81$ $\left ( 10^{n+1}-10+9n \right )/81$ $\left ( 10^{n+1}-10-n \right )/81$ $\left ( 10^{n+1}-10+n \right )/81$
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For $n\geq 1$, the sequence $\left \{ x_{n} \right \}^{\infty }_{n=1},$ where: $x_{n}=1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}}-2\sqrt{n}$ is decreasing increasing constant oscillating
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Define a function: $f\left ( x \right )=\left\{\begin{matrix} x +x^{2} cos\left ( \frac{\pi}{x} \right ), & x\neq 0\\ 0,& x=0. \end{matrix}\right.$ Consider the following statements: ${f}'\left ( 0 \right )$ exists and is equal to $1$ $f$ is not increasing in any ... $f$ is increasing on $\mathbb{R}.$ How many of the above statements is/are true? $0$ $1$ $2$ $3$
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Consider differentiable functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have: $f\left ( b \right )-f\left ( a \right )=\left ( b-a \right ){f}'\left ( \frac{a+b}{2} \right )$ Then which one of the following sentence is true? Every ... $a,b \in \mathbb{R}$
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Let $V$ be an n-dimensional vector space and let $T:V\rightarrow V$ be a linear transformation such that $Rank\:T \leq Rank\:T^{3}$. Then which one of the following statements is necessarily true? Null space$(T)$ = Range$(T)$ Null space$(T)$ $\cap$ Range$(T)$={$0$} There exists a nonzero subspace $W$ of $V$ such that Null space$(T)$ $\cap$ Range$(T)$=$W$ Null space$(T)$ $\subseteq$ Range$(T)$
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The limit $\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$ is equal to $1$ $0$ $+\infty$ $1/2$
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Let $A$ be an $n \times n$ matrix with rank $k$. Consider the following statements: If $A$ has real entries, then $AA^{t}$ necessarily has rank $k$ If $A$ has complex entries, then $AA^{t}$ necessarily has rank $k$. Then (i) and (ii) are true (i) and (ii) are false (i) is true and (ii) is false (i) is false and (ii) is true
Consider the following two statements: $(E)$ Continuous function on $[1,2]$ can be approximated uniformly by a sequence of even polynomials (i.e., polynomials $p\left ( x \right )\in\mathbb{R}\left [ x \right ]$ such that $p\left ( -x \right )=p\left ( x \right )$). $(O)$ Continuous ... both false $(E)$ and $(O)$ are both true $(E)$ is true but $(O)$ is false $(E)$ is false but $(O)$ is true
Let $f:\left ( 0,\infty \right )\rightarrow \mathbb{R}$ be defined by $f\left ( x \right )=\frac{sin\left (x ^{3} \right )}{x}$ . Then $f$ is bounded and uniformly continuous bounded but not uniformly continuous not bounded but uniformly continuous not bounded and not uniformly continuous
Let $S=\left \{ x \in\mathbb{R} \mid x=Trace\:(A) \:for\:some\:A \in M_{4} (\mathbb{R}) such\:that\:A^{2}=A \right\}.$ Then which of the following describes $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$