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Recent questions tagged tifrmaths2019

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1
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 $f\left ( \mathbb{R-Q} \right )\subseteq \mathbb{Q}.$
asked Aug 30 in TIFR soujanyareddy13 12 views
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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.
asked Aug 30 in TIFR soujanyareddy13 38 views
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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.
asked Aug 30 in TIFR soujanyareddy13 14 views
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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.
asked Aug 30 in TIFR soujanyareddy13 10 views
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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 $(ABC)=1$.
asked Aug 30 in TIFR soujanyareddy13 10 views
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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 $M_{n}\left ( \mathbb{R} \right )$.
asked Aug 30 in TIFR soujanyareddy13 17 views
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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.
asked Aug 30 in TIFR soujanyareddy13 15 views
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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 $M_{3}\left ( \mathbb{R} \right )\cong \mathbb{R}^{9}$.
asked Aug 30 in TIFR soujanyareddy13 9 views
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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.
asked Aug 30 in TIFR soujanyareddy13 7 views
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True/False Question : $\mathbb{R}$ and $\mathbb{R}\oplus \mathbb{R}$ are isomorphic as vector spaces over $\mathbb{Q}$.
asked Aug 30 in TIFR soujanyareddy13 7 views
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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 $\left ( 0,\infty \right )$.
asked Aug 30 in TIFR soujanyareddy13 10 views
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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 $x^{2}=x$).
asked Aug 30 in TIFR soujanyareddy13 9 views
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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 $a^{50}{x}'+b^{20}{y}'+c^{15}{z}'=1$.
asked Aug 30 in TIFR soujanyareddy13 7 views
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True/False Question : The ring $\mathbb{R}\left [ x \right ]/\left ( x^{5} +x-3\right )$ is an integral domain.
asked Aug 30 in TIFR soujanyareddy13 9 views
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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 $n$.
asked Aug 30 in TIFR soujanyareddy13 7 views
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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 $N$.
asked Aug 30 in TIFR soujanyareddy13 14 views
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True/False Question : If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is cyclic.
asked Aug 30 in TIFR soujanyareddy13 8 views
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True/False Question : Each solution of the differential equation ${y}''+e^{x}y=0$ remains bounded as $x\rightarrow \infty$.
asked Aug 30 in TIFR soujanyareddy13 16 views
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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.
asked Aug 30 in TIFR soujanyareddy13 10 views
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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 $\frac{\mathrm{dv} }{\mathrm{dt} }\mid _{t=t_{0}}$.
asked Aug 30 in TIFR soujanyareddy13 12 views
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21
The following sum of numbers (expressed in decimal notation) $1+11+111+\cdots +\underset{n}{\underbrace{11\dots1}}$ is equal to $\left ( 10^{n+1}-10-9n \right )/81$ $\left ( 10^{n+1}-10+9n \right )/81$ $\left ( 10^{n+1}-10-n \right )/81$ $\left ( 10^{n+1}-10+n \right )/81$
asked Aug 30 in TIFR soujanyareddy13 12 views
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22
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
asked Aug 30 in TIFR soujanyareddy13 9 views
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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? $0$ $1$ $2$ $3$
asked Aug 30 in TIFR soujanyareddy13 17 views
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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 ... $a,b \in \mathbb{R}$
asked Aug 30 in TIFR soujanyareddy13 10 views
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25
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$(T)$
asked Aug 30 in TIFR soujanyareddy13 14 views
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26
The limit $\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$ is equal to $1$ $0$ $+\infty$ $1/2$
asked Aug 30 in Calculus soujanyareddy13 122 views
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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
asked Aug 30 in TIFR soujanyareddy13 14 views
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28
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
asked Aug 30 in TIFR soujanyareddy13 7 views
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29
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
asked Aug 30 in TIFR soujanyareddy13 6 views
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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 $S$? $S=\left \{ 0,2,4 \right \}$ $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$ $S=\left \{ 0,1,2,3,4 \right \}$ $S=\left \{ 0,4 \right \}$
asked Aug 30 in TIFR soujanyareddy13 29 views
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