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Recent questions tagged true-false
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151
TIFR-2018-Maths-A: 21
True/False Question : A countable group can have only countably many distinct subgroups.
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TIFR
Aug 29, 2020
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soujanyareddy13
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tifrmaths2018
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152
TIFR-2018-Maths-A: 22
True/False Question : There exists a continuous surjection from $\mathbb{R}^{3}-S^{2}$ to $\mathbb{R}^{2}-\left \{ \left ( 0,0 \right ) \right \}$ (here $S^{2}\subset \mathbb{R}^{3}$ denotes the unit sphere defined by the equation $x^{2}+y^{2}+z^{2}=1$).
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
93
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tifrmaths2018
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153
TIFR-2018-Maths-A: 23
True/False Question : The permutation group $S_{10}$ has an element of order $30$.
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
82
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tifrmaths2018
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154
TIFR-2018-Maths-A: 24
True/False Question : Let $G$ be a finite group and $g \in G$ an element of even order. Then we can colour the elements of $G$ with two colours in such a way that $x$ and $gx$ have different colours for each $x \in G$.
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
142
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tifrmaths2018
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155
TIFR-2018-Maths-A: 25
True/False Question : Let $f(x)$ and $g(x)$ be uniformly continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Then their pointwise product $f(x)g(x)$ is uniformly continuous.
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
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tifrmaths2018
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156
TIFR-2019-Maths-B: 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 $f\left ( \mathbb{R-Q} \right )\subseteq \mathbb{Q}.$
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
165
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tifrmaths2019
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157
TIFR-2019-Maths-B: 2
True/False Question : If $A \in M_{10} \left ( \mathbb{R} \right )$ satisfies $A^{2}+A+I=0$, then $A$ is invertible.
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
328
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tifrmaths2019
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158
TIFR-2019-Maths-B: 3
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.
soujanyareddy13
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TIFR
Aug 29, 2020
by
soujanyareddy13
156
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tifrmaths2019
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159
TIFR-2019-Maths-B: 4
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.
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
157
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tifrmaths2019
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160
TIFR-2019-Maths-B: 5
True/False Question : Suppose $A,B,C$ are $3\times3$ real matrices with Rank $A =2$, Rank $B=1$, Rank $C=2$. Then Rank $(ABC)=1$.
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
268
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tifrmaths2019
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161
TIFR-2019-Maths-B: 6
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 $M_{n}\left ( \mathbb{R} \right )$.
soujanyareddy13
asked
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TIFR
Aug 29, 2020
by
soujanyareddy13
135
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tifrmaths2019
true-false
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162
TIFR-2019-Maths-B: 7
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.
soujanyareddy13
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TIFR
Aug 29, 2020
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soujanyareddy13
116
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tifrmaths2019
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163
TIFR-2019-Maths-B: 8
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 $M_{3}\left ( \mathbb{R} \right )\cong \mathbb{R}^{9}$.
soujanyareddy13
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TIFR
Aug 29, 2020
by
soujanyareddy13
162
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tifrmaths2019
true-false
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164
TIFR-2019-Maths-B: 9
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.
soujanyareddy13
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TIFR
Aug 29, 2020
by
soujanyareddy13
137
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tifrmaths2019
true-false
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165
TIFR-2019-Maths-B: 10
True/False Question : $\mathbb{R}$ and $\mathbb{R}\oplus \mathbb{R}$ are isomorphic as vector spaces over $\mathbb{Q}$.
soujanyareddy13
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TIFR
Aug 29, 2020
by
soujanyareddy13
146
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tifrmaths2019
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166
TIFR-2019-Maths-B: 11
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 $\left ( 0,\infty \right )$.
soujanyareddy13
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in
TIFR
Aug 29, 2020
by
soujanyareddy13
160
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tifrmaths2019
true-false
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167
TIFR-2019-Maths-B: 12
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 $x^{2}=x$).
soujanyareddy13
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in
TIFR
Aug 29, 2020
by
soujanyareddy13
161
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tifrmaths2019
true-false
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168
TIFR-2019-Maths-B: 13
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 $a^{50}{x}'+b^{20}{y}'+c^{15}{z}'=1$.
soujanyareddy13
asked
in
TIFR
Aug 29, 2020
by
soujanyareddy13
198
views
tifrmaths2019
true-false
0
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answers
169
TIFR-2019-Maths-B: 14
True/False Question : The ring $\mathbb{R}\left [ x \right ]/\left ( x^{5} +x-3\right )$ is an integral domain.
soujanyareddy13
asked
in
TIFR
Aug 29, 2020
by
soujanyareddy13
131
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tifrmaths2019
true-false
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170
TIFR-2019-Maths-B: 15
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 $n$.
soujanyareddy13
asked
in
TIFR
Aug 29, 2020
by
soujanyareddy13
128
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tifrmaths2019
true-false
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171
TIFR-2019-Maths-B: 16
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 $N$.
soujanyareddy13
asked
in
TIFR
Aug 29, 2020
by
soujanyareddy13
167
views
tifrmaths2019
true-false
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172
TIFR-2019-Maths-B: 17
True/False Question : If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is cyclic.
soujanyareddy13
asked
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TIFR
Aug 29, 2020
by
soujanyareddy13
110
views
tifrmaths2019
true-false
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173
TIFR-2019-Maths-B: 18
True/False Question : Each solution of the differential equation ${y}''+e^{x}y=0$ remains bounded as $x\rightarrow \infty$.
soujanyareddy13
asked
in
TIFR
Aug 29, 2020
by
soujanyareddy13
134
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tifrmaths2019
true-false
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174
TIFR-2019-Maths-B: 19
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.
soujanyareddy13
asked
in
TIFR
Aug 29, 2020
by
soujanyareddy13
106
views
tifrmaths2019
true-false
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175
TIFR-2019-Maths-B: 20
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 $\frac{\mathrm{dv} }{\mathrm{dt} }\mid _{t=t_{0}}$.
soujanyareddy13
asked
in
TIFR
Aug 29, 2020
by
soujanyareddy13
126
views
tifrmaths2019
true-false
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176
TIFR-2020-Maths-B: 1
True/False Question : There exists no monotone function $f:\mathbb{R}\rightarrow \mathbb{R}$ which is discontinuous at every rational number.
soujanyareddy13
asked
in
TIFR
Aug 28, 2020
by
soujanyareddy13
310
views
tifrmaths2020
true-false
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177
TIFR-2020-Maths-B: 2
True/False Question : Let $C\left ( \left [ 0,1 \right ] \right )$ denote the set of continuous real valued functions on $\left [ 0,1 \right ]$, and $\mathbb{R}^{\mathbb{N}}$ the set of all sequences of real numbers. Then there exists an injective map from $C\left ( \left [ 0,1 \right ] \right )$ to $\mathbb{R}^{\mathbb{N}}$ .
soujanyareddy13
asked
in
TIFR
Aug 28, 2020
by
soujanyareddy13
222
views
tifrmaths2020
true-false
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votes
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178
TIFR-2020-Maths-B: 3
True/False Question : Let $\left \{ a_{n} \right \}^{\infty }_{n=1}$ be a bounded sequence of positive real numbers. Then: $\underset{n\rightarrow\infty }{lim sup}\:\frac{1}{a_{n}}=\frac{1}{\underset{n\rightarrow \infty }{lim\:inf \:a_{n}}}.$
soujanyareddy13
asked
in
TIFR
Aug 28, 2020
by
soujanyareddy13
173
views
tifrmaths2020
true-false
0
votes
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answers
179
TIFR-2020-Maths-B: 4
True/False Question : Let $C\left ( \left [ 0,1 \right ] \right )$ denote the metric space of continuous real valued functions on $\left [ 0,1 \right ]$ under the supremum metric - i.e., the distance between $f$ and $g$ in $C\left ( \left [ 0,1 \right ] \right )$ ... the coefficient of $x ^{2}$ is $0$. Then $\text{Q}$ is dense in $C\left ( \left [ 0,1 \right ] \right )$ .
soujanyareddy13
asked
in
TIFR
Aug 28, 2020
by
soujanyareddy13
283
views
tifrmaths2020
true-false
0
votes
0
answers
180
TIFR-2020-Maths-B: 5
True/False Question : If $X$ is a metric space such that every continuous function $f:X\rightarrow \mathbb{R}$ is uniformly continuous, then $X$ is compact.
soujanyareddy13
asked
in
TIFR
Aug 28, 2020
by
soujanyareddy13
150
views
tifrmaths2020
true-false
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